Exact momentum conservation laws for the gyrokinetic Vlasov-Poisson equations
Brizard, Alain J.; Tronko, Natalia
2011-08-15
The exact momentum conservation laws for the nonlinear gyrokinetic Vlasov-Poisson equations are derived by applying the Noether method on the gyrokinetic variational principle [A. J. Brizard, Phys. Plasmas 7, 4816 (2000)]. From the gyrokinetic Noether canonical-momentum equation derived by the Noether method, the gyrokinetic parallel momentum equation and other gyrokinetic Vlasov-moment equations are obtained. In addition, an exact gyrokinetic toroidal angular-momentum conservation law is derived in axisymmetric tokamak geometry, where the transport of parallel-toroidal momentum is related to the radial gyrocenter polarization, which includes contributions from the guiding-center and gyrocenter transformations.
Vlasov-Maxwell and Vlasov-Poisson equations as models of a one-dimensional electron plasma
NASA Technical Reports Server (NTRS)
Klimas, A. J.; Cooper, J.
1983-01-01
The Vlasov-Maxwell and Vlasov-Poisson systems of equations for a one-dimensional electron plasma are defined and discussed. A method for transforming a solution of one system which is periodic over a bounded or unbounded spatial interval to a similar solution of the other is constructed.
Hermeline, F. )
1993-05-01
This paper deals with the approximation of Vlasov-Poisson and Vlasov-Maxwell equations. We present two coupled particle-finite volume methods which use the properties of Delaunay-Voronoi meshes. These methods are applied to benchmark calculations and engineering problems such as simulation of electron injector devices. 42 refs., 13 figs.
Nonlinear gyrokinetic equations for tokamak microturbulence
Hahm, T.S.
1988-05-01
A nonlinear electrostatic gyrokinetic Vlasov equation, as well as Poisson equation, has been derived in a form suitable for particle simulation studies of tokamak microturbulence and associated anomalous transport. This work differs from the existing nonlinear gyrokinetic theories in toroidal geometry, since the present equations conserve energy while retaining the crucial linear and nonlinear polarization physics. In the derivation, the action-variational Lie perturbation method is utilized in order to preserve the Hamiltonian structure of the original Vlasov-Poisson system. Emphasis is placed on the dominant physics of the collective fluctuations in toroidal geometry, rather than on details of particle orbits. 13 refs.
AP-Cloud: Adaptive Particle-in-Cloud method for optimal solutions to Vlasov-Poisson equation
NASA Astrophysics Data System (ADS)
Wang, Xingyu; Samulyak, Roman; Jiao, Xiangmin; Yu, Kwangmin
2016-07-01
We propose a new adaptive Particle-in-Cloud (AP-Cloud) method for obtaining optimal numerical solutions to the Vlasov-Poisson equation. Unlike the traditional particle-in-cell (PIC) method, which is commonly used for solving this problem, the AP-Cloud adaptively selects computational nodes or particles to deliver higher accuracy and efficiency when the particle distribution is highly non-uniform. Unlike other adaptive techniques for PIC, our method balances the errors in PDE discretization and Monte Carlo integration, and discretizes the differential operators using a generalized finite difference (GFD) method based on a weighted least square formulation. As a result, AP-Cloud is independent of the geometric shapes of computational domains and is free of artificial parameters. Efficient and robust implementation is achieved through an octree data structure with 2:1 balance. We analyze the accuracy and convergence order of AP-Cloud theoretically, and verify the method using an electrostatic problem of a particle beam with halo. Simulation results show that the AP-Cloud method is substantially more accurate and faster than the traditional PIC, and it is free of artificial forces that are typical for some adaptive PIC techniques.
NASA Technical Reports Server (NTRS)
Kaup, D. J.; Hansen, P. J.; Choudhury, S. Roy; Thomas, Gary E.
1986-01-01
The equations for the single-particle orbits in a nonneutral high density plasma in the presence of inhomogeneous crossed fields are obtained. Using these orbits, the linearized Vlasov equation is solved as an expansion in the orbital radii in the presence of inhomogeneities and density gradients. A model distribution function is introduced whose cold-fluid limit is exactly the same as that used in many previous studies of the cold-fluid equations. This model function is used to reduce the linearized Vlasov-Poisson equations to a second-order ordinary differential equation for the linearized electrostatic potential whose eigenvalue is the perturbation frequency.
Integer lattice dynamics for Vlasov-Poisson
NASA Astrophysics Data System (ADS)
Mocz, Philip; Succi, Sauro
2017-03-01
We revisit the integer lattice (IL) method to numerically solve the Vlasov-Poisson equations, and show that a slight variant of the method is a very easy, viable, and efficient numerical approach to study the dynamics of self-gravitating, collisionless systems. The distribution function lives in a discretized lattice phase-space, and each time-step in the simulation corresponds to a simple permutation of the lattice sites. Hence, the method is Lagrangian, conservative, and fully time-reversible. IL complements other existing methods, such as N-body/particle mesh (computationally efficient, but affected by Monte Carlo sampling noise and two-body relaxation) and finite volume (FV) direct integration schemes (expensive, accurate but diffusive). We also present improvements to the FV scheme, using a moving-mesh approach inspired by IL, to reduce numerical diffusion and the time-step criterion. Being a direct integration scheme like FV, IL is memory limited (memory requirement for a full 3D problem scales as N6, where N is the resolution per linear phase-space dimension). However, we describe a new technique for achieving N4 scaling. The method offers promise for investigating the full 6D phase-space of collisionless systems of stars and dark matter.
Nonlinear gyrokinetic equations
Dubin, D.H.E.; Krommes, J.A.; Oberman, C.; Lee, W.W.
1983-03-01
Nonlinear gyrokinetic equations are derived from a systematic Hamiltonian theory. The derivation employs Lie transforms and a noncanonical perturbation theory first used by Littlejohn for the simpler problem of asymptotically small gyroradius. For definiteness, we emphasize the limit of electrostatic fluctuations in slab geometry; however, there is a straight-forward generalization to arbitrary field geometry and electromagnetic perturbations. An energy invariant for the nonlinear system is derived, and various of its limits are considered. The weak turbulence theory of the equations is examined. In particular, the wave kinetic equation of Galeev and Sagdeev is derived from an asystematic truncation of the equations, implying that this equation fails to consider all gyrokinetic effects. The equations are simplified for the case of small but finite gyroradius and put in a form suitable for efficient computer simulation. Although it is possible to derive the Terry-Horton and Hasegawa-Mima equations as limiting cases of our theory, several new nonlinear terms absent from conventional theories appear and are discussed.
On the Singularity of the Vlasov-Poisson System
and Hong Qin, Jian Zheng
2013-04-26
The Vlasov-Poisson system can be viewed as the collisionless limit of the corresponding Fokker- Planck-Poisson system. It is reasonable to expect that the result of Landau damping can also be obtained from the Fokker-Planck-Poisson system when the collision frequency v approaches zero. However, we show that the colllisionless Vlasov-Poisson system is a singular limit of the collisional Fokker-Planck-Poisson system, and Landau's result can be recovered only as the approaching zero from the positive side.
On the singularity of the Vlasov-Poisson system
Zheng, Jian; Qin, Hong
2013-09-15
The Vlasov-Poisson system can be viewed as the collisionless limit of the corresponding Fokker-Planck-Poisson system. It is reasonable to expect that the result of Landau damping can also be obtained from the Fokker-Planck-Poisson system when the collision frequency ν approaches zero. However, we show that the collisionless Vlasov-Poisson system is a singular limit of the collisional Fokker-Planck-Poisson system, and Landau's result can be recovered only as the ν approaches zero from the positive side.
Mass-Radius Spirals for Steady State Families of the Vlasov-Poisson System
NASA Astrophysics Data System (ADS)
Ramming, Tobias; Rein, Gerhard
2017-02-01
We consider spherically symmetric steady states of the Vlasov-Poisson system, which describe equilibrium configurations of galaxies or globular clusters. If the microscopic equation of state, i.e., the dependence of the steady state on the particle energy (and angular momentum) is fixed, a one-parameter family of such states is obtained. In the polytropic case the mass of the state along such a one-parameter family is a monotone function of its radius. We prove that for the King, Woolley-Dickens, and related models this mass-radius relation takes the form of a spiral.
NASA Astrophysics Data System (ADS)
Ng, C. S.
2014-10-01
Electrostatic structures have been observed in many regions of space plasmas, including the solar wind, the magnetosphere, the auroral acceleration region. One possible theoretical description of some of these structures is the concept of Bernstein-Greene-Kruskal (BGK) modes, which are exact nonlinear steady-state solutions of the Vlasov-Poisson system of equations in collisionless kinetic theory. We generalize exact solutions of two-dimensional BGK modes in a magnetized plasma with finite magnetic field strength to cases with azimuthal magnetic fields so that these structures carry electric current as well as steady electric and magnetic fields. Such nonlinear solutions now satisfy exactly the Vlasov-Poisson-Ampere system of equations. This work is supported by a National Science Foundation Grant PHY-1004357.
Verification of gyrokinetic microstability codes with an LHD configuration
Mikkelsen, D. R.; Nunami, M.; Sugama, H.; Tanaka, K.; Watanabe, T.-H.
2014-11-15
We extend previous benchmarks of the GS2 and GKV-X codes to verify their algorithms for solving the gyrokinetic Vlasov-Poisson equations for plasma microturbulence. Code benchmarks are the most complete way of verifying the correctness of implementations for the solution of mathematical models for complex physical processes such as those studied here. The linear stability calculations reported here are based on the plasma conditions of an ion-ITB plasma in the LHD configuration. The plasma parameters and the magnetic geometry differ from previous benchmarks involving these codes. We find excellent agreement between the independently written pre-processors that calculate the geometrical coefficients used in the gyrokinetic equations. Grid convergence tests are used to establish the resolution and domain size needed to obtain converged linear stability results. The agreement of the frequencies, growth rates, and eigenfunctions in the benchmarks reported here provides additional verification that the algorithms used by the GS2 and GKV-X codes are correctly finding the linear eigenvalues and eigenfunctions of the gyrokinetic Vlasov-Poisson equations.
Verification of gyrokinetic microstability codes with an LHD configuration
Mikkelsen, D. R.; Nunami, M.; Watanabe, T. -H.; Sugama, H.; Tanaka, K.
2014-11-01
We extend previous benchmarks of the GS2 and GKV-X codes to verify their algorithms for solving the gyrokinetic Vlasov-Poisson equations for plasma microturbulence. Code benchmarks are the most complete way of verifying the correctness of implementations for the solution of mathematical models for complex physical processes such as those studied here. The linear stability calculations reported here are based on the plasma conditions of an ion-ITB plasma in the LHD configuration. The plasma parameters and the magnetic geometry differ from previous benchmarks involving these codes. We find excellent agreement between the independently written pre-processors that calculate the geometrical coefficients used in the gyrokinetic equations. Grid convergence tests are used to establish the resolution and domain size needed to obtain converged linear stability results. The agreement of the frequencies, growth rates and eigenfunctions in the benchmarks reported here provides additional verification that the algorithms used by the GS2 and GKV-X codes are correctly finding the linear eigenvalues and eigenfunctions of the gyrokinetic Vlasov-Poisson equations.
ColDICE: A parallel Vlasov-Poisson solver using moving adaptive simplicial tessellation
NASA Astrophysics Data System (ADS)
Sousbie, Thierry; Colombi, Stéphane
2016-09-01
Resolving numerically Vlasov-Poisson equations for initially cold systems can be reduced to following the evolution of a three-dimensional sheet evolving in six-dimensional phase-space. We describe a public parallel numerical algorithm consisting in representing the phase-space sheet with a conforming, self-adaptive simplicial tessellation of which the vertices follow the Lagrangian equations of motion. The algorithm is implemented both in six- and four-dimensional phase-space. Refinement of the tessellation mesh is performed using the bisection method and a local representation of the phase-space sheet at second order relying on additional tracers created when needed at runtime. In order to preserve in the best way the Hamiltonian nature of the system, refinement is anisotropic and constrained by measurements of local Poincaré invariants. Resolution of Poisson equation is performed using the fast Fourier method on a regular rectangular grid, similarly to particle in cells codes. To compute the density projected onto this grid, the intersection of the tessellation and the grid is calculated using the method of Franklin and Kankanhalli [65-67] generalised to linear order. As preliminary tests of the code, we study in four dimensional phase-space the evolution of an initially small patch in a chaotic potential and the cosmological collapse of a fluctuation composed of two sinusoidal waves. We also perform a "warm" dark matter simulation in six-dimensional phase-space that we use to check the parallel scaling of the code.
The parallel implementation of the one-dimensional Fourier transformed Vlasov Poisson system
NASA Astrophysics Data System (ADS)
Eliasson, Bengt
2005-08-01
A parallel implementation of an algorithm for solving the one-dimensional, Fourier transformed Vlasov-Poisson system of equations is documented, together with the code structure, file formats and settings to run the code. The properties of the Fourier transformed Vlasov-Poisson system is discussed in connection with the numerical solution of the system. The Fourier method in velocity space is used to treat numerical problems arising due the filamentation of the solution in velocity space. Outflow boundary conditions in the Fourier transformed velocity space removes the highest oscillations in velocity space. A fourth-order compact Padé scheme is used to calculate derivatives in the Fourier transformed velocity space, and spatial derivatives are calculated with a pseudo-spectral method. The parallel algorithms used are described in more detail, in particular the parallel solver of the tri-diagonal systems occurring in the Padé scheme. Program summaryTitle of program:vlasov Catalogue identifier:ADVQ Program summary URL:http://cpc.cs.qub.ac.uk/summaries/ADVQ Program obtainable from: CPC Program Library, Queen's University of Belfast, N. Ireland Operating system under which the program has been tested: Sun Solaris; HP-UX; Read Hat Linux Programming language used: FORTRAN 90 with Message Passing Interface (MPI) Computers: Sun Ultra Sparc; HP 9000/785; HP IPF (Itanium Processor Family) ia64 Cluster; PCs cluster Number of lines in distributed program, including test data, etc.:3737 Number of bytes in distributed program, including test data, etc.:18 772 Distribution format: tar.gz Nature of physical problem: Kinetic simulations of collisionless electron-ion plasmas. Method of solution: A Fourier method in velocity space, a pseudo-spectral method in space and a fourth-order Runge-Kutta scheme in time. Memory required to execute with typical data: Uses typically of the order 10 5-10 6 double precision numbers. Restriction on the complexity of the problem: The program uses
A Fully-Implicit, Ion-Electron, Vlasov-Poisson Algorithm
NASA Astrophysics Data System (ADS)
Taitano, William; Knoll, Dana; Chacon, Luis
2010-11-01
The Jacobian-Free-Newton-Krylov method (JFNK) is an advanced non- linear alogorithm that allows solution to a coupled systems of non-linear equations [1]. We put forth a new JFNK-based implicit plasma simulation algorithm. We have studied this algorithm within the context of a two-species Vlasov-Poisson system where the Vlasov equations are solved in an Eulerian frame [2]. We have investigated the route of non-linear-elimination/kinetic- enslavement to reduce the size of block Jacobian matrix in order to solve the field-kinetic system implicitly. The non-linear- elimination/kinetic-enslavement technique allows reduction in the size of non-linear system but still retains high order temporal accuracy and strong non-linear coupling. Our new algorithm make implicit time-dependent, coupled, field-kinetic systems more attractive. As will be shown, a fully implicit run was able to achieve 22 times speed-up compared to the explicit run for our ion-acoustic-showckwave simulation [2].[4pt] [1] D.A. Knoll and D.E. Keyes, J. Comput. Phys. vol. 193 (2004)[0pt] [2] W.T. Taitano, Masters Thesis, Nuclear Engineering, University of Idaho (2010)
A Legendre-Fourier spectral method with exact conservation laws for the Vlasov-Poisson system
NASA Astrophysics Data System (ADS)
Manzini, G.; Delzanno, G. L.; Vencels, J.; Markidis, S.
2016-07-01
We present the design and implementation of an L2-stable spectral method for the discretization of the Vlasov-Poisson model of a collisionless plasma in one space and velocity dimension. The velocity and space dependence of the Vlasov equation are resolved through a truncated spectral expansion based on Legendre and Fourier basis functions, respectively. The Poisson equation, which is coupled to the Vlasov equation, is also resolved through a Fourier expansion. The resulting system of ordinary differential equation is discretized by the implicit second-order accurate Crank-Nicolson time discretization. The non-linear dependence between the Vlasov and Poisson equations is iteratively solved at any time cycle by a Jacobian-Free Newton-Krylov method. In this work we analyze the structure of the main conservation laws of the resulting Legendre-Fourier model, e.g., mass, momentum, and energy, and prove that they are exactly satisfied in the semi-discrete and discrete setting. The L2-stability of the method is ensured by discretizing the boundary conditions of the distribution function at the boundaries of the velocity domain by a suitable penalty term. The impact of the penalty term on the conservation properties is investigated theoretically and numerically. An implementation of the penalty term that does not affect the conservation of mass, momentum and energy, is also proposed and studied. A collisional term is introduced in the discrete model to control the filamentation effect, but does not affect the conservation properties of the system. Numerical results on a set of standard test problems illustrate the performance of the method.
The Cauchy Problem for the 3-D Vlasov-Poisson System with Point Charges
NASA Astrophysics Data System (ADS)
Marchioro, Carlo; Miot, Evelyne; Pulvirenti, Mario
2011-07-01
In this paper we establish global existence and uniqueness of the solution to the three-dimensional Vlasov-Poisson system in the presence of point charges with repulsive interaction. The present analysis extends an analogous two-dimensional result (Caprino and Marchioro in Kinet. Relat. Models 3(2):241-254, 2010).
Criticality in a Vlasov-Poisson system: a fermioniclike universality class.
Ivanov, A V; Vladimirov, S V; Robinson, P A
2005-05-01
A model Vlasov-Poisson system is simulated close to the point of marginal stability, thus assuming only the wave-particle resonant interactions are responsible for saturation, and shown to obey the power-law scaling of a second-order phase transition. The set of critical exponents analogous to those of the Ising universality class is calculated and shown to obey the Widom and Rushbrooke scaling and Josephson's hyperscaling relations at the formal dimensionality d=5 below the critical point at nonzero order parameter. However, the two-point correlation function does not correspond to the propagator of Euclidean quantum field theory, which is the Gaussian model for the Ising universality class. Instead, it corresponds to the propagator for the fermionic vector field and to the upper critical dimensionality d(c) = 2. This suggests criticality of collisionless Vlasov-Poisson systems corresponds to a universality class analogous to that of critical phenomena of a fermionic quantum field description.
Collisional effects on the numerical recurrence in Vlasov-Poisson simulations
Pezzi, Oreste; Valentini, Francesco; Camporeale, Enrico
2016-02-15
The initial state recurrence in numerical simulations of the Vlasov-Poisson system is a well-known phenomenon. Here, we study the effect on recurrence of artificial collisions modeled through the Lenard-Bernstein operator [A. Lenard and I. B. Bernstein, Phys. Rev. 112, 1456–1459 (1958)]. By decomposing the linear Vlasov-Poisson system in the Fourier-Hermite space, the recurrence problem is investigated in the linear regime of the damping of a Langmuir wave and of the onset of the bump-on-tail instability. The analysis is then confirmed and extended to the nonlinear regime through an Eulerian collisional Vlasov-Poisson code. It is found that, despite being routinely used, an artificial collisionality is not a viable way of preventing recurrence in numerical simulations without compromising the kinetic nature of the solution. Moreover, it is shown how numerical effects associated to the generation of fine velocity scales can modify the physical features of the system evolution even in nonlinear regime. This means that filamentation-like phenomena, usually associated with low amplitude fluctuations contexts, can play a role even in nonlinear regime.
Collisional effects on the numerical recurrence in Vlasov-Poisson simulations
NASA Astrophysics Data System (ADS)
Pezzi, Oreste; Camporeale, Enrico; Valentini, Francesco
2016-02-01
The initial state recurrence in numerical simulations of the Vlasov-Poisson system is a well-known phenomenon. Here, we study the effect on recurrence of artificial collisions modeled through the Lenard-Bernstein operator [A. Lenard and I. B. Bernstein, Phys. Rev. 112, 1456-1459 (1958)]. By decomposing the linear Vlasov-Poisson system in the Fourier-Hermite space, the recurrence problem is investigated in the linear regime of the damping of a Langmuir wave and of the onset of the bump-on-tail instability. The analysis is then confirmed and extended to the nonlinear regime through an Eulerian collisional Vlasov-Poisson code. It is found that, despite being routinely used, an artificial collisionality is not a viable way of preventing recurrence in numerical simulations without compromising the kinetic nature of the solution. Moreover, it is shown how numerical effects associated to the generation of fine velocity scales can modify the physical features of the system evolution even in nonlinear regime. This means that filamentation-like phenomena, usually associated with low amplitude fluctuations contexts, can play a role even in nonlinear regime.
A Uniqueness Criterion for Unbounded Solutions to the Vlasov-Poisson System
NASA Astrophysics Data System (ADS)
Miot, Evelyne
2016-09-01
We prove uniqueness for the Vlasov-Poisson system in two and three dimensions under the condition that the L p norms of the macroscopic density grow at most linearly with respect to p. This allows for solutions with logarithmic singularities. We provide explicit examples of initial data that fulfill the uniqueness condition and that exhibit a logarithmic blow-up. In the gravitational two-dimensional case, such states are intimately related to radially symmetric steady solutions of the system. Our method relies on the Lagrangian formulation for the solutions, exploiting the second-order structure of the corresponding ODE.
Energy conserving discontinuous Galerkin spectral element method for the Vlasov-Poisson system
NASA Astrophysics Data System (ADS)
Madaule, Éric; Restelli, Marco; Sonnendrücker, Eric
2014-12-01
We propose a new, energy conserving, spectral element, discontinuous Galerkin method for the approximation of the Vlasov-Poisson system in arbitrary dimension, using Cartesian grids. The method is derived from the one proposed in [4], with two modifications: energy conservation is obtained by a suitable projection operator acting on the solution of the Poisson problem, rather than by solving multiple Poisson problems, and all the integrals appearing in the finite element formulation are approximated with Gauss-Lobatto quadrature, thereby yielding a spectral element formulation. The resulting method has the following properties: exact energy conservation (up to errors introduced by the time discretization), stability (thanks to the use of upwind numerical fluxes), high order accuracy and high locality. For the time discretization, we consider both Runge-Kutta methods and exponential integrators, and show results for 1D and 2D cases (2D and 4D in phase space, respectively).
Fully Electromagnetic Nonlinear Gyrokinetic Equations for Tokamak Edge Turbulence
Hahm, T. S.; Wang, Lu; Madsen, J.
2008-08-01
An energy conserving set of the fully electromagnetic nonlinear gyrokinetic Vlasov equation and Maxwell's equations, which is applicable to both L-mode turbulence with large amplitude and H-mode turbulence in the presence of high E Χ B shear has been derived. The phase-space action variational Lie perturbation method ensures the preservation of the conservation laws of the underlying Vlasov-Maxwell system. Our generalized ordering takes ρ_{i}<< ρ_{θ¡} ~ L_{E} ~ L_{p} << R (here ρ_{i} is the thermal ion Larmor radius and ρ_{θ¡} = B/B_{θ}] ρ_{i}), as typically observed in the tokamak H-mode edge, with LE and Lp being the radial electric field and pressure gradient lengths. We take κ perpendicular to ρ_{i} ~ 1 for generality, and keep the relative fluctuation amplitudes eδφ /Τ_{i} ~ δΒ / Β up to the second order. Extending the electrostatic theory in the presence of high E Χ B shear [Hahm, Phys. Plasmas 3, 4658 (1996)], contributions of electromagnetic fluctuations to the particle charge density and current are explicitly evaluated via pull-back transformation from the gyrocenter distribution function in the gyrokinetic Maxwell's equation.
Diagnosing collisionless energy transfer using field-particle correlations: Vlasov-Poisson plasmas
NASA Astrophysics Data System (ADS)
Howes, Gregory G.; Klein, Kristopher G.; Li, Tak Chu
2017-02-01
Turbulence plays a key role in the conversion of the energy of large-scale fields and flows to plasma heat, impacting the macroscopic evolution of the heliosphere and other astrophysical plasma systems. Although we have long been able to make direct spacecraft measurements of all aspects of the electromagnetic field and plasma fluctuations in near-Earth space, our understanding of the physical mechanisms responsible for the damping of the turbulent fluctuations in heliospheric plasmas remains incomplete. Here we propose an innovative field-particle correlation technique that can be used to measure directly the secular energy transfer from fields to particles associated with collisionless damping of the turbulent fluctuations. Furthermore, this novel procedure yields information about the collisionless energy transfer as a function of particle velocity, providing vital new information that can help to identify the dominant collisionless mechanism governing the damping of the turbulent fluctuations. Kinetic plasma theory is used to devise the appropriate correlation to diagnose Landau damping, and the field-particle correlation technique is thoroughly illustrated using the simplified case of the Landau damping of Langmuir waves in a 1D-1V (one dimension in physical space and one dimension in velocity space) Vlasov-Poisson plasma. Generalizations necessary to apply the field-particle correlation technique to diagnose the collisionless damping of turbulent fluctuations in the solar wind are discussed, highlighting several caveats. This novel field-particle correlation technique is intended to be used as a primary analysis tool for measurements from current, upcoming and proposed spacecraft missions that are focused on the kinetic microphysics of weakly collisional heliospheric plasmas, including the Magnetospheric Multiscale (MMS), Solar Probe Plus, Solar Orbiter and Turbulence Heating ObserveR (THOR) missions.
A fast solver for the gyrokinetic field equation with adiabatic electrons
Borchardt, M.; Kleiber, R.; Hackbusch, W.
2012-07-15
Describing turbulence and microinstabilities in fusion devices is often modelled with the gyrokinetic equation. During the time evolution of the distribution function a field equation for the electrostatic potential needs to be solved. In the case of adiabatic electrons it contains a flux-surface-average term resulting in an integro-differential equation. Its numerical solution is time and memory intensive for three-dimensional configurations. Here a new algorithm is presented which only requires the numerical inversion of a simpler differential operator and a subsequent addition of a correction term. This new procedure is as fast as solving the equation without the surface average.
Geodesic acoustic mode in anisotropic plasmas using double adiabatic model and gyro-kinetic equation
Ren, Haijun; Cao, Jintao
2014-12-15
Geodesic acoustic mode in anisotropic tokamak plasmas is theoretically analyzed by using double adiabatic model and gyro-kinetic equation. The bi-Maxwellian distribution function for guiding-center ions is assumed to obtain a self-consistent form, yielding pressures satisfying the magnetohydrodynamic (MHD) anisotropic equilibrium condition. The double adiabatic model gives the dispersion relation of geodesic acoustic mode (GAM), which agrees well with the one derived from gyro-kinetic equation. The GAM frequency increases with the ratio of pressures, p{sub ⊥}/p{sub ∥}, and the Landau damping rate is dramatically decreased by p{sub ⊥}/p{sub ∥}. MHD result shows a low-frequency zonal flow existing for all p{sub ⊥}/p{sub ∥}, while according to the kinetic dispersion relation, no low-frequency branch exists for p{sub ⊥}/p{sub ∥}≳ 2.
NASA Astrophysics Data System (ADS)
Besse, Nicolas; Coulette, David
2016-08-01
Achieving plasmas with good stability and confinement properties is a key research goal for magnetic fusion devices. The underlying equations are the Vlasov-Poisson and Vlasov-Maxwell (VPM) equations in three space variables, three velocity variables, and one time variable. Even in those somewhat academic cases where global equilibrium solutions are known, studying their stability requires the analysis of the spectral properties of the linearized operator, a daunting task. We have identified a model, for which not only equilibrium solutions can be constructed, but many of their stability properties are amenable to rigorous analysis. It uses a class of solution to the VPM equations (or to their gyrokinetic approximations) known as waterbag solutions which, in particular, are piecewise constant in phase-space. It also uses, not only the gyrokinetic approximation of fast cyclotronic motion around magnetic field lines, but also an asymptotic approximation regarding the magnetic-field-induced anisotropy: the spatial variation along the field lines is taken much slower than across them. Together, these assumptions result in a drastic reduction in the dimensionality of the linearized problem, which becomes a set of two nested one-dimensional problems: an integral equation in the poloidal variable, followed by a one-dimensional complex Schrödinger equation in the radial variable. We show here that the operator associated to the poloidal variable is meromorphic in the eigenparameter, the pulsation frequency. We also prove that, for all but a countable set of real pulsation frequencies, the operator is compact and thus behaves mostly as a finite-dimensional one. The numerical algorithms based on such ideas have been implemented in a companion paper [D. Coulette and N. Besse, "Numerical resolution of the global eigenvalue problem for gyrokinetic-waterbag model in toroidal geometry" (submitted)] and were found to be surprisingly close to those for the original gyrokinetic
NASA Astrophysics Data System (ADS)
Camporeale, E.; Delzanno, G. L.; Bergen, B. K.; Moulton, J. D.
2016-01-01
We describe a spectral method for the numerical solution of the Vlasov-Poisson system where the velocity space is decomposed by means of an Hermite basis, and the configuration space is discretized via a Fourier decomposition. The novelty of our approach is an implicit time discretization that allows exact conservation of charge, momentum and energy. The computational efficiency and the cost-effectiveness of this method are compared to the fully-implicit PIC method recently introduced by Markidis and Lapenta (2011) and Chen et al. (2011). The following examples are discussed: Langmuir wave, Landau damping, ion-acoustic wave, two-stream instability. The Fourier-Hermite spectral method can achieve solutions that are several orders of magnitude more accurate at a fraction of the cost with respect to PIC.
Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria
Frieman, E.A.; Chen, L.
1981-10-01
A nonlinear gyrokinetic formalism for low-frequency (less than the cyclotron frequency) microscopic electromagnetic perturbations in general magnetic field configurations is developed. The nonlinear equations thus derived are valid in the strong-turbulence regime and contain effects due to finite Larmor radius, plasma inhomogeneities, and magentic field geometries. The specific case of axisymmetric tokamaks is then considered, and a model nonlinear equation is derived for electrostatic drift waves. Also, applying the formalism to the shear Alfven wave heating sceme, it is found that nonlinear ion Landau damping of kinetic shear-Alfven waves is modified, both qualitatively and quantitatively, by the diamagnetic drift effects. In particular, wave energy is found to cascade in wavenumber instead of frequency.
From charge motion in general magnetic fields to the non perturbative gyrokinetic equation
Di Troia, C.
2015-04-15
The exact analytical description of non relativistic charge motion in general magnetic fields is, apparently, a simple problem, even if it has not been solved until now, apart for rare cases. The key feature of the present derivation is to adopt a non perturbative magnetic field description to find new solutions of motion. Among all solutions, two are particularly important: guiding particle and gyro-particle solutions. The guiding particle has been characterized to be minimally coupled to the magnetic field; the gyro-particle has been defined to be maximally coupled to the magnetic field and, also, to move on a closed orbit. The generic charged particle motion is shown to be expressed as the sum of such particular solutions. This non perturbative approach corresponds to the description of the particle motion in the gyro-center and/or guiding center reference frame obtained at all the orders of the modern gyro-center transformation. The Boltzmann equation is analyzed with the described exact guiding center coordinates. The obtained gyrokinetic equation is solved for the Boltzmann equation at marginal stability conditions.
Monte Carlo particle-in-cell methods for the simulation of the Vlasov-Maxwell gyrokinetic equations
NASA Astrophysics Data System (ADS)
Bottino, A.; Sonnendrücker, E.
2015-10-01
> The particle-in-cell (PIC) algorithm is the most popular method for the discretisation of the general 6D Vlasov-Maxwell problem and it is widely used also for the simulation of the 5D gyrokinetic equations. The method consists of coupling a particle-based algorithm for the Vlasov equation with a grid-based method for the computation of the self-consistent electromagnetic fields. In this review we derive a Monte Carlo PIC finite-element model starting from a gyrokinetic discrete Lagrangian. The variations of the Lagrangian are used to obtain the time-continuous equations of motion for the particles and the finite-element approximation of the field equations. The Noether theorem for the semi-discretised system implies a certain number of conservation properties for the final set of equations. Moreover, the PIC method can be interpreted as a probabilistic Monte Carlo like method, consisting of calculating integrals of the continuous distribution function using a finite set of discrete markers. The nonlinear interactions along with numerical errors introduce random effects after some time. Therefore, the same tools for error analysis and error reduction used in Monte Carlo numerical methods can be applied to PIC simulations.
Quadratic free energy for the linearized gyrokinetic Vlasov-Maxwell equations
NASA Astrophysics Data System (ADS)
Brizard, Alain
1994-08-01
The motivation for the present work resides in the search for a new formalism for investigating the stability of plasma equilibria perturbed by low-frequency electromagnetic field fluctuations. In this context, the free-energy method of Morrison and Pfirsch [Phys. Fluids B 2, 1105 (1990)] has been extended by deriving an expression for the gyrokinetic free energy. As a check on the gyrokinetic free-energy method, known formulas previously obtained in the electrostatic limit are recovered, while recent (drift-kinetic) formulas obtained by Betti and Freidberg [Phys. Fluids B 4, 1465 (1992)] in the electromagnetic limit are generalized.
Adaptive multiresolution semi-Lagrangian discontinuous Galerkin methods for the Vlasov equations
NASA Astrophysics Data System (ADS)
Besse, N.; Deriaz, E.; Madaule, É.
2017-03-01
We develop adaptive numerical schemes for the Vlasov equation by combining discontinuous Galerkin discretisation, multiresolution analysis and semi-Lagrangian time integration. We implement a tree based structure in order to achieve adaptivity. Both multi-wavelets and discontinuous Galerkin rely on a local polynomial basis. The schemes are tested and validated using Vlasov-Poisson equations for plasma physics and astrophysics.
Bernstein, I.B.; Catto, P.J.
1984-05-01
A nonlinear gyrokinetic formalism is developed which permits mean velocities comparable to thermal speeds in arbitrary magnetic field geometry. The theory is fully electromagnetic and does not employ an eikonal ansatz. The freedom in the theory is exploited to display simply the connection with ideal magnetohydrodynamics.
Gyrokinetic theory and computational methods for electromagnetic perturbations in tokamaks
NASA Astrophysics Data System (ADS)
Qin, Hong
A general gyrokinetic formalism and appropriate computational methods have been developed for electromagnetic perturbations in toroidal plasmas. This formalism and associated numerical code represent the first self-consistent, comprehensive, fully kinetic model for treating both magnetohydrodynamic (MHD) instabilities and electromagnetic drift waves. The gyrokinetic system of equation is derived by phase- space Lagrangian Lie perturbation methods which enable applications to modes with arbitrary wavelength. An important component missing from previous electromagnetic gyrokinetic theories, the gyrokinetic perpendicular dynamics, is identified and developed in the present analysis. This is accomplished by introducing a new ``distribution function'' and an associated governing gyrokinetic equation. Consequently, the compressional Alfvén waves and cyclotron waves can be systematically treated. The new insights into the gyrokinetic perpendicular dynamics uncovered here clarify the understanding of the gyrokinetic approach-the real spirit of the gyrokinetic reduction is to decouple the gyromotion from the guiding center orbital motion, instead of averaging it out. The gyrokinetic perpendicular dynamics is in fact essential to the recovery of the MHD model from a fully kinetic derivation. In particular, it serves to generalize, in gyrokinetic framework, Spitzer's solution of the fluid/particle paradox to a broader regime of applicability. The gyrokinetic system is also shown to be reducible to a simpler form to deal with shear Alfvén waves. This consists of an appropriate form of the gyrokinetic equation governing the distribution function, the gyrokinetic Poisson equation, and a newly derived gyrokinetic moment equation. If all of the kinetic effects are neglected, the gyrokinetic moment equation is shown to recover the ideal MHD equation for shear Alfvén modes. In addition, a gyrokinetic Ohm's law, including both the perpendicular and the parallel components, is
Energetically consistent collisional gyrokinetics
Burby, J. W.; Brizard, A. J.; Qin, H.
2015-10-15
We present a formulation of collisional gyrokinetic theory with exact conservation laws for energy and canonical toroidal momentum. Collisions are accounted for by a nonlinear gyrokinetic Landau operator. Gyroaveraging and linearization do not destroy the operator's conservation properties. Just as in ordinary kinetic theory, the conservation laws for collisional gyrokinetic theory are selected by the limiting collisionless gyrokinetic theory.
NASA Astrophysics Data System (ADS)
Lanti, E.; Dominski, J.; Brunner, S.; McMillan, B. F.; Villard, L.
2016-11-01
This work aims at completing the implementation of a solver for the quasineutrality equation using a Padé approximation in the global gyrokinetic code ORB5. Initially [Dominski, Ph.D. thesis, 2016], the Pade approximation was only implemented for the kinetic electron model. To enable runs with adiabatic or hybrid electron models while using a Pade approximation to the polarization response, the adiabatic response term of the quasi-neutrality equation must be consistently modified. It is shown that the Pade solver is in good agreement with the arbitrary wavelength solver of ORB5 [Dominski, Ph.D. thesis, 2016]. To perform this verification, the linear dispersion relation of an ITG-TEM transition is computed for both solvers and the linear growth rates and frequencies are compared.
Conservation of energy and momentum in nonrelativistic plasmas
Sugama, H.; Watanabe, T.-H.; Nunami, M.
2013-02-15
Conservation laws of energy and momentum for nonrelativistic plasmas are derived from applying Noether's theorem to the action integral for the Vlasov-Poisson-Ampere system [Sugama, Phys. Plasmas 7, 466 (2000)]. The symmetric pressure tensor is obtained from modifying the asymmetric canonical pressure tensor with using the rotational symmetry of the action integral. Differences between the resultant conservation laws and those for the Vlasov-Maxwell system including the Maxwell displacement current are clarified. These results provide a useful basis for gyrokinetic conservation laws because gyrokinetic equations are derived as an approximation of the Vlasov-Poisson-Ampere system.
A Short Introduction to General Gyrokinetic Theory
H. Qin
2005-02-14
Interesting plasmas in the laboratory and space are magnetized. General gyrokinetic theory is about a symmetry, gyro-symmetry, in the Vlasov-Maxwell system for magnetized plasmas. The most general gyrokinetic theory can be geometrically formulated. First, the coordinate-free, geometric Vlasov-Maxwell equations are developed in the 7-D phase space, which is defined as a fiber bundle over the space-time. The Poincar{copyright}-Cartan-Einstein 1-form pullbacked onto the 7-D phase space determines particles' worldlines in the phase space, and realizes the momentum integrals in kinetic theory as fiber integrals. The infinite small generator of the gyro-symmetry is then asymptotically constructed as the base for the gyrophase coordinate of the gyrocenter coordinate system. This is accomplished by applying the Lie coordinate perturbation method to the Poincar{copyright}-Cartan-Einstein 1-form, which also generates the most relaxed condition under which the gyro-symmetry still exists. General gyrokinetic Vlasov-Maxwell equations are then developed as the Vlasov-Maxwell equations in the gyrocenter coordinate system, rather than a set of new equations. Since the general gyrokinetic system-developed is geometrically the same as the Vlasov-Maxwell equations, all the coordinate independent properties of the Vlasov-Maxwell equations, such as energy conservation, momentum conservation, and Liouville volume conservation, are automatically carried over to the general gyrokinetic system. The pullback transformation associated with the coordinate transformation is shown to be an indispensable part of the general gyrokinetic Vlasov-Maxwell equations. Without this vital element, a number of prominent physics features, such as the presence of the compressional Alfven wave and a proper description of the gyrokinetic equilibrium, cannot be readily recovered. Three examples of applications of the general gyrokinetic theory developed in the areas of plasma equilibrium and plasma waves are
Differential formulation of the gyrokinetic Landau operator
NASA Astrophysics Data System (ADS)
Hirvijoki, Eero; Brizard, Alain J.; Pfefferlé, David
2017-02-01
Subsequent to the recent rigorous derivation of an energetically consistent gyrokinetic collision operator in the so-called Landau representation, this paper investigates the possibility of finding a differential formulation of the gyrokinetic Landau collision operator. It is observed that, while a differential formulation is possible in the gyrokinetic phase space, reduction of the resulting system of partial differential equations to five dimensions via gyroaveraging poses a challenge. Based on the present work, it is likely that the gyrocentre analogues of the Rosenbluth-MacDonald-Judd potential functions must be kept gyroangle dependent.
Energetically consistent collisional gyrokinetics
Burby, J. W.; Brizard, A. J.; Qin, H.
2015-10-30
Here, we present a formulation of collisional gyrokinetic theory with exact conservation laws for energy and canonical toroidal momentum. Collisions are accounted for by a nonlinear gyrokinetic Landau operator. Gyroaveraging and linearization do not destroy the operator's conservation properties. Just as in ordinary kinetic theory, the conservation laws for collisional gyrokinetic theory are selected by the limiting collisionless gyrokinetic theory. (C) 2015 AIP Publishing LLC.
Hinton, F. L.; Waltz, R. E.
2006-10-15
Expressions for particle and energy fluxes and heating rates due to turbulence are derived. These fluxes and heating rates are identified from moments of an extended drift-kinetic equation for the equilibrium distribution function. These include neoclassical as well as turbulent diffusion and heating. Phase-space conservation is demonstrated, allowing the drift-kinetic equation to be expressed in conservative form. This facilitates taking moments with few approximations, mainly those consistent with drift kinetics for the equilibrium distribution function and the relative smallness of the fluctuations. The turbulent heating is uniquely defined by choosing the standard gyrokinetic definition for the energy flux. With this definition, most of the heating can be expressed in the form of ohmic heating from turbulent parallel and perpendicular current density perturbations. The latter current is identified with grad-B and curvature drifts, plus terms involving magnetic perturbations (which are smaller for low beta). A small contribution to the heating comes from the divergence of an energy flux that is dependent on the finite gyroradius of the ions. The fluxes and heating rates are expressed in a form that can be easily evaluated from gyrokinetic turbulence simulations.
Electromagnetic nonlinear gyrokinetics with polarization drift
Duthoit, F.-X.; Hahm, T. S.; Wang, Lu
2014-08-15
A set of new nonlinear electromagnetic gyrokinetic Vlasov equation with polarization drift and gyrokinetic Maxwell equations is systematically derived by using the Lie-transform perturbation method in toroidal geometry. For the first time, we recover the drift-kinetic expression for parallel acceleration [R. M. Kulsrud, in Basic Plasma Physics, edited by A. A. Galeev and R. N. Sudan (North-Holland, Amsterdam, 1983)] from the nonlinear gyrokinetic equations, thereby bridging a gap between the two formulations. This formalism should be useful in addressing nonlinear ion Compton scattering of intermediate-mode-number toroidal Alfvén eigenmodes for which the polarization current nonlinearity [T. S. Hahm and L. Chen, Phys. Rev. Lett. 74, 266 (1995)] and the usual finite Larmor radius effects should compete.
Electromagnetic nonlinear gyrokinetics with polarization drift
NASA Astrophysics Data System (ADS)
Duthoit, F.-X.; Hahm, T. S.; Wang, Lu
2014-08-01
A set of new nonlinear electromagnetic gyrokinetic Vlasov equation with polarization drift and gyrokinetic Maxwell equations is systematically derived by using the Lie-transform perturbation method in toroidal geometry. For the first time, we recover the drift-kinetic expression for parallel acceleration [R. M. Kulsrud, in Basic Plasma Physics, edited by A. A. Galeev and R. N. Sudan (North-Holland, Amsterdam, 1983)] from the nonlinear gyrokinetic equations, thereby bridging a gap between the two formulations. This formalism should be useful in addressing nonlinear ion Compton scattering of intermediate-mode-number toroidal Alfvén eigenmodes for which the polarization current nonlinearity [T. S. Hahm and L. Chen, Phys. Rev. Lett. 74, 266 (1995)] and the usual finite Larmor radius effects should compete.
Linear gyrokinetic theory for kinetic magnetohydrodynamic eigenmodes in tokamak plasmas
NASA Astrophysics Data System (ADS)
Qin, H.; Tang, W. M.; Rewoldt, G.
1999-06-01
A two-dimensional (2D) numerical solution method is developed for the recently derived linear gyrokinetic system which describes arbitrary wavelength electromagnetic perturbations in tokamak plasmas. The system consists of the gyrokinetic equation, the gyrokinetic Poisson equation, and the gyrokinetic moment equation. Since familiar magnetohydrodynamic (MHD) results can be recovered entirely from this gyrokinetic model, and all interesting kinetic effects are intrinsically included, this gyrokinetic system offers an approach for kinetic MHD phenomena which is more rigorous, self-consistent, and comprehensive than the previous hybrid models. Meanwhile, drift type microinstabilities can be also investigated systematically in this theoretical framework. The linear gyrokinetic equation is solved for the distribution function in terms of the perturbed fields by integrating along unperturbed particle orbits. The solution is substituted back into the gyrokinetic moment equation and the gyrokinetic Poisson equation. When the boundary conditions are incorporated, an eigenvalue problem is formed. The resulting numerical code, KIN-2DEM, is applied to kinetic ballooning modes, internal kink modes, and toroidal Alfvén eigenmodes (TAEs). The numerical results are benchmarked against the well-established FULL code [G. Rewoldt, W. M. Tang, and M. S. Chance, Phys. Fluids 25, 480 (1982)], the PEST code [J. Manickam, Nucl. Fusion 24, 595 (1984)], and the NOVA-K code [C. Z. Cheng, Phys. Rep. 211, No. 1 (1992)]. More importantly, kinetic effects on MHD modes can be investigated nonperturbatively. In particular, the kinetic effects of the background plasma on internal kink modes and the hot particle destabilization of TAEs are studied numerically.
Second order gyrokinetic theory for particle-in-cell codes
NASA Astrophysics Data System (ADS)
Tronko, Natalia; Bottino, Alberto; Sonnendrücker, Eric
2016-08-01
The main idea of the gyrokinetic dynamical reduction consists in a systematical removal of the fast scale motion (the gyromotion) from the dynamics of the plasma, resulting in a considerable simplification and a significant gain of computational time. The gyrokinetic Maxwell-Vlasov equations are nowadays implemented in for modeling (both laboratory and astrophysical) strongly magnetized plasmas. Different versions of the reduced set of equations exist, depending on the construction of the gyrokinetic reduction procedure and the approximations performed in the derivation. The purpose of this article is to explicitly show the connection between the general second order gyrokinetic Maxwell-Vlasov system issued from the modern gyrokinetic theory and the model currently implemented in the global electromagnetic Particle-in-Cell code ORB5. Necessary information about the modern gyrokinetic formalism is given together with the consistent derivation of the gyrokinetic Maxwell-Vlasov equations from first principles. The variational formulation of the dynamics is used to obtain the corresponding energy conservation law, which in turn is used for the verification of energy conservation diagnostics currently implemented in ORB5. This work fits within the context of the code verification project VeriGyro currently run at IPP Max-Planck Institut in collaboration with others European institutions.
Gyrokinetic Models for Edge Plasmas*
NASA Astrophysics Data System (ADS)
Dimits, Andris
2010-11-01
The use of gyrokinetic equations for the simulation of magnetic fusion edge and scrapeoff-layer plasmas requires that the equations be valid for large relative perturbation amplitudes and, possibly, large flows. The Hamiltonian gyrokinetic theory has therefore been extended to two new orderings [1,2] that are more general than the standard ones in that they allow for potential perturbations or ExB flows of order the thermal levels. These theories both generalize and show that additional terms should have been present some related prior work. Here, full (low-β) electromagnetic toroidal equation sets are presented, and he energy conservation relations are derived using Noether's theorem in a Lagrangian variational approach. Useful subsidiary and reduced orderings are also considered that result in considerable simplification, and methods for the numerical implementation of the new terms in the equations will also be discussed. *This work was performed for US DOE by LLNL under Contract DE-AC52-07NA27344 and is part of the ESL. [4pt] [1] A.M. Dimits et al., Phys. Fluids B4, 274 (1992). [0pt] [2] A.M. Dimits, Phys. Plasmas 17, 055901 (2010).
Global gyrokinetic simulations with strong flows
NASA Astrophysics Data System (ADS)
Collier, J. D.; McMillan, B. F.; Robinson, J. R.
2016-11-01
We report on the investigation of strong toroidal rotation effects in a global tokamak code, ORB5. This includes the implementation of a strong flow gyrokinetic Lagrangian, allowing a complete treatment of centrifugal and Coriolis effects in the laboratory frame. In order to consistently perform the linear analysis in this system, an axisymmetric gyrokinetic equilibrium distribution function is defined using the constants of motion: we show it corresponds to the standard choice in the local limit and is close to the neoclassical solution in the banana regime. The energy and momentum transport equations are presented in an analogous form to those for the weak flow system. Linear studies of Ion Temperature Gradient (ITG) modes in rotating plasmas are performed to determine how the global effects interact with the effects of strong rotation. We also determine the geodesic acoustic mode dispersion with respect to plasma rotation rate in this gyrokinetic model and compare it to MHD theory.
Gyrokinetic Fokker-Planck collision operator.
Li, B; Ernst, D R
2011-05-13
The gyrokinetic linearized exact Fokker-Planck collision operator is obtained in a form suitable for plasma gyrokinetic equations, for arbitrary mass ratio. The linearized Fokker-Planck operator includes both the test-particle and field-particle contributions, and automatically conserves particles, momentum, and energy, while ensuring non-negative entropy production. Finite gyroradius effects in both field-particle and test-particle terms are evaluated. When implemented in gyrokinetic simulations, these effects can be precomputed. The field-particle operator at each time step requires the evaluation of a single two-dimensional integral, and is not only more accurate, but appears to be less expensive to evaluate than conserving model operators.
Continuum Gyrokinetic Edge New Technology
Dorr, M. R.; Hittinger, J. A.; Dorf, M.; Cohen, R.; Ghosh, D.; Lee, W.; Reynolds, C.
2016-05-02
COGENT is a simulation code that models the plasma evolution in the edge region of a tokamak fusion reactor, from the open field line scrape-off layer, across the separatrix, and into the core. The model is based on the 4D gyrokinetic closure of the kinetic equations for a plasma coupled to an electrostatic potential field. The background magnetic field is prescribed either analytically or generated from experimental data, and the grid is aligned with magnetic flux surfaces. Multiple collision operator options are provided, from Krook to fully nonlinear Fokker-Planck.
Effects of collisions on conservation laws in gyrokinetic field theory
NASA Astrophysics Data System (ADS)
Sugama, H.; Watanabe, T.-H.; Nunami, M.
2014-10-01
In gyrokinetic field theory, the gyrokinetic Vlasov equation, Poisson's equation, and Ampere's law are all obtained from the Lagrangian formulation, and conservation laws of energy and momentum for collisionless magnetized plasmas are derived by applying the Noether's theorem. In this work, effects of collisions on conservation laws are investigated by using the gyrokinetic Boltzmann equation which includes Landau's collision operator represented in the gyrocenter coordinates. Particle, energy, and momentum transport equations including collisional transport fluxes are systematically derived by modifying Noether's theorem. Then, the ensemble-averaged transport equations of particles, energy, and toroidal momentum given in the present work are shown to include both collisional and turbulent transport fluxes which agree with those derived from the conventional recursive formulation with the WKB representation.
Free energy balance in gyrokinetic turbulence
Banon Navarro, A.; Morel, P.; Albrecht-Marc, M.; Carati, D.; Merz, F.; Goerler, T.; Jenko, F.
2011-09-15
Free energy plays an important role in gyrokinetic theory, since it is known to be a nonlinear invariant. Its evolution equations are derived and analyzed for the case of ion temperature gradient driven turbulence, using the formalism adopted in the Gene code. In particular, the ion temperature gradient drive, the collisional dissipation as well as entropy/electrostatic energy transfer channels represented by linear curvature and parallel terms are analyzed in detail.
Continuum Edge Gyrokinetic Theory and Simulations
Xu, X Q; Xiong, Z; Dorr, M R; Hittinger, J A; Bodi, K; Candy, J; Cohen, B I; Cohen, R H; Colella, P; Kerbel, G D; Krasheninnikov, S; Nevins, W M; Qin, H; Rognlien, T D; Snyder, P B; Umansky, M V
2007-01-09
The following results are presented from the development and application of TEMPEST, a fully nonlinear (full-f) five dimensional (3d2v) gyrokinetic continuum edge-plasma code. (1) As a test of the interaction of collisions and parallel streaming, TEMPEST is compared with published analytic and numerical results for endloss of particles confined by combined electrostatic and magnetic wells. Good agreement is found over a wide range of collisionality, confining potential, and mirror ratio; and the required velocity space resolution is modest. (2) In a large-aspect-ratio circular geometry, excellent agreement is found for a neoclassical equilibrium with parallel ion flow in the banana regime with zero temperature gradient and radial electric field. (3) The four-dimensional (2d2v) version of the code produces the first self-consistent simulation results of collisionless damping of geodesic acoustic modes and zonal flow (Rosenbluth-Hinton residual) with Boltzmann electrons using a full-f code. The electric field is also found to agree with the standard neoclassical expression for steep density and ion temperature gradients in the banana regime. In divertor geometry, it is found that the endloss of particles and energy induces parallel flow stronger than the core neoclassical predictions in the SOL. (5) Our 5D gyrokinetic formulation yields a set of nonlinear electrostatic gyrokinetic equations that are for both neoclassical and turbulence simulations.
Gyrokinetic Statistical Absolute Equilibrium and Turbulence
Jian-Zhou Zhu and Gregory W. Hammett
2011-01-10
A paradigm based on the absolute equilibrium of Galerkin-truncated inviscid systems to aid in understanding turbulence [T.-D. Lee, "On some statistical properties of hydrodynamical and magnetohydrodynamical fields," Q. Appl. Math. 10, 69 (1952)] is taken to study gyrokinetic plasma turbulence: A finite set of Fourier modes of the collisionless gyrokinetic equations are kept and the statistical equilibria are calculated; possible implications for plasma turbulence in various situations are discussed. For the case of two spatial and one velocity dimension, in the calculation with discretization also of velocity v with N grid points (where N + 1 quantities are conserved, corresponding to an energy invariant and N entropy-related invariants), the negative temperature states, corresponding to the condensation of the generalized energy into the lowest modes, are found. This indicates a generic feature of inverse energy cascade. Comparisons are made with some classical results, such as those of Charney-Hasegawa-Mima in the cold-ion limit. There is a universal shape for statistical equilibrium of gyrokinetics in three spatial and two velocity dimensions with just one conserved quantity. Possible physical relevance to turbulence, such as ITG zonal flows, and to a critical balance hypothesis are also discussed.
Gyrofluid-Gyrokinetic Hybrid Turbulence Model
NASA Astrophysics Data System (ADS)
Dorland, William; Mandell, Noah
2015-11-01
Gyrofluid models of tokamak turbulence are efficient compared to gyrokinetic models, in three senses. First, it is typically easier to develop one's intuition from fluid equations than kinetic equations. Second, because gyrofluid equations are only three-dimensional (instead of 5D or 6D), simulations with gyrofluid models require less memory than kinetic simulations and can therefore more easily fit on highly-optimized computing hardware, such as graphics processors. The third advantage is a result of the first two: one can develop and test ideas quickly with gyrofluid models. The disadvantage of gyrofluid models is their potential lack of physics fidelity. In this poster, we present our attempt to take full advantage of gyrofluid models, without sacrificing physics fidelity. Our approach is encapsulated in the Gryf-X code, which is an implementation of hybrid gyrofluid/gyrokinetic equations. The key improvements that we have brought to bear are: an improved understanding of the cascade of free energy simultaneously in k⊥ and v⊥ an improved model of zonal flow physics; and an implementation of the equations on modern heterogeneous computing platforms, both as a standalone simulation tool and as a component of TRINITY (a transport modeling code for tokamaks).
Advances in stellarator gyrokinetics
NASA Astrophysics Data System (ADS)
Helander, P.; Bird, T.; Jenko, F.; Kleiber, R.; Plunk, G. G.; Proll, J. H. E.; Riemann, J.; Xanthopoulos, P.
2015-05-01
Recent progress in the gyrokinetic theory of stellarator microinstabilities and turbulence simulations is summarized. The simulations have been carried out using two different gyrokinetic codes, the global particle-in-cell code EUTERPE and the continuum code GENE, which operates in the geometry of a flux tube or a flux surface but is local in the radial direction. Ion-temperature-gradient (ITG) and trapped-electron modes are studied and compared with their counterparts in axisymmetric tokamak geometry. Several interesting differences emerge. Because of the more complicated structure of the magnetic field, the fluctuations are much less evenly distributed over each flux surface in stellarators than in tokamaks. Instead of covering the entire outboard side of the torus, ITG turbulence is localized to narrow bands along the magnetic field in regions of unfavourable curvature, and the resulting transport depends on the normalized gyroradius ρ* even in radially local simulations. Trapped-electron modes can be significantly more stable than in typical tokamaks, because of the spatial separation of regions with trapped particles from those with bad magnetic curvature. Preliminary non-linear simulations in flux-tube geometry suggest differences in the turbulence levels in Wendelstein 7-X and a typical tokamak.
NASA Astrophysics Data System (ADS)
Zhang, Chenglong; Gamba, Irene M.
2016-11-01
We propose a deterministic conservative solver for the inhomogeneous Fokker-Planck-Landau equation coupled with Poisson equation. Through time-splitting scheme, a Vlasov-Poisson (collisionless) problem and a homogeneous Landau (collisional) problem are obtained. These two subproblems can be treated separately. We use operator splitting where the transport dynamics for Runge-Kutta Discontinuous Galerkin (RK-DG) method and the collisional dynamics for homogeneous conservative spectral method are adopted respectively. Since two different numerical schemes are applied separately, we have designed a new conservation correction process such that, after projecting the conservative spectral solution onto the DG mesh, there is no loss of moment consvervation. Parallelization is readily implemented. To verify our solver, numerical experiments on linear and nonlinear Landau damping are provided.
The theory of gyrokinetic turbulence: A multiple-scales approach
NASA Astrophysics Data System (ADS)
Plunk, Gabriel Galad
Gyrokinetics is a rich and rewarding playground to study some of the mysteries of modern physics -- such as turbulence, universality, self-organization and dynamic criticality -- which are found in physical systems that are driven far from thermodynamic equilibrium. One such system is of particular importance, as it is central in the development of fusion energy -- this system is the turbulent plasma found in magnetically confined fusion device. In this thesis I present work, motivated by the quest for fusion energy, which seeks to uncover some of the inner workings of turbulence in magnetized plasmas. I present three projects, based on the work of me and my collaborators, which take a tour of different aspects and approaches to the gyrokinetic turbulence problem. I begin with the fundamental theory of gyrokinetics, and a novel formulation of its extension to the equations for mean-scale transport -- the equations which must be solved to determine the performance of Magnetically confined fusion devices. The results of this work include (1) the equations of evolution for the mean scale (equilibrium) density, temperature and magnetic field of the plasma, (2) a detailed Poynting's theorem for the energy balance and (3) the entropy balance equations. The second project presents gyrokinetic secondary instability theory as a mechanism to bring about saturation of the basic instabilities that drive gyrokinetic turbulence. Emphasis is put on the ability for this analytic theory to predict basic properties of the nonlinear state, which can be applied to a mixing length phenomenology of transport. The results of this work include (1) an integral equation for the calculation of the growth rate of the fully gyrokinetic secondary instability with finite Larmor radius (FLR) affects included exactly, (2) the demonstration of the robustness of the secondary instability at fine scales (krhoi for ion temperature gradient (ITG) turbulence and krhoe ≪ 1 for electron temperature
Noiseless Vlasov-Poisson simulations with linearly transformed particles
Pinto, Martin C.; Sonnendrucker, Eric; Friedman, Alex; ...
2014-06-25
We introduce a deterministic discrete-particle simulation approach, the Linearly-Transformed Particle-In-Cell (LTPIC) method, that employs linear deformations of the particles to reduce the noise traditionally associated with particle schemes. Formally, transforming the particles is justified by local first order expansions of the characteristic flow in phase space. In practice the method amounts of using deformation matrices within the particle shape functions; these matrices are updated via local evaluations of the forward numerical flow. Because it is necessary to periodically remap the particles on a regular grid to avoid excessively deforming their shapes, the method can be seen as a development ofmore » Denavit's Forward Semi-Lagrangian (FSL) scheme (Denavit, 1972 [8]). However, it has recently been established (Campos Pinto, 2012 [20]) that the underlying Linearly-Transformed Particle scheme converges for abstract transport problems, with no need to remap the particles; deforming the particles can thus be seen as a way to significantly lower the remapping frequency needed in the FSL schemes, and hence the associated numerical diffusion. To couple the method with electrostatic field solvers, two specific charge deposition schemes are examined, and their performance compared with that of the standard deposition method. Finally, numerical 1d1v simulations involving benchmark test cases and halo formation in an initially mismatched thermal sheet beam demonstrate some advantages of our LTPIC scheme over the classical PIC and FSL methods. Lastly, benchmarked test cases also indicate that, for numerical choices involving similar computational effort, the LTPIC method is capable of accuracy comparable to or exceeding that of state-of-the-art, high-resolution Vlasov schemes.« less
Noiseless Vlasov-Poisson simulations with linearly transformed particles
Pinto, Martin C.; Sonnendrucker, Eric; Friedman, Alex; Grote, David P.; Lund, Steve M.
2014-06-25
We introduce a deterministic discrete-particle simulation approach, the Linearly-Transformed Particle-In-Cell (LTPIC) method, that employs linear deformations of the particles to reduce the noise traditionally associated with particle schemes. Formally, transforming the particles is justified by local first order expansions of the characteristic flow in phase space. In practice the method amounts of using deformation matrices within the particle shape functions; these matrices are updated via local evaluations of the forward numerical flow. Because it is necessary to periodically remap the particles on a regular grid to avoid excessively deforming their shapes, the method can be seen as a development of Denavit's Forward Semi-Lagrangian (FSL) scheme (Denavit, 1972 [8]). However, it has recently been established (Campos Pinto, 2012 [20]) that the underlying Linearly-Transformed Particle scheme converges for abstract transport problems, with no need to remap the particles; deforming the particles can thus be seen as a way to significantly lower the remapping frequency needed in the FSL schemes, and hence the associated numerical diffusion. To couple the method with electrostatic field solvers, two specific charge deposition schemes are examined, and their performance compared with that of the standard deposition method. Finally, numerical 1d1v simulations involving benchmark test cases and halo formation in an initially mismatched thermal sheet beam demonstrate some advantages of our LTPIC scheme over the classical PIC and FSL methods. Lastly, benchmarked test cases also indicate that, for numerical choices involving similar computational effort, the LTPIC method is capable of accuracy comparable to or exceeding that of state-of-the-art, high-resolution Vlasov schemes.
John A. Krommes
2007-10-09
The present state of the theory of fluctuations in gyrokinetic GK plasmas and especially its application to sampling noise in GK particle-in-cell PIC simulations is reviewed. Topics addressed include the Δf method, the fluctuation-dissipation theorem for both classical and GK many-body plasmas, the Klimontovich formalism, sampling noise in PIC simulations, statistical closure for partial differential equations, the theoretical foundations of spectral balance in the presence of arbitrary noise sources, and the derivation of Kadomtsev-type equations from the general formalism.
Gyrokinetic simulations of turbulent transport: size scaling and chaotic behaviour
NASA Astrophysics Data System (ADS)
Villard, L.; Bottino, A.; Brunner, S.; Casati, A.; Chowdhury, J.; Dannert, T.; Ganesh, R.; Garbet, X.; Görler, T.; Grandgirard, V.; Hatzky, R.; Idomura, Y.; Jenko, F.; Jolliet, S.; Khosh Aghdam, S.; Lapillonne, X.; Latu, G.; McMillan, B. F.; Merz, F.; Sarazin, Y.; Tran, T. M.; Vernay, T.
2010-12-01
Important steps towards the understanding of turbulent transport have been made with the development of the gyrokinetic framework for describing turbulence and with the emergence of numerical codes able to solve the set of gyrokinetic equations. This paper presents some of the main recent advances in gyrokinetic theory and computing of turbulence. Solving 5D gyrokinetic equations for each species requires state-of-the-art high performance computing techniques involving massively parallel computers and parallel scalable algorithms. The various numerical schemes that have been explored until now, Lagrangian, Eulerian and semi-Lagrangian, each have their advantages and drawbacks. A past controversy regarding the finite size effect (finite ρ*) in ITG turbulence has now been resolved. It has triggered an intensive benchmarking effort and careful examination of the convergence properties of the different numerical approaches. Now, both Eulerian and Lagrangian global codes are shown to agree and to converge to the flux-tube result in the ρ* → 0 limit. It is found, however, that an appropriate treatment of geometrical terms is necessary: inconsistent approximations that are sometimes used can lead to important discrepancies. Turbulent processes are characterized by a chaotic behaviour, often accompanied by bursts and avalanches. Performing ensemble averages of statistically independent simulations, starting from different initial conditions, is presented as a way to assess the intrinsic variability of turbulent fluxes and obtain reliable estimates of the standard deviation. Further developments concerning non-adiabatic electron dynamics around mode-rational surfaces and electromagnetic effects are discussed.
Cummings, Julian C.
2013-05-15
This project was a collaboration between researchers at the California Institute of Technology and the University of California, Irvine to investigate the utility of a global field-aligned mesh and gyrokinetic field solver for simulations of the tokamak plasma edge region. Mesh generation software from UC Irvine was tested with specific tokamak edge magnetic geometry scenarios and the quality of the meshes and the solutions to the gyrokinetic Poisson equation were evaluated.
A numerical method for solving the Vlasov equation
NASA Technical Reports Server (NTRS)
Satofuka, N.
1982-01-01
A numerical procedure is derived for the solution of the Vlasov-Poisson system of equations in two phase-space variables. Derivatives with respect to the phase-space variables are approximated by a weighted sum of the values of the distribution function at property chosen neighboring points. The resulting set of ordinary differential equations is then solved by using an appropriate time intergration scheme. The accuracy of the proposed method is tested with some simple model problems. The results for the free streaming case, linear Landau damping, and nonlinear Landau damping are investigated and compared with those of the splitting scheme. The proposed method is found to be very accurate and efficient.
Nonlinear gyrokinetic theory for finite-BETA plasmas
Hahm, T.S.; Lee, W.W.; Brizard, A.
1988-02-01
A self-consistent and energy-conserving set of nonlinear gyrokinetic equations, consisting of the averaged Vlasov and Maxwell's equations for finite-..beta.. plasmas, is derived. The method utilized in the present investigation is based on the Hamiltonian formalism and Lie transformation. The resulting formation is valid for arbitrary values of k/perpendicular//rho//sub i/ and, therefore, is most suitable for studying linear and nonlinear evolution of microinstabilities in tokamak plasmas as well as other areas of plasma physics where the finite Larmor radius effects are important. Because the underlying Hamiltonian structure is preserved in the present formalism, these equations are directly applicable to numerical studies based on the existing gyrokinetic particle simulation techniques. 31 refs.
Gyrokinetic Theory and Computational Methods for Electromagnetic Perturbations in Tokamaks
NASA Astrophysics Data System (ADS)
Qin, H.; Tang, W. M.; Rewoldt, G.
1998-11-01
A general gyrokinetic formalism and computational methods have been developed for electromagnetic perturbations in toroidal plasmas. This formalism and the associated numerical code represent the first self-consistent, comprehensive, fully kinetic model for treating both MHD instabilities and electromagnetic drift waves(H. Qin, W. M. Tang, and G. Rewoldt, Phys. Plasmas 5), 1035 (1998). The gyrokinetic system of equations is derived by phase-space Lagrangian Lie perturbation methods. An important component missing from previous gyrokinetic theories, the gyrokinetic perpendicular dynamics, is identified and developed. The corresponding numerical code, KIN-2DEM, has been systematically benchmarked against the high-n FULL code, the PEST code, and the NOVA-K code for kinetic ballooning modes, internal kink modes, and TAEs, respectively. For the internal kink mode, it is found that kinetic effects due to trapped ions can significantly modify the γ vs. q0 curve. For the destabilization of the TAEs by energetic particles, comparisons have been made between the non-perturbative, fully kinetic KIN-2DEM results and the perturbative hybrid NOVA-K results.
Podesta, John J.
2012-08-15
The electric field generated by a time varying point charge in a three-dimensional, unbounded, spatially homogeneous plasma with a uniform background magnetic field and a uniform (static) flow velocity is studied in the electrostatic approximation which is often valid in the near field. For plasmas characterized by Maxwell distribution functions with isotropic temperatures, the linearized Vlasov-Poisson equations may be formulated in terms of an equivalent integral equation in the time domain. The kernel of the integral equation has a relatively simple mathematical form consisting of elementary functions such as exponential and trigonometric functions (sines and cosines), and contains no infinite sums of Bessel functions. Consequently, the integral equation is amenable to numerical solutions and may be useful for the study of the impulse response of magnetized plasmas and, more generally, the response to arbitrary waveforms.
Xiao, Yong; Holod, Ihor; Wang, Zhixuan; Lin, Zhihong; Zhang, Taige
2015-02-15
Developments in gyrokinetic particle simulation enable the gyrokinetic toroidal code (GTC) to simulate turbulent transport in tokamaks with realistic equilibrium profiles and plasma geometry, which is a critical step in the code–experiment validation process. These new developments include numerical equilibrium representation using B-splines, a new Poisson solver based on finite difference using field-aligned mesh and magnetic flux coordinates, a new zonal flow solver for general geometry, and improvements on the conventional four-point gyroaverage with nonuniform background marker loading. The gyrokinetic Poisson equation is solved in the perpendicular plane instead of the poloidal plane. Exploiting these new features, GTC is able to simulate a typical DIII-D discharge with experimental magnetic geometry and profiles. The simulated turbulent heat diffusivity and its radial profile show good agreement with other gyrokinetic codes. The newly developed nonuniform loading method provides a modified radial transport profile to that of the conventional uniform loading method.
A gyrokinetic one-dimensional scrape-off layer model of an edge-localized mode heat pulse
NASA Astrophysics Data System (ADS)
Shi, E. L.; Hakim, A. H.; Hammett, G. W.
2015-02-01
An electrostatic gyrokinetic-based model is applied to simulate parallel plasma transport in the scrape-off layer to a divertor plate. The authors focus on a test problem that has been studied previously, using parameters chosen to model a heat pulse driven by an edge-localized mode in JET. Previous work has used direct particle-in-cell equations with full dynamics, or Vlasov or fluid equations with only parallel dynamics. With the use of the gyrokinetic quasineutrality equation and logical sheath boundary conditions, spatial and temporal resolution requirements are no longer set by the electron Debye length and plasma frequency, respectively. This test problem also helps illustrate some of the physics contained in the Hamiltonian form of the gyrokinetic equations and some of the numerical challenges in developing an edge gyrokinetic code.
A gyrokinetic one-dimensional scrape-off layer model of an edge-localized mode heat pulse
Shi, E. L.; Hakim, A. H.; Hammett, G. W.
2015-02-15
An electrostatic gyrokinetic-based model is applied to simulate parallel plasma transport in the scrape-off layer to a divertor plate. The authors focus on a test problem that has been studied previously, using parameters chosen to model a heat pulse driven by an edge-localized mode in JET. Previous work has used direct particle-in-cell equations with full dynamics, or Vlasov or fluid equations with only parallel dynamics. With the use of the gyrokinetic quasineutrality equation and logical sheath boundary conditions, spatial and temporal resolution requirements are no longer set by the electron Debye length and plasma frequency, respectively. This test problem also helps illustrate some of the physics contained in the Hamiltonian form of the gyrokinetic equations and some of the numerical challenges in developing an edge gyrokinetic code.
Low dimensional gyrokinetic PIC simulation by δf method
NASA Astrophysics Data System (ADS)
Chen, C. M.; Nishimura, Yasutaro; Cheng, C. Z.
2015-11-01
A step by step development of our low dimensional gyrokinetic Particle-in-Cell (PIC) simulation is reported. One dimensional PIC simulation of Langmuir wave dynamics is benchmarked. We then take temporal plasma echo as a test problem to incorporate the δf method. Electrostatic driftwave simulation in one dimensional slab geometry is resumed in the presence of finite density gradients. By carefully diagnosing contour plots of the δf values in the phase space, we discuss the saturation mechanism of the driftwave instabilities. A v∥ formulation is employed in our new electromagnetic gyrokinetic method by solving Helmholtz equation for time derivative of the vector potential. Electron and ion momentum balance equations are employed in the time derivative of the Ampere's law. This work is supported by Ministry of Science and Technology of Taiwan, MOST 103-2112-M-006-007 and MOST 104-2112-M-006-019.
Simulation of neoclassical transport with the continuum gyrokinetic code COGENT
Dorf, M. A.; Cohen, R. H.; Dorr, M.; Rognlien, T.; Hittinger, J.; Compton, J.; Colella, P.; Martin, D.; McCorquodale, P.
2013-01-25
The development of the continuum gyrokinetic code COGENT for edge plasma simulations is reported. The present version of the code models a nonlinear axisymmetric 4D (R, v∥, μ) gyrokinetic equation coupled to the long-wavelength limit of the gyro-Poisson equation. Here, R is the particle gyrocenter coordinate in the poloidal plane, and v∥ and μ are the guiding center velocity parallel to the magnetic field and the magnetic moment, respectively. The COGENT code utilizes a fourth-order finite-volume (conservative) discretization combined with arbitrary mapped multiblock grid technology (nearly field-aligned on blocks) to handle the complexity of tokamak divertor geometry with high accuracy. Furthermore, topics presented are the implementation of increasingly detailed model collision operators, and the results of neoclassical transport simulations including the effects of a strong radial electric field characteristic of a tokamak pedestal under H-mode conditions.
Simulation of neoclassical transport with the continuum gyrokinetic code COGENT
Dorf, M. A.; Cohen, R. H.; Dorr, M.; ...
2013-01-25
The development of the continuum gyrokinetic code COGENT for edge plasma simulations is reported. The present version of the code models a nonlinear axisymmetric 4D (R, v∥, μ) gyrokinetic equation coupled to the long-wavelength limit of the gyro-Poisson equation. Here, R is the particle gyrocenter coordinate in the poloidal plane, and v∥ and μ are the guiding center velocity parallel to the magnetic field and the magnetic moment, respectively. The COGENT code utilizes a fourth-order finite-volume (conservative) discretization combined with arbitrary mapped multiblock grid technology (nearly field-aligned on blocks) to handle the complexity of tokamak divertor geometry with high accuracy.more » Furthermore, topics presented are the implementation of increasingly detailed model collision operators, and the results of neoclassical transport simulations including the effects of a strong radial electric field characteristic of a tokamak pedestal under H-mode conditions.« less
Gyrokinetic treatment of a grazing angle magnetic presheath
NASA Astrophysics Data System (ADS)
Geraldini, A.; Parra, F. I.; Militello, F.
2017-02-01
We develop a gyrokinetic treatment for ions in the magnetic presheath, close to the plasma-wall boundary. We focus on magnetic presheaths with a small magnetic field to wall angle, α \\ll 1 (in radians). Characteristic lengths perpendicular to the wall in such a magnetic presheath scale with the typical ion Larmor orbit size, {ρ }{{i}}. The smallest scale length associated with variations parallel to the wall is taken to be across the magnetic field, and ordered l={ρ }{{i}}/δ , where δ \\ll 1 is assumed. The scale lengths along the magnetic field line are assumed so long that variations associated with this direction are neglected. These orderings are consistent with what we expect close to the divertor target of a tokamak. We allow for a strong component of the electric field {E} in the direction normal to the electron repelling wall, with strong variation in the same direction. The large change of the electric field over an ion Larmor radius distorts the orbit so that it is not circular. We solve for the lowest order orbits by identifying coordinates, which consist of constants of integration, an adiabatic invariant and a gyrophase, associated with periodic ion motion in the system with α =δ =0. By using these new coordinates as variables in the limit α ∼ δ \\ll 1, we obtain a generalised ion gyrokinetic equation. We find another quantity that is conserved to first order and use this to simplify the gyrokinetic equation, solving it in the case of a collisionless magnetic presheath. Assuming a Boltzmann response for the electrons, a form of the quasineutrality equation that exploits the change of variables is derived. The gyrokinetic and quasineutrality equations give the ion distribution function and electrostatic potential in the magnetic presheath if the entrance boundary condition is specified.
Effects of collisions on conservation laws in gyrokinetic field theory
Sugama, H.; Nunami, M.; Watanabe, T.-H.
2015-08-15
Effects of collisions on conservation laws for toroidal plasmas are investigated based on the gyrokinetic field theory. Associating the collisional system with a corresponding collisionless system at a given time such that the two systems have the same distribution functions and electromagnetic fields instantaneously, it is shown how the collisionless conservation laws derived from Noether's theorem are modified by the collision term. Effects of the external source term added into the gyrokinetic equation can be formulated similarly with the collisional effects. Particle, energy, and toroidal momentum balance equations including collisional and turbulent transport fluxes are systematically derived using a novel gyrokinetic collision operator, by which the collisional change rates of energy and canonical toroidal angular momentum per unit volume in the gyrocenter space can be given in the conservative forms. The ensemble-averaged transport equations of particles, energy, and toroidal momentum given in the present work are shown to include classical, neoclassical, and turbulent transport fluxes which agree with those derived from conventional recursive formulations.
Effects of collisions on conservation laws in gyrokinetic field theory
NASA Astrophysics Data System (ADS)
Sugama, H.; Watanabe, T.-H.; Nunami, M.
2015-08-01
Effects of collisions on conservation laws for toroidal plasmas are investigated based on the gyrokinetic field theory. Associating the collisional system with a corresponding collisionless system at a given time such that the two systems have the same distribution functions and electromagnetic fields instantaneously, it is shown how the collisionless conservation laws derived from Noether's theorem are modified by the collision term. Effects of the external source term added into the gyrokinetic equation can be formulated similarly with the collisional effects. Particle, energy, and toroidal momentum balance equations including collisional and turbulent transport fluxes are systematically derived using a novel gyrokinetic collision operator, by which the collisional change rates of energy and canonical toroidal angular momentum per unit volume in the gyrocenter space can be given in the conservative forms. The ensemble-averaged transport equations of particles, energy, and toroidal momentum given in the present work are shown to include classical, neoclassical, and turbulent transport fluxes which agree with those derived from conventional recursive formulations.
Gyrokinetic simulations of the tearing instability
Numata, Ryusuke; Dorland, William; Howes, Gregory G.; Loureiro, Nuno F.; Tatsuno, Tomoya
2011-11-15
Linear gyrokinetic simulations covering the collisional-collisionless transitional regime of the tearing instability are performed. It is shown that the growth rate scaling with collisionality agrees well with that predicted by a two-fluid theory for a low plasma beta case in which ion kinetic dynamics are negligible. Electron wave-particle interactions (Landau damping), finite Larmor radius, and other kinetic effects invalidate the fluid theory in the collisionless regime, in which a general non-polytropic equation of state for pressure (temperature) perturbations should be considered. We also vary the ratio of the background ion to electron temperatures and show that the scalings expected from existing calculations can be recovered, but only in the limit of very low beta.
An asymptotic preserving scheme for the relativistic Vlasov-Maxwell equations in the classical limit
NASA Astrophysics Data System (ADS)
Crouseilles, Nicolas; Einkemmer, Lukas; Faou, Erwan
2016-12-01
We consider the relativistic Vlasov-Maxwell (RVM) equations in the limit when the light velocity c goes to infinity. In this regime, the RVM system converges towards the Vlasov-Poisson system and the aim of this paper is to construct asymptotic preserving numerical schemes that are robust with respect to this limit. Our approach relies on a time splitting approach for the RVM system employing an implicit time integrator for Maxwell's equations in order to damp the higher and higher frequencies present in the numerical solution. A number of numerical simulations are conducted in order to investigate the performances of our numerical scheme both in the relativistic as well as in the classical limit regime. In addition, we derive the dispersion relation of the Weibel instability for the continuous and the discretized problem.
A fully nonlinear characteristic method for gyrokinetic simulation
Parker, S.E.; Lee, W.W.
1992-07-01
We present a new scheme which evolves the perturbed part of the distribution function along a set of characteristics that solves the fully nonlinear gyrokinetic equations. This nonlinear characteristic method for particle simulation is an extension of the partially linear weighting scheme, and may be considered an improvement of existing {delta} f methods. Some of the features of this new method are: the ability to keep all of the nonlinearities, particularly those associated with parallel acceleration; the loading of the physical equilibrium distribution function f{sub o} (e.g., a Maxwellian), with or without the multiple spatial scale approximation; the use of a single of trajectories for the particles; and also, the retention of the conservation properties of the original gyrokinetic system in the numerically converged limit. Therefore, one can take advantage of the low noise property of the weighting scheme together with the quiet start techniques to simulate weak instabilities, with a substantially reduced number of particles than required for a conventional simulation. The new method is used to study a one dimensional drift wave model which isolates the parallel velocity nonlinearity. A mode coupling calculation of the saturation mechanism is given, which is in good agreement with the simulation results and predicts a considerably lower saturation level then the estimate of Sagdeev and Galeev. Finally, we extend the nonlinear characteristic method to the electromagnetic gyrokinetic equations in general geometry.
On push-forward representations in the standard gyrokinetic model
Miyato, N. Yagi, M.; Scott, B. D.
2015-01-15
Two representations of fluid moments in terms of a gyro-center distribution function and gyro-center coordinates, which are called push-forward representations, are compared in the standard electrostatic gyrokinetic model. In the representation conventionally used to derive the gyrokinetic Poisson equation, the pull-back transformation of the gyro-center distribution function contains effects of the gyro-center transformation and therefore electrostatic potential fluctuations, which is described by the Poisson brackets between the distribution function and scalar functions generating the gyro-center transformation. Usually, only the lowest order solution of the generating function at first order is considered to explicitly derive the gyrokinetic Poisson equation. This is true in explicitly deriving representations of scalar fluid moments with polarization terms. One also recovers the particle diamagnetic flux at this order because it is associated with the guiding-center transformation. However, higher-order solutions are needed to derive finite Larmor radius terms of particle flux including the polarization drift flux from the conventional representation. On the other hand, the lowest order solution is sufficient for the other representation, in which the gyro-center transformation part is combined with the guiding-center one and the pull-back transformation of the distribution function does not appear.
Gyrokinetic treatment of GAE modes in cylindrical geometry
Eremin, D.
2006-11-30
Global Alfven eigenmodes (GAEs) are investigated in cylindrical geometry both analytically and numerically. These modes are of particular importance in low-shear magnetic configurations, such as modern stellarators. Analytical treatment starts from the linearised equations of gyrokinetics and yields a generalized dispersion relation for GAE with FLR and kinetic effects taken into account, which is demonstrated to reduce to the well-known MHD counterpart in the appropriate limit. An eigenvalue code is developed to solve the dispersion relation, which is used to investigate the kinetic analogs of GAE modes in various regimes with different beta. On the other hand, GAE modes are simulated with global linear particle-in-cell (PIC) electromagnetic gyrokinetic code following self-consistent time evolution of electromagnetic fields and plasma. GAE modes are observed and their damping rate agrees with predictions made by the eigenvalue code.
Gyro-water-bag approach in nonlinear gyrokinetic turbulence
Besse, Nicolas Bertrand, Pierre
2009-06-20
Turbulent transport is a key issue for controlled thermonuclear fusion based on magnetic confinement. The thermal confinement of a magnetized fusion plasma is essentially determined by the turbulent heat conduction across the equilibrium magnetic field. It has long been acknowledged, that the prediction of turbulent transport requires to solve Vlasov-type gyrokinetic equations. Although the kinetic description is more accurate than fluid models (MHD, gyro-fluid), because among other things it takes into account nonlinear resonant wave-particle interaction, kinetic modeling has the drawback of a huge computer resource request. An unifying approach consists in considering water-bag-like weak solutions of kinetic collisionless equations, which allow to reduce the full kinetic Vlasov equation into a set of hydrodynamic equations, while keeping its kinetic behaviour. As a result this exact reduction induces a multi-fluid numerical resolution cost. Therefore finding water-bag-like weak solutions of the gyrokinetic equations leads to the birth of the gyro-water-bag model. This model is suitable for studying linear and nonlinear low-frequency micro-instabilities and the associated anomalous transport in magnetically-confined plasmas. The present paper addresses the derivation of the nonlinear gyro-water-bag model, its quasilinear approximation and their numerical approximations by Runge-Kutta semi-Lagrangian methods and Runge-Kutta discontinuous Galerkin schemes respectively.
Gyrokinetic Turbulence Simulations for Stellarators
NASA Astrophysics Data System (ADS)
Merz, F.; Xanthopoulos, P.; Gorler, T.; Jenko, F.; Mikkelsen, D.
2007-11-01
While there is an abundance of publications on plasma microturbulence in tokamaks, not much is presently known about its character in nonaxisymmetric devices. The present work constitutes the first attempt to investigate turbulent transport in modern stellarators, using the gyrokinetic turbulence code Gene and realistic magnetic equilibria. First, linear and nonlinear gyrokinetic simulations of ion-temperature-gradient (ITG) and trapped electron modes are presented for the optimized stellarator Wendelstein 7-X which is currently under construction at Greifswald, Germany. The newly developed code Tracer -- based on field line tracing -- is employed to extract the required geometric information from the MHD equilibria [Phys. Plasmas 13, 092301 (2006)]. Extensive linear studies reveal substantial differences with respect to axisymmetric geometry [Phys. Plasmas 14, 042501 (2007)]. Nonlinear ITG simulations are also presented [Phys. Rev. Lett., in print]. Several fundamental features are discussed, including the role of zonal flows for turbulence saturation, the resulting flux-gradient relationship and the co-existence of ITG modes with trapped ion modes in the saturated state. Similar studies will be presented for the stellarator experiment NCSX at PPPL with the aim to comprehend the effects of quasi-axisymmetric geometry on the properties - both linear and nonlinear - of various microinstabilities.
Deng, Zhao; Waltz, R. E.
2015-05-15
This paper presents numerical simulations of the nonlinear cyclokinetic equations in the cyclotron harmonic representation [R. E. Waltz and Zhao Deng, Phys. Plasmas 20, 012507 (2013)]. Simulations are done with a local flux-tube geometry and with the parallel motion and variation suppressed using a newly developed rCYCLO code. Cyclokinetic simulations dynamically follow the high-frequency ion gyro-phase motion which is nonlinearly coupled into the low-frequency drift-waves possibly interrupting and suppressing gyro-averaging and increasing the transport over gyrokinetic levels. By comparing the more fundamental cyclokinetic simulations with the corresponding gyrokinetic simulations, the breakdown of gyrokinetics at high turbulence levels is quantitatively tested over a range of relative ion cyclotron frequency 10 < Ω*{sup }< 100 where Ω*{sup }= 1/ρ*, and ρ* is the relative ion gyroradius. The gyrokinetic linear mode rates closely match the cyclokinetic low-frequency rates for Ω*{sup }> 5. Gyrokinetic transport recovers cyclokinetic transport at high relative ion cyclotron frequency (Ω*{sup }≥ 50) and low turbulence level as required. Cyclokinetic transport is found to be lower than gyrokinetic transport at high turbulence levels and low-Ω* values with stable ion cyclotron (IC) modes. The gyrokinetic approximation is found to break down when the density perturbations exceed 20%. For cyclokinetic simulations with sufficiently unstable IC modes and sufficiently low Ω*{sup }∼ 10, the high-frequency component of cyclokinetic transport level can exceed the gyrokinetic transport level. However, the low-frequency component of the cyclokinetic transport and turbulence level does not exceed that of gyrokinetics. At higher and more physically relevant Ω*{sup }≥ 50 values and physically realistic IC driving rates, the low-frequency component of the cyclokinetic transport and turbulence level is still smaller than that of
Gyrokinetic modeling: A multi-water-bag approach
Morel, P.; Gravier, E.; Besse, N.; Klein, R.; Ghizzo, A.; Bertrand, P.; Garbet, X.; Ghendrih, P.; Grandgirard, V.; Sarazin, Y.
2007-11-15
Predicting turbulent transport in nearly collisionless fusion plasmas requires one to solve kinetic (or, more precisely, gyrokinetic) equations. In spite of considerable progress, several pending issues remain; although more accurate, the kinetic calculation of turbulent transport is much more demanding in computer resources than fluid simulations. An alternative approach is based on a water-bag representation of the distribution function that is not an approximation but rather a special class of initial conditions, allowing one to reduce the full kinetic Vlasov equation into a set of hydrodynamic equations while keeping its kinetic character. The main result for the water-bag model is a lower cost in the parallel velocity direction since no differential operator associated with some approximate numerical scheme has to be carried out on this variable v{sub parallel}. Indeed, a small bag number is sufficient to correctly describe the ion temperature gradient instability.
Gyrokinetic Simulation of Reverse Shear Alfven Eigenmodes in DIII-D Plasmas
NASA Astrophysics Data System (ADS)
Chen, Yang; Parker, Scott; Fu, Guo-Yong
2012-03-01
We present simulation results of the beam driven Reverse Shear Alfven Eigenmodes (RSAE) observed in DIII-D discharge 142111 using the Particle-in-Cell gyrokinetic code GEM [1]. Bulk ions and energetic particles are gyrokinetic, but electrons are described by a mass-less fluid model. Two schemes for obtaining the electric potential are implemented, one by solving the gyrokinetic Poisson equation for φ directly, the other by solving the gyrokinetic moment (GKM) equation for φ/t and then integrating in time. The GKM approach is found to be more robust for linear simulations (allowing larger time steps) but less robust for nonlinear simulations. Previous simulations reproduced the chirping in frequency as seen in the experiment. Recently it has been reported by other simulation codes (GTC, GYRO and TAEFL) that the shearing direction of the mode structure in the poloidal plane disagrees with observation. We found that the mode structure, including the shearing in the poloidal plane, is in general sensitive to the beam distribution. By changing the radial profile of the beam density while keeping the velocity dependence fixed, both shearing directions can be produced in the simulation. [4pt] [1] Y. Chen and S. E. Parker, J. Comp. Phys. 220, 839 (2007)
A gyrokinetic one-dimensional scrape-off layer model of an edge-localized mode heat pulse
Shi, E. L.; Hakim, A. H.; Hammett, G. W.
2015-02-03
An electrostatic gyrokinetic-based model is applied to simulate parallel plasma transport in the scrape-off layer to a divertor plate. We focus on a test problem that has been studied previously, using parameters chosen to model a heat pulse driven by an edge-localized mode in JET. Previous work has used direct particle-in-cellequations with full dynamics, or Vlasov or fluid equations with only parallel dynamics. With the use of the gyrokinetic quasineutrality equation and logical sheathboundary conditions, spatial and temporal resolution requirements are no longer set by the electron Debye length and plasma frequency, respectively. Finally, this test problem also helps illustratemore » some of the physics contained in the Hamiltonian form of the gyrokineticequations and some of the numerical challenges in developing an edge gyrokinetic code.« less
A gyrokinetic one-dimensional scrape-off layer model of an edge-localized mode heat pulse
Shi, E. L.; Hakim, A. H.; Hammett, G. W.
2015-02-03
An electrostatic gyrokinetic-based model is applied to simulate parallel plasma transport in the scrape-off layer to a divertor plate. We focus on a test problem that has been studied previously, using parameters chosen to model a heat pulse driven by an edge-localized mode in JET. Previous work has used direct particle-in-cellequations with full dynamics, or Vlasov or fluid equations with only parallel dynamics. With the use of the gyrokinetic quasineutrality equation and logical sheathboundary conditions, spatial and temporal resolution requirements are no longer set by the electron Debye length and plasma frequency, respectively. Finally, this test problem also helps illustrate some of the physics contained in the Hamiltonian form of the gyrokineticequations and some of the numerical challenges in developing an edge gyrokinetic code.
Gyrokinetic theory of the screw pinch
NASA Astrophysics Data System (ADS)
Stéfant, Robert J.
1989-06-01
The gyrokinetic differential equation for waves propagating in a hot collisionless current-carrying plasma is derived in cylindrical geometry. It is shown that the averaging of the wave electric field over the ion Larmor circle leads to a transcendental differential equation (d.e.) of infinite order in the radial derivative. This reduces to a d.e. of sixth order when the scale length of the plasma inhomogeneity Ln is greater than the ion gyroradius ρi by a factor (M/m)1/2, where M and m are, respectively, the ion and electron mass. This sixth-order d.e. describes the properties of the two (compressional and torsional) Alfvén modes and the ion acoustic mode. When Ln <(M/m)1/2ρi, the plasma can only support modes of the magnetokinetic (short wavelength) type. In the absence of a finite Larmor radius (FLR) effect, shear, and equilibrium current, we find that the correct equation to start with is the Hain and Lust [Z. Naturforsch. A 13, 936 (1958)] d.e. of second order that is singular at the Alfvén resonance layer (ARL). The ARL behaves in that case like a Budden absorption layer that traps the global Alfvén eigenmodes (GAEM) inside the plasma cavity where they are damped by transit time magnetic pumping (TTMP). The logarithmic singularity does not disappear with the introduction of the FLR effect in the Hain and Lust d.e., but only with the TTMP damping term. There is no mode conversion between the fast magnetosonic mode and the shear or magnetokinetic mode at the ARL or anywhere in the plasma. In the presence of shear and equilibrium current, the correct equation to use is a d.e. of fourth (or greater) order whose solutions descibe the shear Alfvén mode in the long wavelength limit or the magnetokinetic mode at shorter wavelengths. In the true magnetohydrodynamic (MHD) limit, both modes become degenerate. It is shown that the slow Alfvén eigenmodes are (almost) completely decoupled from the fast magnetosonic wave and therefore the growth rates show no
Gyrokinetic Studies of Magnetic Reconnection
NASA Astrophysics Data System (ADS)
Pueschel, Moritz J.; Jenko, Frank; Told, Daniel; Buechner, Joerg
2011-10-01
Collisionless magnetic reconnection constitutes an effective mechanism for particle acceleration in astrophysical plasmas, in particular the solar corona. In addition, it is also of relevance to fusion experiments. Gyrokinetic simulations with the GENE code are performed to explore the temporal evolution of current sheets in two-dimensional slab geometry with a strong guide field. After successful code-code benchmarking, Extensive parameter studies are performed, covering a wide range of physical scenarios. In particular, differing findings regarding the influence of the ion temperature are explained. In its nonlinear phase, the characteristics of the reconnection process depend on whether the system is driven or decaying. Decaying turbulence sees an inverse cascade, and all energy is ultimately transferred to the largest radial scale. If driven by a Krook-type term, the system develops into a turbulent, quasi-stationary state. An important quantity to investigate in nonlinear simulations is the parallel electric field which is able to accelerate particles along the background magnetic field. The spatial structure of this field is studied for the different nonlinear cases, and its amplitude reported as a function of the drive frequency.
Dynamic procedure for filtered gyrokinetic simulations
Morel, P.; Banon Navarro, A.; Albrecht-Marc, M.; Carati, D.; Merz, F.; Goerler, T.; Jenko, F.
2012-01-15
Large eddy simulations (LES) of gyrokinetic plasma turbulence are investigated as interesting candidates to decrease the computational cost. A dynamic procedure is implemented in the gene code, allowing for dynamic optimization of the free parameters of the LES models (setting the amplitudes of dissipative terms). Employing such LES methods, one recovers the free energy and heat flux spectra obtained from highly resolved direct numerical simulations. Systematic comparisons are performed for different values of the temperature gradient and magnetic shear, parameters which are of prime importance in ion temperature gradient driven turbulence. Moreover, the degree of anisotropy of the problem, which can vary with parameters, can be adapted dynamically by the method that shows gyrokinetic large eddy simulation to be a serious candidate to reduce numerical cost of gyrokinetic solvers.
The nonlinear gyro-kinetic flux tube code GKW
NASA Astrophysics Data System (ADS)
Peeters, A. G.; Camenen, Y.; Casson, F. J.; Hornsby, W. A.; Snodin, A. P.; Strintzi, D.; Szepesi, G.
2009-12-01
A new nonlinear gyro-kinetic flux tube code (GKW) for the simulation of micro instabilities and turbulence in magnetic confinement plasmas is presented in this paper. The code incorporates all physics effects that can be expected from a state of the art gyro-kinetic simulation code in the local limit: kinetic electrons, electromagnetic effects, collisions, full general geometry with a coupling to a MHD equilibrium code, and E×B shearing. In addition the physics of plasma rotation has been implemented through a formulation of the gyro-kinetic equation in the co-moving system. The gyro-kinetic model is five-dimensional and requires a massive parallel approach. GKW has been parallelised using MPI and scales well up to 8192+ cores. The paper presents the set of equations solved, the numerical methods, the code structure, and the essential benchmarks. Program summaryProgram title: GKW Catalogue identifier: AEES_v1_0 Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEES_v1_0.html Program obtainable from: CPC Program Library, Queen's University, Belfast, N. Ireland Licensing provisions: GNU GPL v3 No. of lines in distributed program, including test data, etc.: 29 998 No. of bytes in distributed program, including test data, etc.: 206 943 Distribution format: tar.gz Programming language: Fortran 95 Computer: Not computer specific Operating system: Any for which a Fortran 95 compiler is available Has the code been vectorised or parallelised?: Yes. The program can efficiently utilise 8192+ processors, depending on problem and available computer. 128 processors is reasonable for a typical nonlinear kinetic run on the latest x86-64 machines. RAM:˜128 MB-1 GB for a linear run; 25 GB for typical nonlinear kinetic run (30 million grid points) Classification: 19.8, 19.9, 19.11 External routines: None required, although the functionality of the program is somewhat limited without a MPI implementation (preferably MPI-2) and the FFTW3 library. Nature of problem: Five
Self-consistent modeling of multiscale gyrokinetics and transport
NASA Astrophysics Data System (ADS)
Parker, Jeffrey; Lodestro, Lynda; Told, Daniel; Jenko, Frank
2016-10-01
In the core of tokamak plasmas, a separation of timescales between turbulence and transport makes direct simulation of both processes computationally expensive. A workable, practical method to exploit the separation of timescales will be a key component in enabling the self-consistent solution of macroscopic profiles of density and temperature. We report on progress to implement the LoDestro scheme coupled with the gyrokinetic code GENE to perform for the first time coupled turbulence and transport simulations using a global gyrokinetic code. One of the advantages of the LoDestro scheme, which is essentially a method of solving an implicitly advanced nonlinear transport problem, is that it does not use Newton iteration and hence avoids difficulties that arise from calculating Jacobians or Jacobian-vector products in the presence of noisy fluxes. Instead, the implicit timestep equation is solved with an iteration scheme by representing the turbulent flux as the sum of diffusive and convective pieces, after which Picard iteration is used to converge to the self-consistent solution. Preliminary results will be presented. Prepared by LLNL under Contract DE-AC52-07NA27344.
Gyrokinetic theory and simulation of angular momentum transport
Waltz, R. E.; Staebler, G. M.; Candy, J.; Hinton, F. L.
2007-12-15
A gyrokinetic theory of turbulent toroidal angular momentum transport as well as modifications to neoclassical poloidal rotation from turbulence is formulated starting from the fundamental six-dimensional kinetic equation. The gyro-Bohm scaled transport is evaluated from toroidal delta-f gyrokinetic simulations using the GYRO code [Candy and Waltz, J. Comput. Phys. 186, 545 (2003)]. The simulations recover two pinch mechanisms in the radial transport of toroidal angular momentum: The slab geometry ExB shear pinch [Dominguez and Staebler, Phys. Fluids B 5, 387 (1993)] and the toroidal geometry 'Coriolis' pinch [Peeters, Angioni, and Strintzi, Phys. Rev. Lett. 98, 265003 (2007)]. The pinches allow the steady state null stress (or angular momentum transport flow) condition required to understand intrinsic (or spontaneous) toroidal rotation in heated tokamak without an internal source of torque [Staebler, Kinsey, and Waltz, Bull. Am. Phys. Soc. 46, 221 (2001)]. A predicted turbulent shift in the neoclassical poloidal rotation [Staebler, Phys. Plasmas 11, 1064 (2004)] appears to be small at the finite relative gyroradius (rho-star) of current experiments.
Gyrokinetic simulation of microturbulence in EAST tokamak
NASA Astrophysics Data System (ADS)
Xiao, Yong; Zhang, Taige; Zhao, Chen
2014-10-01
A complete understanding of anomalous transport is critical for designing future magnetic fusion reactors. It is generally accepted that the micro-scale turbulence leads to anomalous transport. For low beta toroidal plasmas, the electrostatic modes may dominate and ion temperature gradient (ITG) mode and trapped electron mode (TEM) are two very important candidates accounting for ion and electron turbulent transport respectively. Recently the massively parallel gyrokinetic simulation has emerged as a major tool to investigate the nonlinear physics of the turbulent transport. The newly-developed capabilities enable the gyrokinetic code GTC to simulate the turbulent transport for real tokamak plasma shape and profiles. These capabilities include a new gyrokinetic Poisson solver and zonal flow solver suitable for general plasma shape and profiles, improvements on the conventional four-point gyroaverage and newly-developed nonuniform initial marker loading. The GTC code is now able to import experimental plasma profiles and equilibrium magnetic field that come from the EFIT or TRANSP equilibrium reconstruction. Linear and nonlinear gyrokinetic simulations are carried out with the new capabilities in GTC for the electron coherent mode (ECM) recently observed in the EAST tokamak (EAST shot # 38300). We found that in the pedestal region with strong electron temperature gradient, the unstable waves propagate in the electron diamagnetic direction, showing a trapped electron mode (TEM) feature. It is also found in the collisionless limit, the linear mode frequency is higher than that from the experiment.
3D hybrid simulations with gyrokinetic particle ions and fluid electrons
Belova, E.V.; Park, W.; Fu, G.Y.; Strauss, H.R.; Sugiyama, L.E.
1998-12-31
The previous hybrid MHD/particle model (MH3D-K code) represented energetic ions as gyrokinetic (or drift-kinetic) particles coupled to MHD equations using the pressure or current coupling scheme. A small energetic to bulk ion density ratio was assumed, n{sub h}/n{sub b} {much_lt} 1, allowing the neglect of the energetic ion perpendicular inertia in the momentum equation and the use of MHD Ohm`s law E = {minus}v{sub b} {times} B. A generalization of this model in which all ions are treated as gyrokinetic/drift-kinetic particles and fluid description is used for the electron dynamics is considered in this paper.
Gyrokinetic simulations of ETG Turbulence*
NASA Astrophysics Data System (ADS)
Nevins, William
2005-10-01
Recent gyrokinetic simulations of electron temperature gradient (ETG) turbulence [1,2] produced different results despite similar plasma parameters. Ref.[1] differs from Ref.[2] in that [1] eliminates magnetically trapped particles ( r/R=0 ), while [2] retains magnetically trapped particles ( r/R 0.18 ). Differences between [1] and [2] have been attributed to insufficient phase-space resolution and novel physics associated with toroidicity and/or global simulations[2]. We have reproduced the results reported in [2] using a flux-tube, particle-in-cell (PIC) code, PG3EQ[3], thereby eliminating global effects as the cause of the discrepancy. We observe late-time decay of ETG turbulence and the steady-state heat transport in agreement with [2], and show this results from discrete particle noise. Discrete particle noise is a numerical artifact, so both the PG3EQ simulations reported here and those reported in Ref.[2] have little to say about steady-state ETG turbulence and the associated anomalous electron heat transport. Our attempts to benchmark PIC and continuum[4] codes at the plasma parameters used in Ref.[2] produced very large, intermittent transport. We will present an alternate benchmark point for ETG turbulence, where several codes reproduce the same transport levels. Parameter scans about this new benchmark point will be used to investigate the parameter dependence of ETG transport and to elucidate saturation mechanisms proposed in Refs.[1,2] and elsewhere[5-7].*In collaboration with A. Dimits (LLNL), J. Candy, C. Estrada-Mila (GA), W. Dorland (U of MD), F. Jenko, T. Dannert (Max-Planck Institut), and G. Hammett (PPPL). Work at LLNL performed for US DOE under Contract W7405-ENG-48.[1] F. Jenko and W. Dorland, PRL 89, 225001 (2002).[2] Z. Lin et al, 2004 Sherwood Mtg.; 2004 TTF Mtg.; Fusion Energy 2004 (IAEA, Vienna, 2005); Bull. Am. Phys. Soc. (November, 2004); 2005 TTF Mtg.; 2005 Sherwood Mtg.; Z. Lin, et al, Phys. Plasmas 12, 056125 (2005). [3] A.M. Dimits
Status of Continuum Edge Gyrokinetic Code Physics Development
Xu, X Q; Xiong, Z; Dorr, M R; Hittinger, J A; Kerbel, G D; Nevins, W M; Cohen, B I; Cohen, R H
2005-05-31
We are developing an edge gyro-kinetic continuum simulation code to study the boundary plasma over a region extending from inside the H-mode pedestal across the separatrix to the divertor plates. A 4-D ({psi}, {theta}, {epsilon}, {mu}) version of this code is presently being implemented, en route to a full 5-D version. A set of gyrokinetic equations[1] are discretized on computational grid which incorporates X-point divertor geometry. The present implementation is a Method of Lines approach where the phase-space derivatives are discretized with finite differences and implicit backwards differencing formulas are used to advance the system in time. A fourth order upwinding algorithm is used for particle cross-field drifts, parallel streaming, and acceleration. Boundary conditions at conducting material surfaces are implemented on the plasma side of the sheath. The Poisson-like equation is solved using GMRES with multi-grid preconditioner from HYPRE. A nonlinear Fokker-Planck collision operator from STELLA[2] in ({nu}{sub {parallel}},{nu}{sub {perpendicular}}) has been streamlined and integrated into the gyro-kinetic package using the same implicit Newton-Krylov solver and interpolating F and dF/dt|{sub coll} to/from ({epsilon}, {mu}) space. With our 4D code we compute the ion thermal flux, ion parallel velocity, self-consistent electric field, and geo-acoustic oscillations, which we compare with standard neoclassical theory for core plasma parameters; and we study the transition from collisional to collisionless end-loss. In the real X-point geometry, we find that the particles are trapped near outside midplane and in the X-point regions due to the magnetic configurations. The sizes of banana orbits are comparable to the pedestal width and/or the SOL width for energetic trapped particles. The effect of the real X-point geometry and edge plasma conditions on standard neoclassical theory will be evaluated, including a comparison of our 4D code with other kinetic
Benchmarking finite- β ITG gyrokinetic simulations
NASA Astrophysics Data System (ADS)
Nevins, W. M.; Dimits, A. M.; Candy, J.; Holland, C.; Howard, N.
2016-10-01
We report the results of an electromagnetic gyrokinetic-simulation benchmarking study based on a well-diagnosed ion-temperature-gradient (ITG)-turbulence dominated experimental plasma. We compare the 4x3 matrix of transport/transfer quantities for each plasma species; namely the (a) particle flux, Γa, (b) momentum flux, Πa, (c) energy flux, Qa, and (d) anomalous heat exchange, Sa, with each transport coefficient broken down into: (1) electrostatic (δφ) (2) transverse electromagnetic (δA∥) , and (3) compressional electromagnetic, (δB∥) contributions. We compare realization-independent quantities (correlation functions, spectral densities, etc.), which characterize the fluctuating fields from various gyrokinetic simulation codes. Prepared for US DOE by LLNL under Contract DE-AC52-07NA27344 and by GA under Contract DE-FG03-95ER54309. This work was supported by the U.S. DOE, Office of Science, Fusion Energy Sciences.
Chasing Hamiltonian structure in gyrokinetic theory
Burby, J. W.
2015-09-01
Hamiltonian structure is pursued and uncovered in collisional and collisionless gyrokinetic theory. A new Hamiltonian formulation of collisionless electromagnetic theory is presented that is ideally suited to implementation on modern supercomputers. The method used to uncover this structure is described in detail and applied to a number of examples, where several well-known plasma models are endowed with a Hamiltonian structure for the first time. The first energy- and momentum-conserving formulation of full-F collisional gyrokinetics is presented. In an effort to understand the theoretical underpinnings of this result at a deeper level, a stochastic Hamiltonian modeling approach is presented and applied to pitch angle scattering. Interestingly, the collision operator produced by the Hamiltonian approach is equal to the Lorentz operator plus higher-order terms, but does not exactly conserve energy. Conversely, the classical Lorentz collision operator is provably not Hamiltonian in the stochastic sense.
Neoclassical simulation of tokamak plasmas using the continuum gyrokinetic code TEMPEST.
Xu, X Q
2008-07-01
We present gyrokinetic neoclassical simulations of tokamak plasmas with a self-consistent electric field using a fully nonlinear (full- f ) continuum code TEMPEST in a circular geometry. A set of gyrokinetic equations are discretized on a five-dimensional computational grid in phase space. The present implementation is a method of lines approach where the phase-space derivatives are discretized with finite differences, and implicit backward differencing formulas are used to advance the system in time. The fully nonlinear Boltzmann model is used for electrons. The neoclassical electric field is obtained by solving the gyrokinetic Poisson equation with self-consistent poloidal variation. With a four-dimensional (psi,theta,micro) version of the TEMPEST code, we compute the radial particle and heat fluxes, the geodesic-acoustic mode, and the development of the neoclassical electric field, which we compare with neoclassical theory using a Lorentz collision model. The present work provides a numerical scheme for self-consistently studying important dynamical aspects of neoclassical transport and electric field in toroidal magnetic fusion devices.
Proposal of a brand-new gyrokinetic algorithm for global MHD simulation
NASA Astrophysics Data System (ADS)
Naitou, Hiroshi; Kobayashi, Kenichi; Hashimoto, Hiroki; Andachi, Takehisa; Lee, Wei-Li; Tokuda, Shinji; Yagi, Masatoshi
2009-11-01
A new algorithm for the gyrokinetic PIC code is proposed. The basic equations are energy conserving and composed of (1) the gyrokinetic Vlasov (GKV) equation, (2) the Vortex equation, and (3) the generalized Ohm's law along the magnetic field. Equation (2) is used to advance electrostatic potential in time. Equation (3) is used to advance longitudinal component of vector potential in time as well as estimating longitudinal induced electric field to accelerate charged particles. The particle information is used to estimate pressure terms in equation (3). The idea was obtained in the process of reviewing the split-weight-scheme formalism. This algorithm was incorporated in the Gpic-MHD code. Preliminary results for the m=1/n=1 internal kink mode simulation in the cylindrical geometry indicate good energy conservation, quite low noise due to particle discreteness, and applicability to larger spatial scale and higher beta regimes. The advantage of new Gpic-MHD is that the lower order moments of the GKV equation are estimated by the moment equation while the particle information is used to evaluate the second order moment.
Generalized Covariant Gyrokinetic Dynamics of Magnetoplasmas
Cremaschini, C.; Tessarotto, M.; Nicolini, P.; Beklemishev, A.
2008-12-31
A basic prerequisite for the investigation of relativistic astrophysical magnetoplasmas, occurring typically in the vicinity of massive stellar objects (black holes, neutron stars, active galactic nuclei, etc.), is the accurate description of single-particle covariant dynamics, based on gyrokinetic theory (Beklemishev et al., 1999-2005). Provided radiation-reaction effects are negligible, this is usually based on the assumption that both the space-time metric and the EM fields (in particular the magnetic field) are suitably prescribed and are considered independent of single-particle dynamics, while allowing for the possible presence of gravitational/EM perturbations driven by plasma collective interactions which may naturally arise in such systems. The purpose of this work is the formulation of a generalized gyrokinetic theory based on the synchronous variational principle recently pointed out (Tessarotto et al., 2007) which permits to satisfy exactly the physical realizability condition for the four-velocity. The theory here developed includes the treatment of nonlinear perturbations (gravitational and/or EM) characterized locally, i.e., in the rest frame of a test particle, by short wavelength and high frequency. Basic feature of the approach is to ensure the validity of the theory both for large and vanishing parallel electric field. It is shown that the correct treatment of EM perturbations occurring in the presence of an intense background magnetic field generally implies the appearance of appropriate four-velocity corrections, which are essential for the description of single-particle gyrokinetic dynamics.
Hybrid Gyrofluid/Gyrokinetic Modeling of Tokamak Turbulence with GryfX
NASA Astrophysics Data System (ADS)
Mandell, Noah; Dorland, Bill; Highcock, Edmund; Hammett, Greg
2016-10-01
Gyrofluid models are more efficient than gyrokinetic models, but have a disadvantage in their potential lack of physics fidelity. Here we present three major improvements to the physics fidelity and speed of gyrofluid models, which we encapsulate in the GryfX gyrofluid turbulence code. First, we implement a new nonlinear closure to model the cascade of free energy simultaneously in k⊥ and v⊥ via nonlinear phase-mixing (NLPM). Second, we use a hybrid algorithm that improves zonal flow physics by simulating zonal flow modes with a fully gyrokinetic model. These two improvements bring heat flux predictions from nonlinear GryfX simulations into agreement with the gyrokinetic code GS2. Third, we implement the equations on modern heterogeneous computing platforms, both as a standalone simulation tool that exploits the power of GPUs and as a component of TRINITY (a transport modeling code for tokamaks). GryfX has a roughly 1,200 times performance advantage over GS2 due to the combination of GPU acceleration and the reduction of hundreds of velocity space grid points to six gyrofluid moments. This makes GryfX ideal for large parameter scans, and enables the use of the TRINITY-GryfX system for efficient multi-scale analysis of tokamak turbulence on transport time scales. Present address: Chalmers University, Gothenburg, Sweden.
A conservative semi-Lagrangian HWENO method for the Vlasov equation
NASA Astrophysics Data System (ADS)
Cai, Xiaofeng; Qiu, Jianxian; Qiu, Jing-Mei
2016-10-01
In this paper, we propose a high order conservative semi-Lagrangian (SL) finite difference Hermite weighted essentially non-oscillatory (HWENO) method for the Vlasov equation based on dimensional splitting. HWENO was first proposed for solving nonlinear hyperbolic problems by evolving both function values and its first derivative values (Qiu and Shu (2004) [23]). The major advantage of HWENO, compared with the original WENO, lies in its compactness in reconstruction stencils. There are several new ingredients in this paper. Firstly we propose a mass-conservative SL HWENO scheme for a 1-D equation by working with a flux-difference form, following the work of Qiu and Christlieb (2010) [25]. Secondly, we propose a proper splitting for equations of partial derivatives in HWENO framework to ensure local mass conservation. The proposed fifth order SL HWENO scheme with dimensional splitting has been tested to work well in capturing filamentation structures without oscillations when the time step size is within the Eulerian CFL constraint. However, when the time stepping size becomes larger, numerical oscillations are observed for the 'mass conservative' dimensional splitting HWENO scheme, as there are extra source terms in equations of partial derivatives. In this case, we introduce WENO limiters to control oscillations. Classical numerical examples on linear passive transport problems, as well as the nonlinear Vlasov-Poisson system, have been tested to demonstrate the performance of the proposed scheme.
Gyrokinetic Simulation of TAE in Fusion plasmas
NASA Astrophysics Data System (ADS)
Wang, Zhixuan
Linear gyrokinetic simulation of fusion plasmas finds a radial localization of the toroidal Alfvén eigenmodes (TAE) due to the non-perturbative energetic particles (EP) contribution. The EP-driven TAE has a radial mode width much smaller than that predicted by the magnetohydrodynamic (MHD) theory. The TAE radial position stays around the strongest EP pressure gradients when the EP profile evolves. The non-perturbative EP contribution is also the main cause for the breaking of the radial symmetry of the ballooning mode structure and for the dependence of the TAE frequency on the toroidal mode number. These phenomena are beyond the picture of the conventional MHD theory. Linear gyrokinetic simulation of the electron cyclotron heating (ECH) experiments on DIII-D successfully recover the TAE and RSAE. The EP profile, rather than the electron temperature, is found to be the key factor determining whether TAE or RSAE is the dominant mode in the system in our simulation. Investigation on the nonlinear gyrokinetic simulation model reveals a missing nonlinear term which has important contributions to the zonal magnetic fields. A new fluid-electron hybrid model is proposed to keep this nonlinear term in the lowest order fluid part. Nonlinear simulation of TAE using DIII-D parameters confirms the importance of this new term for the zonal magnetic fields. It is also found that zonal structures dominated by zonal electric fields are forced driven at about twice the linear growth rate of TAE in the linear phase. The zonal flows then limit the nonlinear saturation level by tearing the eigenmode structures apart. In the nonlinear phase of the simulation, the major frequency in the system chirps down by about 30% and stays there.
A very general electromagnetic gyrokinetic formalism
NASA Astrophysics Data System (ADS)
McMillan, B. F.; Sharma, A.
2016-09-01
We derive a gyrokinetic formalism which is very generally valid: the ordering allows both large inhomogeneities in plasma flow and magnetic field at long wavelength, such as typical drift-kinetic theories, as well as fluctuations at the gyro-scale. The underlying approach is to order the vorticity to be small, and to assert that the timescales in the local plasma frame are long compared to the gyrofrequency. Unlike most other derivations, we do not treat the long and short wavelength components of the fluctuating fields separately; the single-field description defines the particle motion and their interaction with the electromagnetic field at small-scale, the system-scale, and intermediate length scales in a unified fashion. As in earlier literature, the work consists of identifying a coordinate system where the gyroangle-dependent terms are small, and using a near-unity transform to systematically find a set of coordinates where the gyroangle dependence vanishes. We derive a gyrokinetic Lagrangian which is valid where the vorticity | ∇ × ( E × B / B ) | is small compared to the gyrofrequency Ω, and the magnetic field scale length is long compared to the gyroradius; we also require that time variation be slow in an appropriately chosen reference frame. This appears to be a minimum set of constraints on a gyrokinetic theory and is substantially more general than earlier approaches. It is the general-geometry electromagnetic extension of Dimits, Phys. Plasmas 17, 055901 (2010) (which is an electrostatic formalism with a homogeneous background magnetic field). This approach also does not require a separate treatment of fluctuating and background components of the magnetic field, unlike much of the previous literature. As a consequence, the "cross terms" due to a combination of long- and short-wavelength variation, which were ignored in the earlier work (but derived in a more restrictive ordering in Parra and Calvo, Plasma Phys. Controlled Fusion 53, 045001 (2011
Gyrokinetic simulation of current-driven instabilities
NASA Astrophysics Data System (ADS)
McClenaghan, Joseph
The gyrokinetic toroidal code(GTC) capability has been extended for simulating current-driven instabilities in magnetized plasmas such as kink and resistive tearing modes with kinetic effects. This new gyrokinetic capability enables first-principles, integrated simulations of macroscopic magnetohydrodynamic(MHD) modes, which limit the performance of burning plasmas and threaten the integrity of fusion devices. The excitation and evolution of macroscopic MHD modes often depend on the kinetic effects at microscopic scales and the nonlinear coupling of multiple physical processes. GTC simulation in the fluid limit of the internal kink modes in cylindrical geometry has been verified by benchmarking with an MHD eigenvalue code. The global simulation domain covers the magnetic axis which is necessary for simulating the macroscopic MHD modes. Gyrokinetic simulations of the internal kink modes in the toroidal geometry find that ion kinetic effects significantly reduce the growth rate even when the banana orbit width is much smaller than the radial width of the perturbed current layer at the mode rational surface. This new GTC capability for current-driven instability has now been extended to simulate fishbone instabilities excited by energetic particles and resistive tearing modes. GTC has also been applied to study the internal kink modes in astrophysical jets that are formed around supermassive black holes. Linear simulations find that the internal kink modes in astrophysical jets are unstable with a broad eigenmode. Nonlinear saturation amplitude of these kink modes is observed to be small, suggesting that the jets can remain collimated even in the presence of the internal kink modes. Generation of a mean parallel electric field by the nonlinear dynamics of internal kink modes and the potential implication of this field on particle acceleration in jets has been examined.
Gyrokinetic simulations of ion and impurity transport
NASA Astrophysics Data System (ADS)
Estrada-Mila, C.; Candy, J.; Waltz, R. E.
2005-02-01
A systematic study of turbulent particle and energy transport in both pure and multicomponent plasmas is presented. In this study, gyrokinetic results from the GYRO code [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] are supplemented with those from the GLF23 [R. E. Waltz, G. M. Staebler, W. Dorland et al., Phys. Plasmas 4, 2482 (1997)] transport model, as well as from quasilinear theory. Various results are obtained. The production of a particle pinch driven by temperature gradients (a thermal pinch) is demonstrated, and further shown to be weakened by finite electron collisionality. Helium transport and the effects of helium density gradient and concentration in a deuterium plasma are examined. Interestingly, it is found that the simple D-v (diffusion versus convective velocity) model of impurity flow is consistent with results obtained from nonlinear gyrokinetic simulations. Also studied is the transport in a 50-50 deuterium-tritium plasma, where a symmetry breaking is observed indicating the potential for fuel separation in a burning plasma. Quasilinear theory together with linear simulations shows that the symmetry breaking which enhances the tritium confinement arises largely from finite-Larmor-radius effects. To justify the numerical methods used in the paper, a variety of linear benchmarks and nonlinear grid refinement studies are detailed.
Gyrokinetic simulation of internal kink modes
Naitou, Hiroshi; Tsuda, Kenji; Lee, W.W.; Sydora, R.D.
1995-05-01
Internal disruption in a tokamak has been simulated using a three-dimensional magneto-inductive gyrokinetic particle code. The code operates in both the standard gyrokinetic mode (total-f code) and the fully nonlinear characteristic mode ({delta}f code). The latter, a recent addition, is a quiet low noise algorithm. The computational model represents a straight tokamak with periodic boundary conditions in the toroidal direction. The plasma is initially uniformly distributed in a square cross section with perfectly conducting walls. The linear mode structure of an unstable m = 1 (poloidal) and n = 1 (toroidal) kinetic internal kink mode is clearly observed, especially in the {delta}f code. The width of the current layer around the x-point, where magnetic reconnection occurs, is found to be close to the collisionless electron skin depth. This is consistent with the theory in which electron inertia has a dominant role. The nonlinear behavior of the mode is found to be quite similar for both codes. Full reconnection in the Alfven time scale is observed along with the electrostatic potential structures created during the full reconnection phase. The E x B drift due to this electrostatic potential dominates the nonlinear phase of the development after the full reconnection.
Gyrokinetic particle simulation of fast-electron driven beta-induced Aflvén eigenmode
NASA Astrophysics Data System (ADS)
Cheng, Junyi; Zhang, Wenlu; Lin, Zhihong; Holod, Ihor; Li, Ding; Chen, Yang; Cao, Jintao
2016-05-01
The fast-electron driven beta-induced Alfvén eigenmode (e-BAE) in toroidal plasmas is investigated for the first time using global gyrokinetic particle simulations, where the fast electron is described by the drift kinetic equation. The simulation shows that the e-BAE propagates in the fast electron diamagnetic direction and its polarization is close to an ideal MHD mode. The phase space structure shows that only the fast electron processional resonance is responsible for the e-BAE excitations while fast-ion driven BAE can be excited through all the channels, including transit, bounce, and processional resonance.
Gyrokinetic Study of L-H Transition with Profile Evolution
NASA Astrophysics Data System (ADS)
Xie, Hua-Sheng; GTC Team
2015-11-01
Recent simulations based on gyrokinetic toroidal code (GTC) and theory based on model eigen equation (H. S. Xie and Y. Xiao, arXiv:1503.04440) have found that the eigenstates of mirco-instabilities (trapped electron mode TEM or ion temperature gradient mode ITG) under strong and weak gradients are not the same. Under weak gradient, the most unstable mode is on the ground state, with conventional ballooning mode structure. When the gradient exceed a critical value, the most unstable mode jump to non-ground state. The mode structures of non-ground state are rich and unconventional, and thus can reduced the transport level, which can provide a explanation to the H-mode in the mirco-scale aspect. Nonlinear simulations (H. S. Xie, Y. Xiao and Z. Lin, 9th West Lake International Symposium on Plasma Simulation, May. 18-21, 2015, Hangzhou, China) verified this and have also found a turning point of the gradient. The turbulent transport coefficient would decrease with the gradient increasing when the gradient exceed a critical value. This provide a new route for the L to H transition without invoking shear flow or zonal flow. In the above works, the profiles are fixed. In this work, we will give some preliminary results on self-consistent simulations of L-H transition including the evolution of the radial plasma profiles. Collaboration with GTC team.
Full f gyrokinetic method for particle simulation of tokamak transport
Heikkinen, J.A. Janhunen, S.J.; Kiviniemi, T.P.; Ogando, F.
2008-05-10
A gyrokinetic particle-in-cell approach with direct implicit construction of the coefficient matrix of the Poisson equation from ion polarization and electron parallel nonlinearity is described and applied in global electrostatic toroidal plasma transport simulations. The method is applicable for calculation of the evolution of particle distribution function f including as special cases strong plasma pressure profile evolution by transport and formation of neoclassical flows. This is made feasible by full f formulation and by recording the charge density changes due to the ion polarization drift and electron acceleration along the local magnetic field while particles are advanced. The code has been validated against the linear predictions of the unstable ion temperature gradient mode growth rates and frequencies. Convergence and saturation in both turbulent and neoclassical limit of the ion heat conductivity is obtained with numerical noise well suppressed by a sufficiently large number of simulation particles. A first global full f validation of the neoclassical radial electric field in the presence of turbulence for a heated collisional tokamak plasma is obtained. At high Mach number (M{sub p}{approx}1) of the poloidal flow, the radial electric field is significantly enhanced over the standard neoclassical prediction. The neoclassical radial electric field together with the related GAM oscillations is found to regulate the turbulent heat and particle diffusion levels particularly strongly in a large aspect ratio tokamak at low plasma current.
The next-generation ESL continuum gyrokinetic edge code
NASA Astrophysics Data System (ADS)
Cohen, R.; Dorr, M.; Hittinger, J.; Rognlien, T.; Collela, P.; Martin, D.
2009-05-01
The Edge Simulation Laboratory (ESL) project is developing continuum-based approaches to kinetic simulation of edge plasmas. A new code is being developed, based on a conservative formulation and fourth-order discretization of full-f gyrokinetic equations in parallel-velocity, magnetic-moment coordinates. The code exploits mapped multiblock grids to deal with the geometric complexities of the edge region, and utilizes a new flux limiter [P. Colella and M.D. Sekora, JCP 227, 7069 (2008)] to suppress unphysical oscillations about discontinuities while maintaining high-order accuracy elsewhere. The code is just becoming operational; we will report initial tests for neoclassical orbit calculations in closed-flux surface and limiter (closed plus open flux surfaces) geometry. It is anticipated that the algorithmic refinements in the new code will address the slow numerical instability that was observed in some long simulations with the existing TEMPEST code. We will also discuss the status and plans for physics enhancements to the new code.
Benchmarking gyrokinetic simulations in a toroidal flux-tube
Chen, Y.; Parker, S. E.; Wan, W.; Bravenec, R.
2013-09-15
A flux-tube model is implemented in the global turbulence code GEM [Y. Chen and S. E. Parker, J. Comput. Phys. 220, 839 (2007)] in order to facilitate benchmarking with Eulerian codes. The global GEM assumes the magnetic equilibrium to be completely given. The initial flux-tube implementation simply selects a radial location as the center of the flux-tube and a radial size of the flux-tube, sets all equilibrium quantities (B, ∇B, etc.) to be equal to the values at the center of the flux-tube, and retains only a linear radial profile of the safety factor needed for boundary conditions. This implementation shows disagreement with Eulerian codes in linear simulations. An alternative flux-tube model based on a complete local equilibrium solution of the Grad-Shafranov equation [J. Candy, Plasma Phys. Controlled Fusion 51, 105009 (2009)] is then implemented. This results in better agreement between Eulerian codes and the particle-in-cell (PIC) method. The PIC algorithm based on the v{sub ||}-formalism [J. Reynders, Ph.D. dissertation, Princeton University, 1992] and the gyrokinetic ion/fluid electron hybrid model with kinetic electron closure [Y. Chan and S. E. Parker, Phys. Plasmas 18, 055703 (2011)] are also implemented in the flux-tube geometry and compared with the direct method for both the ion temperature gradient driven modes and the kinetic ballooning modes.
Non-Maxwellian background effects in gyrokinetic simulations with GENE
NASA Astrophysics Data System (ADS)
Di Siena, A.; Görier, T.; Doerk, H.; Citrin, J.; Johnson, T.; Schneider, M.; Poli, E.; Contributors, JET
2016-11-01
The interaction between fast particles and core turbulence has been established as a central issue for a tokamak reactor. Recent results predict significant enhancement of electromagnetic stabilisation of ITG turbulence in the presence of fast ions. However, most of these simulations were performed with the assumption of equivalent Maxwellian distributed particles, whereas to rigorously model fast ions, a non-Maxwellian background distribution function is needed. To this aim, the underlying equations in the gyrokinetic code GENE have been re-derived and implemented for a completely general background distribution function. After verification studies, a previous investigation on a particular JET plasma has been revised with linear simulations. The plasma is composed by Deuterium, electron, Carbon impurities, NBI fast Deuterium and ICRH 3He. Fast particle distributions have been modelled with a number of different analytic choices in order to study the impact of non-Maxwellian distributions on the plasma turbulence: slowing down and anisotropic Maxwellian. Linear growth rates are studied as a function of the wave number and compared with those obtained using an equivalent Maxwellian. Generally, the choice of the 3He distribution seems to have a stronger impact on the microinstabilities than that of the fast Deuterium.
Monte-Carlo finite elements gyrokinetic simulations of Alfven modes in tokamaks
NASA Astrophysics Data System (ADS)
Bottino, Alberto; Biancalani, Alessandro; Palermo, Francesco; Tronko, Natalia
2016-10-01
The global gyrokinetic code ORB5 can simultaneously include electromagnetic perturbations, general ideal MHD axisymmetric equilibria, zonal-flow preserving sources, collisions, and the ability to solve the full core plasma including the magnetic axis. In this work, a Monte Carlo Particle In Cell Finite Element model, starting from a gyrokinetic discrete Lagrangian, is derived and implemented into the ORB5 code. The variations of the Lagrangian are used to obtain the time continuous equations of motion for the particles and the Finite Element approximation of the field equations. The Noether theorem for the semi-discretised system, implies a certain number of conservation properties for the final set of equation. Linear and nonlinear results, concerning Alfvén instabilities, in the presence of an energetic particle population, and microinstabilities, such as electromagnetic ion temperature gradient (ITG) driven modes and kinetic ballooning modes (KBM), will be presented and discussed. Due to losses of energetic particles, Alfvén instabilities can not only affect plasma stability and damage the walls, but also strongly impact the heating efficiency of a fusion reactor and ultimately the possibility of reaching ignition.
Nonlinear electromagnetic gyrokinetic simulations of tokamak plasmas
NASA Astrophysics Data System (ADS)
Jenko, F.; Dorland, W.
2001-12-01
One of the central physics issues currently targeted by nonlinear gyrokinetic simulations is the role of finite-β effects. The latter change the MHD equilibrium, introduce new dynamical space and time scales, alter and enlarge the zoo of electrostatic microinstabilities and saturation mechanisms, and lead to turbulent transport along fluctuating magnetic field lines. It is shown that the electromagnetic effects on primarily electrostatic microinstabilities are generally weakly or moderately stabilizing. However, the saturation of these modes and hence the determination of the transport level in the quasi-stationary turbulent state can be dominated by nonlinear electromagnetic effects and yield surprising results. Despite this, the induced transport is generally electrostatic in nature well below the ideal ballooning limit.
Petascale Parallelization of the Gyrokinetic Toroidal Code
Ethier, Stephane; Adams, Mark; Carter, Jonathan; Oliker, Leonid
2010-05-01
The Gyrokinetic Toroidal Code (GTC) is a global, three-dimensional particle-in-cell application developed to study microturbulence in tokamak fusion devices. The global capability of GTC is unique, allowing researchers to systematically analyze important dynamics such as turbulence spreading. In this work we examine a new radial domain decomposition approach to allow scalability onto the latest generation of petascale systems. Extensive performance evaluation is conducted on three high performance computing systems: the IBM BG/P, the Cray XT4, and an Intel Xeon Cluster. Overall results show that the radial decomposition approach dramatically increases scalability, while reducing the memory footprint - allowing for fusion device simulations at an unprecedented scale. After a decade where high-end computing (HEC) was dominated by the rapid pace of improvements to processor frequencies, the performance of next-generation supercomputers is increasingly differentiated by varying interconnect designs and levels of integration. Understanding the tradeoffs of these system designs is a key step towards making effective petascale computing a reality. In this work, we examine a new parallelization scheme for the Gyrokinetic Toroidal Code (GTC) [?] micro-turbulence fusion application. Extensive scalability results and analysis are presented on three HEC systems: the IBM BlueGene/P (BG/P) at Argonne National Laboratory, the Cray XT4 at Lawrence Berkeley National Laboratory, and an Intel Xeon cluster at Lawrence Livermore National Laboratory. Overall results indicate that the new radial decomposition approach successfully attains unprecedented scalability to 131,072 BG/P cores by overcoming the memory limitations of the previous approach. The new version is well suited to utilize emerging petascale resources to access new regimes of physical phenomena.
NASA Astrophysics Data System (ADS)
Ku, Seung-Hoe; Hager, R.; Chang, C. S.; Chacon, L.; Chen, G.; EPSI Team
2016-10-01
The cancelation problem has been a long-standing issue for long wavelengths modes in electromagnetic gyrokinetic PIC simulations in toroidal geometry. As an attempt of resolving this issue, we implemented a fully implicit time integration scheme in the full-f, gyrokinetic PIC code XGC1. The new scheme - based on the implicit Vlasov-Darwin PIC algorithm by G. Chen and L. Chacon - can potentially resolve cancelation problem. The time advance for the field and the particle equations is space-time-centered, with particle sub-cycling. The resulting system of equations is solved by a Picard iteration solver with fixed-point accelerator. The algorithm is implemented in the parallel velocity formalism instead of the canonical parallel momentum formalism. XGC1 specializes in simulating the tokamak edge plasma with magnetic separatrix geometry. A fully implicit scheme could be a way to accurate and efficient gyrokinetic simulations. We will test if this numerical scheme overcomes the cancelation problem, and reproduces the dispersion relation of Alfven waves and tearing modes in cylindrical geometry. Funded by US DOE FES and ASCR, and computing resources provided by OLCF through ALCC.
T.S. Hahm; Z. Lin; P.H. Diamond; G. Rewoldt; W.X. Wang; S. Ethier; O. Gurcan; W.W. Lee; W.M. Tang
2004-12-21
An integrated program of gyrokinetic particle simulation and theory has been developed to investigate several outstanding issues in both turbulence and neoclassical physics. Gyrokinetic particle simulations of toroidal ion temperature gradient (ITG) turbulence spreading using the GTC code and its related dynamical model have been extended to the case with radially increasing ion temperature gradient, to study the inward spreading of edge turbulence toward the core. Due to turbulence spreading from the edge, the turbulence intensity in the core region is significantly enhanced over the value obtained from simulations of the core region only. Even when the core gradient is within the Dimits shift regime (i.e., self-generated zonal flows reduce the transport to a negligible value), a significant level of turbulence and transport is observed in the core due to spreading from the edge. The scaling of the turbulent front propagation speed is closer to the prediction from our nonlinear diffusion model than one based on linear toroidal coupling. A calculation of ion poloidal rotation in the presence of sharp density and toroidal angular rotation frequency gradients from the GTC-Neo particle simulation code shows that the results are significantly different from the conventional neoclassical theory predictions. An energy conserving set of a fully electromagnetic nonlinear gyrokinetic Vlasov equation and Maxwell's equations, which is applicable to edge turbulence, is being derived via the phase-space action variational Lie perturbation method. Our generalized ordering takes the ion poloidal gyroradius to be on the order of the radial electric field gradient length.
Verification of Gyrokinetic codes: theoretical background and applications
NASA Astrophysics Data System (ADS)
Tronko, Natalia
2016-10-01
In fusion plasmas the strong magnetic field allows the fast gyro motion to be systematically removed from the description of the dynamics, resulting in a considerable model simplification and gain of computational time. Nowadays, the gyrokinetic (GK) codes play a major role in the understanding of the development and the saturation of turbulence and in the prediction of the consequent transport. We present a new and generic theoretical framework and specific numerical applications to test the validity and the domain of applicability of existing GK codes. For a sound verification process, the underlying theoretical GK model and the numerical scheme must be considered at the same time, which makes this approach pioneering. At the analytical level, the main novelty consists in using advanced mathematical tools such as variational formulation of dynamics for systematization of basic GK code's equations to access the limits of their applicability. The indirect verification of numerical scheme is proposed via the Benchmark process. In this work, specific examples of code verification are presented for two GK codes: the multi-species electromagnetic ORB5 (PIC), and the radially global version of GENE (Eulerian). The proposed methodology can be applied to any existing GK code. We establish a hierarchy of reduced GK Vlasov-Maxwell equations using the generic variational formulation. Then, we derive and include the models implemented in ORB5 and GENE inside this hierarchy. At the computational level, detailed verification of global electromagnetic test cases based on the CYCLONE are considered, including a parametric β-scan covering the transition between the ITG to KBM and the spectral properties at the nominal β value.
Extended gyrokinetic field theory for time-dependent magnetic confinement fields
NASA Astrophysics Data System (ADS)
Sugama, H.; Watanabe, T.-H.; Nunami, M.
2014-01-01
A gyrokinetic system of equations for turbulent toroidal plasmas in time-dependent axisymmetric background magnetic fields is derived from the variational principle. Besides governing equations for gyrocenter distribution functions and turbulent electromagnetic fields, the conditions which self-consistently determine the background magnetic fields varying on a transport time scale are obtained by using the Lagrangian, which includes the constraint on the background fields. Conservation laws for energy and toroidal angular momentum of the whole system in the time-dependent background magnetic fields are naturally derived by applying Noether's theorem. It is shown that the ensemble-averaged transport equations of particles, energy, and toroidal momentum given in the present work agree with the results from the conventional recursive formulation with the WKB representation except that collisional effects are disregarded here.
Extended gyrokinetic field theory for time-dependent magnetic confinement fields
Sugama, H.; Watanabe, T.-H.; Nunami, M.
2014-01-15
A gyrokinetic system of equations for turbulent toroidal plasmas in time-dependent axisymmetric background magnetic fields is derived from the variational principle. Besides governing equations for gyrocenter distribution functions and turbulent electromagnetic fields, the conditions which self-consistently determine the background magnetic fields varying on a transport time scale are obtained by using the Lagrangian, which includes the constraint on the background fields. Conservation laws for energy and toroidal angular momentum of the whole system in the time-dependent background magnetic fields are naturally derived by applying Noether's theorem. It is shown that the ensemble-averaged transport equations of particles, energy, and toroidal momentum given in the present work agree with the results from the conventional recursive formulation with the WKB representation except that collisional effects are disregarded here.
Accuracy of momentum and gyrodensity transport in global gyrokinetic particle-in-cell simulations
McMillan, B. F.; Villard, L.
2014-05-15
Gyrokinetic Particle-In-Cell (PIC) simulations based on conservative Lagrangian formalisms admit transport equations for conserved quantities such as gyrodensity and toroidal momentum, and these can be derived for arbitrary wavelength, even though previous applications have used the long-wavelength approximation. In control-variate PIC simulations, a consequence of the different treatment of the background (f{sub 0}) and perturbed parts (δf), when a splitting f = f{sub 0} + δf is performed, is that analytical transport relations for the relevant fluxes and moments are only reproduced in the large marker number limit. The transport equations for f can be used to write the inconsistency in the perturbed quantities explicitly in terms of the sampling of the background distribution f{sub 0}. This immediately allows estimates of the error in consistency of momentum transport in control-variate PIC simulations. This inconsistency tends to accumulate secularly and is not directly affected by the sources and noise control in the system. Although physical tokamaks often rotate quite strongly, the standard gyrokinetic formalism assumes weak perpendicular flows, comparable to the drift speed. For systems with such weak flows, maintaining acceptably small relative errors requires that a number of markers scale with the fourth power of the linear system size to consistently resolve long-wavelength evolution. To avoid this unfavourable scaling, an algorithm for exact gyrodensity transport has been developed, and this is shown to allow accurate simulations with an order of magnitude fewer markers.
NASA Astrophysics Data System (ADS)
Chen, Yang; Fu, Guo-Yong; Parker, Scott
2016-10-01
We report simulation of simultaneous excitation of multiple Reverse Shear Alfven eigenmodes in DIII-D plasmas (discharge #142111), using the gyrokinetic ion/fluid electron hybrid model of GEM. Thermal ions and beam ions are gyrokinetic, electrons are fluid with finite-mass correction in the Ohm's law. The vorticity equation is solved instead of the quasi-neutrality condition. This improves numerical stability. We extend previous single-n nonlinear simulation to simultaneous excitation of toroidal modes with n = 0 and 2 < n < 15 . Both the zonal n = 0 mode and the n = 8 mode are observed to be force driven by the linearly dominant n = 4 mode coupled to itself, with a growth rate twice that of the n = 4 mode. The zonal mode (including the surface averaged ϕ and A∥) significantly reduces the initial saturation level of the n = 4 mode. Evolution of all the other modes are also dominated by nonlinear coupling to the n = 4 mode. The mechanism of zonal structure generation will be examined by comparing various terms in the vorticity equation, including the Reynolds stress, the magnetic stress and the beam ion nonlinear effect.
Gyrokinetic particle simulation of a field reversed configuration
Fulton, D. P. Lau, C. K.; Holod, I.; Lin, Z.; Dettrick, S.
2016-01-15
Gyrokinetic particle simulation of the field-reversed configuration (FRC) has been developed using the gyrokinetic toroidal code (GTC). The magnetohydrodynamic equilibrium is mapped from cylindrical coordinates to Boozer coordinates for the FRC core and scrape-off layer (SOL), respectively. A field-aligned mesh is constructed for solving self-consistent electric fields using a semi-spectral solver in a partial torus FRC geometry. This new simulation capability has been successfully verified and driftwave instability in the FRC has been studied using the gyrokinetic simulation for the first time. Initial GTC simulations find that in the FRC core, the ion-scale driftwave is stabilized by the large ion gyroradius. In the SOL, the driftwave is unstable on both ion and electron scales.
Gyrokinetic particle simulation of beta-induced Alfven eigenmode
Zhang, H. S.; Lin, Z.; Holod, I.; Xiao, Y.; Wang, X.; Zhang, W. L.
2010-11-15
The beta-induced Alfven eigenmode (BAE) in toroidal plasmas is studied using global gyrokinetic particle simulations. The BAE real frequency and damping rate measured in the initial perturbation simulation and in the antenna excitation simulation agree well with each other. The real frequency is slightly higher than the ideal magnetohydrodynamic (MHD) accumulation point frequency due to the kinetic effects of thermal ions. Simulations with energetic particle density gradient show exponential growth of BAE with a growth rate sensitive to the energetic particle temperature and density. The nonperturbative contributions by energetic particles modify the mode structure and reduce the frequency relative to the MHD theory. The finite Larmor radius effects of energetic particles reduce the BAE growth rate. Benchmarks between gyrokinetic particle simulation and hybrid MHD-gyrokinetic simulation show good agreement in BAE real frequency and mode structure.
Status of the 5D gyrokinetic code COGENT and its initial applications
NASA Astrophysics Data System (ADS)
Lee, Wonjae; Dorf, M.; Dorr, M.; Cohen, R.; Ghosh, D.; Rognlien, T.; Hittinger, J.; Umansky, M.; Krasheninnikov, S.
2016-10-01
We report recent progress with the development of the 5D (3D configuration and 2D velocity space) version of the full-f continuum gyrokinetic code COGENT. The original 2D configuration space has been successfully extended to 3D, with the Cartesian (slab) geometry chosen for verification and initial applications. The code has been successfully verified with drift-wave simulations including drift-kinetic equations for both electrons and ions coupled to the long-wavelength limit of the Gyro-Poisson equation. The initial application of the 5D COGENT is focused on addressing kinetic effects of drift-wave instabilities (e.g., universal instability) on blob dynamics in tokamak edge plasmas. Work performed for USDOE, at UCSD under Grants DE-FG02-04ER54739 and DE-SC0010413, and at LLNL under contract DE-AC52-07NA27344 and under Livermore Graduate Scholar Program.
A verification of the gyrokinetic microstability codes GEM, GYRO, and GS2
Bravenec, R. V.; Chen, Y.; Wan, W.; Parker, S.; Candy, J.
2013-10-15
A previous publication [R. V. Bravenec et al., Phys. Plasmas 18, 122505 (2011)] presented favorable comparisons of linear frequencies and nonlinear fluxes from the Eulerian gyrokinetic codes gyro[J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and gs2[W. Dorland et al., Phys. Rev. Lett. 85, 5579 (2000)]. The motivation was to verify the codes, i.e., demonstrate that they correctly solve the gyrokinetic-Maxwell equations. The premise was that it is highly unlikely for both codes to yield the same incorrect results. In this work, we add the Lagrangian particle-in-cell code gem[Y. Chen and S. Parker, J. Comput. Phys. 220, 839 (2007)] to the comparisons, not simply to add another code, but also to demonstrate that the codes' algorithms do not matter. We find good agreement of gem with gyro and gs2 for the plasma conditions considered earlier, thus establishing confidence that the codes are verified and that ongoing validation efforts for these plasma parameters are warranted.
Fully Nonlinear Edge Gyrokinetic Simulations of Kinetic Geodesic-Acoustic Modes and Boundary Flows
Xu, X Q; Belli, E; Bodi, K; Candy, J; Chang, C S; Cohen, B I; Cohen, R H; Colella, P; Dimits, A M; Dorr, M R; Gao, Z; Hittinger, J A; Ko, S; Krasheninnikov, S; McKee, G R; Nevins, W M; Rognlien, T D; Snyder, P B; Suh, J; Umansky, M V
2008-09-18
We present edge gyrokinetic neoclassical simulations of tokamak plasmas using the fully nonlinear (full-f) continuum code TEMPEST. A nonlinear Boltzmann model is used for the electrons. The electric field is obtained by solving the 2D gyrokinetic Poisson Equation. We demonstrate the following: (1) High harmonic resonances (n > 2) significantly enhance geodesic-acoustic mode (GAM) damping at high-q (tokamak safety factor), and are necessary to explain both the damping observed in our TEMPEST q-scans and experimental measurements of the scaling of the GAM amplitude with edge q{sub 95} in the absence of obvious evidence that there is a strong q dependence of the turbulent drive and damping of the GAM. (2) The kinetic GAM exists in the edge for steep density and temperature gradients in the form of outgoing waves, its radial scale is set by the ion temperature profile, and ion temperature inhomogeneity is necessary for GAM radial propagation. (3) The development of the neoclassical electric field evolves through different phases of relaxation, including GAMs, their radial propagation, and their long-time collisional decay. (4) Natural consequences of orbits in the pedestal and scrape-off layer region in divertor geometry are substantial non-Maxwellian ion distributions and flow characteristics qualitatively like those observed in experiments.
Poloidal tilting symmetry of high order tokamak flux surface shaping in gyrokinetics
NASA Astrophysics Data System (ADS)
Ball, Justin; Parra, Felix I.; Barnes, Michael
2016-04-01
A poloidal tilting symmetry of the local nonlinear δ f gyrokinetic model is demonstrated analytically and verified numerically. This symmetry shows that poloidally rotating all the flux surface shaping effects with large poloidal mode number by a single tilt angle has an exponentially small effect on the transport properties of a tokamak. This is shown using a generalization of the Miller local equilibrium model to specify an arbitrary flux surface geometry. With this geometry specification we find that, when performing an expansion in large flux surface shaping mode number, the governing equations of gyrokinetics are symmetric in the poloidal tilt of the high order shaping effects. This allows us to take the fluxes from a single configuration and calculate the fluxes in any configuration that can be produced by tilting the large mode number shaping effects. This creates a distinction between tokamaks with mirror symmetric flux surfaces and tokamaks without mirror symmetry, which is expected to have important consequences for generating toroidal rotation using up-down asymmetry.
Nonlinear Gyrokinetics: A Powerful Tool for the Description of Microturbulence in Magnetized Plasmas
John E. Krommes
2010-09-27
Gyrokinetics is the description of low-frequency dynamics in magnetized plasmas. In magnetic-confinement fusion, it provides the most fundamental basis for numerical simulations of microturbulence; there are astrophysical applications as well. In this tutorial, a sketch of the derivation of the novel dynamical system comprising the nonlinear gyrokinetic (GK) equation (GKE) and the coupled electrostatic GK Poisson equation will be given by using modern Lagrangian and Lie perturbation methods. No background in plasma physics is required in order to appreciate the logical development. The GKE describes the evolution of an ensemble of gyrocenters moving in a weakly inhomogeneous background magnetic field and in the presence of electromagnetic perturbations with wavelength of the order of the ion gyroradius. Gyrocenters move with effective drifts, which may be obtained by an averaging procedure that systematically, order by order, removes gyrophase dependence. To that end, the use of the Lagrangian differential one-form as well as the content and advantages of Lie perturbation theory will be explained. The electromagnetic fields follow via Maxwell's equations from the charge and current density of the particles. Particle and gyrocenter densities differ by an important polarization effect. That is calculated formally by a "pull-back" (a concept from differential geometry) of the gyrocenter distribution to the laboratory coordinate system. A natural truncation then leads to the closed GK dynamical system. Important properties such as GK energy conservation and fluctuation noise will be mentioned briefly, as will the possibility (and diffculties) of deriving nonlinear gyro fluid equations suitable for rapid numerical solution -- although it is probably best to directly simulate the GKE. By the end of the tutorial, students should appreciate the GKE as an extremely powerful tool and will be prepared for later lectures describing its applications to physical problems.
Phase space structures in gyrokinetic simulations of fusion plasma turbulence
NASA Astrophysics Data System (ADS)
Ghendrih, Philippe; Norscini, Claudia; Cartier-Michaud, Thomas; Dif-Pradalier, Guilhem; Abiteboul, Jérémie; Dong, Yue; Garbet, Xavier; Gürcan, Ozgür; Hennequin, Pascale; Grandgirard, Virginie; Latu, Guillaume; Morel, Pierre; Sarazin, Yanick; Storelli, Alexandre; Vermare, Laure
2014-10-01
Gyrokinetic simulations of fusion plasmas give extensive information in 5D on turbulence and transport. This paper highlights a few of these challenging physics in global, flux driven simulations using experimental inputs from Tore Supra shot TS45511. The electrostatic gyrokinetic code GYSELA is used for these simulations. The 3D structure of avalanches indicates that these structures propagate radially at localised toroidal angles and then expand along the field line at sound speed to form the filaments. Analysing the poloidal mode structure of the potential fluctuations (at a given toroidal location), one finds that the low modes m = 0 and m = 1 exhibit a global structure; the magnitude of the m = 0 mode is much larger than that of the m = 1 mode. The shear layers of the corrugation structures are thus found to be dominated by the m = 0 contribution, that are comparable to that of the zonal flows. This global mode seems to localise the m = 2 mode but has little effect on the localisation of the higher mode numbers. However when analysing the pulsation of the latter modes one finds that all modes exhibit a similar phase velocity, comparable to the local zonal flow velocity. The consequent dispersion like relation between the modes pulsation and the mode numbers provides a means to measure the zonal flow. Temperature fluctuations and the turbulent heat flux are localised between the corrugation structures. Temperature fluctuations are found to exhibit two scales, small fluctuations that are localised by the corrugation shear layers, and appear to bounce back and forth radially, and large fluctuations, also readily observed on the flux, which are associated to the disruption of the corrugations. The radial ballistic velocity of both avalanche events if of the order of 0.5ρ∗c0 where ρ∗ = ρ0/a, a being the tokamak minor radius and ρ0 being the characteristic Larmor radius, ρ0 = c0/Ω0. c0 is the reference ion thermal velocity and Ω0 = qiB0/mi the reference
A basic plasma test for gyrokinetics: GDC turbulence in LAPD
NASA Astrophysics Data System (ADS)
Pueschel, M. J.; Rossi, G.; Told, D.; Terry, P. W.; Jenko, F.; Carter, T. A.
2017-02-01
Providing an important step towards validating gyrokinetics under comparatively little-explored conditions, simulations of pressure-gradient-driven plasma turbulence in the Large Plasma Device (LAPD) are compared with experimental observations. The corresponding signatures confirm the existence of a novel regime of turbulence, based on the recently-discovered gradient-driven drift coupling (GDC) instability, which is thus confirmed as a candidate mechanism for turbulence in basic, space and astrophysical plasmas. Despite the limitations of flux-tube gyrokinetics for this scenario, when accounting for box size scaling by applying a scalar factor η =6, agreement between simulations and experiment improves to within a factor of two for key observables: compressional magnetic, density, and temperature fluctuations, both in amplitude and structure. Thus, a first, strong indication is presented that the GDC instability seen in gyrokinetics appears to operate in the experiment and that the essential instability physics is present in the numerical model. Overall, the gyrokinetic framework and its numerical implementation in the Gene code therefore perform well for LAPD plasmas very different from their brethren in fusion experiments.
Comparison of Implicit Multiscale Full Kinetics to Gyrokinetics
NASA Astrophysics Data System (ADS)
Parker, Scott; Sturdevant, Benjamin; Chen, Yang
2016-10-01
Recent progress has been made developing full kinetic Lorentz force ion dynamics using implicit multiscale techniques. It is now possible to capture low-frequency physics along with finite Larmor radius (FLR) effects with a fully kinetic multiscale delta-f particle simulation. The utility of such a model is to be able to verify gyrokinetics in situations where the smallness of the ordering parameters are under question. Additionally, such a model can help identify what higher order terms in gyrokinetics might be important. Orbit averaging and sub-cycling are utilized with an implicit particle time advance based on variational principles. This produces stable and accurate ion trajectories on long time scales. Excellent agreement with the gyrokinetic dispersion relation is obtained including full FLR effects. Ion Bernstein waves are easily suppressed with the implicit time advance. We have developed a global toroidal electrostatic adiabatic electron Lorentz ion code. We will report our linear results benchmarking Lorentz ions with gyrokinetics for the Cyclone base case. We will also present our progress on ion including drift-kinetic electrons and electromagnetic perturbations.
NASA Astrophysics Data System (ADS)
Bénisti, Didier
2016-10-01
This paper addresses the linear and nonlinear three-dimensional propagation of an electron wave in a collisionless plasma that may be inhomogeneous, nonstationary, anisotropic, and even weakly magnetized. The wave amplitude, together with any hydrodynamic quantity characterizing the plasma (density, temperature, etc.) is supposed to vary very little within one wavelength or one wave period. Hence, the geometrical optics limit is assumed, and the wave propagation is described by a first order differential equation. This equation explicitly accounts for three-dimensional effects, plasma inhomogeneity, Landau damping, and the collisionless dissipation and electron acceleration due to trapping. It is derived by mixing results obtained from a direct resolution of the Vlasov-Poisson system and from a variational formalism involving a nonlocal Lagrangian density. In a one-dimensional situation, abrupt transitions are predicted in the coefficients of the wave equation. They occur when the state of the electron plasma wave changes, from a linear wave to a wave with trapped electrons. In a three dimensional geometry, the transitions are smoother, especially as regards the nonlinear Landau damping rate, for which a very simple effective and accurate analytic expression is provided.
Multiscale gyrokinetics for rotating tokamak plasmas: fluctuations, transport and energy flows.
Abel, I G; Plunk, G G; Wang, E; Barnes, M; Cowley, S C; Dorland, W; Schekochihin, A A
2013-11-01
This paper presents a complete theoretical framework for studying turbulence and transport in rapidly rotating tokamak plasmas. The fundamental scale separations present in plasma turbulence are codified as an asymptotic expansion in the ratio ε = ρi/α of the gyroradius to the equilibrium scale length. Proceeding order by order in this expansion, a set of coupled multiscale equations is developed. They describe an instantaneous equilibrium, the fluctuations driven by gradients in the equilibrium quantities, and the transport-timescale evolution of mean profiles of these quantities driven by the interplay between the equilibrium and the fluctuations. The equilibrium distribution functions are local Maxwellians with each flux surface rotating toroidally as a rigid body. The magnetic equilibrium is obtained from the generalized Grad-Shafranov equation for a rotating plasma, determining the magnetic flux function from the mean pressure and velocity profiles of the plasma. The slow (resistive-timescale) evolution of the magnetic field is given by an evolution equation for the safety factor q. Large-scale deviations of the distribution function from a Maxwellian are given by neoclassical theory. The fluctuations are determined by the 'high-flow' gyrokinetic equation, from which we derive the governing principle for gyrokinetic turbulence in tokamaks: the conservation and local (in space) cascade of the free energy of the fluctuations (i.e. there is no turbulence spreading). Transport equations for the evolution of the mean density, temperature and flow velocity profiles are derived. These transport equations show how the neoclassical and fluctuating corrections to the equilibrium Maxwellian act back upon the mean profiles through fluxes and heating. The energy and entropy conservation laws for the mean profiles are derived from the transport equations. Total energy, thermal, kinetic and magnetic, is conserved and there is no net turbulent heating. Entropy is produced
Gyrokinetic turbulence cascade via predator-prey interactions between different scales
Kobayashi, Sumire Gurcan, Ozgur D.
2015-05-15
Gyrokinetic simulations in a closed fieldline geometry are presented to explore the physics of nonlinear transfer in plasma turbulence. As spontaneously formed zonal flows and small-scale turbulence demonstrate “predator-prey” dynamics, a particular cascade spectrum emerges. The electrostatic potential and the density spectra appear to be in good agreement with the simple theoretical prediction based on Charney-Hasegawa-Mima equation | ϕ{sup ~}{sub k} |{sup 2}∼| n{sup ~}{sub k} |{sup 2}∝k{sup −3}/(1+k{sup 2}){sup 2}, with the spectra becoming anisotropic at small scales. The results indicate that the disparate scale interactions, in particular, the refraction and shearing of larger scale eddies by the self-consistent zonal flows, dominate over local interactions, and contrary to the common wisdom, the comprehensive scaling relation is created even within the energy injection region.
Piattella, O.F.; Rodrigues, D.C.; Fabris, J.C.; Pacheco, J.A. de Freitas E-mail: davi.rodrigues@ufes.br E-mail: pacheco@oca.eu
2013-11-01
We discuss solutions of Vlasov-Einstein equation for collisionless dark matter particles in the context of a flat Friedmann universe. We show that, after decoupling from the primordial plasma, the dark matter phase-space density indicator Q = ρ/(σ{sub 1D}{sup 2}){sup 3/2} remains constant during the expansion of the universe, prior to structure formation. This well known result is valid for non-relativistic particles and is not ''observer dependent'' as in solutions derived from the Vlasov-Poisson system. In the linear regime, the inclusion of velocity dispersion effects permits to define a physical Jeans length for collisionless matter as function of the primordial phase-space density indicator: λ{sub J} = (5π/G){sup 1/2}Q{sup −1/3}ρ{sub dm}{sup −1/6}. The comoving Jeans wavenumber at matter-radiation equality is smaller by a factor of 2-3 than the comoving wavenumber due to free-streaming, contributing to the cut-off of the density fluctuation power spectrum at the lowest scales. We discuss the physical differences between these two scales. For dark matter particles of mass equal to 200 GeV, the derived Jeans mass is 4.3 × 10{sup −6}M{sub ⊙}.
Gyrokinetic Simulation of Low-n Tearing Modes
NASA Astrophysics Data System (ADS)
Chen, Yang
2015-11-01
Direct gyrokinetic simulation of the low-n tearing mode in a tokamak plasma has been a great computational challenge, for two reasons. First, low-n tearing modes, unlike the micro-tearing modes, have very small growth rates and very fine mode structure in the tearing layer, which requires a large number of radial grid cells and fine control of numerical dissipation. Second, kinetic electron effects are needed in the tearing layer. Here, we first present linear gyrokinetic simulation of the low-n tearing mode in cylindrical geometry. Ions are gyrokinetic and electrons are either drift kinetic or fluid. New field solvers have been developed in the gyrokinetic code GEM [Chen and Parker, J. Comput. Phys. 220, 839 (2007)] to simulate low-n modes. For the fluid electron model, an eigenmode analysis with finite Larmor radius effects has been developed to study the linear resistive tearing mode. Excellent agreement between eigenmode analysis and initial value gyrokinetic simulation is obtained. The mode growth rate is shown to scale with resistivity as η 1 / 3, the same as the semi-collisional regime in previous kinetic treatments. Simulation of the collisionless and semi-collisional tearing mode with drift kinetic electrons has been carried out with GEM's direct split-weight control-variate algorithm. It is found that a full torus simulation of the m=2, n=1 tearing mode in a present day large tokamak is still difficult with kinetic electrons, but a generalized matching technique can be used to ameliorate the problem. The radial dimension is divided into an external region and the tearing region, with the external region described by a reduced model that gives the boundary condition for the tearing region. The size of the tearing region is small compared with the minor radius, but not arbitrarily small as done in the standard asymptotic matching approach. Gyrokinetic simulation verifies the collisionless tearing mode growth rate with finite electron mass, the semi
The energetic coupling of scales in gyrokinetic plasma turbulence
NASA Astrophysics Data System (ADS)
Teaca, Bogdan; Navarro, Alejandro Bañón; Jenko, Frank
2014-07-01
In magnetized plasma turbulence, the couplings of perpendicular spatial scales that arise due to the nonlinear interactions are analyzed from the perspective of the free-energy exchanges. The plasmas considered here, with appropriate ion or electron adiabatic electro-neutrality responses, are described by the gyrokinetic formalism in a toroidal magnetic geometry. Turbulence develops due to the electrostatic fluctuations driven by temperature gradient instabilities, either ion temperature gradient (ITG) or electron temperature gradient (ETG). The analysis consists in decomposing the system into a series of scale structures, while accounting separately for contributions made by modes possessing special symmetries (e.g., the zonal flow modes). The interaction of these scales is analyzed using the energy transfer functions, including a forward and backward decomposition, scale fluxes, and locality functions. The comparison between the ITG and ETG cases shows that ETG turbulence has a more pronounced classical turbulent behavior, exhibiting a stronger energy cascade, with implications for gyrokinetic turbulence modeling.
Gyrokinetic investigation of ITG modes in helical RFPs
NASA Astrophysics Data System (ADS)
Predebon, I.; Xanthopoulos, P.; Terranova, D.
2013-10-01
Micro-instabilities in the RFP have been investigated in the last years from several viewpoints and with various numerical tools. The strongest underlying assumption for all of these approaches is that the magnetic equilibrium does not deviate significantly from axisymmetry. Contrary to this, in RFX-mod, the physical conditions more favorable for the onset of electrostatic/electromagnetic turbulence emerge when magnetic surfaces are helical, i.e., during the single helicity states of the RFP. In general, we wish to systematically revisit the existing gyrokinetic studies of microturbulence focusing on the novel 3D feature. The RFP equilibria are now derived using the VMEC code and subsequently applied to the nonlinear gyrokinetic code GENE with the aid of the GIST interface code. The physical problem we address here is the occurrence of ITG instability in single helicity plasmas, and its distinct properties compared to the axisymmetric geometry.
A Numerical Instability in an ADI Algorithm for Gyrokinetics
E.A. Belli; G.W. Hammett
2004-12-17
We explore the implementation of an Alternating Direction Implicit (ADI) algorithm for a gyrokinetic plasma problem and its resulting numerical stability properties. This algorithm, which uses a standard ADI scheme to divide the field solve from the particle distribution function advance, has previously been found to work well for certain plasma kinetic problems involving one spatial and two velocity dimensions, including collisions and an electric field. However, for the gyrokinetic problem we find a severe stability restriction on the time step. Furthermore, we find that this numerical instability limitation also affects some other algorithms, such as a partially implicit Adams-Bashforth algorithm, where the parallel motion operator v{sub {parallel}} {partial_derivative}/{partial_derivative}z is treated implicitly and the field terms are treated with an Adams-Bashforth explicit scheme. Fully explicit algorithms applied to all terms can be better at long wavelengths than these ADI or partially implicit algorithms.
The energetic coupling of scales in gyrokinetic plasma turbulence
Teaca, Bogdan; Jenko, Frank
2014-07-15
In magnetized plasma turbulence, the couplings of perpendicular spatial scales that arise due to the nonlinear interactions are analyzed from the perspective of the free-energy exchanges. The plasmas considered here, with appropriate ion or electron adiabatic electro-neutrality responses, are described by the gyrokinetic formalism in a toroidal magnetic geometry. Turbulence develops due to the electrostatic fluctuations driven by temperature gradient instabilities, either ion temperature gradient (ITG) or electron temperature gradient (ETG). The analysis consists in decomposing the system into a series of scale structures, while accounting separately for contributions made by modes possessing special symmetries (e.g., the zonal flow modes). The interaction of these scales is analyzed using the energy transfer functions, including a forward and backward decomposition, scale fluxes, and locality functions. The comparison between the ITG and ETG cases shows that ETG turbulence has a more pronounced classical turbulent behavior, exhibiting a stronger energy cascade, with implications for gyrokinetic turbulence modeling.
A geometry interface for gyrokinetic microturbulence investigations in toroidal configurations
NASA Astrophysics Data System (ADS)
Xanthopoulos, P.; Cooper, W. A.; Jenko, F.; Turkin, Yu.; Runov, A.; Geiger, J.
2009-08-01
The GENE/GIST code package is developed for the investigation of plasma microturbulence, suitable for both stellarator and tokamak configurations. The geometry module is able to process typical equilibrium files and create the interface for the gyrokinetic solver. The analytical description of the method for constructing the geometric elements is documented, together with several numerical evaluation tests. As a concrete application of this product, a cross-machine comparison of the anomalous ion heat diffusivity is presented.
Plasma Simulation Using Gyrokinetic-Gyrofluid Hybrid Models
Scott Parker
2009-04-09
We are developing kinetic ion models for the simulation of extended MHD phenomena. The model they have developed uses full Lorentz force ions, and either drift-kinetic or gyro-kinetic electrons. Quasi-neutrality is assumed and the displacement current is neglected. They are also studying alpha particle driven Toroidal Alfven Eigenmodes (TAE) in the GEM gyrokinetic code [Chen 07]. The basic kinetic ion MHD model was recently reported in an invited talk given by Dan Barnes at the 2007 American Physical Society - Division of Plasma Physics (APS-DPP) and it has been published [Jones 04, Barnes 08]. The model uses an Ohm's law that includes the Hall term, pressure term and the electron inertia [Jones 04]. These results focused on the ion physics and assumed an isothermal electron closure. It is found in conventional gyrokinetic turbulence simulations that the timestep cannot be made much greater than the ion cyclotron period. However, the kinetic ion MHD model has the compressional mode, which further limits the timestep. They have developed an implicit scheme to avoid this timestep constraint. They have also added drift kinetic electrons. This model has been benchmarked linearly. Waves investigated where shear and compressional Alfven, whisterl, ion acoustic, and drift waves, including the kinetic damping rates. This work is ongoing and was first reported at the 2008 Sherwood Fusion Theory Conference [Chen 08] and they are working on a publication. They have also formulated an integrated gyrokinetic electron model, which is of interest for studying electron gradient instabilities and weak guide-field magnetic reconnection.
Morrison, P.J.; Pfirsch, D.
1992-04-01
Expressions for the energy content of one-dimensional electrostatic perturbations about homogeneous equilibria are revisited. The well-known dielectric energy, {var_epsilon}{sub D}, is compared with the exact plasma free energy expression, {delta}{sup 2}F, that is conserved by the Vlasov-Poisson system. The former is an expression in terms of the perturbed electric field amplitude, while the latter is determined by a generating function, which describes perturbations of the distribution function that respect the important constraint of dynamical accessibility of the system. Thus the comparison requires solving the Vlasov equation for such a perturbations of the distribution function in terms of the electric field. This is done for neutral modes of oscillation that occur for equilibria with stationary inflection points, and it is seen that for these special modes {delta}{sup 2}F = {var_epsilon}{sub D}. In the case of unstable and corresponding damped modes it is seen that {delta}{sup 2}F {ne} {var_epsilon}{sub D}; in fact {delta}{sup 2}F {equivalent_to} 0. This failure of the dielectric energy expression persists even for arbitrarily small growth and damping rates since {var_epsilon}{sub D} is nonzero in this limit, whereas {delta}{sup 2}F remains zero. The connection between the new exact energy expression and the at-best approximate {var_epsilon}{sub D} is described. The new expression motivates natural definitions of Hamiltonian action variables and signature. A general linear integral transform is introduced that maps the linear version of the noncanonical Hamiltonian structure, which describes the Vlasov equation, to action-angle (diagonal) form.
Morrison, P.J. . Inst. for Fusion Studies); Pfirsch, D. )
1992-04-01
Expressions for the energy content of one-dimensional electrostatic perturbations about homogeneous equilibria are revisited. The well-known dielectric energy, {var epsilon}{sub D}, is compared with the exact plasma free energy expression, {delta}{sup 2}F, that is conserved by the Vlasov-Poisson system. The former is an expression in terms of the perturbed electric field amplitude, while the latter is determined by a generating function, which describes perturbations of the distribution function that respect the important constraint of dynamical accessibility of the system. Thus the comparison requires solving the Vlasov equation for such a perturbations of the distribution function in terms of the electric field. This is done for neutral modes of oscillation that occur for equilibria with stationary inflection points, and it is seen that for these special modes {delta}{sup 2}F = {var epsilon}{sub D}. In the case of unstable and corresponding damped modes it is seen that {delta}{sup 2}F {ne} {var epsilon}{sub D}; in fact {delta}{sup 2}F {equivalent to} 0. This failure of the dielectric energy expression persists even for arbitrarily small growth and damping rates since {var epsilon}{sub D} is nonzero in this limit, whereas {delta}{sup 2}F remains zero. The connection between the new exact energy expression and the at-best approximate {var epsilon}{sub D} is described. The new expression motivates natural definitions of Hamiltonian action variables and signature. A general linear integral transform is introduced that maps the linear version of the noncanonical Hamiltonian structure, which describes the Vlasov equation, to action-angle (diagonal) form.
NASA Astrophysics Data System (ADS)
Hakim, Ammar; Hammett, Greg; Shi, Eric
2016-10-01
We have developed a hybrid discontinuous/continuous Galerkin scheme for gyrokinetic equations. Our scheme solves the equations in the Poisson bracket formulation and, with a careful choice of basis functions, conserves energy exactly. We use a sparse grid representation to reduce cost. We have developed a novel form of sheath boundary conditions, going beyond logical-sheath BCs, that allows current flow into the boundaries, yet retains overall charge continuity as well as conserves energy. First applications to a simplified model of the NSTX-like scrape-off-layer (SOL) are presented. We treat the SOL in simplified geometry, retaining magnetic curvature effects, and study the turbulent spreading of particle and heat flux. Extensions to include magnetic fluctuations and collisions are discussed. Supported by the Max-Planck/Princeton Center for Plasma Physics, the SciDAC Center for the Study of Plasma Microturbulence, and DOE Contract DE-AC02-09CH11466.
NASA Astrophysics Data System (ADS)
Ohana, N.; Jocksch, A.; Lanti, E.; Tran, T. M.; Brunner, S.; Gheller, C.; Hariri, F.; Villard, L.
2016-11-01
With the aim of enabling state-of-the-art gyrokinetic PIC codes to benefit from the performance of recent multithreaded devices, we developed an application from a platform called the “PIC-engine” [1, 2, 3] embedding simplified basic features of the PIC method. The application solves the gyrokinetic equations in a sheared plasma slab using B-spline finite elements up to fourth order to represent the self-consistent electrostatic field. Preliminary studies of the so-called Particle-In-Fourier (PIF) approach, which uses Fourier modes as basis functions in the periodic dimensions of the system instead of the real-space grid, show that this method can be faster than PIC for simulations with a small number of Fourier modes. Similarly to the PIC-engine, multiple levels of parallelism have been implemented using MPI+OpenMP [2] and MPI+OpenACC [1], the latter exploiting the computational power of GPUs without requiring complete code rewriting. It is shown that sorting particles [3] can lead to performance improvement by increasing data locality and vectorizing grid memory access. Weak scalability tests have been successfully run on the GPU-equipped Cray XC30 Piz Daint (at CSCS) up to 4,096 nodes. The reduced time-to-solution will enable more realistic and thus more computationally intensive simulations of turbulent transport in magnetic fusion devices.
NASA Astrophysics Data System (ADS)
Manas, P.; Hornsby, W. A.; Angioni, C.; Camenen, Y.; Peeters, A. G.
2017-03-01
The impact of the neoclassical background on turbulent impurity transport is investigated by means of gyrokinetic simulations supported by fluid equations. The latter are derived, using a Laguerre polynomials expansion of the first order neoclassical distribution function, and analytical expressions of the turbulent momentum flux and impurity transport coefficients are assessed. Comparisons of gyrokinetic simulations including this neoclassical background (coupling between the codes GKW and NEO) and the fluid model are used to identify the main mechanisms behind the modification of the turbulent transport channels and benchmark the numerical implementation. These mechanisms include a modification of the parallel dynamics of the main ions and direct contributions stemming from the asymmetry in the parallel velocity space of the neoclassical distribution function. The latter which is found dominant for turbulent impurity transport, increases with increasing collisionality, R/{L}{Ti}, R/{L}n, impurity mass, safety factor and aspect ratio. These contributions to momentum and impurity fluxes are also found to depend on the directions of the toroidal magnetic field and plasma current.
Predictive Gyrokinetic Transport Simulations and Application of Synthetic Diagnostics
NASA Astrophysics Data System (ADS)
Candy, J.
2009-11-01
In this work we make use of the gyrokinetic transport solver TGYRO [1] to predict kinetic plasma profiles consistent with energy and particle fluxes in the DIII-D tokamak. TGYRO uses direct nonlinear and neoclassical fluxes calculated by the GYRO and NEO codes, respectively, to solve for global, self-consistent temperature and density profiles via Newton iteration. Previous work has shown that gyrokinetic simulation results for DIII-D discharge 128913 match experimental data rather well in the plasma core, but with a discrepancy in both fluxes and fluctuation levels emerging closer to the edge (r/a > 0.8). The present work will expand on previous results by generating model predictions across the entire plasma core, rather than at isolated test radii. We show that TGYRO predicts temperature and density profiles in good agreement with experimental observations which simultaneously yield near-exact (to within experimental uncertainties) agreement with power balance calculations of the particle and energy fluxes for r/a <=0.8. Moreover, we use recently developed synthetic diagnostic algorithms [2] to show that TGYRO also predicts density and electron temperature fluctuation levels in close agreement with experimental measurements across the simulated plasma volume. 8pt [1] J. Candy, C. Holland, R.E. Waltz, M.R. Fahey, and E. Belli, ``Tokamak profile prediction using direct gyrokinetic and neoclassical simulation," Phys. Plasmas 16, 060704 (2009). [2] C. Holland, A.E. White, G.R. McKee, M.W. Shafer, J. Candy, R.E. Waltz, L. Schmitz, and G.R. Tynan, ``Implementation and application of two synthetic diagnostics for validating simulations of core tokamak turbulence," Phys. Plasmas 16, 052301 (2009).
Optimized Loading for Particle-in-cell Gyrokinetic Simulations
J.L.V. Lewandowski
2004-05-13
The problem of particle loading in particle-in-cell gyrokinetic simulations is addressed using a quadratic optimization algorithm. Optimized loading in configuration space dramatically reduces the short wavelength modes in the electrostatic potential that are partly responsible for the non-conservation of total energy; further, the long wavelength modes are resolved with good accuracy. As a result, the conservation of energy for the optimized loading is much better that the conservation of energy for the random loading. The method is valid for any geometry and can be coupled to optimization algorithms in velocity space.
Gyrokinetic simulation of isotope scaling in tokamak plasmas
Lee, W.W.; Santoro, R.A.
1995-07-01
A three-dimensional global gyrokinetic particle code in toroidal geometry has been used for investigating the transport properties of ion temperature gradient (ITG) drift instabilities in tokamak plasmas. Using the isotopes of hydrogen (H{sup +}), deuterium (D{sup +}) and tritium (T{sup +}), we have found that, under otherwise identical conditions, there exists a favorable isotope scaling for the ion thermal diffusivity, i.e., Xi decreases with mass. Such a scaling, which exists both at the saturation of the instability and also at the nonlinear steady state, can be understood from the resulting wavenumber and frequency spectra.
Applications of large eddy simulation methods to gyrokinetic turbulence
Bañón Navarro, A. Happel, T.; Teaca, B. [Applied Mathematics Research Centre, Coventry University, Coventry CV1 5FB; Max-Planck für Sonnensystemforschung, Max-Planck-Str. 2, D-37191 Katlenburg-Lindau; Max-Planck Jenko, F. [Max-Planck-Institut für Plasmaphysik, EURATOM Association, D-85748 Garching; Max-Planck Hammett, G. W. [Max-Planck Collaboration: ASDEX Upgrade Team
2014-03-15
The large eddy simulation (LES) approach—solving numerically the large scales of a turbulent system and accounting for the small-scale influence through a model—is applied to nonlinear gyrokinetic systems that are driven by a number of different microinstabilities. Comparisons between modeled, lower resolution, and higher resolution simulations are performed for an experimental measurable quantity, the electron density fluctuation spectrum. Moreover, the validation and applicability of LES is demonstrated through a series of diagnostics based on the free energetics of the system.
Gyrokinetic simulation and theory for kinetic ballooning mode
NASA Astrophysics Data System (ADS)
Li, Yue-Yan; Xiao, Yong
2016-10-01
The kinetic ballooning mode (KBM) plays an important role in H mode formation and edge-localized mode (ELM) physics and internal transport barrier. A thorough understanding of the linear KBM physics is crucial to understanding the nonlinear edge physics in tokamaks. The second stability regime in small shear and sufficiently large pressure gradient has been predicted by the ballooning MHD theory. In the present work, a kinetic ballooning mode is found for the second stability regime in s-alpha space. This KBM mode is characterized by a broad-spread eigenfunction in the ballooning space, and destabilized dramatically by the ion temperature gradient. Such KBM mode almost exists in the entire second stability regime. Also KBM has been found in the sufficiently small shear even with negative shear. The gyrokinetic code GTC is employed to study the KBM physics, and finds that the linear KBM growth rate and frequency are very sensitive to the equilibrium profile. The effect of parallel current and density has been investigated, and comparisons between gyrokinetic simulation and analytic theory are carried out. The results shows that parallel current response have a strong effect in stabilize KBM instability, which could have a large effect on nonlinear electromagnetic turbulent transport.
Direct identification of predator-prey dynamics in gyrokinetic simulations
Kobayashi, Sumire Gürcan, Özgür D; Diamond, Patrick H.
2015-09-15
The interaction between spontaneously formed zonal flows and small-scale turbulence in nonlinear gyrokinetic simulations is explored in a shearless closed field line geometry. It is found that when clear limit cycle oscillations prevail, the observed turbulent dynamics can be quantitatively captured by a simple Lotka-Volterra type predator-prey model. Fitting the time traces of full gyrokinetic simulations by such a reduced model allows extraction of the model coefficients. Scanning physical plasma parameters, such as collisionality and density gradient, it was observed that the effective growth rates of turbulence (i.e., the prey) remain roughly constant, in spite of the higher and varying level of primary mode linear growth rates. The effective growth rate that was extracted corresponds roughly to the zonal-flow-modified primary mode growth rate. It was also observed that the effective damping of zonal flows (i.e., the predator) in the parameter range, where clear predator-prey dynamics is observed, (i.e., near marginal stability) agrees with the collisional damping expected in these simulations. This implies that the Kelvin-Helmholtz-like instability may be negligible in this range. The results imply that when the tertiary instability plays a role, the dynamics becomes more complex than a simple Lotka-Volterra predator prey.
Properties of Discontinuous Galerkin Algorithms and Implications for Edge Gyrokinetics
NASA Astrophysics Data System (ADS)
Hammett, G. W.; Hakim, A.; Shi, E. L.; Abel, I. G.; Stoltzfus-Dueck, T.
2015-11-01
The continuum gyrokinetic code Gkeyll uses Discontinuous Galerkin (DG) algorithms, which have a lot of flexibility in the choice of basis functions and inner product norm that can be useful in designing algorithms for particular problems. Rather than use regular polynomial basis functions, we consider here Maxwellian-weighted basis functions (which have similarities to Gaussian radial basis functions). The standard Galerkin approach loses particle and energy conservation, but this can be restored with a particular weight for the inner product (this is equivalent to a Petrov-Galerkin method). This allows a full- F code to have some benefits similar to the Gaussian quadrature used in gyrokinetic δf codes to integrate Gaussians times some polynomials exactly. In tests of Gkeyll for electromagnetic fluctuations, we found it is important to use consistent basis functions where the potential is in a higher-order continuity subspace of the space for the vector potential A| |. A regular projection method to this subspace is a non-local operation, while we show a self-adjoint averaging operator that can preserve locality and energy conservation. This does not introduce damping, but like gyro-averaging involves only the reactive part of the dynamics. Supported by the Max-Planck/Princeton Center for Plasma Physics, the SciDAC Center for the Study of Plasma Microturbulence, and DOE Contract DE-AC02-09CH11466.
Gyrokinetic modelling of stationary electron and impurity profiles in tokamaks
Skyman, A. Tegnered, D. Nordman, H. Strand, P.
2014-09-15
Particle transport due to Ion Temperature Gradient (ITG)/Trapped Electron Mode (TEM) turbulence is investigated using the gyrokinetic code GENE. Both a reduced quasilinear treatment and nonlinear simulations are performed for typical tokamak parameters corresponding to ITG dominated turbulence. The gyrokinetic results are compared and contrasted with results from a computationally efficient fluid model. A selfconsistent treatment is used, where the stationary local profiles are calculated corresponding to zero particle flux simultaneously for electrons and trace impurities. The scaling of the stationary profiles with magnetic shear, safety factor, electron-to-ion temperature ratio, collisionality, toroidal sheared rotation, plasma β, triangularity, and elongation is investigated. In addition, the effect of different main ion mass on the zero flux condition is discussed. The electron density gradient can significantly affect the stationary impurity profile scaling. It is therefore expected that a selfconsistent treatment will yield results more comparable to experimental results for parameter scans where the stationary background density profile is sensitive. This is shown to be the case in scans over magnetic shear, collisionality, elongation, and temperature ratio, for which the simultaneous zero flux electron and impurity profiles are calculated. A slight asymmetry between hydrogen, deuterium, and tritium with respect to profile peaking is obtained, in particular, for scans in collisionality and temperature ratio.
Linear signatures in nonlinear gyrokinetics: interpreting turbulence with pseudospectra
NASA Astrophysics Data System (ADS)
Hatch, D. R.; Jenko, F.; Bañón Navarro, A.; Bratanov, V.; Terry, P. W.; Pueschel, M. J.
2016-07-01
A notable feature of plasma turbulence is its propensity to retain features of the underlying linear eigenmodes in a strongly turbulent state—a property that can be exploited to predict various aspects of the turbulence using only linear information. In this context, this work examines gradient-driven gyrokinetic plasma turbulence through three lenses—linear eigenvalue spectra, pseudospectra, and singular value decomposition (SVD). We study a reduced gyrokinetic model whose linear eigenvalue spectra include ion temperature gradient driven modes, stable drift waves, and kinetic modes representing Landau damping. The goal is to characterize in which ways, if any, these familiar ingredients are manifest in the nonlinear turbulent state. This pursuit is aided by the use of pseudospectra, which provide a more nuanced view of the linear operator by characterizing its response to perturbations. We introduce a new technique whereby the nonlinearly evolved phase space structures extracted with SVD are linked to the linear operator using concepts motivated by pseudospectra. Using this technique, we identify nonlinear structures that have connections to not only the most unstable eigenmode but also subdominant modes that are nonlinearly excited. The general picture that emerges is a system in which signatures of the linear physics persist in the turbulence, albeit in ways that cannot be fully explained by the linear eigenvalue approach; a non-modal treatment is necessary to understand key features of the turbulence.
Gyrokinetic simulations of microturbulence in DIII-D tokamak pedestal
NASA Astrophysics Data System (ADS)
Holod, Ihor; Fulton, Daniel; Taimourzadeh, Sam; Lin, Zhihong; Nazikian, Raffi; Spong, Donald
2015-11-01
The characteristics of H-mode pedestal are generally believed to be constrained by current-driven peeling-ballooning modes and pressure-driven instabilities, such as kinetic ballooning mode (KBM). In this work we use global gyrokinetic code (GTC) to identify and study the edge pressure-driven instabilities in the H-mode pedestal using realistic geometry and plasma profiles of DIII-D shot 131997. In our simulations we observe the KBM mode marginally dominant in the steep gradient region (ψN = 0 . 98), in the range of kθ ~ 1 cm-1 which corresponds to the most unstable mode number in the nonlinearly saturated state. For shorter wavelengths the trapped electron mode becomes dominant since its linear growth rate increases with the mode number, while the KBM gets saturated. In the pedestal top region (ψN = 0 . 95) the ITG dominates. Resonant magnetic perturbations (RMP) are widely applied for ELM mitigation. During RMP suppression, the increase of edge turbulence is often observed. To understand this phenomena we use gyrokinetic simulations to address the direct effect of magnetic perturbations on the microturbulence. Simulations with 3D equilibrium reconstructed by VMEC code have been compared with toroidally averaged equilibrium, using identical pressure profiles. Work supported by DOE grant DE-SC0010416 and by General Atomics subcontract.
Gyrokinetic linear stability analysis of NSTX L-mode plasmas
NASA Astrophysics Data System (ADS)
Han, Ke; Ren, Yang
2016-10-01
NSTX offered unique opportunities in studying transport and turbulence with low aspect ratio, strong plasma shaping and strong E ×B shear. NSTX L-mode plasmas have some favorable properties to facilitate the study of the relation between microturbulence and thermal transport: easier to obtain stationary profiles; easier to maintain MHD quiescence; no complications from edge transport barrier. Studies of NSTX RF/NBI-heated L-mode plasmas have provided new insight into the role of ion and electron-scale turbulence in driving anomalous transport. Here we present linear stability analysis of some NSTX L-mode plasmas with GS2 gyrokinetic code. GS2 is an initial value gyrokinetic code which, in its linear mode, finds the fastest growing mode for a given pair of poloidal and radial wavenumbers. The linear simulations used local Miller equilibria and plasma parameters derived from measured experimental profiles with electromagnetic effects, electron and ion collisions and carbon impurity. The work is supported by DOE, China Scholarship Council, the Natural Science Foundation of China (61402138) and the Natural Science Foundation of Heilongjiang Province (E201452).
Multi-code benchmark of global gyrokinetic electromagnetic instabilities
NASA Astrophysics Data System (ADS)
Goerler, Tobias; Bottino, Alberto; Hornsby, William A.; Kleiber, Ralf; Tronko, Natalia; Grandgirard, Virginie; Norscini, Claudia; Sonnendruecker, Eric
2015-11-01
Considering the recent major extensions of global gyrokinetic codes towards a comprehensive and self-consistent treatment of electromagnetic (EM) effects, corresponding verification tests are obvious and necessary steps to be taken. While a number of (semi-)analytic test cases and benchmarks exist in the axisymmetric limit, microinstabilities and particularly EM turbulence are rarely addressed. In order to remedy this problem, a hierarchical linear gyrokinetic benchmark study is presented starting with electrostatic ion temperature gradient microinstabilities with adiabatic electron response and progressing finally to the characterization of fully EM instabilities as a function of β. The inter-code comparison involves contributions from Eulerian Vlasov, Lagrangian PIC, and Semi-Lagrange codes at least in one level of this verification exercise, thus confirming a high degree of reliability for the implementations that has rarely been achieved before in this context. Additionally, possible extensions of this benchmark into the physically more relevant nonlinear turbulence regime will be discussed, e.g., relaxation problems or gradient-driven setups. This work has been carried out within the framework of the EUROfusion Consortium.
A new hybrid-Lagrangian numerical scheme for gyrokinetic simulation of tokamak edge plasma
Ku, S.; Hager, R.; Chang, C.S.; Kwon, J.M.; Parker, S.E.
2016-06-15
In order to enable kinetic simulation of non-thermal edge plasmas at a reduced computational cost, a new hybrid-Lagrangian δf scheme has been developed that utilizes the phase space grid in addition to the usual marker particles, taking advantage of the computational strengths from both sides. The new scheme splits the particle distribution function of a kinetic equation into two parts. Marker particles contain the fast space-time varying, δf, part of the distribution function and the coarse-grained phase-space grid contains the slow space-time varying part. The coarse-grained phase-space grid reduces the memory-requirement and the computing cost, while the marker particles provide scalable computing ability for the fine-grained physics. Weights of the marker particles are determined by a direct weight evolution equation instead of the differential form weight evolution equations that the conventional delta-f schemes use. The particle weight can be slowly transferred to the phase space grid, thereby reducing the growth of the particle weights. The non-Lagrangian part of the kinetic equation – e.g., collision operation, ionization, charge exchange, heat-source, radiative cooling, and others – can be operated directly on the phase space grid. Deviation of the particle distribution function on the velocity grid from a Maxwellian distribution function – driven by ionization, charge exchange and wall loss – is allowed to be arbitrarily large. The numerical scheme is implemented in the gyrokinetic particle code XGC1, which specializes in simulating the tokamak edge plasma that crosses the magnetic separatrix and is in contact with the material wall.
A new hybrid-Lagrangian numerical scheme for gyrokinetic simulation of tokamak edge plasma
Ku, S.; Hager, R.; Chang, C. S.; Kwon, J. M.; Parker, S. E.
2016-06-01
In order to enable kinetic simulation of non-thermal edge plasmas at a reduced computational cost, a new hybrid-Lagrangian δf scheme has been developed that utilizes the phase space grid in addition to the usual marker particles, taking advantage of the computational strengths from both sides. The new scheme splits the particle distribution function of a kinetic equation into two parts. Marker particles contain the fast space-time varying, δf, part of the distribution function and the coarse-grained phase-space grid contains the slow space-time varying part. The coarse-grained phase-space grid reduces the memory-requirement and the computing cost, while the marker particles provide scalable computing ability for the fine-grained physics. Weights of the marker particles are determined by a direct weight evolution equation instead of the differential form weight evolution equations that the conventional delta-f schemes use. The particle weight can be slowly transferred to the phase space grid, thereby reducing the growth of the particle weights. The non-Lagrangian part of the kinetic equation – e.g., collision operation, ionization, charge exchange, heat-source, radiative cooling, and others – can be operated directly on the phase space grid. Deviation of the particle distribution function on the velocity grid from a Maxwellian distribution function – driven by ionization, charge exchange and wall loss – is allowed to be arbitrarily large. The numerical scheme is implemented in the gyrokinetic particle code XGC1, which specializes in simulating the tokamak edge plasma that crosses the magnetic separatrix and is in contact with the material wall.
Nonlinear theory of drift-cyclotron kinetics and the possible breakdown of gyro-kinetics
Waltz, R. E.; Deng Zhao
2013-01-15
A nonlinear theory of drift-cyclotron kinetics (termed cyclo-kinetics here) is formulated to test the breakdown of the gyro-kinetic approximations. Six dimensional cyclo-kinetics can be regarded as an extension of five dimensional gyro-kinetics to include high-frequency cyclotron waves, which can interrupt the low-frequency gyro-averaging in the (sixth velocity grid) gyro-phase angle. Nonlinear cyclo-kinetics has no limit on the amplitude of the perturbations. Formally, there is no gyro-averaging when all cyclotron (gyro-phase angle) harmonics of the perturbed distribution function (delta-f) are retained. Retaining only the (low frequency) zeroth cyclotron harmonic in cyclo-kinetics recovers both linear and nonlinear gyro-kinetics. Simple recipes are given for converting continuum nonlinear delta-f gyro-kinetic transport simulation codes to cyclo-kinetics codes by retaining (at least some) higher cyclotron harmonics.
NASA Astrophysics Data System (ADS)
Kleiber, R.; Hatzky, R.; Könies, A.; Mishchenko, A.; Sonnendrücker, E.
2016-03-01
A new algorithm for electromagnetic gyrokinetic simulations, the so called "pullback transformation scheme" proposed by Mishchenko et al. [Phys. Plasmas 21, 092110 (2014)] is motivated as an explicit time integrator reset after each full timestep and investigated in detail. Using a numerical dispersion relation valid in slab geometry, it is shown that the linear properties of the scheme are comparable to those of an implicit v∥ -scheme. A nonlinear extension of the mixed variable formulation, derived consistently from a field Lagrangian, is proposed. The scheme shows excellent numerical properties with a low statistical noise level and a large time step especially for MHD modes. The example of a nonlinear slab tearing mode simulation is used to illustrate the properties of different formulations of the physical model equations.
On the Existence of Canonical Gyrokinetic Variables for Chaotic Magnetic Fields
Nicolini, Piero; Tessarotto, Massimo
2008-12-31
The gyrokinetic description of particle dynamics faces a basic difficulty when a special type of canonical variables is sought, i.e., the so-called gyrokinetic canonical variables. These are defined in such a way that two of them are respectively identified with the gyrophase-angle, describing the fast particle gyration motion around magnetic field lines, and its canonically conjugate momentum. In this paper we intend to discuss the conditions of existence for these variables.
Stability of drift waves with the integral eigenmode equation
Chen, L.; Ke, F.J.; Xu, M.J.; Tsai, S.T.; Lee, Y.C.; Antonsen, T.M. Jr.
1981-11-01
An analytical theory on the stability properties of drift-wave eigenmodes in a slab plasma with finite magnetic shear is presented. The corresponding eigenmode equation is the integral equation first given by Coppi, Rosenbluth, and Sagdeev (1967) and rederived here, in a relatively simpler fashion, via the gyrokinetic equation. It is then proved that the universal drift-wave eigenmodes remain absolutely stable and finite electron temperature gradients do not alter the stability.
NASA Astrophysics Data System (ADS)
Martin, Mike; Landreman, Matt; Mandell, Noah; Dorland, William
2016-10-01
GryfX is a delta-f code that evolves the gyrofluid set of equations using sophisticated nonlinear closures, with the option to evolve zonal flows (ky =0) kinetically. Since fluid models require less memory to store than a kinetic model, GryfX is ideally suited and thus written to run on a Graphics Processing Unit (GPU), yielding about a 1,200 times performance advantage over GS2. Here we present the first stellarator simulations using GryfX. Results compare linear growth rates of the Ion Temperature Gradient (ITG) mode between GryfX and the gyrokinetic code, GS2, using stellarator geometries from the National Compact Stellarator Experiment (NCSX) and Wendelstein 7-X (W7X). Strong agreement of <10% for maximum growth rates is observed between GS2 and GryfX for temperature gradients away from marginal stability for both NCSX and W7X geometries. Nonlinear stellarator results using GS2/GryfX are also presented.
The direct method for gyrokinetic simulation with kinetic electrons and magnetic perturbations
NASA Astrophysics Data System (ADS)
Chen, Yang; Parker, Scott
2016-10-01
Over the past fifteen years we have developed two Particle-in-Cell (PIC) algorithms for gyrokinetic simulation of tokamak plasmas with kinetic electrons and magnetic perturbations. The first, called the Direct Method, uses an adjustable split-weight scheme and a control-variate method to properly handle the ``cancellation problem''. The second algorithm is the Closure Scheme, which solves the vorticity equation and the generalized Ohm's law, closing with an electron pressure calculated from delta-f PIC electrons. The extention of the Direct Method to handle low-n (long wavelength) fluctuations will be discussed. We will explain the algorithm, describe the low-n geometrical implementation, and present numerical observations in applying the Direct Method to various waves, including shear Alfven waves, ITGs and the n = 1 tearing mode. These observations suggest that, whereas the cancellation problem appears to be unavoidable and can be solved by the controlled variate method, the split-weight scheme is primarily a technique for numerical stability, and can probably be replaced by other techniques (such as the mixed-variable approach.
Tummel, K.; Chen, L.; Wang, Z.; Wang, X. Y.; Lin, Y.
2014-05-15
A kinetic electrostatic eigenvalue equation for the lower-hybrid drift instability (LHDI) in a thin Harris current sheet with a guide field is derived based on the gyrokinetic electron and fully kinetic ion(GeFi) description. Three-dimensional nonlocal analyses are carried out to investigate the influence of a guide field on the stabilization of the LHDI by finite parallel wavenumber, k{sub ∥}. Detailed stability properties are first analyzed locally, and then as a nonlocal eigenvalue problem. Our results indicate that at large equilibrium drift velocities, the LHDI is further destabilized by finite k{sub ∥} in the short-wavelength domain. This is demonstrated in a local stability analysis and confirmed by the peak in the eigenfunction amplitude. We find the most unstable modes localized at the current sheet edges, and our results agree well with simulations employing the GeFi code developed by Lin et al. [Plasma Phys. Controlled Fusion 47, 657 (2005); Plasma Phys. Controlled Fusion 53, 054013 (2011)].
Gyrokinetic studies of microinstabilities in the reversed field pinch
Carmody, D.; Pueschel, M. J.; Terry, P. W.
2013-05-15
An analytic equilibrium, the Toroidal Bessel Function Model, is used in conjunction with the gyrokinetic code GYRO to investigate the nature of microinstabilities in a reversed field pinch plasma. The effect of the normalized electron plasma pressure β on the characteristics of the microinstabilities is studied. At a β of 4.5%, a transition between an ion temperature gradient (ITG) and a microtearing mode is observed. Suppression of the ITG mode occurs as in the tokamak, through coupling to shear Alfvén waves, with a critical β for stability higher than its tokamak equivalent due to a shorter parallel connection length. A steep dependence of the microtearing growth rate on the temperature gradient suggests high profile stiffness. There is evidence for a collisionless microtearing mode. The properties of this mode are investigated, and it is found that electron curvature drift plays an important role in the instability.
Implications of advanced collision operators for gyrokinetic simulation
NASA Astrophysics Data System (ADS)
Belli, E. A.; Candy, J.
2017-04-01
In this work, we explore both the potential improvements and pitfalls that arise when using advanced collision models in gyrokinetic simulations of plasma microinstabilities. Comparisons are made between the simple-but-standard electron Lorentz operator and specific variations of the advanced Sugama operator. The Sugama operator describes multi-species collisions including energy diffusion, momentum and energy conservation terms, and is valid for arbitrary wavelength. We report scans over collision frequency for both low and high {k}θ {ρ }s modes, with relevance for multiscale simulations that couple ion and electron scale physics. The influence of the ion–ion collision terms—not retained in the electron Lorentz model—on the damping of zonal flows is also explored. Collision frequency scans for linear and nonlinear simulations of ion-temperature-gradient instabilities including impurity ions are presented. Finally, implications for modeling turbulence in the highly collisional edge are discussed.
Gyrokinetic Theory for Arbitrary Wavelength Electromagnetic Modes in Tokamaks
Qin, H.; Rewoldt, G.; Tang, W.M.
1997-10-01
A linear gyrokinetic system for arbitrary wavelength electromagnetic modes is developed. A wide range of modes in inhomogeneous plasmas, such as the internal kink modes, the toroidal Alfvén eigenmode (TAE) modes, and the drift modes, can be recovered from this system. The inclusion of most of the interesting physical factors into a single framework enables us to look at many familiar modes simultaneously and thus to study the modifications of and the interactions between them in a systematic way. Especially, we are able to investigate self-consistently the kinetic MHD phenomena entirely from the kinetic side. Phase space Lagrangian Lie perturbation methods and a newly developed computer algebra package for vector analysis in general coordinate system are utilized in the analytical derivation. In tokamak geometries, a 2D finite element code has been developed and tested. In this paper, we present the basic theoretical formalism and some of the preliminary results.
Gyrokinetic turbulence under near-separatrix or nonaxisymmetric conditionsa)
NASA Astrophysics Data System (ADS)
Jenko, F.; Told, D.; Xanthopoulos, P.; Merz, F.; Horton, L. D.
2009-05-01
Linear and nonlinear gyrokinetic simulations with the GENE code [F. Jenko et al., Phys. Plasmas 7, 1904 (2000)] for tokamak edge plasmas as well as for stellarator core plasmas are presented, shedding light on the behavior of plasma microturbulence under near-separatrix or nonaxisymmetric conditions. To this aim, the required geometric coefficients are inferred directly from the magnetohydrodynamic equilibria of three different devices via the newly developed GIST code. It is found that the residual electron heat transport level in the H-mode edge can be explained in terms of high-wave-number fluctuations driven by electron temperature gradient modes. Moreover, the study of adiabatic ion temperature gradient turbulence in optimized stellarators points to the possibility of a systematic geometric optimization with respect to anomalous transport in nonaxisymmetric devices.
Gyrokinetic studies of stellarator turbulent transport via Gene
NASA Astrophysics Data System (ADS)
Mynick, H. E.; Xanthopoulos, P.; Boozer, A. H.
2009-05-01
We study the turbulence and turbulent transport in stellarators (and tokamaks), via analysis of simulation results from the 3D gyrokinetic code Gene with particular emphasis on the configuration-dependence of turbulence characteristics. Present day stellarator designs seek to optimize stellarator neoclassical transport. With the advent of simulation codes like Gene, one can now seek to characterize and then optimize designs for TOTAL transport. The comparison between different configurations in stellarator parameter space is of 2 types, global and local. Global comparisons look at changes in plasma performance (eg, levels of turbulent fluxes and zonal flows, dependence of these on plasma gradients) between very different designs (eg, a QA versus a QI/QO design). Local comparisons look at the changes in performance between slight variants of the same design. Both aid in gaining insight into which geometric features (curvature, local and global shear, etc) are important in determining the turbulent characteristics. P. Xanthopoulos, F. Jenko, Phys.Plasmas 13, 092301 (2006).
Gyrokinetic Simulation of Global Turbulent Transport Properties in Tokamak Experiments
Wang, W.X.; Lin, Z.; Tang, W.M.; Lee, W.W.; Ethier, S.; Lewandowski, J.L.V.; Rewoldt, G.; Hahm, T.S.; Manickam, J.
2006-01-01
A general geometry gyro-kinetic model for particle simulation of plasma turbulence in tokamak experiments is described. It incorporates the comprehensive influence of noncircular cross section, realistic plasma profiles, plasma rotation, neoclassical (equilibrium) electric fields, and Coulomb collisions. An interesting result of global turbulence development in a shaped tokamak plasma is presented with regard to nonlinear turbulence spreading into the linearly stable region. The mutual interaction between turbulence and zonal flows in collisionless plasmas is studied with a focus on identifying possible nonlinear saturation mechanisms for zonal flows. A bursting temporal behavior with a period longer than the geodesic acoustic oscillation period is observed even in a collisionless system. Our simulation results suggest that the zonal flows can drive turbulence. However, this process is too weak to be an effective zonal flow saturation mechanism.
New variational principle for the Vlasov-Maxwell equations.
Brizard, A J
2000-06-19
A new Eulerian variational principle is presented for the Vlasov-Maxwell equations. This principle uses constrained variations for the Vlasov distribution in eight-dimensional extended phase space. The standard energy-momentum conservation law is then derived explicitly by the Noether method. This new variational principle can be applied to various reduced Vlasov-Maxwell equations in which fast time scales have been asymptotically eliminated (e.g., low-frequency gyrokinetic theory).
Full-f gyrokinetic simulation over a confinement time
Idomura, Yasuhiro
2014-02-15
A long time ion temperature gradient driven turbulence simulation over a confinement time is performed using the full-f gyrokinetic Eulerian code GT5D. The convergence of steady temperature and rotation profiles is examined, and it is shown that the profile relaxation can be significantly accelerated when the simulation is initialized with linearly unstable temperature profiles. In the steady state, the temperature profile and the ion heat diffusivity are self-consistently determined by the power balance condition, while the intrinsic rotation profile is sustained by complicated momentum transport processes without momentum input. The steady turbulent momentum transport is characterized by bursty non-diffusive fluxes, and the resulting turbulent residual stress is consistent with the profile shear stress theory [Y. Camenen et al., “Consequences of profile shearing on toroidal momentum transport,” Nucl. Fusion 51, 073039 (2011)] in which the residual stress depends not only on the profile shear and the radial electric field shear but also on the radial electric field itself. Based on the toroidal angular momentum conservation, it is found that in the steady null momentum transport state, the turbulent residual stress is cancelled by the neoclassical counterpart, which is greatly enhanced in the presence of turbulent fluctuations.
Shear-Alfv'en Waves in Gyrokinetic Particle Simulation
NASA Astrophysics Data System (ADS)
Dickerson, Thomas D.; Startsev, Edward A.; Lee, W. W.
2012-10-01
Numerical properties of shear-Alfv'en waves in slab geometry have been studied using a Particle-in-Cell code implementing the recently developed double split-weight scheme [1]. This scheme separates the non-adiabatic response of the particles from both their adiabatic responses and the field-line bending effects arising from the background density and temperature gradients of both the electrons and the ions. This scheme is an improvement over the original split-weight scheme [2] in the presence of the zeroth-order inhomogeneities. The present studies consist of testing numerical restrictions on temporal resolution in the simulation of these waves in one and two dimensions, and on spatial resolutions on the formation of shear Alfv'en eigenmodes in two dimensional sheared slab simulations. For example, it is found that the correct behavior of ion temperature gradient modes in terms of frequencies and growth rates can be maintained with time steps larger than the limit imposed by the shear-Alfven waves. Details will be reported.[4pt] [1] E. A. Startsev and W. W. Lee, ``Finite-Beta Simulation of Microinstabilities,'' manuscript in preparation (2012). [0pt] [2] W. W. Lee, J. L. V. Lewandowski, T. S. Hahm and Z. Lin, ``Shear-Alf'en Waves in Gyrokinetic Plasmas,'' Phys. Plasmas 10, 4435 (2001).
A gyrokinetic perspective on the JET-ILW pedestal
NASA Astrophysics Data System (ADS)
Hatch, D. R.; Kotschenreuther, M.; Mahajan, S.; Valanju, P.; Liu, X.
2017-03-01
JET has been unable to recover historical confinement levels when operating with an ITER-like wall (ILW) due largely to the inaccessibility of high pedestal temperatures. Finding a path to overcome this challenge is of utmost importance for both a prospective JET DT campaign and for future ITER operation. Gyrokinetic simulations (using the Gene code) quantitatively capture experimental transport levels for a representative experimental discharge and qualitatively recover the major experimental trends. Microtearing turbulence is a major transport mechanisms for the low-temperature pedestals characteristic of unseeded JET-ILW discharges. At higher temperatures and/or lower {ρ\\ast} , we identify electrostatic ITG transport of a type that is strongly shear-suppressed on smaller machines. Consistent with observations, this transport mechanism is strongly reduced by the presence of a low-Z impurity (e.g. carbon or nitrogen at the level of {{Z}\\text{eff}}∼ 2 ), recovering the accessibility of high pedestal temperatures. Notably, simulations based on dimensionless {ρ\\ast} scans recover historical scaling behavior except in the unique JET-ILW parameter regime where ITG turbulence becomes important. Our simulations also elucidate the observed degradation of confinement caused by gas puffing, emphasizing the important role of the density pedestal structure. This study maps out important regions of parameter space, providing insights that may point to optimal physical regimes that can enable the recovery of high pedestal temperatures on JET.
Global gyrokinetic simulations of microturbulence for TCV-relevant plasmas
NASA Astrophysics Data System (ADS)
Merlo, G.; Brunner, S.; Coda, S.; Huang, Z.; Sauter, O.; Villard, L.; Görler, T.; Jenko, F.; Told, D.; Camenen, Y.; Marinoni, A.
2015-11-01
Due to significant global effects, in smaller-sized tokamaks such as TCV local (flux-tube) microturbulence simulations are unable to fully reproduce experimental transport levels. We will therefore present results obtained with the global version of the gyrokinetic code GENE aiming at addressing two observations made on TCV. 1) Effect of negative triangularity: it has been experimentally demonstrated that half the heating power is required to maintain the same electron temperature profile when the sign of triangularity of the Last Closed Flux Surface is reversed from δ=0.4 to -0.4 . Local simulations fail at reproducing both the actual transport level and positive/negative δ flux ratio. Therefore global simulations have been carried out with the aim of recovering the experimental results. 2) GAM physics: a complete multi-diagnostic characterization of the Geodesic Acoustic Mode has been reported from TCV. In particular the dependency of frequency, radial location and wave vector on plasma parameters have been experimentally investigated. Global runs modeling these TCV conditions will be discussed and simulations compared to experiments with the help of synthetic diagnostics.
Effects of Plasma Shaping on Nonlinear Gyrokinetic Turbulence
E. A. Belli; Hammett, G. W.; Dorland, W.
2008-08-01
The effects of flux surface shape on the gyrokinetic stability and transport of tokamak plasmas are studied using the GS2 code [M. Kotschenreuther, G. Rewoldt, and W.M. Tang, Comput. Phys. Commun. 88, 128 (1995); W. Dorland, F. Jenko, M. Kotschenreuther, and B.N. Rogers, Phys. Rev. Lett. 85, 5579 (2000)]. Studies of the scaling of nonlinear turbulence with shaping parameters are performed using analytic equilibria based on interpolations of representative shapes of the Joint European Torus (JET) [P.H. Rebut and B.E. Keen, Fusion Technol. 11, 13 (1987)]. High shaping is found to be a stabilizing influence on both the linear ion-temperature-gradient (ITG) instability and the nonlinear ITG turbulence. For the parameter regime studied here, a scaling of the heat flux with elongation of χ ~ κ^{-1.5} or κ^{-2.0}, depending on the triangularity, is observed at fixed average temperature gradient. While this is not as strong as empirical elongation scalings, it is also found that high shaping results in a larger Dimits upshift of the nonlinear critical temperature gradient due to an enhancement of the Rosenbluth-Hinton residual zonal flows.
Uncertainty estimation and a stopping rule in nonlinear gyrokinetic simulations
NASA Astrophysics Data System (ADS)
Oberparleiter, Michael; Nordman, Hans; Verdoolaege, Geert; Jenko, Frank
2016-11-01
We present a method to estimate the mean and uncertainty of fluctuating quantities, such as spatially averaged density and temperature fluctuations or radial fluxes, from initial value simulations of the Eulerian gyrokinetic code GENE[1, 2]. Since the time series are autocorrelated in time, the data is grouped into batches based on the autocorrelation time and their means form the sample for further statistical treatment, such as calculating the standard error of the mean.Based on this uncertainty estimate we develop a stopping rule for a nonlinear simulation: First, regression tests ensure that it has reached a stationary (quasisteady) state and data before this point is discarded. Then the previously described estimate is calculated. If the estimated relative error is below a prescribed threshold, the simulation is stopped. This scheme is applied to several previously performed GENE simulations ranging from simple benchmarks to modelling of JET and ASDEX discharges. It can be demonstrated that a number of simulations could be around 30% shorter if a maximal statistical relative uncertainty of 5% is desired for all monitored quantities.
An Efficient Method for Verifying Gyrokinetic Microstability Codes
NASA Astrophysics Data System (ADS)
Bravenec, R.; Candy, J.; Dorland, W.; Holland, C.
2009-11-01
Benchmarks for gyrokinetic microstability codes can be developed through successful ``apples-to-apples'' comparisons among them. Unlike previous efforts, we perform the comparisons for actual discharges, rendering the verification efforts relevant to existing experiments and future devices (ITER). The process requires i) assembling the experimental analyses at multiple times, radii, discharges, and devices, ii) creating the input files ensuring that the input parameters are faithfully translated code-to-code, iii) running the codes, and iv) comparing the results, all in an organized fashion. The purpose of this work is to automate this process as much as possible: At present, a python routine is used to generate and organize GYRO input files from TRANSP or ONETWO analyses. Another routine translates the GYRO input files into GS2 input files. (Translation software for other codes has not yet been written.) Other python codes submit the multiple GYRO and GS2 jobs, organize the results, and collect them into a table suitable for plotting. (These separate python routines could easily be consolidated.) An example of the process -- a linear comparison between GYRO and GS2 for a DIII-D discharge at multiple radii -- will be presented.
Toroidal universal drift instability: A global gyrokinetic study
Chowdhury, J.; Ganesh, R.; Brunner, S.; Vaclavik, J.; Villard, L.
2010-10-15
An electron density gradient driven instability identified as the toroidal branch of the universal drift instability is studied using a global gyrokinetic model treating both electrons and ions fully nonadiabatically and valid at all orders in the ratio of the Larmor radius to the wavelength. The physics of the magnetic drift resonance, Landau resonance and transit resonance, which are considered to be important for the toroidal universal mode, are kept for both species. A systematic parametric study is carried out for the mode. The toroidal universal drift mode is observed to sustain finite temperature gradient and can thus coexist with the temperature gradient driven modes and may contribute to the observed particle transport along with other drift modes. Especially at intermediate scales between the ion temperature gradient driven mode and electron temperature gradient driven mode, this branch of the drift instability can also be a plausible candidate for the observed particle loss. The effect of magnetic fluctuations on the mode is also investigated. In contrast to the slab mode, the toroidal branch of the universal drift mode is found to be strongly stabilized by electromagnetic effects at finite plasma {beta}. Finally, the effect of trapped electrons on the universal mode is studied and compared with the other possible modes in the same parameter regime, namely, ion temperature gradient mode in the presence of trapped electrons and pure trapped electron modes.
ADVANCES IN COMPREHENSIVE GYROKINETIC SIMULATIONS OF TRANSPORT IN TOKAMAKS
WALTZ,R.E; CANDY,J; HINTON,F.L; ESTRADA-MILA,C; KINSEY,J.E
2004-10-01
A continuum global gyrokinetic code GYRO has been developed to comprehensively simulate core turbulent transport in actual experimental profiles and enable direct quantitative comparisons to the experimental transport flows. GYRO not only treats the now standard ion temperature gradient (ITG) mode turbulence, but also treats trapped and passing electrons with collisions and finite {beta}, equilibrium ExB shear stabilization, and all in real tokamak geometry. Most importantly the code operates at finite relative gyroradius ({rho}{sub *}) so as to treat the profile shear stabilization and nonlocal effects which can break gyroBohm scaling. The code operates in either a cyclic flux-tube limit (which allows only gyroBohm scaling) or globally with physical profile variation. Bohm scaling of DIII-D L-mode has been simulated with power flows matching experiment within error bars on the ion temperature gradient. Mechanisms for broken gyroBohm scaling, neoclassical ion flows embedded in turbulence, turbulent dynamos and profile corrugations, are illustrated.
Propagation of global shear Alfven waves in gyrokinetic tokamak plasmas
NASA Astrophysics Data System (ADS)
Nishimura, Y.; Lin, Z.; Holod, I.; Chen, L.; Decyk, V.; Klasky, S.; Ma, K.; Adams, M.; Ethier, S.; Hahm, T.; Lee, W.; Lewandowski, J.; Rewoldt, G.; Wang, W.
2006-04-01
Employing the electromagnetic gyrokinetic simulation models, Alfven wave dynamics in global tokamak geometry is studied. Based on a small parameter expansion by the square-root of the electron-ion mass ratio, the fluid-kinetic hybrid electron model solves the adiabatic response in the lowest order and solves the kinetic response in the higher orders. We verify the propagation of shear Alfven waves in the absence of drives or damping mechanisms by perturbing the magnetic field lines at t=0 in a global eigenmode structure. The Alfven wave experiences continuum damping. In the presence of energetic particles, excitations of toroidal Alfven eigenmode (TAE) is expected within the frequency gap. With the ηi gradient drive, at a critical β value, the kinetic ballooning mode (KBM) is excited below the ideal MHD limit. W.W.Lee et al., Phys. Plasmas 8, 4435 (2001). Z.Lin and L.Chen, Phys. Plasmas 8, 1447 (2001). J.A.Tataronis and W. Grossman, Z. Phys. 14, 203 (1973). C.Z.Cheng, L.Chen, and M.S.Chance, Ann.Phys. 161, 21 (1984). C.Z.Cheng, Nucl. Fusion 22, 773 (1982).
Developing Experimentally Relevant Benchmarks for Gyrokinetic Microstability Codes
NASA Astrophysics Data System (ADS)
Bravenec, R.; Candy, J.; Dorland, W.; Ernst, D.; Staebler, G.; Waltz, R.
2008-11-01
A few nonlinear gyrokinetic microstability codes are now capable of simulating tokamak plasmas to an unprecedented level of complexity. Verification of these ``experimentally relevant'' simulations is difficult, however, because no benchmarks exist with which the codes can compare. This work describes the development of such benchmarks through ``apples-to-apples'' comparisons among codes, i.e., comparisons for the same plasma containing the same physics and having sufficient temporal, spatial, pitch-angle, and energy resolutions. A single utility code is used to extract experimental data from analysis by TRANSP, ONETWO, etc., and to produce input files for all the codes. The codes are first run linearly and, if differences in the mode frequencies are found, the computations are simplified by removing shaping, collisions, etc., one at a time, until agreement is reached. This process pinpoints the source(s) of the disagreement which the code developers attempt to resolve. Next, nonlinear runs are undertaken for the same cases and the procedure is repeated. The final results are both linear and nonlinear benchmarks at various levels of complexity by which other codes may be verified.
Gyrokinetic Simulation of Microturbulent Saturation at Finite β
NASA Astrophysics Data System (ADS)
Terry, P. W.; Pueschel, M. J.; Carmody, D.; Whelan, G. G.
2014-10-01
Saturation and zonal flow physics for microturbulence is investigated for tokamaks and the RFP using gyrokinetic computation to understand scalings with respect magnetic shear and β. Modeling an MST discharge shows that the critical instability gradient for TEM is higher than the tokamak threshold by the aspect ratio (R / a) . This factor is rooted in the shorter magnetic field scale length of the RFP. Nonlinear simulations show strong zonal flows and a large Dimits shift exceeding the tokamak shift by a factor of order (R / a) . The non zonal transition (NZT), a critical β for which zonal flows are disabled by flutter-induced charge loss is also considered. The critical β occurs when the radial displacement of a magnetic field line over a half connection length is equal to the radial correlation length. These quantities scale with the connection length and magnetic drift scale lengths entering the instability threshold and quasilinear diffusivities, making the RFP critical β for NZT higher than the tokamak value by (R / a) 1 . 5 times tokamak q. These results are consistent with magnetic shear and q dependence in the kinetic ballooning threshold, indicating that β effects will only arise at high β relative to typical RFP operation.
Electromagnetic gyrokinetic turbulence in finite-beta helical plasmas
Ishizawa, A.; Watanabe, T.-H.; Sugama, H.; Nakajima, N.; Maeyama, S.
2014-05-15
A saturation mechanism for microturbulence in a regime of weak zonal flow generation is investigated by means of electromagnetic gyrokinetic simulations. The study identifies a new saturation process of the kinetic ballooning mode (KBM) turbulence originating from the spatial structure of the KBM instabilities in a finite-beta Large Helical Device (LHD) plasma. Specifically, the most unstable KBM in LHD has an inclined mode structure with respect to the mid-plane of a torus, i.e., it has a finite radial wave-number in flux tube coordinates, in contrast to KBMs in tokamaks as well as ion-temperature gradient modes in tokamaks and helical systems. The simulations reveal that the growth of KBMs in LHD is saturated by nonlinear interactions of oppositely inclined convection cells through mutual shearing as well as by the zonal flow. The saturation mechanism is quantitatively investigated by analysis of the nonlinear entropy transfer that shows not only the mutual shearing but also a self-interaction with an elongated mode structure along the magnetic field line.
ADVANCES IN COMPREHENSIVE GYROKINETIC SIMULATIONS OF TRANSPORT IN TOKAMAKS
WALTZ RE; CANDY J; HINTON FL; ESTRADA-MILA C; KINSEY JE
2004-10-01
A continuum global gyrokinetic code GYRO has been developed to comprehensively simulate core turbulent transport in actual experimental profiles and enable direct quantitative comparisons to the experimental transport flows. GYRO not only treats the now standard ion temperature gradient (ITG) mode turbulence, but also treats trapped and passing electrons with collisions and finite {beta}, equilibrium ExB shear stabilization, and all in real tokamak geometry. Most importantly the code operates at finite relative gyroradius ({rho}{sub *}) so as to treat the profile shear stabilization and nonlocal effects which can break gyroBohm scaling. The code operates in either a cyclic flux-tube limit (which allows only gyroBohm scaling) or a globally with physical profile variation. Rohm scaling of DIII-D L-mode has been simulated with power flows matching experiment within error bars on the ion temperature gradient. Mechanisms for broken gyroBohm scaling, neoclassical ion flows embedded in turbulence, turbulent dynamos and profile corrugations, plasma pinches and impurity flow, and simulations at fixed flow rather than fixed gradient are illustrated and discussed.
The rotation effect on tokamak stability in the gyrokinetic description
NASA Astrophysics Data System (ADS)
Zheng, L.-J.; Tessarotto, M.
1996-11-01
The combined eigenvalue and initial value problems for both ion temperature gradient (ITG) and ballooning modes are solved analytically, with full gyrokinetic and toroidal effects taken into account. The results show that the sonic rotation can be harmful for both ITG and ballooning stabilities, while subsonic rotation plays always a stabilizing role. The flow shear damping on the ballooning modes is found to be much weaker than that on the ITG ones. This suggests that the ballooning instabilities are more likely to develop than the ITG modes in the rotating plasmas. The existence of the flow shear leads the time evolution of the fluctuation signals to become the type containing two timescales --- fast oscillating (with either fast or slow growing) signals with their amplitudes modulated by the slow oscillating and damping envelops, looking as if fish-bone like fluctuations. The comparison with the experimental observations are also discussed, in addressing the issues that which type of modes: the electrostatic drift or the MHD ones is likely to be responsible for the anomalous transport in tokamaks, and how to suppress them based on the current analyses.
A multi-species collisional operator for full-F gyrokinetics
Estève, D.; Garbet, X.; Sarazin, Y.; Grandgirard, V.; Cartier-Michaud, T.; Dif-Pradalier, G.; Ghendrih, P.; Latu, G.; Norscini, C.
2015-12-15
A linearized multi-species collision operator has been developed for an efficient implementation in gyrokinetic codes. This operator satisfies the main expected properties: particle, momentum, and energy conservation, and existence of an H-theorem. A gyrokinetic version is then calculated, which involves derivatives with respect to the gyrocenter position, parallel velocity, and magnetic momentum. An isotropic version in the velocity space can be constructed for the specific problem of trace impurities colliding with a main species. A simpler version that involves derivatives with parallel velocity only has been developed. This reduced version has been implemented in the GYSELA gyrokinetic code, and is shown to comply with particle, momentum, and energy conservation laws. Moreover, the interspecies relaxation rates for momentum and energy agree very well with the theoretical values.
Gyrokinetic particle simulation of beta-induced Alfvén eigenmode
NASA Astrophysics Data System (ADS)
Zhang, H. S.; Lin, Z.; Holod, I.; Wang, X.; Xiao, Y.; Zhang, W. L.
2010-11-01
The beta-induced Alfvén eigenmode (BAE) in toroidal plasmas is studied using global gyrokinetic particle simulations. The BAE real frequency and damping rate measured in the initial perturbation simulation and in the antenna excitation simulation agree well with each other. The real frequency is slightly higher than the ideal magnetohydrodynamic (MHD) accumulation point frequency due to the kinetic effects of thermal ions. Simulations with energetic particle density gradient show exponential growth of BAE with a growth rate sensitive to the energetic particle temperature and density. The nonperturbative contributions by energetic particles modify the mode structure and reduce the frequency relative to the MHD theory. The finite Larmor radius effects of energetic particles reduce the BAE growth rate. Benchmarks between gyrokinetic particle simulation and hybrid MHD-gyrokinetic simulation show good agreement in BAE real frequency and mode structure.
Hatch, David R.; Del-Castillo-Negrete, Diego B; Terry, P.W.
2012-01-01
Higher order singular value decomposition (HOSVD) is explored as a tool for analyzing and compressing gyrokinetic data. An efficient numerical implementation of an HOSVD algorithm is described. HOSVD is used to analyze the full six-dimensional (three spatial, two velocity space, and time dimensions) gyrocenter distribution function from gyrokinetic simulations of ion temperature gradient, electron temperature gradient, and trapped electron mode driven turbulence. The HOSVD eigenvalues for the velocity space coordinates decay very rapidly, indicating that only a few structures in velocity space can capture the most important dynamics. In almost all of the cases studied, HOSVD extracts parallel velocity space structures which are very similar to orthogonal polynomials. HOSVD is also used to compress gyrokinetic datasets, an application in which it is shown to significantly outperform the more commonly used singular value decomposition. It is shown that the effectiveness of the HOSVD compression improves as the dimensionality of the dataset increases. (C) 2012 Elsevier Inc. All rights reserved.
NASA Astrophysics Data System (ADS)
Dominski, J.; McMillan, B. F.; Brunner, S.; Merlo, G.; Tran, T.-M.; Villard, L.
2017-02-01
The influence of the fine layers of the non-adiabatic passing electron response on electrostatic turbulent transport, previously studied systematically in flux tube geometry [Dominski et al., Phys. Plasmas 22, 062303 (2015)], is pursued in global geometry in conditions relevant for the TCV tokamak with a deuterium plasma (mi/me = 3672). The spectral organization of the passing electron turbulent flux and its dependence on the radial profile of the safety factor are revealed. A radially dependent toroidal spectral analysis of the turbulent fluxes led to the key result that the particle and heat diffusivities of passing-electrons are proportional to the local density of low-order mode rational surfaces. To permit this study of the short radial scales associated with the passing electron dynamics, a new field solver valid at an arbitrary wavelength is implemented in ORB5, for the gyrokinetic quasi-neutrality equation. A benchmark is conducted against the global version of the gyrokinetic code GENE, showing very good agreement.
Linear dispersion relation for the mirror instability in context of the gyrokinetic theory
Porazik, Peter; Johnson, Jay R.
2013-10-15
The linear dispersion relation for the mirror instability is discussed in context of the gyrokinetic theory. The objective is to provide a coherent view of different kinetic approaches used to derive the dispersion relation. The method based on gyrocenter phase space transformations is adopted in order to display the origin and ordering of various terms.
Global gyrokinetic particle-in-cell simulations of internal kink instabilities
Mishchenko, Alexey; Zocco, Alessandro
2012-12-15
Internal kink instabilities have been studied in straight tokamak geometry employing an electromagnetic gyrokinetic particle-in-cell (PIC) code. The ideal-MHD internal kink mode and the collisionless m=1 tearing mode have been successfully simulated with the PIC code. Diamagnetic effects on the internal kink modes have also been investigated.
Performance of particle in cell methods on highly concurrent computational architectures
NASA Astrophysics Data System (ADS)
Adams, M. F.; Ethier, S.; Wichmann, N.
2007-07-01
Particle in cell (PIC) methods are effective in computing Vlasov-Poisson system of equations used in simulations of magnetic fusion plasmas. PIC methods use grid based computations, for solving Poisson's equation or more generally Maxwell's equations, as well as Monte-Carlo type methods to sample the Vlasov equation. The presence of two types of discretizations, deterministic field solves and Monte-Carlo methods for the Vlasov equation, pose challenges in understanding and optimizing performance on today large scale computers which require high levels of concurrency. These challenges arises from the need to optimize two very different types of processes and the interactions between them. Modern cache based high-end computers have very deep memory hierarchies and high degrees of concurrency which must be utilized effectively to achieve good performance. The effective use of these machines requires maximizing concurrency by eliminating serial or redundant work and minimizing global communication. A related issue is minimizing the memory traffic between levels of the memory hierarchy because performance is often limited by the bandwidths and latencies of the memory system. This paper discusses some of the performance issues, particularly in regard to parallelism, of PIC methods. The gyrokinetic toroidal code (GTC) is used for these studies and a new radial grid decomposition is presented and evaluated. Scaling of the code is demonstrated on ITER sized plasmas with up to 16K Cray XT3/4 cores.
Toroidal Alfvén eigenmodes with nonlinear gyrokinetic and fluid hybrid models
NASA Astrophysics Data System (ADS)
Cole, M. D. J.; Biancalani, A.; Bottino, A.; Kleiber, R.; Könies, A.; Mishchenko, A.
2017-02-01
Alfvén eigenmodes may be important in driving fast particle transport in magnetic confinement fusion devices, with potentially deleterious results. To explain and predict this behaviour, numerical simulations are necessary. In order to predict transport, modes must be simulated through to their nonlinear saturated state. In this work, the first simulations of non-linear wave-particle interaction between an energetic particle population and a Toroidal Alfvén Eigenmode are performed in which fluctuations responding self-consistently to modification of the fast particle profile are calculated with gyrokinetic treatment of all plasma species. Results from two such gyrokinetic codes are compared with new results from non-perturbative and perturbative fluid-gyrokinetic hybrid codes. There is a power-law relationship between the saturated magnetic perturbation amplitude, δB/B0, and the linear mode growth rate, γL. All models show a transition from a higher to a lower exponent regime with increasing γL. Measured values of the higher exponent from different codes fall in a range between 1.45 and 1.79, while the lower exponent falls in a range between 0.47 and 0.79. There is a consistent difference of 1.0 between the higher and lower exponents independent of the model. The absolute level of saturated δB/B0 is determined by the damping rate. In the fluid-gyrokinetic hybrid codes, an ad-hoc damping is applied, while in the gyrokinetic case the measured damping is consistent with the estimated rate of physical electron Landau damping.
Linear and nonlinear verification of gyrokinetic microstability codes
NASA Astrophysics Data System (ADS)
Bravenec, R. V.; Candy, J.; Barnes, M.; Holland, C.
2011-12-01
Verification of nonlinear microstability codes is a necessary step before comparisons or predictions of turbulent transport in toroidal devices can be justified. By verification we mean demonstrating that a code correctly solves the mathematical model upon which it is based. Some degree of verification can be accomplished indirectly from analytical instability threshold conditions, nonlinear saturation estimates, etc., for relatively simple plasmas. However, verification for experimentally relevant plasma conditions and physics is beyond the realm of analytical treatment and must rely on code-to-code comparisons, i.e., benchmarking. The premise is that the codes are verified for a given problem or set of parameters if they all agree within a specified tolerance. True verification requires comparisons for a number of plasma conditions, e.g., different devices, discharges, times, and radii. Running the codes and keeping track of linear and nonlinear inputs and results for all conditions could be prohibitive unless there was some degree of automation. We have written software to do just this and have formulated a metric for assessing agreement of nonlinear simulations. We present comparisons, both linear and nonlinear, between the gyrokinetic codes GYRO [J. Candy and R. E. Waltz, J. Comput. Phys. 186, 545 (2003)] and GS2 [W. Dorland, F. Jenko, M. Kotschenreuther, and B. N. Rogers, Phys. Rev. Lett. 85, 5579 (2000)]. We do so at the mid-radius for the same discharge as in earlier work [C. Holland, A. E. White, G. R. McKee, M. W. Shafer, J. Candy, R. E. Waltz, L. Schmitz, and G. R. Tynan, Phys. Plasmas 16, 052301 (2009)]. The comparisons include electromagnetic fluctuations, passing and trapped electrons, plasma shaping, one kinetic impurity, and finite Debye-length effects. Results neglecting and including electron collisions (Lorentz model) are presented. We find that the linear frequencies with or without collisions agree well between codes, as do the time averages of
Leerink, S.; Heikkinen, J. A.; Janhunen, S. J.; Kiviniemi, T. P.; Nora, M.; Ogando, F.
2008-09-15
The ELMFIRE gyrokinetic simulation code has been used to perform full f simulations of the FT-2 tokamak. The dynamics of the radial electric field and the creation of poloidal velocity in the presence of turbulence are presented.
Chen, Yang
2012-03-07
in the hybrid model we have studied a kinetic electron closure scheme for the fluid electron model. The most important element of the closure scheme is a complete Ohm’s law for the parallel electric field E{sub ||}, derived by combining the quasi-neutrality condition, the Ampere’s equation and the v{sub ||} moment of the gyrokinetic equations. A discretization method for the closure scheme is studied in detail for a three-dimensional shear-less slab plasma. It is found that for long-wavelength shear Alfven waves the kinetic closure scheme is both more accurate and robust than the previous GEM algorithm using the split-scheme, whereas for the ion-gradient-driven instability the previous algorithm is more efficient. This kinetic electron closure scheme will be implemented in GEM in the future. We have studied the beam driven Reverse Shear Alfven Eigenmodes (RSAE) observed in DIII-D discharge 142111. For this purpose a new scheme for obtaining the electric potential is implemented, i.e., by solving the gyrokinetic moment (GKM) equation, which is essentially the equation for {partial_derivative}{phi}}/{partial_derivative}t used in GEM’s split-weight scheme, and then integrating in time. Due to charge-neutrality the ExB motions of the equilibrium densities of all species cancel each other and do not cause charge separation if there is no finite Larmor radius effect. The advantage of solving the GKM equation is that this lowest-order cancellation can be made explicit. The GKM approach is found to be more accurate and robust. GEM simulations have reproduced many features of RSAE seen in the experiment, such as frequency chirping and the chirping range. It has been reported by other simulation codes that the shearing direction of the mode structure in the poloidal plane disagrees with observation. We found that the mode structure, including the shearing in the poloidal plane, is in general sensitive to the beam distribution. Using the same beam density profile as in
NASA Astrophysics Data System (ADS)
Deng, Zhao
2014-10-01
Gyrokinetic simulations of L-mode near edge tokamak plasmas with the GYRO code underpredict both the transport and the turbulence levels by 5 to 10 fold, which suggest either some important mechanism is missing from current gyrokinetic codes like GYRO or the gyrokinetic approximation itself is breaking down. It is known that GYRO drift-kinetic simulations with gyro-averaging suppressed recover most of the missing transport. With these motivations, we developed a flux tube nonlinear cyclokinetic code rCYCLO with the parallel motion and variation suppressed. rCYCLO dynamically follows the high frequency ion gyro-phase motion (with no averaging) which is nonlinearly coupled into the low frequency drift-waves thereby interrupting and possibly suppressing the gyro-averaging. By comparison with the corresponding gyrokinetic simulations, we can test the conditions for the breakdown of gyrokinetics. rCYCLO nonlinearly couples ∇B driven ion temperature gradient (ITG) modes and collisional fluid electron drift modes to ion cyclotron (IC) modes. As required, rCYCLO cyclokinetic transport recovers gyrokinetics at high relative ion cyclotron frequency (Ω*) and low turbulence levels. However, because the IC modes are stable and act as a turbulence sink, we have found that at high turbulence levels and low-Ω* cyclokinetic transport is lower (not higher) than gyrokinetic transport. Work is in progress with unstable IC modes to explore the possibility of driving cyclokinetic transport higher than gyrokinetic transport. Supported by the CSC, NSFC No. 1126114032, No. 10975012 ITER-CN No. 2013GB112006 and the US DOE under DE-FG02-95ER54309.
A combined discontinuous Galerkin and finite volume scheme for multi-dimensional VPFP system
Asadzadeh, M.; Bartoszek, K.
2011-05-20
We construct a numerical scheme for the multi-dimensional Vlasov-Poisson-Fokker-Planck system based on a combined finite volume (FV) method for the Poisson equation in spatial domain and the streamline diffusion (SD) and discontinuous Galerkin (DG) finite element in time, phase-space variables for the Vlasov-Fokker-Planck equation.
GYSELA, a full-f global gyrokinetic Semi-Lagrangian code for ITG turbulence simulations
Grandgirard, V.; Sarazin, Y.; Garbet, X.; Dif-Pradalier, G.; Ghendrih, Ph.; Besse, N.; Bertrand, P.
2006-11-30
This work addresses non-linear global gyrokinetic simulations of ion temperature gradient (ITG) driven turbulence with the GYSELA code. The particularity of GYSELA code is to use a fixed grid with a Semi-Lagrangian (SL) scheme and this for the entire distribution function. The 4D non-linear drift-kinetic version of the code already showns the interest of such a SL method which exhibits good properties of energy conservation in non-linear regime as well as an accurate description of fine spatial scales. The code has been upgrated to run 5D simulations of toroidal ITG turbulence. Linear benchmarks and non-linear first results prove that semi-lagrangian codes can be a credible alternative for gyrokinetic simulations.
Advanced methods in global gyrokinetic full f particle simulation of tokamak transport
Ogando, F.; Heikkinen, J. A.; Henriksson, S.; Janhunen, S. J.; Kiviniemi, T. P.; Leerink, S.
2006-11-30
A new full f nonlinear gyrokinetic simulation code, named ELMFIRE, has been developed for simulating transport phenomena in tokamak plasmas. The code is based on a gyrokinetic particle-in-cell algorithm, which can consider electrons and ions jointly or separately, as well as arbitrary impurities. The implicit treatment of the ion polarization drift and the use of full f methods allow for simulations of strongly perturbed plasmas including wide orbit effects, steep gradients and rapid dynamic changes. This article presents in more detail the algorithms incorporated into ELMFIRE, as well as benchmarking comparisons to both neoclassical theory and other codes.Code ELMFIRE calculates plasma dynamics by following the evolution of a number of sample particles. Because of using an stochastic algorithm its results are influenced by statistical noise. The effect of noise on relevant magnitudes is analyzed.Turbulence spectra of FT-2 plasma has been calculated with ELMFIRE, obtaining results consistent with experimental data.
Gyrokinetic particle simulation of the beta-induced Alfven eigen mode
NASA Astrophysics Data System (ADS)
Zhang, Huasen; Lin, Zhihong; Holod, Ihor; Wang, Xin; Xiao, Yong; Zhang, Wenlu
2010-11-01
The beta-induced Alfven eigen mode (BAE) is studied using the global gyrokinetic particle code GTC. In our simulation, BAE is successfully excited by antenna and energetic particle density gradient. Through the antenna frequency scan, we can measure the BAE frequency and damping rate by numerical fitting the saturation amplitude. BAE excitation by energetic particles shows that the BAE propagates in the ion diamagnetic direction and the frequency has a little downshift, which is due to modification of the energetic particles. The frequency and growth rate in gyrokinetic simulation is a little different from drift kinetic simulation, which is expected due to the finite larmor radius effect. We also find that the BAE frequency is related to the wavelength and the plasma beta while the growth rate is sensitive to the energetic particle properties. Benchmarks between GTC and HMGC are also done through initial perturbation, antenna excitation and energetic particle excitation. The simulation results agree with each other very well.
Gyrokinetic Simulation of Energetic Particles Turbulence and Transport in Fusion Plasmas
NASA Astrophysics Data System (ADS)
Zhang, Wenlu; Lin, Zhihong; Holod, Ihor; Xiao, Yong; Bierwage, Andreas; Spong, Donald; Chu, Ming
2009-05-01
The confinement of the energetic particles (EP) is a critical issue in the International Thermonuclear Experimental Reactor (ITER), since that ignition relies on the self-heating by the fusion products. Shear Alfven wave excitations by EP in toroidal systems, for example Toroidal Alfven Eigenmode (TAE) and Energetic Particle Mode (EPM) have been investigated as primary candidate for fluctuation-induced transport of EP in fusion plasma. In this work, TAE excitations by energetic particles are investigated in large scale first-principle simulations of fusion plasmas using the global gyrokinetic toroidal code (GTC) [Lin, Science 1998]. Comprehensive linear benchmarking results are reported between GTC, GYRO, fluid code TAEFL, and Magnetohydrodynamic-gyrokinetic hybrid code HMGC.
NASA Astrophysics Data System (ADS)
Taimourzadeh, Sam; Holod, Ihor; Lin, Zhihong; Nazikian, Raffi
2015-11-01
It has been demonstrated that edge localized modes (ELMs) can be fully suppressed in DIII-D H-mode plasmas with the application of resonant magnetic perturbations (RMPs), and that there is a corresponding reduction of pedestal gradients, changes in rotation, and changes in the radial electric field (Er) profile. However, with the application of RMPs there is also an increase in short wavelength, electrostatic turbulence on top of the pedestal, as observed with BES, DBS, and other fluctuation diagnostics. The effects of Er shear on this turbulence, is investigated using gyrokinetic simulations via the gyrokinetic toroidal code (GTC) for in DIII-D shot 158103, at times 3750 ms (RMP on, ELM active) and 3050 ms (RMP on, ELM suppressed).
Gyrokinetic studies of core turbulence features in ASDEX Upgrade H-mode plasmas
Navarro, A. Bañón Told, D.; Happel, T.; Görler, T.; Abiteboul, J.; Bustos, A.; Doerk, H.; Jenko, F.
2015-04-15
Gyrokinetic validation studies are crucial for developing confidence in the model incorporated in numerical simulations and thus improving their predictive capabilities. As one step in this direction, we simulate an ASDEX Upgrade discharge with the GENE code, and analyze various fluctuating quantities and compare them to experimental measurements. The approach taken is the following. First, linear simulations are performed in order to determine the turbulence regime. Second, the heat fluxes in nonlinear simulations are matched to experimental fluxes by varying the logarithmic ion temperature gradient within the expected experimental error bars. Finally, the dependence of various quantities with respect to the ion temperature gradient is analyzed in detail. It is found that density and temperature fluctuations can vary significantly with small changes in this parameter, thus making comparisons with experiments very sensitive to uncertainties in the experimental profiles. However, cross-phases are more robust, indicating that they are better observables for comparisons between gyrokinetic simulations and experimental measurements.
Gyrokinetic simulations of collisionless reconnection in turbulent non-uniform plasmas
Kobayashi, Sumire; Rogers, Barrett N.; Numata, Ryusuke
2014-04-15
We present nonlinear gyrokinetic simulations of collisionless magnetic reconnection with non-uniformities in the plasma density, the electron temperature, and the ion temperature. The density gradient can stabilize reconnection due to diamagnetic effects but destabilize driftwave modes that produce turbulence. The electron temperature gradient triggers microtearing modes that drive rapid small-scale reconnection and strong electron heat transport. The ion temperature gradient destabilizes ion temperature gradient modes that, like the driftwaves, may enhance reconnection in some cases.
Gyrokinetic simulation of edge blobs and divertor heat-load footprint
NASA Astrophysics Data System (ADS)
Chang, C. S.; Ku, S.; Hager, R.; Churchill, M.; D'Azevedo, E.; Worley, P.
2015-11-01
Gyrokinetic study of divertor heat-load width Lq has been performed using the edge gyrokinetic code XGC1. Both neoclassical and electrostatic turbulence physics are self-consistently included in the simulation with fully nonlinear Fokker-Planck collision operation and neutral recycling. Gyrokinetic ions and drift kinetic electrons constitute the plasma in realistic magnetic separatrix geometry. The electron density fluctuations from nonlinear turbulence form blobs, as similarly seen in the experiments. DIII-D and NSTX geometries have been used to represent today's conventional and tight aspect ratio tokamaks. XGC1 shows that the ion neoclassical orbit dynamics dominates over the blob physics in setting Lq in the sample DIII-D and NSTX plasmas, re-discovering the experimentally observed 1/Ip type scaling. Magnitude of Lq is in the right ballpark, too, in comparison with experimental data. However, in an ITER standard plasma, XGC1 shows that the negligible neoclassical orbit excursion effect makes the blob dynamics to dominate Lq. Differently from Lq 1mm (when mapped back to outboard midplane) as was predicted by simple-minded extrapolation from the present-day data, XGC1 shows that Lq in ITER is about 1 cm that is somewhat smaller than the average blob size. Supported by US DOE and the INCITE program.
A flux-matched gyrokinetic analysis of DIII-D L-mode turbulence
Görler, T. Told, D.; White, A. E.; Jenko, F.; Holland, C.; Rhodes, T. L.
2014-12-15
Previous nonlinear gyrokinetic simulations of specific DIII-D L-mode cases have been found to significantly underpredict the ion heat transport and associated density and temperature fluctuation levels by up to almost one of order of magnitude in the outer-core domain, i.e., roughly in the last third of the minor radius. Since then, this so-called shortfall issue has been subject to various speculations on possible reasons and furthermore motivation for a number of dedicated comparisons for L-mode plasmas in comparable machines. However, only a rather limited number of simulations and gyrokinetic codes has been applied to the original scenario, thus calling for further dedicated investigations in order to broaden the scientific basis. The present work contributes along these lines by employing another well-established gyrokinetic code in a numerically and physically comprehensive manner. Contrary to the previous studies, only a mild underprediction is observed at the outer radial positions which can furthermore be overcome by varying the ion temperature gradient within the error bars associated with the experimental measurement. The significance and reliability of these simulations are demonstrated by benchmarks, numerical convergence tests, and furthermore by extensive validation studies. The latter involve cross-phase and cross-power spectra analyses of various fluctuating quantities and confirm a high degree of realism. The code discrepancies come as a surprise since the involved software packages had been benchmarked repeatedly and very successfully in the past. Further collaborative effort in identifying the underlying difference is hence required.
A flux-matched gyrokinetic analysis of DIII-D L-mode turbulence
NASA Astrophysics Data System (ADS)
Görler, T.; White, A. E.; Told, D.; Jenko, F.; Holland, C.; Rhodes, T. L.
2014-12-01
Previous nonlinear gyrokinetic simulations of specific DIII-D L-mode cases have been found to significantly underpredict the ion heat transport and associated density and temperature fluctuation levels by up to almost one of order of magnitude in the outer-core domain, i.e., roughly in the last third of the minor radius. Since then, this so-called shortfall issue has been subject to various speculations on possible reasons and furthermore motivation for a number of dedicated comparisons for L-mode plasmas in comparable machines. However, only a rather limited number of simulations and gyrokinetic codes has been applied to the original scenario, thus calling for further dedicated investigations in order to broaden the scientific basis. The present work contributes along these lines by employing another well-established gyrokinetic code in a numerically and physically comprehensive manner. Contrary to the previous studies, only a mild underprediction is observed at the outer radial positions which can furthermore be overcome by varying the ion temperature gradient within the error bars associated with the experimental measurement. The significance and reliability of these simulations are demonstrated by benchmarks, numerical convergence tests, and furthermore by extensive validation studies. The latter involve cross-phase and cross-power spectra analyses of various fluctuating quantities and confirm a high degree of realism. The code discrepancies come as a surprise since the involved software packages had been benchmarked repeatedly and very successfully in the past. Further collaborative effort in identifying the underlying difference is hence required.
Global gyrokinetic models for energetic particle driven Alfvén instabilities in 3D equilibria
NASA Astrophysics Data System (ADS)
Spong, Don; Holod, Ihor
2015-11-01
The GTC global gyrokinetic PIC model has been adapted to 3D VMEC equilibria and provides a new method for the analysis of Alfvénic instabilities in stellarators, 3D tokamaks, and helical RFP states. The gyrokinetic orderings (k||/k⊥ << 1, ω/Ωci << 1, ρEP/L << 1) are applicable to a range of energetic particle driven instabilities that have been observed in 3D configurations. Applications of this model to stellarators have indicated that a variety of different Alfvén instabilities can be excited, depending on the toroidal mode number, fast ion average energy and fast ion density profile. Both an LHD discharge where bursting n = 1 Alfvén activity in the TAE gap was observed and a W7-X case have been examined. TAE,/EAE/GAE modes have been found in the simulations, depending on the mode family and fast ion profiles used. The dynamical evolution of the instabilities shows the field period coupling between n and n + Nfp expected for a stellarator. The development of gyrofluid reduced models that can capture relevant physics aspects of the gyrokinetic models will also be discussed. Research sponsored by the U.S. Department of Energy under Contract DE-AC05-00OR22725 with UT-Battelle, LLC and the GSEP SciDAC Center.
Fluid electron, gyrokinetic ion simulations of linear internal kink and energetic particle modes
NASA Astrophysics Data System (ADS)
Cole, Michael; Mishchenko, Alexey; Könies, Axel; Kleiber, Ralf; Borchardt, Matthias
2014-07-01
The internal kink mode is an important plasma instability responsible for a broad class of undesirable phenomena in tokamaks, including the sawtooth cycle and fishbones. To predict and discover ways to mitigate this behaviour in current and future devices, numerical simulations are necessary. The internal kink mode can be modelled by reduced magnetohydrodynamics (MHD). Fishbone modes are an inherently kinetic and non-linear phenomenon based on the n = 1 Energetic Particle Mode (EPM), and have been studied using hybrid codes that combine a reduced MHD bulk plasma model with a kinetic treatment of fast ions. In this work, linear simulations are presented using a hybrid model which couples a fluid treatment of electrons with a gyrokinetic treatment of both bulk and fast ions. Studies of the internal kink mode in geometry relevant to large tokamak experiments are presented and the effect of gyrokinetic ions is considered. Interaction of the kink with gyrokinetic fast ions is also considered, including the destabilisation of the linear n = 1 EPM underlying the fishbone.
Fluid electron, gyrokinetic ion simulations of linear internal kink and energetic particle modes
Cole, Michael Mishchenko, Alexey; Könies, Axel; Kleiber, Ralf; Borchardt, Matthias
2014-07-15
The internal kink mode is an important plasma instability responsible for a broad class of undesirable phenomena in tokamaks, including the sawtooth cycle and fishbones. To predict and discover ways to mitigate this behaviour in current and future devices, numerical simulations are necessary. The internal kink mode can be modelled by reduced magnetohydrodynamics (MHD). Fishbone modes are an inherently kinetic and non-linear phenomenon based on the n = 1 Energetic Particle Mode (EPM), and have been studied using hybrid codes that combine a reduced MHD bulk plasma model with a kinetic treatment of fast ions. In this work, linear simulations are presented using a hybrid model which couples a fluid treatment of electrons with a gyrokinetic treatment of both bulk and fast ions. Studies of the internal kink mode in geometry relevant to large tokamak experiments are presented and the effect of gyrokinetic ions is considered. Interaction of the kink with gyrokinetic fast ions is also considered, including the destabilisation of the linear n = 1 EPM underlying the fishbone.
Comparison of two-fluid and gyrokinetic models for kinetic Alfvén waves in solar and space plasmas
Yang, L.; Wu, D. J.; Wang, S. J.; Lee, L. C.
2014-09-01
An analytical comparative study of a two-fluid and a gyrokinetic model of kinetic Alfvén waves (KAWs) is presented for various solar and space plasma environments. Based on the linear KAW dispersion relation for gyrokinetics (Howes et al. 2006), the wave group velocity and electromagnetic polarizations are obtained analytically. Then the gyrokinetic wave properties are compared with those of the two-fluid model. The results show that both models agree well with each other not only in the long wavelength regime (>> the ion gyroradius ρ {sub i}) for all cases considered, but also in wavelengths ∼ρ {sub i} and <<ρ {sub i} (still much larger than the electron gyroscale) for a moderate or low (≲ 1) and a high (>>1) ion/electron temperature ratio T {sub 0i}/T {sub 0e}, respectively. However, the fluid model calculations deviate strongly from the gyrokinetic model at scales <ρ {sub i} for a relatively low T {sub 0i}/T {sub 0e} due to the electron gyroradius effect. Meanwhile, the plasma β {sub i} can make the gyrokinetic dispersion relation of KAWs become complex and sometimes have an oscillation-like structure. With the inherent simplicity of the fluid theory, these results may improve our understanding of the applicability of the two-fluid model, and may have important implications for computer simulation studies of KAWs in the solar and space plasma surroundings.
NASA Astrophysics Data System (ADS)
White, A. E.; Howard, N. T.; Creely, A. J.; Chilenski, M. A.; Greenwald, M.; Hubbard, A. E.; Hughes, J. W.; Marmar, E.; Rice, J. E.; Sierchio, J. M.; Sung, C.; Walk, J. R.; Whyte, D. G.; Mikkelsen, D. R.; Edlund, E. M.; Kung, C.; Holland, C.; Candy, J.; Petty, C. C.; Reinke, M. L.; Theiler, C.
2015-05-01
For the first time, nonlinear gyrokinetic simulations of I-mode plasmas are performed and compared with experiment. I-mode is a high confinement regime, featuring energy confinement similar to H-mode, but without enhanced particle and impurity particle confinement [D. G. Whyte et al., Nucl. Fusion 50, 105005 (2010)]. As a consequence of the separation between heat and particle transport, I-mode exhibits several favorable characteristics compared to H-mode. The nonlinear gyrokinetic code GYRO [J. Candy and R. E. Waltz, J Comput. Phys. 186, 545 (2003)] is used to explore the effects of E × B shear and profile stiffness in I-mode and compare with L-mode. The nonlinear GYRO simulations show that I-mode core ion temperature and electron temperature profiles are more stiff than L-mode core plasmas. Scans of the input E × B shear in GYRO simulations show that E × B shearing of turbulence is a stronger effect in the core of I-mode than L-mode. The nonlinear simulations match the observed reductions in long wavelength density fluctuation levels across the L-I transition but underestimate the reduction of long wavelength electron temperature fluctuation levels. The comparisons between experiment and gyrokinetic simulations for I-mode suggest that increased E × B shearing of turbulence combined with increased profile stiffness are responsible for the reductions in core turbulence observed in the experiment, and that I-mode resembles H-mode plasmas more than L-mode plasmas with regards to marginal stability and temperature profile stiffness.
Gyrokinetic Particle Simulation of Turbulent Transport in Burning Plasmas (GPS - TTBP) Final Report
Chame, Jacqueline
2011-05-27
The goal of this project is the development of the Gyrokinetic Toroidal Code (GTC) Framework and its applications to problems related to the physics of turbulence and turbulent transport in tokamaks,. The project involves physics studies, code development, noise effect mitigation, supporting computer science efforts, diagnostics and advanced visualizations, verification and validation. Its main scientific themes are mesoscale dynamics and non-locality effects on transport, the physics of secondary structures such as zonal flows, and strongly coherent wave-particle interaction phenomena at magnetic precession resonances. Special emphasis is placed on the implications of these themes for rho-star and current scalings and for the turbulent transport of momentum. GTC-TTBP also explores applications to electron thermal transport, particle transport; ITB formation and cross-cuts such as edge-core coupling, interaction of energetic particles with turbulence and neoclassical tearing mode trigger dynamics. Code development focuses on major initiatives in the development of full-f formulations and the capacity to simulate flux-driven transport. In addition to the full-f -formulation, the project includes the development of numerical collision models and methods for coarse graining in phase space. Verification is pursued by linear stability study comparisons with the FULL and HD7 codes and by benchmarking with the GKV, GYSELA and other gyrokinetic simulation codes. Validation of gyrokinetic models of ion and electron thermal transport is pursed by systematic stressing comparisons with fluctuation and transport data from the DIII-D and NSTX tokamaks. The physics and code development research programs are supported by complementary efforts in computer sciences, high performance computing, and data management.
Verification of a magnetic island in gyro-kinetics by comparison with analytic theory
Zarzoso, D. Casson, F. J.; Poli, E.; Hornsby, W. A.; Peeters, A. G.
2015-02-15
A rotating magnetic island is imposed in the gyrokinetic code GKW, when finite differences are used for the radial direction, in order to develop the predictions of analytic tearing mode theory and understand its limitations. The implementation is verified against analytics in sheared slab geometry with three numerical tests that are suggested as benchmark cases for every code that imposes a magnetic island. The convergence requirements to properly resolve physics around the island separatrix are investigated. In the slab geometry, at low magnetic shear, binormal flows inside the island can drive Kelvin-Helmholtz instabilities which prevent the formation of the steady state for which the analytic theory is formulated.
J. Squire, H. Qin and W.M. Tang
2012-09-25
We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in Ref. 1. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with the Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincare theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models and Casimir type stability methods. __________________________________________________
Gyrokinetic simulations in general geometry and applications to collisional damping of zonal flows
Lin, Z.; Hahm, T.S.; Lee, W.W.; Tang, W.M.; White, R.B.
2000-02-15
A fully three-dimensional gyrokinetic particle code using magnetic coordinates for general geometry has been developed and applied to the investigation of zonal flows dynamics in toroidal ion-temperature-gradient turbulence. Full torus simulation results support the important conclusion that turbulence-driven zonal flows significantly reduce the turbulent transport. Linear collisionless simulations for damping of an initial poloidal flow perturbation exhibit an asymptotic residual flow. The collisional damping of this residual causes the dependence of ion thermal transport on the ion-ion collision frequency even in regimes where the instabilities are collisionless.
Effects of the magnetic equilibrium on gyrokinetic simulations of tokamak microinstabilities
Wan, Weigang; Chen, Yang; Parker, Scott E.; Groebner, Richard J.
2015-06-15
The general geometry of the experimental tokamak magnetic equilibrium is implemented in the global gyrokinetic simulation code GEM. Compared to the general geometry, the well used Miller parameterization of the magnetic equilibrium is a good approximation in the core region and up to the top of the pedestal. Linear simulations indicate that results with the two geometries agree for r/a ≤ 0.9. However, in the edge region, the instabilities are sensitive to the magnetic equilibrium in both the L-mode and the H-mode plasmas. A small variation of the plasma shaping parameters leads to large changes to the edge instability.
Global gyrokinetic stability of collisionless microtearing modes in large aspect ratio tokamaks
Swamy, Aditya K.; Ganesh, R.; Chowdhury, J.; Brunner, S.; Vaclavik, J.; Villard, L.
2014-08-15
Linear full radius gyrokinetic calculations show the existence of unstable microtearing modes (MTMs) in purely collisionless, high temperature, large aspect ratio tokamak plasmas. The present study takes into account fully gyrokinetic highly passing ions and electrons. The global 2-D structures of the collisionless mode with full radius coupling of the poloidal modes is obtained and compared with another electromagnetic mode, namely, the Alfvén Ion Temperature Gradient (AITG) mode (or Kinetic Ballooning Mode, KBM) for the same equilibrium profile. Several important characteristics of the modes are brought out and compared, such as a clear signature in the symmetry properties of the two modes, the plasma–β dependence, and radial and poloidal length scales of the electrostatic and magnetic vector potential fluctuations. Extensive parameter scans for this collisionless microtearing mode reveal the scaling of the growth rate with β and the electron temperature gradient η{sub e}. Scans at different β values show an inverse relationship between the η{sub e} threshold and β, leading to a stability diagram, and implying that the mode might exist at moderate to strong temperature gradients for finite β plasmas in large aspect ratio tokamaks. In contrast to small aspect ratio tokamaks where the trapped electron magnetic drift resonance is found to be important, in large aspect ratio tokamaks, a strong destabilization due to the magnetic drift resonance of passing electrons is observed and is identified as a possible collisionless drive mechanism for the collisionless MTM.
Numerical Instability in a 2D Gyrokinetic Code Caused by Divergent E × B Flow
NASA Astrophysics Data System (ADS)
Byers, J. A.; Dimits, A. M.; Matsuda, Y.; Langdon, A. B.
1994-12-01
In this paper, a numerical instability first observed in a 2D electrostatic gyrokinetic code is described. The instability should also be present in some form in many versons of particle-in-cell simulation codes that employ guiding center drifts. A perturbation analysis of the instability is given and its results agree quantitatively with the observations from the gyrokinetic code in all respects. The basic mechanism is a false divergence of the E × B flow caused by the interpolation between the grid and the particles as coupled with the specific numerical method for calculating E - ∇φ. Stability or instability depends in detail on the specific choice of particle interpolation method and field method. One common interpolation method, subtracted dipole, is stable. Other commonly used interpolation methods, linear and quadratic, are unstable when combined with a finite difference for the electric field. Linear and quadratic interpolation can be rendered stable if combined with another method for the electric field, the analytic differential of the interpolated potential.
Weir, G. M.; Faber, B. J.; Likin, K. M.; Talmadge, J. N.; Anderson, D. T.; Anderson, F. S. B.
2015-05-15
Stiffness measurements are presented in the quasi-helically symmetric experiment (HSX), in which the neoclassical transport is comparable to that in a tokamak and turbulent transport dominates throughout the plasma. Electron cyclotron emission is used to measure the local electron temperature response to modulated electron cyclotron resonant heating. The amplitude and phase of the heat wave through the steep electron temperature gradient (ETG) region of the plasma are used to determine a transient electron thermal diffusivity that is close to the steady-state diffusivity. The low stiffness in the region between 0.2 ≤ r/a ≤ 0.4 agrees with the scaling of the steady-state heat flux with temperature gradient in this region. These experimental results are compared to gyrokinetic calculations in a flux-tube geometry using the gyrokinetic electromagnetic numerical experiment code with two kinetic species. Linear simulations show that the ETG mode may be experimentally relevant within r/a ≤ 0.2, while the Trapped Electron Mode (TEM) is the dominant long-wavelength microturbulence instability across most of the plasma. The TEM is primarily driven by the density gradient. Non-linear calculations of the saturated heat flux driven by the TEM and ETG bracket the experimental heat flux.
Gyrokinetic characterization of the isotope effect in turbulent transport at the FT-2 tokamak
NASA Astrophysics Data System (ADS)
Niskala, P.; Gurchenko, A. D.; Gusakov, E. Z.; Altukhov, A. B.; Esipov, L. A.; Kantor, M. Yu; Kiviniemi, T. P.; Kouprienko, D.; Korpilo, T.; Lashkul, S. I.; Leerink, S.; Perevalov, A. A.; Rochford, R.
2017-04-01
Isotope effect allows fusion devices to perform better when heavier hydrogen isotopes are used as fuel, but the reason for this improvement is not yet understood. We present the first direct evidence of the isotope effect on particle confinement in the FT-2 tokamak and investigate it via gyrokinetic simulations. Experimental measurements for comparable hydrogen and deuterium discharges show that the particle confinement time increases by 40% for the heavier isotope species. The isotope effect on particle flux is reproduced in global and local gyrokinetic simulations. Global ELMFIRE simulations demonstrate a systemic reduction in particle fluxes across the radial range, showing a ratio of fluxes {{{Γ }}}{{H}}/{{{Γ }}}{{D}}=1.3 at the edge and {{{Γ }}}{{H}}/{{{Γ }}}{{D}}=1.4 at r/a=0.6. Local GENE simulations agree qualitatively with the result. Besides the fluctuation level, smaller scales and a favorable shift in the cross-phase between the turbulent fluctuations are found to contribute to the isotope effect in the simulations.
The linear tearing instability in three dimensional, toroidal gyro-kinetic simulations
Hornsby, W. A. Migliano, P.; Buchholz, R.; Kroenert, L.; Weikl, A.; Peeters, A. G.; Zarzoso, D.; Poli, E.; Casson, F. J.
2015-02-15
Linear gyro-kinetic simulations of the classical tearing mode in three-dimensional toroidal geometry were performed using the global gyro-kinetic turbulence code, GKW. The results were benchmarked against a cylindrical ideal MHD and analytical theory calculations. The stability, growth rate, and frequency of the mode were investigated by varying the current profile, collisionality, and the pressure gradients. Both collisionless and semi-collisional tearing modes were found with a smooth transition between the two. A residual, finite, rotation frequency of the mode even in the absence of a pressure gradient is observed, which is attributed to toroidal finite Larmor-radius effects. When a pressure gradient is present at low collisionality, the mode rotates at the expected electron diamagnetic frequency. However, the island rotation reverses direction at high collisionality. The growth rate is found to follow a η{sup 1∕7} scaling with collisional resistivity in the semi-collisional regime, closely following the semi-collisional scaling found by Fitzpatrick. The stability of the mode closely follows the stability analysis as performed by Hastie et al. using the same current and safety factor profiles but for cylindrical geometry, however, here a modification due to toroidal coupling and pressure effects is seen.
What happens to full-f gyrokinetic transport and turbulence in a toroidal wedge simulation?
Kim, Kyuho; Chang, C. S.; Seo, Janghoon; ...
2017-01-24
Here, in order to save the computing time or to fit the simulation size into a limited computing hardware in a gyrokinetic turbulence simulation of a tokamak plasma, a toroidal wedge simulation may be utilized in which only a partial toroidal section is modeled with a periodic boundary condition in the toroidal direction. The most severe restriction in the wedge simulation is expected to be in the longest wavelength turbulence, i.e., ion temperature gradient (ITG) driven turbulence. The global full-f gyrokinetic code XGC1 is used to compare the transport and turbulence properties from a toroidal wedge simulation against the fullmore » torus simulation in an ITG unstable plasma in a model toroidal geometry. It is found that (1) the convergence study in the wedge number needs to be conducted all the way down to the full torus in order to avoid a false convergence, (2) a reasonably accurate simulation can be performed if the correct wedge number N can be identified, (3) the validity of a wedge simulation may be checked by performing a wave-number spectral analysis of the turbulence amplitude |δΦ| and assuring that the variation of δΦ between the discrete kθ values is less than 25% compared to the peak |δΦ|, and (4) a frequency spectrum may not be used for the validity check of a wedge simulation.« less
Gyrokinetic GDC turbulence simulations: confirming a new instability regime in LAPD plasmas
NASA Astrophysics Data System (ADS)
Pueschel, M. J.; Rossi, G.; Told, D.; Terry, P. W.; Jenko, F.; Carter, T. A.
2016-10-01
Recent high-beta experiments at the LArge Plasma Device have found significant parallel magnetic fluctuations in the region of large pressure gradients. Linear gyrokinetic simulations show the dominant instability at these radii to be the gradient-driven drift coupling (GDC) mode, a non-textbook mode driven by pressure gradients and destabilized by the coupling of ExB and grad-B∥ drifts. Unlike in previous studies, the large parallel extent of the device allows for finite-kz versions of this instability in addition to kz = 0 . The locations of maximum linear growth match very well with experimentally observed peaks of B∥ fluctuations. Local nonlinear simulations reproduce many features of the observations fairly well, with the exception of Bperp fluctuations, for which experimental profiles suggest a source unrelated to pressure gradients. In toto, the results presented here show that turbulence and transport in these experiments are driven by the GDC instability, that important characteristics of the linear instability carry over to nonlinear simulations, and - in the context of validation - that the gyrokinetic framework performs surprisingly well far outside its typical area of application, increasing confidence in its predictive abilities. Supported by U.S. DOE.
Fluid and gyrokinetic modelling of particle transport in plasmas with hollow density profiles
NASA Astrophysics Data System (ADS)
Tegnered, D.; Oberparleiter, M.; Nordman, H.; Strand, P.
2016-11-01
Hollow density profiles occur in connection with pellet fuelling and L to H transitions. A positive density gradient could potentially stabilize the turbulence or change the relation between convective and diffusive fluxes, thereby reducing the turbulent transport of particles towards the center, making the fuelling scheme inefficient. In the present work, the particle transport driven by ITG/TE mode turbulence in regions of hollow density profiles is studied by fluid as well as gyrokinetic simulations. The fluid model used, an extended version of the Weiland transport model, Extended Drift Wave Model (EDWM), incorporates an arbitrary number of ion species in a multi-fluid description, and an extended wavelength spectrum. The fluid model, which is fast and hence suitable for use in predictive simulations, is compared to gyrokinetic simulations using the code GENE. Typical tokamak parameters are used based on the Cyclone Base Case. Parameter scans in key plasma parameters like plasma β, R/LT , and magnetic shear are investigated. It is found that β in particular has a stabilizing effect in the negative R/Ln region, both nonlinear GENE and EDWM show a decrease in inward flux for negative R/Ln and a change of direction from inward to outward for positive R/Ln . This might have serious consequences for pellet fuelling of high β plasmas.
What happens to full-f gyrokinetic transport and turbulence in a toroidal wedge simulation?
NASA Astrophysics Data System (ADS)
Kim, Kyuho; Chang, C. S.; Seo, Janghoon; Ku, S.; Choe, W.
2017-01-01
In order to save the computing time or to fit the simulation size into a limited computing hardware in a gyrokinetic turbulence simulation of a tokamak plasma, a toroidal wedge simulation may be utilized in which only a partial toroidal section is modeled with a periodic boundary condition in the toroidal direction. The most severe restriction in the wedge simulation is expected to be in the longest wavelength turbulence, i.e., ion temperature gradient (ITG) driven turbulence. The global full-f gyrokinetic code XGC1 is used to compare the transport and turbulence properties from a toroidal wedge simulation against the full torus simulation in an ITG unstable plasma in a model toroidal geometry. It is found that (1) the convergence study in the wedge number needs to be conducted all the way down to the full torus in order to avoid a false convergence, (2) a reasonably accurate simulation can be performed if the correct wedge number N can be identified, (3) the validity of a wedge simulation may be checked by performing a wave-number spectral analysis of the turbulence amplitude |δΦ| and assuring that the variation of δΦ between the discrete kθ values is less than 25% compared to the peak |δΦ| , and (4) a frequency spectrum may not be used for the validity check of a wedge simulation.
NASA Astrophysics Data System (ADS)
Bañón Navarro, A.; Bardóczi, L.; Carter, T. A.; Jenko, F.; Rhodes, T. L.
2017-03-01
Neoclassical tearing modes have deleterious effects on plasma confinement and, if they grow large enough, they can lead to discharge termination. Therefore, they impose a major barrier in the development of operating scenarios of present-day tokamaks. Gyrokinetics offers a path toward studying multi-scale interactions with turbulence and the effect on plasma confinement. As a first step toward this goal, we have implemented static magnetic islands in nonlinear gyrokinetic simulations with the GENE code. We investigate the effect of the islands on profiles, flows, turbulence and transport and the scaling of these effects with respect to island size. We find a clear threshold island width, below which the islands have little or no effect while beyond this point the islands significantly perturb flows, increase turbulence and transport. Additionally, we study the effect of radially asymmetric islands on shear flows for the first time. We find that island induced shear flows can regulate turbulent fluctuation levels in the vicinity of the island separatrices. Throughout this work, we focus on experimentally relevant quantities, such as rms levels of density and electron temperature fluctuations, as well as amplitude and phasing of turbulence modulation. These simulations aim to provide guidelines for interpreting experimental results by comparing qualitative trends in the simulations with those obtained in tokamak experiments.
Storelli, A. Vermare, L.; Hennequin, P.; Gürcan, Ö. D.; Singh, Rameswar; Morel, P.; Dif-Pradalier, G.; Sarazin, Y.; Garbet, X.; Grandgirard, V.; Ghendrih, P.; Görler, T.
2015-06-15
In a dedicated collisionality scan in Tore Supra, the geodesic acoustic mode (GAM) is detected and identified with the Doppler backscattering technique. Observations are compared to the results of a simulation with the gyrokinetic code GYSELA. We found that the GAM frequency in experiments is lower than predicted by simulation and theory. Moreover, the disagreement is higher in the low collisionality scenario. Bursts of non harmonic GAM oscillations have been characterized with filtering techniques, such as the Hilbert-Huang transform. When comparing this dynamical behaviour between experiments and simulation, the probability density function of GAM amplitude and the burst autocorrelation time are found to be remarkably similar. In the simulation, where the radial profile of GAM frequency is continuous, we observed a phenomenon of radial phase mixing of the GAM oscillations, which could influence the burst autocorrelation time.
NASA Astrophysics Data System (ADS)
Yi, Sumin; Jhang, Hogun; Kwon, J. M.
2016-10-01
We report the results of a gyrokinetic simulation study elucidating the characteristics of the current driven by electron temperature gradient (ETG) turbulence in toroidal geometry. We examined the amount of the ETG turbulence-driven current for different turbulence levels, which were obtained by varying the relative electron gyroradius ρ* = ρe/a. Simulations show that the amount of the ETG turbulence-driven current increases with ρ* due to the gyro-Bohm scaling of turbulence intensity. A perturbation of the equilibrium q-profile by the ETG turbulence-driven current becomes noticeable when ρ* > 1/4000. Even in a small ρ* case, the proportional relation between the ETG turbulence-driven current and turbulence intensity suggests that a considerable intrinsic current can be driven inside an edge pedestal where a steep gradient of the electron temperature profile can excite ETG turbulence in a narrow region.
Structure of Plasma Heating in Gyrokinetic Alfvénic Turbulence
NASA Astrophysics Data System (ADS)
Navarro, Alejandro Bañón; Teaca, Bogdan; Told, Daniel; Groselj, Daniel; Crandall, Paul; Jenko, Frank
2016-12-01
We analyze plasma heating in weakly collisional kinetic Alfvén wave turbulence using high resolution gyrokinetic simulations spanning the range of scales between the ion and the electron gyroradii. Real space structures that have a higher than average heating rate are shown not to be confined to current sheets. This novel result is at odds with previous studies, which use the electromagnetic work in the local electron fluid frame, i.e., J .(E +ve×B ) , as a proxy for turbulent dissipation to argue that heating follows the intermittent spatial structure of the electric current. Furthermore, we show that electrons are dominated by parallel heating while the ions prefer the perpendicular heating route. We comment on the implications of the results presented here.
Collision-dependent power law scalings in two dimensional gyrokinetic turbulence
Cerri, S. S. Bañón Navarro, A.; Told, D.; Jenko, F.
2014-08-15
Nonlinear gyrokinetics provides a suitable framework to describe short-wavelength turbulence in magnetized laboratory and astrophysical plasmas. In the electrostatic limit, this system is known to exhibit a free energy cascade towards small scales in (perpendicular) real and/or velocity space. The dissipation of free energy is always due to collisions (no matter how weak the collisionality), but may be spread out across a wide range of scales. Here, we focus on freely decaying two dimensional electrostatic turbulence on sub-ion-gyroradius scales. An existing scaling theory for the turbulent cascade in the weakly collisional limit is generalized to the moderately collisional regime. In this context, non-universal power law scalings due to multiscale dissipation are predicted, and this prediction is confirmed by means of direct numerical simulations.
Search for gyrokinetic dependencies in helium transport at Alcator C-Mod
NASA Astrophysics Data System (ADS)
Liao, Kenneth; Rowan, William; Hatch, David; Bespamyatnov, Igor; Horton, Wendell
2013-10-01
Helium-3 and helium-4 impurity transport measurements and density profile measurements have been obtained on Alcator C-Mod in a variety of discharge conditions, using the core Charge Exchange Recombination Spectroscopy (CXRS) diagnostic. The helium concentrations range from trace (< 2 %) to large minority (~ 20 %). L-mode, H-mode, and I-mode results are included, with Ohmic heated, ICRF heated, and LH heated plasmas. Helium profiles are observed to vary with plasma current, and also change in time during ICRF shots. Linear and nonlinear gyrokinetic simulations are performed for some shots using the GENE code. Sensitivity scans are done for magnetic shear, impurity density, and other plasma parameters and transport scalings are compared with experimental results. Simulated transport flux is compared with experimentally derived D and v parameters. Supported by USDoE awards DE-FG03-96ER-54373 and DE-FC02-99ER54512.
NASA Astrophysics Data System (ADS)
Brunner, S.; Fivaz, M.; Tran, T. M.; Vaclavik, J.
1998-11-01
A solution to the full two-dimensional eigenvalue problem of electrostatic microinstabilities in a tokamak plasma is presented in the framework of gyrokinetic theory. The approach is the generalization of methods previously developed for a cylindrical system [S. Brunner and J. Vaclavik, Phys. Plasmas 5, 365 (1998)]. By solving the spectral problem in a special Fourier space adapted to the curved geometry, orbit width as well as Larmor radius can be kept to all orders. For a first numerical implementation, a large aspect ratio plasma with circular concentric magnetic surfaces is considered. A root finding algorithm for identifying the eigenfrequencies, based on a higher order Nyquist method, enables straightforward implementation on a parallel computer. Illustrative results for ion temperature gradient-related instabilities are presented. These include scaling studies of the radial width, and toroidicity and magnetic shear scans, as well as the effects of nonadiabatic trapped electron dynamics.
Grid-based Parallel Data Streaming Implemented for the Gyrokinetic Toroidal Code
S. Klasky; S. Ethier; Z. Lin; K. Martins; D. McCune; R. Samtaney
2003-09-15
We have developed a threaded parallel data streaming approach using Globus to transfer multi-terabyte simulation data from a remote supercomputer to the scientist's home analysis/visualization cluster, as the simulation executes, with negligible overhead. Data transfer experiments show that this concurrent data transfer approach is more favorable compared with writing to local disk and then transferring this data to be post-processed. The present approach is conducive to using the grid to pipeline the simulation with post-processing and visualization. We have applied this method to the Gyrokinetic Toroidal Code (GTC), a 3-dimensional particle-in-cell code used to study microturbulence in magnetic confinement fusion from first principles plasma theory.
Squire, J.; Tang, W. M.; Qin, H.; Chandre, C.
2013-02-15
We present a new variational principle for the gyrokinetic system, similar to the Maxwell-Vlasov action presented in H. Cendra et al., [J. Math. Phys. 39, 3138 (1998)]. The variational principle is in the Eulerian frame and based on constrained variations of the phase space fluid velocity and particle distribution function. Using a Legendre transform, we explicitly derive the field theoretic Hamiltonian structure of the system. This is carried out with a modified Dirac theory of constraints, which is used to construct meaningful brackets from those obtained directly from Euler-Poincare theory. Possible applications of these formulations include continuum geometric integration techniques, large-eddy simulation models, and Casimir type stability methods.
Chang, C S; Ku, Seung-Hoe; Diamond, P. H.; Adams, Mark; Tchoua, Roselyne B; Chen, Yang; Cummings, J.; D'Azevedo, Ed F; Dif-Pradalier, Guilhem; Ethier, Stephane; Greengard, Leslie; Hahm, Taik Soo; Hinton, Fred; Keyes, David E; Klasky, Scott A; Lin, Z.; Lofstead, J.; Park, G.; Podhorszki, Norbert; Schwan, Karsten; Shoshani, A.; Silver, D.; Wolf, M.; Worley, Patrick H; Zorin, Denis
2009-01-01
Performance prediction for ITER is based upon the ubiquitous experimental observation that the plasma energy confinement in the device core is strongly coupled to the edge confinement for an unknown reason. The coupling time-scale is much shorter than the plasma transport time-scale. In order to understand this critical observation, a multi-scale turbulence-neoclassical simulation of integrated edge-core plasma in a realistic diverted geometry is a necessity, but has been a formidable task. Thanks to the recent development in high performance computing, we have succeeded in the integrated multiscale gyrokinetic simulation of the ion-temperature-gradient driven turbulence in realistic diverted tokamak geometry for the first time. It is found that modification of the self-organized criticality in the core plasma by nonlocal core-edge coupling of ITG turbulence can be responsible for the core-edge confinement coupling.
Flux- and gradient-driven global gyrokinetic simulation of tokamak turbulence
Goerler, Tobias; Jenko, Frank; Marcus, Patrick; Merz, Florian; Told, Daniel; Lapillonne, Xavier; Brunner, Stephan; Aghdam, Sohrab Khosh; McMillan, Ben F.; Sauter, Olivier; Villard, Laurent; Dannert, Tilman
2011-05-15
The Eulerian gyrokinetic turbulence code gene has recently been extended to a full torus code. Moreover, it now provides Krook-type sources for gradient-driven simulations where the profiles are maintained on average as well as localized heat sources for a flux-driven type of operation. Careful verification studies and benchmarks are performed successfully. This setup is applied to address three related transport issues concerning nonlocal effects. First, it is confirmed that in gradient-driven simulations, the local limit can be reproduced--provided that finite aspect ratio effects in the geometry are treated carefully. In this context, it also becomes clear that the profile widths (not the device width) may constitute a more appropriate measure for finite-size effects. Second, the nature and role of heat flux avalanches are discussed in the framework of both local and global, flux- and gradient-driven simulations. Third, simulations dedicated to discharges with electron internal barriers are addressed.
Flux- and gradient-driven global gyrokinetic simulation of tokamak turbulencea)
NASA Astrophysics Data System (ADS)
Görler, Tobias; Lapillonne, Xavier; Brunner, Stephan; Dannert, Tilman; Jenko, Frank; Aghdam, Sohrab Khosh; Marcus, Patrick; McMillan, Ben F.; Merz, Florian; Sauter, Olivier; Told, Daniel; Villard, Laurent
2011-05-01
The Eulerian gyrokinetic turbulence code gene has recently been extended to a full torus code. Moreover, it now provides Krook-type sources for gradient-driven simulations where the profiles are maintained on average as well as localized heat sources for a flux-driven type of operation. Careful verification studies and benchmarks are performed successfully. This setup is applied to address three related transport issues concerning nonlocal effects. First, it is confirmed that in gradient-driven simulations, the local limit can be reproduced—provided that finite aspect ratio effects in the geometry are treated carefully. In this context, it also becomes clear that the profile widths (not the device width) may constitute a more appropriate measure for finite-size effects. Second, the nature and role of heat flux avalanches are discussed in the framework of both local and global, flux- and gradient-driven simulations. Third, simulations dedicated to discharges with electron internal barriers are addressed.
Gyrokinetic Simulations of RMP Effects on DIII-D Edge Turbulence
NASA Astrophysics Data System (ADS)
Taimourzadeh, Sam; Holod, Ihor; Lin, Zhihong; Nazikian, Raffi; Wingen, Andreas
2016-10-01
It has been demonstrated that edge localized modes (ELMs) can be fully suppressed in DIII-D H-mode plasmas with the application of resonant magnetic perturbations (RMPs), and that there is a corresponding reduction of pedestal gradients, changes in rotation, and changes in the radial electric field (Er) profile. However, with the application of RMPs there is also an increase in long wavelength, electrostatic turbulence on top of the pedestal, as observed with BES, DBS, and other fluctuation diagnostics. Using the Gyrokinetic Toroidal Code (GTC), DIII-D shots 158103, at times 3750 ms (ELMing w/ RMP) and 3050 ms (ELM suppressed w/ RMP), and shot 158104.1350 (ELMing) are investigated, and a link between increased pedestal top turbulence, during the ELM suppressed phase, and a shift in the Er profile is demonstrated.
Gyrokinetic simulations of solar wind turbulence from ion to electron scales.
Howes, G G; TenBarge, J M; Dorland, W; Quataert, E; Schekochihin, A A; Numata, R; Tatsuno, T
2011-07-15
A three-dimensional, nonlinear gyrokinetic simulation of plasma turbulence resolving scales from the ion to electron gyroradius with a realistic mass ratio is presented, where all damping is provided by resolved physical mechanisms. The resulting energy spectra are quantitatively consistent with a magnetic power spectrum scaling of k(-2.8) as observed in in situ spacecraft measurements of the "dissipation range" of solar wind turbulence. Despite the strongly nonlinear nature of the turbulence, the linear kinetic Alfvén wave mode quantitatively describes the polarization of the turbulent fluctuations. The collisional ion heating is measured at subion-Larmor radius scales, which provides evidence of the ion entropy cascade in an electromagnetic turbulence simulation.
Faber, B. J.; Pueschel, M. J.; Terry, P. W.; Proll, J. H. E.; Hegna, C. C.; Weir, G. M.; Likin, K. M.; Talmadge, J. N.
2015-07-15
Gyrokinetic simulations of plasma microturbulence in the Helically Symmetric eXperiment are presented. Using plasma profiles relevant to experimental operation, four dominant drift wave regimes are observed in the ion wavenumber range, which are identified as different flavors of density-gradient-driven trapped electron modes. For the most part, the heat transport exhibits properties associated with turbulence driven by these types of modes. Additionally, long-wavelength, radially localized, nonlinearly excited coherent structures near the resonant central flux surface, not predicted by linear simulations, can further enhance flux levels. Integrated heat fluxes are compatible with experimental observations in the corresponding density gradient range. Despite low shearing rates, zonal flows are observed to regulate turbulence but can be overwhelmed at higher density gradients by the long-wavelength coherent structures.
NASA Astrophysics Data System (ADS)
Kraus, Michael; Kormann, Katharina; Sonnendrücker, Eric; Morrison, Philip
2016-10-01
In this talk we will describe recent work on the development of geometric particle-in-cell methods for the Vlasov-Maxwell system and gyrokinetics. We present a novel framework for particle-in-cell methods based on the discretization of the underlying Hamiltonian structure of the Vlasov-Maxwell system. We derive semi-discrete Poisson brackets which satisfy the Jacobi identity and apply Hamiltonian splitting schemes for time integration. Techniques from Finite Element Exterior Calculus and spline differential forms ensure conservation of the divergence of the magnetic field and Gauss' law as well as stability of the field solver. The resulting methods are gauge-invariant, feature exact charge conservation show excellent long-time energy behaviour. The talk will be concluded with an outline of how to extend these techniques towards gyrokinetics.
Hager, Robert; Chang, C. S.
2016-04-08
As a follow-up on the drift-kinetic study of the non-local bootstrap current in the steep edge pedestal of tokamak plasma by Koh et al. [Phys. Plasmas 19, 072505 (2012)], a gyrokinetic neoclassical study is performed with gyrokinetic ions and drift-kinetic electrons. Besides the gyrokinetic improvement of ion physics from the drift-kinetic treatment, a fully non-linear Fokker-Planck collision operator—that conserves mass, momentum, and energy—is used instead of Koh et al.'s linearized collision operator in consideration of the possibility that the ion distribution function is non-Maxwellian in the steep pedestal. An inaccuracy in Koh et al.'s result is found in the steepmore » edge pedestal that originated from a small error in the collisional momentum conservation. The present study concludes that (1) the bootstrap current in the steep edge pedestal is generally smaller than what has been predicted from the small banana-width (local) approximation [e.g., Sauter et al., Phys. Plasmas 6, 2834 (1999) and Belli et al., Plasma Phys. Controlled Fusion 50, 095010 (2008)], (2) the plasma flow evaluated from the local approximation can significantly deviate from the non-local results, and (3) the bootstrap current in the edge pedestal, where the passing particle region is small, can be dominantly carried by the trapped particles in a broad trapped boundary layer. In conclusion, a new analytic formula based on numerous gyrokinetic simulations using various magnetic equilibria and plasma profiles with self-consistent Grad-Shafranov solutions is constructed.« less
NASA Astrophysics Data System (ADS)
Hager, Robert; Chang, C. S.
2016-04-01
As a follow-up on the drift-kinetic study of the non-local bootstrap current in the steep edge pedestal of tokamak plasma by Koh et al. [Phys. Plasmas 19, 072505 (2012)], a gyrokinetic neoclassical study is performed with gyrokinetic ions and drift-kinetic electrons. Besides the gyrokinetic improvement of ion physics from the drift-kinetic treatment, a fully non-linear Fokker-Planck collision operator—that conserves mass, momentum, and energy—is used instead of Koh et al.'s linearized collision operator in consideration of the possibility that the ion distribution function is non-Maxwellian in the steep pedestal. An inaccuracy in Koh et al.'s result is found in the steep edge pedestal that originated from a small error in the collisional momentum conservation. The present study concludes that (1) the bootstrap current in the steep edge pedestal is generally smaller than what has been predicted from the small banana-width (local) approximation [e.g., Sauter et al., Phys. Plasmas 6, 2834 (1999) and Belli et al., Plasma Phys. Controlled Fusion 50, 095010 (2008)], (2) the plasma flow evaluated from the local approximation can significantly deviate from the non-local results, and (3) the bootstrap current in the edge pedestal, where the passing particle region is small, can be dominantly carried by the trapped particles in a broad trapped boundary layer. A new analytic formula based on numerous gyrokinetic simulations using various magnetic equilibria and plasma profiles with self-consistent Grad-Shafranov solutions is constructed.
Hager, Robert; Chang, C. S.
2016-04-08
As a follow-up on the drift-kinetic study of the non-local bootstrap current in the steep edge pedestal of tokamak plasma by Koh et al. [Phys. Plasmas 19, 072505 (2012)], a gyrokinetic neoclassical study is performed with gyrokinetic ions and drift-kinetic electrons. Besides the gyrokinetic improvement of ion physics from the drift-kinetic treatment, a fully non-linear Fokker-Planck collision operator—that conserves mass, momentum, and energy—is used instead of Koh et al.'s linearized collision operator in consideration of the possibility that the ion distribution function is non-Maxwellian in the steep pedestal. An inaccuracy in Koh et al.'s result is found in the steep edge pedestal that originated from a small error in the collisional momentum conservation. The present study concludes that (1) the bootstrap current in the steep edge pedestal is generally smaller than what has been predicted from the small banana-width (local) approximation [e.g., Sauter et al., Phys. Plasmas 6, 2834 (1999) and Belli et al., Plasma Phys. Controlled Fusion 50, 095010 (2008)], (2) the plasma flow evaluated from the local approximation can significantly deviate from the non-local results, and (3) the bootstrap current in the edge pedestal, where the passing particle region is small, can be dominantly carried by the trapped particles in a broad trapped boundary layer. In conclusion, a new analytic formula based on numerous gyrokinetic simulations using various magnetic equilibria and plasma profiles with self-consistent Grad-Shafranov solutions is constructed.
Gyrokinetic particle-in-cell simulations of Alfvén eigenmodes in presence of continuum effects
Mishchenko, Alexey Könies, Axel; Hatzky, Roman
2014-05-15
First-principle gyrokinetic particle-in-cell simulations of a global Toroidal Alfvén Eigenmode (TAE) are undertaken in the presence of a strong coupling with the continuum. Effects of the bulk plasma temperature on the interplay between the TAE and Kinetic Alfvén Waves (KAWs) are investigated. A global TAE-KAW structure is identified which appears to be more unstable with respect to the fast ions than a simple (fluid-like) TAE mode.
Static cylindrically symmetric spacetimes
NASA Astrophysics Data System (ADS)
Fjällborg, Mikael
2007-05-01
We prove the existence of static solutions to the cylindrically symmetric Einstein Vlasov system, and we show that the matter cylinder has finite extension in two of the three spatial dimensions. The same results are also proved for a quite general class of equations of state for perfect fluids coupled to the Einstein equations, extending the class of equations of state considered by Bicak et al (2004 Class. Quantum Grav.21 1583). We also obtain this result for the Vlasov Poisson system.
Search for the Missing L-mode Edge Transport and Possible Breakdown of Gyrokinetics
NASA Astrophysics Data System (ADS)
Waltz, R. E.
2012-10-01
While GYRO simulations of typical core (0 < r/a < 0.7) DIII-D L-modes seems to be in good agreement with experiment, simulated low-k (kθρs< 1) transport and turbulence intensity is more than 5-fold lower than experimentally inferred levels in the near edge L-mode (r/a=0.7-0.95) DIII-D shot 128913 [1]. Global edge slice GYRO simulations of this and the well-studied discharge 101391 [2] are presented here to document the shortfall. TGLF transport code simulations over a large L-mode database indicate this short fall is not atypical so that L-mode edges transit to H-like pedestal profiles contrary to experiment. High edge e-i collisionality stabilizes the TEM modes so that diffusivities (χ) decrease like T^7/2/n to the cold edge. The very high magnetic shear and density gradients stabilize the ITG despite the very high temperature gradient drive and high q. High-k ETG can make-up for the shortfall in the electron but increases ion transport very little. Near L-edge transport is highly local. Focusing on local simulations at r/a=0.9, the ion channel short fall can exceed 10-fold. An artificial 10-fold increase in collisionality is needed to reach the expected resistive g-mode scaling with χ increasing like nT-1/2. Identical GYRO drift kinetic ion simulations (suppressing the gyroaverage) are close to experiment levels suggesting a possible breakdown of low-frequency gyrokinetics. Formulation of a nonlinear theory of 6D drift-cyclotron kinetics following the fast time scale of the gyrophase to test the breakdown of 5D gyrokinetics with reduced model simulations is presented. 6pt [1] C. Holland, A.E. White, et al., Phys. Plasmas 16, 052301 (2009). [2] R.E. Waltz, J. Candy, C.C. Petty, Phys. Plasmas 13, 072304 (2006).
Understanding TCV L-mode plasmas via global gyrokinetic GENE simulations
NASA Astrophysics Data System (ADS)
Merlo, Gabriele; Brunner, Stephan; Coda, Stefano; Goerler, Tobias; Huang, Zhouji; Jenko, Frank; Told, Daniel; Sauter, Olivier; Villard, Laurent
2016-10-01
It is known that global effects can have a significant influence on turbulent transport driven by microintabilities, especially for small size machines like the TCV tokamak. The global version of the gyrokinetic GENE code has been extensively used to model TCV plasmas for which finite ρ* effects are expected to be crucial in order to recover the experimentally observed behaviour. We will address in particular: (i) The effect of negative triangularity, which has been experimentally observed to lower up to a factor of two the heat flux through the lectron channel at all radial locations. Global effects and the inclusion of carbon impurities turn out to be the key elements required in order to match experiments and simulation results. (ii) The formation of either radially coherent or dispersive axisymmetric density fluctuations, experimentally interpreted as Geodesic Acoustic Modes. GENE simulations reproduce the observed behaviour and allow to conclude that the modification of safety factor alone cannot explain the transition between these two different fluctuation regimes.
Comparisons between global and local gyrokinetic simulations of an ASDEX Upgrade H-mode plasma
NASA Astrophysics Data System (ADS)
Navarro, Alejandro Bañón; Told, Daniel; Jenko, Frank; Görler, Tobias; Happel, Tim
2016-04-01
We investigate by means of local and global nonlinear gyrokinetic GENE simulations an ASDEX Upgrade H-mode plasma. We find that for the outer core positions (i.e., ρ tor ≈ 0.5 - 0.7 ), nonlocal effects are important. For nominal input parameters local simulations over-predict the experimental heat fluxes by a large factor, while a good agreement is found with global simulations. This was a priori not expected, since the values of 1 / ρ ⋆ were large enough that global and local simulations should have been in accordance. Nevertheless, due to the high sensitivity of the heat fluxes with respect to the input parameters, it is still possible to match the heat fluxes in local simulations with the experimental and global results by varying the ion temperature gradient within the experimental uncertainties. In addition to that, once an agreement in the transport quantities between local (flux-matched) and global simulations is achieved, an agreement for other quantities, such as density and temperature fluctuations, is also found. The case presented here clearly shows that even in the presence of global size-effects, the local simulation approach is still a valid and accurate approach.
Gyrokinetic Particle Simulation of Fast Electron Driven Beta-induced Alfven Eigenmodes
NASA Astrophysics Data System (ADS)
Zhang, Wenlu; Cheng, Junyi; Lin, Zhihong
2016-10-01
The fast electron driven beta induced Alfven eigenmode (e-BAE) has been routinely observed in HL-2A tokamak. We study e-BAE for the first time using global gyrokinetic GTC simulation, where the fast electrons are described by the drift kinetic model. Frequency chirping is observed in nonlinear simulations in the absence of sources and sinks, which provide a new nonlinear paradigm beyond the standard ``bump-on-tail'' model. For weakly driven case, nonlinear frequency is observed to be in phase with particle flux, and nonlinear mode structure is almost the same as linear stage. In the strongly driven case, BAAE is also unstable and co-exists with BAE after the BAE saturation. Analysis of nonlinear wave-particle interactions shows that the frequency chirping is induced by the nonlinear evolution of the coherent structures in the fast electron phase space, where the dynamics of the coherent structure is controlled by the formation and destruction of phrase space islands in the canonical variables. Zonal fields are found to affect wave-particle resonance in the nonlinear e-BAE simulations.
A study of self organized criticality in ion temperature gradient mode driven gyrokinetic turbulence
Mavridis, M.; Isliker, H.; Vlahos, L.; Görler, T.; Jenko, F.; Told, D.
2014-10-15
An investigation on the characteristics of self organized criticality (Soc) in ITG mode driven turbulence is made, with the use of various statistical tools (histograms, power spectra, Hurst exponents estimated with the rescaled range analysis, and the structure function method). For this purpose, local non-linear gyrokinetic simulations of the cyclone base case scenario are performed with the GENE software package. Although most authors concentrate on global simulations, which seem to be a better choice for such an investigation, we use local simulations in an attempt to study the locally underlying mechanisms of Soc. We also study the structural properties of radially extended structures, with several tools (fractal dimension estimate, cluster analysis, and two dimensional autocorrelation function), in order to explore whether they can be characterized as avalanches. We find that, for large enough driving temperature gradients, the local simulations exhibit most of the features of Soc, with the exception of the probability distribution of observables, which show a tail, yet they are not of power-law form. The radial structures have the same radial extent at all temperature gradients examined; radial motion (transport) though appears only at large temperature gradients, in which case the radial structures can be interpreted as avalanches.
Gyrokinetic Particle Simulation of Compressible Electromagnetic Turbulence in High-β Plasmas
Lin, Zhihong
2014-03-13
Supported by this award, the PI and his research group at the University of California, Irvine (UCI) have carried out computational and theoretical studies of instability, turbulence, and transport in laboratory and space plasmas. Several massively parallel, gyrokinetic particle simulation codes have been developed to study electromagnetic turbulence in space and laboratory plasmas. In space plasma projects, the simulation codes have been successfully applied to study the spectral cascade and plasma heating in kinetic Alfven wave turbulence, the linear and nonlinear properties of compressible modes including mirror instability and drift compressional mode, and the stability of the current sheet instabilities with finite guide field in the context of collisionless magnetic reconnection. The research results have been published in 25 journal papers and presented at many national and international conferences. Reprints of publications, source codes, and other research-related information are also available to general public on the PI’s webpage (http://phoenix.ps.uci.edu/zlin/). Two PhD theses in space plasma physics are highlighted in this report.
Continuum Gyrokinetic Simulations of Turbulence in Open-Field-Line Plasmas
NASA Astrophysics Data System (ADS)
Shi, E. L.; Hammett, G. W.; Stoltzfus-Dueck, T.; Hakim, A.
2016-10-01
We have performed our first 3D2V gyrokinetic simulations of electrostatic plasma turbulence in open-field-line geometries using the full-F discontinuous-Galerkin code Gkeyll. These simulations include the basic elements of a scrape-off layer: localized sources to model plasma outflow from the core, cross-field turbulent transport, parallel flow along magnetic field lines, and parallel losses at the limiter or divertor with sheath boundary conditions. The set of boundary conditions used in our model allows currents to flow through the walls and satisfies energy conservation. In addition to details of our numerical approach, we will present results from flux-tube simulations of devices containing straight-field lines (such as LAPD) and helical-field-lines (such as the TORPEX simple magnetized torus). Preliminary results show turbulent fluctuation levels similar to fluid simulations, which are comparable to the observed fluctuation level in LAPD but somewhat smaller than observed in TORPEX. This research was supported by the Max-Planck/Princeton Center for Plasma Physics, the SciDAC Center for the Study of Plasma Microturbulence, and U.S. DOE contract DE-AC02-09CH11466.
Joiner, N.; Hirose, A.
2008-08-15
The kinetic ballooning mode (KBM) has been shown in previous work to be unstable within the magnetohydrodynamic (MHD) region (in s-{alpha} space) of second stability [Hirose et al., Phys. Rev. Lett. 72, 3993 (2004)]. In this work we verify this result using the gyrokinetic code GS2 [Kotschenreuther et al., Comput. Phys. Commun. 88, 128 (1996)] treating both ions and electrons as kinetic species and retaining the magnetosonic perturbation B{sub parallel}. Growth rates calculated using GS2 differ significantly from the previous differential/shooting code analysis. Calculations without B{sub parallel} find the stability region is preserved, while the addition of B{sub parallel} causes the mode to be more unstable than previously calculated within the region of MHD second stability. The inclusion of parallel ion current and B{sub parallel} into the shooting code does not account for the GS2 results. The evidence presented in this paper leads us to the conclusion that the adiabatic electron approximation employed in previous studies is found to be unsuitable for this type of instability. Based on the findings of this work, the KBM becomes an interesting instability in the context of internal transport barriers, where {alpha} is often large and magnetic shear is small (positive or negative)
Gyrokinetic electromagnetic isotope effect in ITER-hybrid plasmas and validation
NASA Astrophysics Data System (ADS)
Goerler, Tobias; Garcia, Jeronimo; Jenko, Frank
2015-11-01
A number of high-realism simulations with the gyrokinetic turbulence code GENE have been performed recently for comparison with experimental measurements in, e.g., ASDEX Upgrade and DIII-D. Some of these successful validation studies will be reviewed briefly as basis for subsequent predictive simulations for a particular ITER hybrid scenario. Here, comprehensive local GENE simulations have been employed considering the multi-component character of such plasmas including impurities, fuel ions, helium ash, up to two fast ion species as well as electromagnetic fluctuations, inter- and intra-species collisions, and external shear effects. The fluxes are in general in good agreement with those in the above ITER study performed with the CRONOS code suite. A particular subject of interest is the turbulent transport comparison between deuterium-tritium (DT) plasmas and pure deuterium (DD) fuel as mostly used in present-day experiments. Here, a strong heat flux drop from DD to DT plasmas can be observed which is in line with experimental evidence found at TFTR and JET. This contribution may hence help to gain a better understanding of this so-called isotope effect and improve projections for future ITER DD- and DT-plasma studies.
Particle-in-cell δf gyrokinetic simulations of the microtearing mode
Chowdhury, J.; Chen, Yang; Wan, Weigang; Parker, Scott E.; Guttenfelder, W.; Canik, J. M.
2016-01-15
The linear stability properties of the microtearing mode are investigated in the edge and core regimes of the National Spherical Torus Experiment (NSTX) using the particle-in-cell method based gyrokinetic code GEM. The dependence of the mode on various equilibrium quantities in both regions is compared. While the microtearing mode in the core depends upon the electron-ion collisions, in the edge region, it is found to be weakly dependent on the collisions and exists even when the collision frequency is zero. The electrostatic potential is non-negligible in each of the cases. It plays opposite roles in the core and edge of NSTX. While the microtearing mode is partially stabilized by the electrostatic potential in the core, it has substantial destabilizing effect in the edge. In addition to the spherical tokamak, we also study the microtearing mode for parameters relevant to the core of a standard tokamak. The fundamental characteristics of the mode remain the same; however, the electrostatic potential in this case is destabilizing as opposed to the core of NSTX. The velocity dependence of the collision frequency, which is crucial for the mode to grow in slab calculations, is not required to destabilize the mode in toroidal devices.
Comparative study between ion-scale turbulence measurements and gyrokinetic simulations
NASA Astrophysics Data System (ADS)
Lee, W.; Ko, S. H.; Choi, M. J.; Ko, W. H.; Lee, K. D.; Leem, J.; Yun, G. S.; Park, H. K.; Wang, W. X.; Budny, R. V.; Park, Y. S.; Luhmann, N. C., Jr.; Kim, K. W.; Kstar Team
2016-10-01
Ion gyroscale density fluctuations were measured with a microwave imaging reflectometer (MIR) in neutral beam injected L-mode plasmas on KSTAR. The spatial and temporal characteristic scales of the measured fluctuations were studied by comparing with the local equilibrium parameters relevant to the ion-scale turbulence. Linear and nonlinear gyrokinetic simulations predicted unstable modes with poloidal wavenumbers of 3 cm-1 (or kθρs 0.4) and the wavenumbers were also identified from the measured fluctuations. The poloidal wavenumber can be derived from the measured mode frequency and poloidal velocity. The dominant mode frequency and poloidal velocity were obtained from cross correlations among 16 poloidal channels. Both the mode frequency and poloidal velocity mostly are primarily due to the E x B flow velocity in fast rotating plasmas with neutral beam injection. Work supported by NRF Korea under Grant Number NRF-2014M1A7A1A03029865 and Korean Ministry of Science, ICT, and Future Planning under the KSTAR project contract.
NASA Astrophysics Data System (ADS)
Toda, S.; Nakata, M.; Nunami, M.; Ishizawa, A.; Watanabe, T.-H.; Sugama, H.
2016-10-01
A reduced model of the turbulent ion heat diffusivity is proposed by the gyrokinetic simulation code (GKV-X) with the adiabatic electrons for the high-Ti Large Helical Device discharge. The plasma parameter region of the short poloidal wavelength is studied, where the ion temperature gradient mode becomes unstable. The ion heat diffusivity by the nonlinear simulation with the kinetic electrons is found to be several times larger than the simulation results using the adiabatic electrons in the radial region 0.46 <= r / a <= 0.80 . The electromagnetic contribution is about a several percent in the ion energy flux. The model of the turbulent diffusivity is derived as the function of the squared electrostatic potential fluctuation and the squared zonal flow potential. Next, the squared electrostatic potential fluctuation is approximated with the mixing length estimate. The squared zonal flow potential fluctuation is shown as the linear zonal flow response function. The reduced model of the turbulent diffusivity is derived as the function of the physical parameters by the linear GKV-X simulation with the kinetic electrons. This reduced model is applied to the transport code with the same procedure as.
Gyrokinetic simulation of I-mode C-Mod pedestal using GENE
NASA Astrophysics Data System (ADS)
Liu, Xing; David, H.; Kotschenreuther, M.; Mahajan, S.; Huges, J.; Hubbard, A.; Valanju, P.
2016-10-01
Naturally stable to ELMs, and with widths larger than EPED predictions, the I-modes are an excellent laboratory for investigating the role of drift microinstabilities in pedestal formation since I-mode pedestal are not ``limited'' by MHD instabilities-Peeling Ballooning or the Kinetic Ballooning. Because the Weakly Coherent Mode (WCM) is shown to be correlated, primarily, to particle transport, the pedestal heat transport, in some sense, must be controlled by drift-type modes. We present here a study based on gyrokinetic simulations (using GENE) to model heat transport in the I-mode pedestals in C-Mod. Nonlinear ETG simulations, found to be streamer-dominated, can match experimental heat flux with profile adjustment well within experimental error bars. The ETG simulations reveal very notable fine-scale structure (in the parallel direction) of the eigenfunctions in both linear and nonlinear simulations. Simulations, varying impurity level (Zeff) and temperature and density profiles (within experimental error bars), are used to probe the sensitivity of ETG heat transport to the most important input parameters. Efforts to identify an instability corresponding to the WCM will also be discussed. Work supported by USDOE Grant DE-FG02-04ER54742.
Bounce-Averaged Gyrokinetic Simulation of Current-Collection Feedback in a Laboratory Magnetosphere
NASA Astrophysics Data System (ADS)
Roberts, T. M.; Garnier, D.; Kesner, J.; Mauel, M. E.
2014-10-01
A self-consistent, nonlinear simulation of interchange dynamics including the bounce-averaged gyro-kinetics of trapped electrons was previously used to understand frequency sweeping and the turbulent cascades observed in dipole-confined plasmas. Through adjustment of the particle and heat sources this code reproduces dynamics that resemble the turbulence measured experimentally, both in spectral power-law trends and in the onset of a steepened density profile. Time stepping is performed in an explicit leap-frog manner and a flux-corrected transport algorithm is implemented. In this presentation, we discuss the physics and numerical methods of the simulations as well as plans for including the effects of a biasing electrode which can collect or inject electrons. By varying this source/sink of electrons at the electrode location based on the potential fluctuations occurring elsewhere, we study the effects of current-collection feedback to compare to recent experiments observed to regulate interchange turbulence. Supported by NSF-DOE Partnership for Plasma Science and DOE Grant DE-FG02-00ER54585 and NSF Award PHY-1201896.
Three-dimensional gyrokinetic simulation of the relaxation of a magnetized temperature filament
Sydora, R. D.; Morales, G. J.; Maggs, J. E.; Van Compernolle, B.
2015-10-15
An electromagnetic, 3D gyrokinetic particle code is used to study the relaxation of a magnetized electron temperature filament embedded in a large, uniform plasma of lower temperature. The study provides insight into the role played by unstable drift-Alfvén waves observed in a basic electron heat transport experiment [D. C. Pace et al., Phys. Plasmas 15, 122304 (2008)] in which anomalous cross-field transport has been documented. The simulation exhibits the early growth of temperature-gradient-driven, drift-Alfvén fluctuations that closely match the eigenmodes predicted by linear theory. At the onset of saturation, the unstable fluctuations display a spiral spatial pattern, similar to that observed in the laboratory, which causes the rearrangement of the temperature profile. After saturation of the linear instability, the system exhibits a markedly different behavior depending on the inclusion in the computation of modes without variation along the magnetic field, i.e., k{sub z} = 0. In their absence, the initial filament evolves into a broadened temperature profile, self-consistent with undamped, finite amplitude drift-Alfvén waves. But the inclusion of k{sub z} = 0 modes causes the destruction of the filament and damping of the drift-Alfvén modes leading to a final state consisting of undamped convective cells and multiple, smaller-scale filaments.
Spong, D. A.; Bass, E. M.; Deng, W.; Heidbrink, W. W.; Lin, Z.; Tobias, B.; Van Zeeland, M. A.; Austin, M. E.; Domier, C. W.; Luhmann, N. C. Jr.
2012-08-15
A verification and validation study is carried out for a sequence of reversed shear Alfven instability time slices. The mode frequency increases in time as the minimum (q{sub min}) in the safety factor profile decreases. Profiles and equilibria are based upon reconstructions of DIII-D discharge (no. 142111) in which many such frequency up-sweeping modes were observed. Calculations of the frequency and mode structure evolution from two gyrokinetic codes, GTC and GYRO, and a gyro-Landau fluid code TAEFL are compared. The experimental mode structure of the instability was measured using time-resolved two-dimensional electron cyclotron emission imaging. The three models reproduce the frequency upsweep event within {+-}10% of each other, and the average of the code predictions is within {+-}8% of the measurements; growth rates are predicted that are consistent with the observed spectral line widths. The mode structures qualitatively agree with respect to radial location and width, dominant poloidal mode number, ballooning structure, and the up-down asymmetry, with some remaining differences in the details. Such similarities and differences between the predictions of the different models and the experimental results are a valuable part of the verification/validation process and help to guide future development of the modeling efforts.
Collisional Ion and Electron Scale Gyrokinetic Simulations in the Tokamak Pedestal
NASA Astrophysics Data System (ADS)
Belli, E. A.; Candy, J.; Snyder, P. B.
2016-10-01
A new gyrokinetic solver, CGYRO, has been developed for precise studies of high collisionality regimes, such as the H-mode pedestal and L-mode edge. Building on GYRO and NEO, CGYRO uses the same velocity-space coordinates as NEO to optimize the accuracy of the collision dynamics and allow for advanced operators beyond the standard Lorentz pitch-angle scattering model. These advanced operators include energy diffusion and finite-FLR collisional effects. The code is optimized for multiscale (coupled electron and ion turbulence scales) simulations, employing a new spatial discretization and array distribution scheme that targets scalability on next-generation (exascale) HPC systems. In this work, CGYRO is used to study the complex spectrum of modes in the pedestal region. The onset of the linear KBM with full collisional effects is assessed to develop an improved KBM/RBM model for EPED. The analysis is extended to high k to explore the role of electron-scale (ETG-range) physics. Comparisons with new analytic collisional theories are made. Inclusion of sonic toroidal rotation (including full centrifugal effects) for studies including heavy wall impurities is also reported. Work supported in part by the US DOE under DE-FC02-06ER54873 and DE-FC02-08ER54963.
Leerink, S; Bulanin, V V; Gurchenko, A D; Gusakov, E Z; Heikkinen, J A; Janhunen, S J; Lashkul, S I; Altukhov, A B; Esipov, L A; Kantor, M Yu; Kiviniemi, T P; Korpilo, T; Kuprienko, D V; Petrov, A V
2012-10-19
Direct measurements of micro-, meso-, and macroscale transport phenomena in the FT-2 tokamak are shown to be quantitatively reproduced by global full f nonlinear gyrokinetic simulation predictions. A detailed agreement with mean equilibrium E×B flows, oscillating fine-scale zonal flows, and turbulence spectra observed by a set of sophisticated microwave backscattering techniques as well as a good fit of the thermal diffusivity data are demonstrated. A clear influence of the impurity ions on the fluctuating radial electric field is observed.
Angioni, C.
2015-10-15
A gyrokinetic study based on numerical and analytical calculations is presented, which computes the dependence of the turbulent diffusion of highly charged impurities on the ratio of the electron to the ion heat flux of the plasma. Nonlinear simulations show that the size of the turbulent diffusion of heavy impurities can vary by one order of magnitude with fixed total heat flux and is an extremely sensitive function of the electron to ion heat flux ratio. Numerical linear calculations are found to reproduce the nonlinear results. Thereby, a quasi-linear analytical approach is used to explain the origin of this dependence.
Krommes, J. A.
2013-12-15
Some physical interpretations are given of the well-known second-order gyrokinetic Hamiltonian in the magnetohydrodynamic limit. Its relations to the conservation of the true (Galilean-invariant) magnetic moment and fluid nonlinearities are described. Subtleties about its derivation as a cold-ion limit are explained; it is important to take that limit in the frame moving with the E×B velocity. The discussion also provides some geometric understanding of certain well-known Lie generating functions, and it makes contact with general discussions of ponderomotive potentials and the thermodynamics of dielectric media.
Liu, Dongjian; Bao, Jian; Han, Tao; Wang, Jiaqi; Lin, Zhihong
2016-02-15
A finite-mass electron fluid model for low frequency electromagnetic fluctuations, particularly the collisionless tearing mode, has been implemented in the gyrokinetic toroidal code. Using this fluid model, linear properties of the collisionless tearing mode have been verified. Simulations verify that the linear growth rate of the single collisionless tearing mode is proportional to D{sub e}{sup 2}, where D{sub e} is the electron skin depth. On the other hand, the growth rate of a double tearing mode is proportional to D{sub e} in the parameter regime of fusion plasmas.
Guttenfelder, W.; Kaye, S. M.; Ren, Y.; Solomon, W.; Bell, R. E.; Candy, J.; Gerhardt, S. P.; LeBlanc, B. P.; Yuh, H.
2016-05-11
This paper presents quasi-linear gyrokinetic predictions of the Coriolis momentum pinch for low aspect-ratio National Spherical Torus Experiment (NSTX) H-modes where previous experimental measurements were focused. Local, linear calculations predict that in the region of interest (just outside the mid-radius) of these relatively high-beta plasmas, profiles are most unstable to microtearing modes that are only effective in transporting electron energy. However, sub-dominant electromagnetic and electrostaticballooning modes are also unstable, which are effective at transporting energy, particles, and momentum. The quasi-linear prediction of transport from these weaker ballooning modes, assuming they contribute transport in addition to that from microtearing modes in a nonlinear turbulent state, leads to a very small or outward convection of momentum, inconsistent with the experimentally measured inward pinch, and opposite to predictions in conventional aspect ratio tokamaks. Additional predictions of a low beta L-mode plasma, unstable to more traditional electrostatic ion temperature gradient-trapped electron mode instability, show that the Coriolis pinch is inward but remains relatively weak and insensitive to many parameter variations. The weak or outward pinch predicted in NSTX plasmas appears to be at least partially correlated to changes in the parallel mode structure that occur at a finite beta and low aspect ratio, as discussed in previous theories. The only conditions identified where a stronger inward pinch is predicted occur either in the purely electrostatic limit or if the aspect ratio is increased. Lastly, as the Coriolis pinch cannot explain the measured momentum pinch, additional theoretical momentum transport mechanisms are discussed that may be potentially important.
Lapillonne, X.; Brunner, S.; Dannert, T.; Jolliet, S.; Marinoni, A.; Villard, L.; Goerler, T.; Jenko, F.; Merz, F.
2009-03-15
In the context of gyrokinetic flux-tube simulations of microturbulence in magnetized toroidal plasmas, different treatments of the magnetic equilibrium are examined. Considering the Cyclone DIII-D base case parameter set [Dimits et al., Phys. Plasmas 7, 969 (2000)], significant differences in the linear growth rates, the linear and nonlinear critical temperature gradients, and the nonlinear ion heat diffusivities are observed between results obtained using either an s-{alpha} or a magnetohydrodynamic (MHD) equilibrium. Similar disagreements have been reported previously [Redd et al., Phys. Plasmas 6, 1162 (1999)]. In this paper it is shown that these differences result primarily from the approximation made in the standard implementation of the s-{alpha} model, in which the straight field line angle is identified to the poloidal angle, leading to inconsistencies of order {epsilon} ({epsilon}=a/R is the inverse aspect ratio, a the minor radius and R the major radius). An equilibrium model with concentric, circular flux surfaces and a correct treatment of the straight field line angle gives results very close to those using a finite {epsilon}, low {beta} MHD equilibrium. Such detailed investigation of the equilibrium implementation is of particular interest when comparing flux tube and global codes. It is indeed shown here that previously reported agreements between local and global simulations in fact result from the order {epsilon} inconsistencies in the s-{alpha} model, coincidentally compensating finite {rho}{sup *} effects in the global calculations, where {rho}{sup *}={rho}{sub s}/a with {rho}{sub s} the ion sound Larmor radius. True convergence between local and global simulations is finally obtained by correct treatment of the geometry in both cases, and considering the appropriate {rho}{sup *}{yields}0 limit in the latter case.
Guttenfelder, W.; Kaye, S. M.; Ren, Y.; ...
2016-05-11
This paper presents quasi-linear gyrokinetic predictions of the Coriolis momentum pinch for low aspect-ratio National Spherical Torus Experiment (NSTX) H-modes where previous experimental measurements were focused. Local, linear calculations predict that in the region of interest (just outside the mid-radius) of these relatively high-beta plasmas, profiles are most unstable to microtearing modes that are only effective in transporting electron energy. However, sub-dominant electromagnetic and electrostaticballooning modes are also unstable, which are effective at transporting energy, particles, and momentum. The quasi-linear prediction of transport from these weaker ballooning modes, assuming they contribute transport in addition to that from microtearing modes inmore » a nonlinear turbulent state, leads to a very small or outward convection of momentum, inconsistent with the experimentally measured inward pinch, and opposite to predictions in conventional aspect ratio tokamaks. Additional predictions of a low beta L-mode plasma, unstable to more traditional electrostatic ion temperature gradient-trapped electron mode instability, show that the Coriolis pinch is inward but remains relatively weak and insensitive to many parameter variations. The weak or outward pinch predicted in NSTX plasmas appears to be at least partially correlated to changes in the parallel mode structure that occur at a finite beta and low aspect ratio, as discussed in previous theories. The only conditions identified where a stronger inward pinch is predicted occur either in the purely electrostatic limit or if the aspect ratio is increased. Lastly, as the Coriolis pinch cannot explain the measured momentum pinch, additional theoretical momentum transport mechanisms are discussed that may be potentially important.« less
Baumgaertel, J. A.; Redi, M. H.; Budny, R. V.; Rewoldt, G.; Dorland, W.
2005-10-19
Insight into plasma microturbulence and transport is being sought using linear simulations of drift waves on the National Spherical Torus Experiment (NSTX), following a study of drift wave modes on the Alcator C-Mod Tokamak. Microturbulence is likely generated by instabilities of drift waves, which cause transport of heat and particles. Understanding this transport is important because the containment of heat and particles is required for the achievement of practical nuclear fusion. Microtearing modes may cause high heat transport through high electron thermal conductivity. It is hoped that microtearing will be stable along with good electron transport in the proposed low collisionality International Thermonuclear Experimental Reactor (ITER). Stability of the microtearing mode is investigated for conditions at mid-radius in a high density NSTX high performance (H-mode) plasma, which is compared to the proposed ITER plasmas. The microtearing mode is driven by the electron temperature gradient, and believed to be mediated by ion collisions and magnetic shear. Calculations are based on input files produced by TRXPL following TRANSP (a time-dependent transport analysis code) analysis. The variability of unstable mode growth rates is examined as a function of ion and electron collisionalities using the parallel gyrokinetic computational code GS2. Results show the microtearing mode stability dependence for a range of plasma collisionalities. Computation verifies analytic predictions that higher collisionalities than in the NSTX experiment increase microtearing instability growth rates, but that the modes are stabilized at the highest values. There is a transition of the dominant mode in the collisionality scan to ion temperature gradient character at both high and low collisionalities. The calculations suggest that plasma electron thermal confinement may be greatly improved in the low-collisionality ITER.
Waltz, R. E.; Staebler, G. M.; Solomon, W. M.
2011-04-15
Residual stress refers to the remaining toroidal angular momentum (TAM) flux (divided by major radius) when the shear in the equilibrium fluid toroidal velocity (and the velocity itself) vanishes. Previously [Waltz et al., Phys. Plasmas 14, 122507 (2007); errata 16, 079902 (2009)], we demonstrated with GYRO [Candy and Waltz, J. Comp. Phys. 186, 545 (2003)] gyrokinetic simulations that TAM pinching from (ion pressure gradient supported or diamagnetic level) equilibrium ExB velocity shear could provide some of the residual stress needed to support spontaneous toroidal rotation against normal diffusive loss. Here we show that diamagnetic level shear in the intrinsic drift wave velocities (or ''profile shear'' in the ion and electron density and temperature gradients) provides a comparable residual stress. The individual signed contributions of these small (rho-star level) ExB and profile velocity shear rates to the turbulence level and (rho-star squared) ion energy transport stabilization are additive if the rates are of the same sign. However because of the additive stabilization effect, the contributions to the small (rho-star cubed) residual stress is not always simply additive. If the rates differ in sign, the residual stress from one can buck out that from the other (and in some cases reduce the stabilization.) The residual stress from these diamagnetic velocity shear rates is quantified by the ratio of TAM flow to ion energy (power) flow (M/P) in a global GYRO core simulation of a ''null'' toroidal rotation DIII-D [Mahdavi and Luxon, Fusion Sci. Technol. 48, 2 (2005)] discharge by matching M/P profiles within experimental uncertainty. Comparison of global GYRO (ion and electron energy as well as particle) transport flow balance simulations of TAM transport flow in a high-rotation DIII-D L-mode quantifies and isolates the ExB shear and parallel velocity (Coriolis force) pinching components from the larger ''diffusive'' parallel velocity shear driven component and
NASA Astrophysics Data System (ADS)
Pochelon, Antoine
2010-11-01
Due to turbulence, core energy transport in tokamaks generally exceeds collisional transport by at least an order of magnitude. It is therefore crucial to understand the instabilities driving the turbulent state and to find ways to control them. Shaping the plasma is one of these fundamental tools. In low collisionality plasmas, such as in a reactor, changing triangularity from positive (delta=+0.4) to negative triangularity (delta=-0.4) is shown on TCV to reduce the energy transport by a factor two. This opens the possibility of having H-mode-like confinement time within an L-mode edge, or reduced ELMs. An optimum triangularity can be sought between steep edge barriers (delta>0), plagued by large ELMs, and improved core confinement (delta<0). Recent correlation ECE measurements show that the reduction of transport at negative delta is reflected in a reduction by a factor of two of both the amplitude of temperature fluctuations in the broadband frequency range 30-150 kHz, and the fluctuation correlation length, measured at mid-radius. In addition, the fluctuations amplitude is reduced with increasing collisionality, consistent with a reduction of the Trapped Electron Modes (TEM) drive. The effect of negative triangularity on turbulence and transport is compared to gyrokinetic code results: First, global linear simulations predict shorter radial TEM wavelength, consistent with the shorter radial turbulence correlation length observed. Second, at least close to the strongly shaped plasma boundary, local nonlinear simulations predict lower TEM induced transport with decreased triangularity. Calculations are now being extended to global nonlinear simulations.
Mikkelsen, D. R. Bitter, M.; Delgado-Aparicio, L.; Hill, K. W.; Greenwald, M.; Howard, N. T.; Hughes, J. W.; Rice, J. E.; Reinke, M. L.; Podpaly, Y.; Ma, Y.; Candy, J.; Waltz, R. E.
2015-06-15
Peaked density profiles in low-collisionality AUG and JET H-mode plasmas are probably caused by a turbulently driven particle pinch, and Alcator C-Mod experiments confirmed that collisionality is a critical parameter. Density peaking in reactors could produce a number of important effects, some beneficial, such as enhanced fusion power and transport of fuel ions from the edge to the core, while others are undesirable, such as lower beta limits, reduced radiation from the plasma edge, and consequently higher divertor heat loads. Fundamental understanding of the pinch will enable planning to optimize these impacts. We show that density peaking is predicted by nonlinear gyrokinetic turbulence simulations based on measured profile data from low collisionality H-mode plasma in Alcator C-Mod. Multiple ion species are included to determine whether hydrogenic density peaking has an isotope dependence or is influenced by typical levels of low-Z impurities, and whether impurity density peaking depends on the species. We find that the deuterium density profile is slightly more peaked than that of hydrogen, and that experimentally relevant levels of boron have no appreciable effect on hydrogenic density peaking. The ratio of density at r/a = 0.44 to that at r/a = 0.74 is 1.2 for the majority D and minority H ions (and for electrons), and increases with impurity Z: 1.1 for helium, 1.15 for boron, 1.3 for neon, 1.4 for argon, and 1.5 for molybdenum. The ion temperature profile is varied to match better the predicted heat flux with the experimental transport analysis, but the resulting factor of two change in heat transport has only a weak effect on the predicted density peaking.
SciDAC Center for Gyrokinetic Particle Simulation of Turbulent Transport in Burning Plasmas
Lin, Zhihong
2013-12-18
During the first year of the SciDAC gyrokinetic particle simulation (GPS) project, the GPS team (Zhihong Lin, Liu Chen, Yasutaro Nishimura, and Igor Holod) at the University of California, Irvine (UCI) studied the tokamak electron transport driven by electron temperature gradient (ETG) turbulence, and by trapped electron mode (TEM) turbulence and ion temperature gradient (ITG) turbulence with kinetic electron effects, extended our studies of ITG turbulence spreading to core-edge coupling. We have developed and optimized an elliptic solver using finite element method (FEM), which enables the implementation of advanced kinetic electron models (split-weight scheme and hybrid model) in the SciDAC GPS production code GTC. The GTC code has been ported and optimized on both scalar and vector parallel computer architectures, and is being transformed into objected-oriented style to facilitate collaborative code development. During this period, the UCI team members presented 11 invited talks at major national and international conferences, published 22 papers in peer-reviewed journals and 10 papers in conference proceedings. The UCI hosted the annual SciDAC Workshop on Plasma Turbulence sponsored by the GPS Center, 2005-2007. The workshop was attended by about fifties US and foreign researchers and financially sponsored several gradual students from MIT, Princeton University, Germany, Switzerland, and Finland. A new SciDAC postdoc, Igor Holod, has arrived at UCI to initiate global particle simulation of magnetohydrodynamics turbulence driven by energetic particle modes. The PI, Z. Lin, has been promoted to the Associate Professor with tenure at UCI.
Multi-channel transport experiments at Alcator C-Mod and comparison with gyrokinetic simulations
White, A. E.; Howard, N. T.; Greenwald, M.; Reinke, M. L.; Sung, C.; Baek, S.; Barnes, M.; Dominguez, A.; Ernst, D.; Gao, C.; Hubbard, A. E.; Hughes, J. W.; Lin, Y.; Parra, F.; Porkolab, M.; Rice, J. E.; Walk, J.; Wukitch, S. J.; Team, Alcator C-Mod; Candy, J.; and others
2013-05-15
Multi-channel transport experiments have been conducted in auxiliary heated (Ion Cyclotron Range of Frequencies) L-mode plasmas at Alcator C-Mod [Marmar and Alcator C-Mod Group, Fusion Sci. Technol. 51(3), 3261 (2007)]. These plasmas provide good diagnostic coverage for measurements of kinetic profiles, impurity transport, and turbulence (electron temperature and density fluctuations). In the experiments, a steady sawtoothing L-mode plasma with 1.2 MW of on-axis RF heating is established and density is scanned by 20%. Measured rotation profiles change from peaked to hollow in shape as density is increased, but electron density and impurity profiles remain peaked. Ion or electron heat fluxes from the two plasmas are the same. The experimental results are compared directly to nonlinear gyrokinetic theory using synthetic diagnostics and the code GYRO [Candy and Waltz, J. Comput. Phys. 186, 545 (2003)]. We find good agreement with experimental ion heat flux, impurity particle transport, and trends in the fluctuation level ratio (T(tilde sign){sub e}/T{sub e})/(ñ{sub e}/n{sub e}), but underprediction of electron heat flux. We find that changes in momentum transport (rotation profiles changing from peaked to hollow) do not correlate with changes in particle transport, and also do not correlate with changes in linear mode dominance, e.g., Ion Temperature Gradient versus Trapped Electron Mode. The new C-Mod results suggest that the drives for momentum transport differ from drives for heat and particle transport. The experimental results are inconsistent with present quasilinear models, and the strong sensitivity of core rotation to density remains unexplained.
Tangri, Varun; Terry, P. W.; Waltz, R. E.
2011-05-15
A simple large-aspect-ratio (R{sub 0}/r) circular equilibrium model is developed for low-beta reversed field pinch (RFP) geometry. The model is suitable for treating small scale instability and turbulent transport driven by ion temperature gradient (ITG) and related electron drift modes in gyrokinetic simulations. The equilibrium model is an RFP generalization of the common tokamak s-{alpha} model to small safety factor (q), where the poloidal field dominates the toroidal field. The model accommodates the RFP toroidal field reversal (where q vanishes) by generalizing the cylindrical force-free Bessel function model (BFM) [J. B. Taylor, Phys. Rev. Lett. 33, 1139 (1974)] to toroidal geometry. The global equilibrium can be described in terms of the RFP field reversal and pinch parameters [F,{Theta}]. This new toroidal Bessel function model (TBFM) has been incorporated into the gyrokinetic code GYRO [J. Candy and R. E. Waltz, J.Comput. Phys. 186, 545 (2003)] and used here to explore local electrostatic ITG adiabatic electron instability rates for typical low-q RFP parameters.
Coury, M.; Guttenfelder, W.; Mikkelsen, D. R.; Canik, J. M.; Canal, G. P.; Diallo, A.; Kaye, S.; Kramer, G. J.; Maingi, R.
2016-06-30
Linear (local) gyrokinetic predictions of edge microinstabilities in highly shaped, lithiated and non-lithiated NSTX discharges are reported using the gyrokinetic code GS2. Microtearing modes dominate the non-lithiated pedestal top. The stabilization of these modes at the lithiated pedestal top enables the electron temperature pedestal to extend further inwards, as observed experimentally. Kinetic ballooning modes are found to be unstable mainly at the mid-pedestal of both types of discharges, with un- stable trapped electron modes nearer the separatrix region. At electron wavelengths, ETG modes are found to be unstable from mid-pedestal outwards for η_{e, exp} ~2.2 with higher growth rates for the lithiated discharge. Near the separatrix, the critical temperature gradient for driving ETG modes is reduced in the presence of lithium, re ecting the reduction of the lithiated density gradients observed experimentally. A preliminary linear study in the edge of non-lithiated discharges shows that the equilibrium shaping alters the electrostatic modes stability, found more unstable at high plasma shaping.
Coury, M.; Guttenfelder, W.; Mikkelsen, D. R.; ...
2016-06-30
Linear (local) gyrokinetic predictions of edge microinstabilities in highly shaped, lithiated and non-lithiated NSTX discharges are reported using the gyrokinetic code GS2. Microtearing modes dominate the non-lithiated pedestal top. The stabilization of these modes at the lithiated pedestal top enables the electron temperature pedestal to extend further inwards, as observed experimentally. Kinetic ballooning modes are found to be unstable mainly at the mid-pedestal of both types of discharges, with un- stable trapped electron modes nearer the separatrix region. At electron wavelengths, ETG modes are found to be unstable from mid-pedestal outwards for ηe, exp ~2.2 with higher growth rates formore » the lithiated discharge. Near the separatrix, the critical temperature gradient for driving ETG modes is reduced in the presence of lithium, re ecting the reduction of the lithiated density gradients observed experimentally. A preliminary linear study in the edge of non-lithiated discharges shows that the equilibrium shaping alters the electrostatic modes stability, found more unstable at high plasma shaping.« less
NASA Astrophysics Data System (ADS)
Holod, Ihor; Lin, Zhihong; Taimourzadeh, Sam; Nazikian, Raffi; Spong, Donald; Wingen, Andreas
2016-10-01
Vacuum Resonant Magnetic Perturbation (RMP) applied to otherwise axisymmetric plasmas for the purpose of ELM mitigation produce in general a combination of non resonant effects preserving closed flux surfaces (kink response) and resonant effects that introduce magnetic islands. The effect of the plasma kink response on the stability and transport of edge turbulence is studied using the gyrokinetic code GTC for a DIII-D discharge with applied n=2 vacuum RMP. Three reference equilibria were modeled using VMEC code, based on DIII-D shot 158103: axisymmetric (no RMP) equilibrium, n=2 RMP, and artificially amplified RMPx10 equilibria. Gyrokinetic simulations reveal no increase of growth rates for electrostatic driftwave instability and electromagnetic kinetic-ballooning mode in the presence of the RMP. The effect of RMP on zonal flow damping is found to be insufficient to modify turbulent transport. Therefore, the plasma kink response to the RMP cannot account for the change in the turbulence level seen in experiments with suppressed ELMs. These results demonstrate that other physics must be controlling the transition in confinement responsible for ELM suppression. Work is supported by General Atomics subcontract 4500055243, U.S. DOE Grant DE-SC0010416 and DE-SC0013804, and by General Atomics collaboration agreement under DOE Grant DE-FG03-94ER54271.
NASA Astrophysics Data System (ADS)
Leboeuf, Jean-Noel; Dimits, Andris; Shumaker, Dan
2005-10-01
Development of the PG3EQ/NC module within the SUMMIT gyrokinetic PIC FORTRAN 90 framework is largely completed. It provides SUMMIT with the capability of performing 3D nonlinear toroidal gyrokinetic computations of ion turbulence in real DIII-D geometry. PG3EQ/NC uses local, field line following, quasi-ballooning coordinates and direct interface with DIII-D equilibrium data via the EFIT and ONETWO codes. In addition, Holger Saint John's PLOTEQ code is used to determine the (r,z) position of each flux surface. The thus initialized SUMMIT computations have been carried out for shot /118561 at times 01450 and 02050 at many of the 51 flux surfaces from the core to the edge. Linear SUMMIT results will be compared to available data from calculations with the GKS code for the same discharges. Nonlinear SUMMIT results will also be compared with scattering measurements of turbulence, as well as with accessible measurements of fluctuation amplitudes and spectra from other diagnostics.
NASA Astrophysics Data System (ADS)
Angioni, C.; Bilato, R.; Casson, F. J.; Fable, E.; Mantica, P.; Odstrcil, T.; Valisa, M.; ASDEX Upgrade Team; Contributors, JET
2017-02-01
In tokamaks, the role of turbulent transport of heavy impurities, relative to that of neoclassical transport, increases with increasing size of the plasma, as clarified by means of general scalings, which use the ITER standard scenario parameters as reference, and by actual results from a selection of discharges from ASDEX Upgrade and JET. This motivates the theoretical investigation of the properties of the turbulent convection of heavy impurities by nonlinear gyrokinetic simulations in the experimentally relevant conditions of comparable ion and electron heat fluxes. These conditions also correspond to an intermediate regime between dominant ion temperature gradient turbulence and trapped electron mode turbulence. At moderate plasma toroidal rotation, the turbulent convection of heavy impurities, computed with nonlinear gyrokinetic simulations, is found to be directed outward, in contrast to that obtained by quasi-linear calculations based on the most unstable linear mode, which is directed inward. In this mixed turbulence regime, with comparable electron and ion heat fluxes, the nonlinear results of the impurity transport can be explained by the coexistence of both ion temperature gradient and trapped electron modes in the turbulent state, both contributing to the turbulent convection and diffusion of the impurity. The impact of toroidal rotation on the turbulent convection is also clarified.
Complete fluid equations for low-n singular modes in axisymmetric toroidal plasmas
Glasser, A.H.
1990-01-01
The goal of this work is to develop a complete linear theory of the singular region, including all important dynamical effects. The present phase of the work treats the more collision fluid regime. A later phase will treat the less collisional gyrokinetic regime. This paper concerns the derivation and form of the fluid equations for the singular region of low-n modes. Later work will treat high-n ballooning modes. In addition, the ordering in the present work must be amended before it is applicable to the neighborhood of the field reversal surface of the RFP.
Fahey, Mark R.; Candy, Jeff
2013-11-07
This project initiated the development of TGYRO ? a steady-state Gyrokinetic transport code (SSGKT) that integrates micro-scale GYRO turbulence simulations into a framework for practical multi-scale simulation of conventional tokamaks as well as future reactors. Using a lightweight master transport code, multiple independent (each massively parallel) gyrokinetic simulations are coordinated. The capability to evolve profiles using the TGLF model was also added to TGYRO and represents a more typical use-case for TGYRO. The goal of the project was to develop a steady-state Gyrokinetic transport code (SSGKT) that integrates micro-scale gyrokinetic turbulence simulations into a framework for practical multi-scale simulation of a burning plasma core ? the International Thermonuclear Experimental Reactor (ITER) in particular. This multi-scale simulation capability will be used to predict the performance (the fusion energy gain, Q) given the H-mode pedestal temperature and density. At present, projections of this type rely on transport models like GLF23, which are based on rather approximate fits to the results of linear and nonlinear simulations. Our goal is to make these performance projections with precise nonlinear gyrokinetic simulations. The method of approach is to use a lightweight master transport code to coordinate multiple independent (each massively parallel) gyrokinetic simulations using the GYRO code. This project targets the practical multi-scale simulation of a reactor core plasma in order to predict the core temperature and density profiles given the H-mode pedestal temperature and density. A master transport code will provide feedback to O(16) independent gyrokinetic simulations (each massively parallel). A successful feedback scheme offers a novel approach to predictive modeling of an important national and international problem. Success in this area of fusion simulations will allow US scientists to direct the research path of ITER over the next two
Theory of nonmonotonic double layers
Kim, K.Y.
1987-12-01
A simple graphic method of solving the Vlasov--Poisson system associated with nonlinear eigenvalue conditions for arbitrary potential structures is presented. A general analytic formulation for nonmonotonic double layers is presented and illustrated with some particular closed form solutions. This class of double layers satisfies the time stationary Vlasov--Poisson system while requiring a Sagdeev potential, which is a double-valued function of the physical potential. It follows that any distribution function having a density representation as any integer or noninteger power series of potential can never satisfy the nonmonotonic double-layer boundary conditions. A Korteweg--de Vries-like equation is found showing a relationship among the speed of the nonmonotonic double layer, its scale length, and its degree of asymmetry.
NASA Astrophysics Data System (ADS)
Wang, Weixing
2008-11-01
Global gyrokinetic simulations using the GTS code [1] have found that a large inward flux of toroidal momentum is driven robustly in the post saturation phase of ion temperature gradient (ITG) turbulence. As a consequence, core plasma rotation spins up resulting in δu a few percent of vth in the case with no momentum source at the edge. The underlying physics for the inward flux is identified to be the generation of residual stress due to the k symmetry breaking [2] induced by self-generated zonal flow shear which is quasi-stationary in global simulations. The elatively low level momentum flux in the long- time steady state appears to be approximately diffusive, with effective χφ/χi on the order of unity, in broad agreement with experimental observations and theory predictions for ITG turbulence [3]. Neoclassical simulations using the GTC- NEO code [4] also show that the ion temperature gradient can drive a significant inward nondiffusive momentum flux. However, the overall neoclassical contribution to the momentum transport is negligibly small compared to experimental levels for NSTX and DIII-D plasmas. It is also found that finite residual turbulence can survive strong mean ExB shear flow induced damping. This residual turbulence in the presence of strong ExB shear may drive an insignificant ion heat flux reasonably close to the neoclassical value, and a finite momentum flux significantly higher than the neoclassical level. Moreover, the equilibrium ExB flow shear is found to reduce the turbulence driven transport for energy more efficiently than for momentum. These findings may offer an explanation for rather peculiar observations of near neoclassical ion heat and anomalous momentum transport in experiments, which has been often observed in various machines, but with little theoretical understanding. [1] W.X. Wang et al., Phys. Plasmas 14, 072306 (2007). [2] O.D. Gurcan et al., Phys. Plasmas 14, 042306 (2007). [3] N. Mattor and P.H. Diamond, Phys. Fluids 31
NASA Astrophysics Data System (ADS)
White, Anne
2014-10-01
Understanding transport in high performance ELM-suppressed tokamak plasmas is of great interest for ITER and other future experiments. `I-mode' regime on Alcator C-Mod, also known as `improved L-mode' on ASDEX Upgrade, has several favorable characteristics: pedestals in electron and ion temperature, with ITER98y2 H-factors similar to and exceeding H-mode [Hubbard et al., Phys. Plasmas 18, 056115 (2011)], but without a density pedestal and without impurity accumulation and without ELMs. Most research on I-mode focuses on changes in edge and pedestal turbulence/transport and stability. In this work, transport in I-mode is probed by measuring changes in core turbulence across L-I transitions at Alcator C-Mod and comparing with nonlinear gyrokinetic simulations. Long wavelength (kθρs < 0.5) density fluctuation levels decrease from L-mode levels by up to 30% in I-mode, and long wavelength (kθρs < 0.3) electron temperature fluctuation levels decrease by up to 70%, reaching the instrumental sensitivity limit. Gyrokinetic simulation results suggest that ExB shear in the core of these intrinsically rotating plasmas can reduce the fluctuation amplitude in I-mode. As the pedestal temperature increases across slow L-I transitions, core density fluctuations (0.40 < ρ <0.95) are reduced prior to the onset of the edge-localized (0.99 < ρ < 1.0) weakly coherent mode (WCM) and prior to the reduction of low-frequency turbulence in the edge/pedestal region (0.99 < ρ < 1.0), which suggests that effects of profile stiffness across the radius can also lead to reduced core turbulence. By comparing experimental measurements from Alcator C-Mod to nonlinear gyrokinetic simulations and to different models of profile stiffness, this talk will explore the impact of core turbulence and transport on overall I-mode confinement and on the separation of particle and heat transport in I-mode. This work was supported by U.S. Department of Energy Contract DE-FC02-99ER54512-CMOD.
Multi-Channel Validation of Nonlinear Gyrokinetic Simulations in Alcator C-Mod I-mode Plasmas
NASA Astrophysics Data System (ADS)
Creely, A. J.
2016-10-01
New multi-channel validation of nonlinear gyrokinetic simulations (GYRO) is carried out for I-mode plasmas on Alcator C-Mod, utilizing heat fluxes, profile stiffness, and density and temperature fluctuations. I-mode plasmas are characterized by high energy confinement, similar to H-mode, but with L-mode-like particle confinement, making them favorable for reactors due to natural absence of ELMs, but without impurity accumulation [Whyte NF 2010]. At C-Mod, I-mode plasmas have been obtained across a wide range of plasma currents (Ip = 0.55-1.2MA) and magnetic fields (Bt = 2.8-8.0T). I-mode is also actively studied at ASDEX Upgrade, DIII-D and other tokamaks [Hubbard NF 2016]. Open questions remain regarding core transport in I-mode compared to L and H-mode, making validation studies in I-mode of great interest. Previous work at C-Mod found that ITG/TEM-scale GYRO simulations can match both electron and ion heat fluxes within error bars in I-mode [White PoP 2015], suggesting that multi-scale, cross-scale coupling effects [Howard PoP 2016] may be less important in I-mode than in L-mode. Adding the constraint of experimental perturbative heat diffusivity, however, revealed that ITG/TEM scale simulations do not adequately capture the high profile stiffness in I-mode [Creely NF 2016]. These results motivated more comprehensive comparisons of gyrokinetic simulations with I-mode plasmas. This talk expands upon past I-mode GYRO validation work to simultaneously constrain nonlinear gyrokinetic simulations with experimental electron and ion heat fluxes, electron temperature fluctuations measured with Correlation ECE, density fluctuations measured with Phase Contrast Imaging and reflectometry, and the temperature profile stiffness measured using partial sawtooth heat pulses. This work is supported by the US DOE under Grants DE-SC0006419 and DEFC02-99ER54512-CMOD.
NASA Astrophysics Data System (ADS)
Lee, W. W.; White, Roscoe
2016-10-01
A novel mechanism for producing the equilibrium potential well near the edge of a tokamak has been recently proposed. Briefly, because of the difference in gyroradii between electrons and ions, an equilibrium electrostatic potential is generated in the presence of spatial inhomogeneity of the background plasma, which, in turn, produces a well associated with the radial electric field, Er, as observed at the edge of many tokamak experiments. Specifically, this theoretically predicted Er field, which can be regarded as producing a long radial wave length zonal flow, agrees well with recent experimental measurements on JET, NSTX and C-Mod. A possible verification of this new mechanism using a proposed procedure involving global gyrokinetic particle simulation codes and equilibrium MHD codes will be discussed. The approach is iteratively to decouple the transport problem from the equilibrium problem, so that each may be treated accurately, and, then couple them through parameter exchanges. DoE-Grant DE-AC02-09CH11466.
Toroidal Electromagnetic Particle-in-Cell Code with Gyro-kinetic Electron and Fully-kinetic ion
NASA Astrophysics Data System (ADS)
Lin, Jingbo; Zhang, Wenlu; Liu, Pengfei; Li, Ding
2016-10-01
A kinetic simulation model has been developed using gyro-kinetic electron and fully-kinetic ion by removing fast gyro motion of electrons using the Lie-transform perturbation theory. A particle-in-cell kinetic code is developed based on this model in general magnetic flux coordinate systems, which is particularly suitable for simulations of toroidally confined plasma. Single particle motion and field solver are successfully verified respectively. Integrated electrostatic benchmark, for example the lower-hybrid wave (LHW) and ion Bernstein wave (IBW), shows a good agreement with theoretical results. Preliminary electromagnetic benchmark of fast wave at lower hybrid frequency range is also presented. This code can be a first-principal tool to investigate high frequency nonlinear phenomenon, such as parametric decay instability, during lower-hybrid current drive (LHCD) and ion cyclotron radio frequency heating (ICRF) with complex geometry effect included. Supported by National Special Research Program of China For ITER and National Natural Science Foundation of China.
NASA Astrophysics Data System (ADS)
Ku, S.; Chang, C.-S.; Adams, M.; Cummings, J.; Hinton, F.; Keyes, D.; Klasky, S.; Lee, W.; Lin, Z.; Parker, S.; CPES Team
2006-09-01
A gyrokinetic neoclassical solution for a diverted tokamak edge plasma has been obtained for the first time using the massively parallel Jaguar XT3 computer at Oak Ridge National Laboratory. The solutions show similar characteristics to the experimental observations: electric potential is positive in the scrape-off layer and negative in the H-mode layer, and the parallel rotation is positive in the scrape-off layer and at the inside boundary of the H-mode layer. However, the solution also makes a new physical discovery that there is a strong ExB convective flow in the scrape-off plasma. A general introduction to the edge simulation problem is also presented.
Adams, Mark; Chang, C. S.; Cummings, J.; Hinton, F.; Keyes, David E; Klasky, Scott A; Ku, S.; Lee, W. W.; Lin, Z.; Parker, Scott; CPES Team, the
2006-01-01
A gyrokinetic neoclassical solution for a diverted tokamak edge plasma has been obtained for the first time using the massively parallel Jaguar XT3 computer at Oak Ridge National Laboratory. The solutions show similar characteristics to the experimental observations: electric potential is positive in the scrape-off layer and negative in the H-mode layer, and the parallel rotation is positive in the scrape-off layer and at the inside boundary of the H-mode layer. However, the solution also makes a new physical discovery that there is a strong ExB convective flow in the scrape-off plasma. A general introduction to the edge simulation problem is also presented.
Xu, Yingfeng Dai, Zongliang; Wang, Shaojie
2014-04-15
The nonlinear gyrokinetic theory in the tokamak configuration based on the two-step transform is developed; in the first step, we transform the magnetic potential perturbation to the Hamiltonian part, and in the second step, we transform away the gyroangle-dependent part of the perturbed Hamiltonian. Then the I-transform method is used to decoupled the perturbation part of the motion from the unperturbed motion. The application of the I-transform method to the computation of the guiding-center orbit and the guiding-center distribution function in tokamaks is presented. It is demonstrated that the I-transform method of the orbit computation which involves integrating only along the unperturbed orbit agrees with the conventional method which integrates along the full orbit. A numerical code based on the I-transform method is developed and two numerical examples are given to verify the new method.
Bass, E. M.; Waltz, R. E.
2013-01-15
The unstable spectrum of Alfven eigenmodes (AEs) driven by neutral beam-sourced energetic particles (EPs) in a benchmark DIII-D discharge (142111) is calculated in a fully gyrokinetic model using the GYRO code's massively parallel linear eigenvalue solver. One cycle of the slow (equilibrium scale) frequency sweep of the reverse shear Alfven eigenmode (RSAE) at toroidal mode number n=3 is mapped. The RSAE second harmonic and an unstable beta-induced Alfven eigenmode (BAE) are simultaneously tracked alongside the primary RSAE. An observed twist in the eigenmode pattern, caused mostly by shear in the driving EP profile, is shown through artificially varying the E Multiplication-Sign B rotational velocity shear to depend generally on shear in the local wave phase velocity. Coupling to the BAE and to the toroidal Alfven eigenmode limit the RSAE frequency sweeps at the lower and upper end, respectively. While the present fully gyrokinetic model (including thermal ions and electrons) constitutes the best treatment of compressibility physics available, the BAE frequency is overpredicted by about 20% against experiment here and is found to be sensitive to energetic beam ion pressure. The RSAE frequency is more accurately matched except when it is limited by the BAE. Simulations suggest that the experiment is very close to marginal AE stability at points of RSAE-BAE coupling. A recipe for comparing the radial profile of quasilinear transport flux from local modes to that from global modes paves the way for the development of a stiff (critical gradient) local AE transport model based on local mode stability thresholds.
Center for Gyrokinetic Particle Simulations of Turbulent Transport in Burning Plasmas
Scott, Parker
2011-05-02
This is the Final Technical Report for University of Colorado's portion of the SciDAC project 'Center for Gyrokinetic Particle Simulation of Turbulent Transport.' This is funded as a multi-institutional SciDAC Center and W.W. Lee at the Princeton Plasma Physics Laboratory is the lead Principal Investigator. Scott Parker is the local Principal Investigator for University of Colorado and Yang Chen is a Co-Principal Investigator. This is Cooperative Agreement DE-FC02-05ER54816. Research personnel include Yang Chen (Senior Research Associate), Jianying Lang (Graduate Research Associate, Ph.D. Physics Student) and Scott Parker (Associate Professor). Research includes core microturbulence studies of NSTX, simulation of trapped electron modes, development of efficient particle-continuum hybrid methods and particle convergence studies of electron temperature gradient driven turbulence simulations. Recently, the particle-continuum method has been extended to five-dimensions in GEM. We find that actually a simple method works quite well for the Cyclone base case with either fully kinetic or adiabatic electrons. Particles are deposited on a 5D phase-space grid using nearest-grid-point interpolation. Then, the value of delta-f is reset, but not the particle's trajectory. This has the effect of occasionally averaging delta-f of nearby (in the phase space) particles. We are currently trying to estimate the dissipation (or effective collision operator). We have been using GEM to study turbulence and transport in NSTX with realistic equilibrium density and temperature profiles, including impurities, magnetic geometry and ExB shear flow. Greg Rewoldt, PPPL, has developed a TRANSP interface for GEM that specifies the equilibrium profiles and parameters needed to run realistic NSTX cases. Results were reported at the American Physical Society - Division of Plasma Physics, and we are currently running convergence studies to ensure physical results. We are also studying the effect of
Gyrokinetic Electron and Fully Kinetic Ion Particle Simulation of Collisionless Plasma Dynamics
Yu Lin; Xueyi Wang; Liu Chen; Zhihong Lin
2009-08-11
Fully kinetic-particle simulations and hybrid simulations have been utilized for decades to investigate various fundamental plasma processes, such as magnetic reconnection, fast compressional waves, and wave-particle interaction. Nevertheless, due to disparate temporal and spatial scales between electrons and ions, existing fully kinetic-particle codes have to employ either unrealistically high electron-to-ion mass ratio, me/mi, or simulation domain limited to a few or a few ten's of the ion Larmor radii, or/and time much less than the global Alfven time scale in order to accommodate available computing resources. On the other hand, in the hybrid simulation, the ions are treated as fully kinetic particles but the electrons are treated as a massless fluid. The electron kinetic effects, e.g., wave-particle resonances and finite electron Larmor radius effects, are completely missing. Important physics, such as the electron transit time damping of fast compressional waves or the triggering mechanism of magnetic reconnection in collisionless plasmas is absent in the hybrid codes. Motivated by these considerations and noting that dynamics of interest to us has frequencies lower than the electron gyrofrequency, we planned to develop an innovative particle simulation model, gyrokinetic (GK) electrons and fully kinetic (FK) ions. In the GK-electron and FK-ion (GKe/FKi) particle simulation model, the rapid electron cyclotron motion is removed, while keeping finite electron Larmor radii, realistic me/mi ratio, wave-particle interactions, and off-diagonal components of electron pressure tensor. The computation power can thus be significantly improved over that of the full-particle codes. As planned in the project DE-FG02-05ER54826, we have finished the development of the new GK-electron and FK-ion scheme, finished its benchmark for a uniform plasma in 1-D, 2-D, and 3-D systems against linear waves obtained from analytical theories, and carried out a further convergence test
Comparison of Semi-Lagrangian Algorithms for Solving Vlasov-type Equations
NASA Astrophysics Data System (ADS)
Brunner, Stephan
2005-10-01
In view of pursuing CRPP's effort in carrying out gyrokinetic simulations using an Eulerian-type approach [M. Brunetti et. al, Comp. Phys. Comm. 163, 1 (2004)], different alternative algorithms have been considered. The issue is to identify the most appropriate time-stepping scheme, both from a point of view of numerical accuracy and numerical efficiency. Our efforts have concentrated on two semi-Lagrangian approaches: The widely used cubic B-spline interpolation scheme, based on the original work of Cheng and Knorr [C. Z. Cheng and G. Knorr, J. Comp. Phys. 22, 330 (1976)], as well as the Cubic Interpolation Propagation (CIP) scheme, based on cubic Hermite interpolation, which has only more recently been applied for solving Vlasov-type equations [T. Nakamura and T. Yabe, Comp. Phys. Comm. 120, 122 (1999)]. The systematic comparison of these algorithms with respect to their basic spectral (diffusion/dispersion) properties, as well as their ability to avoid the overshoot (Gibbs) problem, is first presented. Results from solving a guiding-center model of the two-dimensional Kelvin-Helmholtz instability are then compared. This test problem enables to address some of the key technical issues also met with the more complex gyrokinetic-type equations.
Brizard, Alain J
2009-12-31
Final Technical Report for U.S. Department of Energy Grant No. DE-FG02-09ER55005 Nonlinear FLR Effects in Reduced Fluid Models Alain J. Brizard, Saint Michael's College The above-mentioned DoE grant was used to support research activities by the PI during a sabbatical leave from Saint Michael's College in 2009. The major focus of the work was the role played by guiding-center and gyrocenter (linear and nonlinear) polarization and magnetization effects in understanding transport processes in turbulent magnetized plasmas. The theoretical tools used for this work include Lie-transform perturbation methods and Lagrangian (variational) methods developed by the PI in previous work. The present final technical report lists (I) the peer-reviewed publications that were written based on work funded by the Grant; (II) invited and contributed conference presentations during the period funded by the Grant; and (III) seminars presented during the period funded by the Grant. I. Peer-reviewed Publications A.J. Brizard and N. Tronko, 2011, Exact momentum conservation for the gyrokinetic Vlasov- Poisson equations, Physics of Plasmas 18 , 082307:1-14 [http://dx.doi.org/10.1063/1.3625554 ]. J. Decker, Y. Peysson, A.J. Brizard, and F.-X. Duthoit, 2010, Orbit-averaged guiding-center Fokker-Planck operator for numerical applications, Physics of Plasmas 17, 112513:1-12 [http://dx.doi.org/10.1063/1.3519514]. A.J. Brizard, 2010, Noether derivation of exact conservation laws for dissipationless reduced fluid models, Physics of Plasmas 17, 112503:1-8 [http://dx.doi.org/10.1063/1.3515303]. F.-X. Duthoit, A.J. Brizard, Y. Peysson, and J. Decker, 2010, Perturbation analysis of trapped particle dynamics in axisymmetric dipole geometry, Physics of Plasmas 17, 102903:1-9 [http://dx.doi.org/10.1063/1.3486554]. A.J. Brizard, 2010, Exact energy conservation laws for full and truncated nonlinear gyrokinetic equations, Physics of Plasmas 17, 042303:1-11 [http://dx.doi.org/10.1063/1.3374428]. A
NASA Astrophysics Data System (ADS)
Prentis, Jeffrey J.
1996-05-01
One of the most challenging goals of a physics teacher is to help students see that the equations of physics are connected to each other, and that they logically unfold from a small number of basic ideas. Derivations contain the vital information on this connective structure. In a traditional physics course, there are many problem-solving exercises, but few, if any, derivation exercises. Creating an equation poem is an exercise to help students see the unity of the equations of physics, rather than their diversity. An equation poem is a highly refined and eloquent set of symbolic statements that captures the essence of the derivation of an equation. Such a poetic derivation is uncluttered by the extraneous details that tend to distract a student from understanding the essential physics of the long, formal derivation.
Young, C.W.
1997-10-01
In 1967, Sandia National Laboratories published empirical equations to predict penetration into natural earth materials and concrete. Since that time there have been several small changes to the basic equations, and several more additions to the overall technique for predicting penetration into soil, rock, concrete, ice, and frozen soil. The most recent update to the equations was published in 1988, and since that time there have been changes in the equations to better match the expanding data base, especially in concrete penetration. This is a standalone report documenting the latest version of the Young/Sandia penetration equations and related analytical techniques to predict penetration into natural earth materials and concrete. 11 refs., 6 tabs.
A time-dependent gyro-kinetic model of thermal ion upflows in the high-latitude F region
NASA Technical Reports Server (NTRS)
Loranc, M.; St.-Maurice, J.-P.
1994-01-01
Ample evidence supports the significance of the high-latitude ionospheric contribution to magnetospheric plasma. Assuming flux conservation along a flux tube, the upward field-aligned ion flows observed in the magnetosphere require high-latitude ionospheric field-aligned ion upflows of the order of 10(exp 8) to 10(exp 9)/sq cm/s. Since radar and satellite observations of high-latitude F region flows at times exceed this flux requirement by an order of magnitude, the thermal ionospheric upflows are not simply the ionospheric response to a magnetospheric flux requirement. Several ionospheric ion upflow mechanisms have been proposed, but simulations based on fluid theory do not reproduce all the observed features of ionospheric ion upflows. Certain asymmetries in the statistical morphology of high-latitude F region ion upflows suggest that the ion upflows may be generated by ion-neutral frictional heating. We developed a single-component (O(+)), time-dependent gyro-kinetic model of the high-latitude F region response to frictional heating in which the neutral exobase is a discontinuous boundary between fully collisional and collisionless plasmas. The concept of a discontinuous neutreal exobase and the assumption of a constant and uniform polarization electric field reduce the ion velocity distribution function, from which we can compute the ion density, parallel velocity, parallel and perpendicular temperature, and parallel flux. Using our model, we simulated the response of a convecting flux tube between 500 km and 2500 km to various frictional heating inputs; the results were both qualitatively and quantitatively different from fluid model results, which may indicate an inadequacy of the fluid theory approach. The gyro-kinetic frictional heating model responses to the various simulations were qualitatively similar: (1) initial perturbations of all the modeled parameters propagated rapidly up the flux tube, (2) transient values of the ion parallel velocity
A simple derivation of relativistic full-wave equations at electron cyclotron resonance
NASA Astrophysics Data System (ADS)
McDonald, D. C.; Cairns, R. A.; Lashmore-Davies, C. N.
1994-10-01
When a wave passes through an electron gyroresonance, in a plasma in the presence of a magnetic field gradient, there is a small spread in the resonance due to the electron's Larmor radius. Mathematically this is represented by the inclusion of the so called gyrokinetic term in the resonance condition, Lashmore-Davies and Dendy. The smallness of this term, compared with other effects such as relativistic broadening, suggests that it should be negligible. However, we shall show here, by extending the method of Cairns et al., into the relativistic regime, that its inclusion is vital for producing self consistent full-wave equations which describe electron gyroresonance. The method is considerably simpler than those used previously by Maroli et al., Petrillo et al., and Lampis et al., for obtaining similar equations. As an example we include a calculation for the O-Mode passing perpendicularly through the fundamental.
Wang, Zhenyu; Lin, Yu; Wang, Xueyi; ...
2016-07-07
The eigenmode stability properties of three-dimensional lower-hybrid-drift-instabilities (LHDI) in a Harris current sheet with a small but finite guide magnetic field have been systematically studied by employing the gyrokinetic electron and fully kinetic ion (GeFi) particle-in-cell (PIC) simulation model with a realistic ion-to-electron mass ratio mi/me. In contrast to the fully kinetic PIC simulation scheme, the fast electron cyclotron motion and plasma oscillations are systematically removed in the GeFi model, and hence one can employ the realistic mi/me. The GeFi simulations are benchmarked against and show excellent agreement with both the fully kinetic PIC simulation and the analytical eigenmode theory. Our studies indicate that, for small wavenumbers, ky, along the current direction, the most unstable eigenmodes are peaked at the location wheremore » $$\\vec{k}$$• $$\\vec{B}$$ =0, consistent with previous analytical and simulation studies. Here, $$\\vec{B}$$ is the equilibrium magnetic field and $$\\vec{k}$$ is the wavevector perpendicular to the nonuniformity direction. As ky increases, however, the most unstable eigenmodes are found to be peaked at $$\\vec{k}$$ •$$\\vec{B}$$ ≠0. Additionally, the simulation results indicate that varying mi/me, the current sheet width, and the guide magnetic field can affect the stability of LHDI. Simulations with the varying mass ratio confirm the lower hybrid frequency and wave number scalings.« less
NASA Astrophysics Data System (ADS)
Jolliet, S.; McMillan, B. F.; Vernay, T.; Villard, L.; Hatzky, R.; Bottino, A.; Angelino, P.
2009-07-01
In this paper, the influence of the parallel nonlinearity on zonal flows and heat transport in global particle-in-cell ion-temperature-gradient simulations is studied. Although this term is in theory orders of magnitude smaller than the others, several authors [L. Villard, P. Angelino, A. Bottino et al., Plasma Phys. Contr. Fusion 46, B51 (2004); L. Villard, S. J. Allfrey, A. Bottino et al., Nucl. Fusion 44, 172 (2004); J. C. Kniep, J. N. G. Leboeuf, and V. C. Decyck, Comput. Phys. Commun. 164, 98 (2004); J. Candy, R. E. Waltz, S. E. Parker et al., Phys. Plasmas 13, 074501 (2006)] found different results on its role. The study is performed using the global gyrokinetic particle-in-cell codes TORB (theta-pinch) [R. Hatzky, T. M. Tran, A. Könies et al., Phys. Plasmas 9, 898 (2002)] and ORB5 (tokamak geometry) [S. Jolliet, A. Bottino, P. Angelino et al., Comput. Phys. Commun. 177, 409 (2007)]. In particular, it is demonstrated that the parallel nonlinearity, while important for energy conservation, affects the zonal electric field only if the simulation is noise dominated. When a proper convergence is reached, the influence of parallel nonlinearity on the zonal electric field, if any, is shown to be small for both the cases of decaying and driven turbulence.
NASA Astrophysics Data System (ADS)
Spong, D. A.; Bass, E. M.; Deng, W.; Heidbrink, W. W.; Lin, Z.; Tobias, B.; Van Zeeland, M. A.; Austin, M. E.; Domier, C. W.; Luhmann, N. C.
2012-08-01
A verification and validation study is carried out for a sequence of reversed shear Alfvén instability time slices. The mode frequency increases in time as the minimum (qmin) in the safety factor profile decreases. Profiles and equilibria are based upon reconstructions of DIII-D discharge (#142111) in which many such frequency up-sweeping modes were observed. Calculations of the frequency and mode structure evolution from two gyrokinetic codes, GTC and GYRO, and a gyro-Landau fluid code TAEFL are compared. The experimental mode structure of the instability was measured using time-resolved two-dimensional electron cyclotron emission imaging. The three models reproduce the frequency upsweep event within ±10% of each other, and the average of the code predictions is within ±8% of the measurements; growth rates are predicted that are consistent with the observed spectral line widths. The mode structures qualitatively agree with respect to radial location and width, dominant poloidal mode number, ballooning structure, and the up-down asymmetry, with some remaining differences in the details. Such similarities and differences between the predictions of the different models and the experimental results are a valuable part of the verification/validation process and help to guide future development of the modeling efforts.
NASA Astrophysics Data System (ADS)
Wang, Zhenyu; Lin, Yu; Wang, Xueyi; Tummel, Kurt; Chen, Liu
2016-07-01
The eigenmode stability properties of three-dimensional lower-hybrid-drift-instabilities (LHDI) in a Harris current sheet with a small but finite guide magnetic field have been systematically studied by employing the gyrokinetic electron and fully kinetic ion (GeFi) particle-in-cell (PIC) simulation model with a realistic ion-to-electron mass ratio mi/me . In contrast to the fully kinetic PIC simulation scheme, the fast electron cyclotron motion and plasma oscillations are systematically removed in the GeFi model, and hence one can employ the realistic mi/me . The GeFi simulations are benchmarked against and show excellent agreement with both the fully kinetic PIC simulation and the analytical eigenmode theory. Our studies indicate that, for small wavenumbers, ky, along the current direction, the most unstable eigenmodes are peaked at the location where k →.B → =0 , consistent with previous analytical and simulation studies. Here, B → is the equilibrium magnetic field and k → is the wavevector perpendicular to the nonuniformity direction. As ky increases, however, the most unstable eigenmodes are found to be peaked at k →.B → ≠0 . In addition, the simulation results indicate that varying mi/me , the current sheet width, and the guide magnetic field can affect the stability of LHDI. Simulations with the varying mass ratio confirm the lower hybrid frequency and wave number scalings.
Benchmark Studies of the Gyro-Landau-Fluid code and Gyro-kinetic Codes on Kinetic Ballooning Modes
NASA Astrophysics Data System (ADS)
Tang, Tengfei; Xu, Xueqiao; Ma, Chenhao; Holland, Chris; Candy, Jeff
2015-11-01
A Gyro-Landau-Fluid (GLF) 3 +1 model has been implemented in BOUT + + framework recently, which contains full Finite-Larmor-Radius (FLR) effects, Landau damping and toroidal resonance. A linear global beta scan has been done using the cbm18 series equilibriums, showing that the unstable modes are kinetic ballooning modes (KBMs). In this work, we use the GYRO code, which is a gyrokinetic continuum code widely used for simulation of the plasma microturbulence, to benchmark with GLF 3 +1 code on KBMs. As the modes locate in peak pressure gradient region, a linear local beta scan using the same set of equilibriums has been done at this position for comparison. With the drift kinetic electron module in the GYRO code by including small electron-ion collision to damp electron modes, GYRO generated mode structures and parity suggest that they are kinetic ballooning modes, and the growth rate is comparable to the GLF results. However, a radial scan of the pedestal for a particular cbm18 equilibrium shows that the growth rate of the most unstable mode shifts outward to the bottom of pedestal and the real frequency of what was originally the KBMs steadily approaches and crosses over to the electron diamagnetic drift direction. Prepared by LLNL under Contract DE-AC52-07NA27344.
NASA Astrophysics Data System (ADS)
Leboeuf, Jean-Noel; Rhodes, Terry; Dimits, Andris; Shumaker, Dan
2006-10-01
The PG3EQ_NC module within the SUMMIT Gyrokinetic PIC FORTRAN90 Framework makes possible 3D nonlinear toroidal computations of ion turbulence in the real geometry of DIII-D discharges. This is accomplished with the use of local, field line following, quasi-ballooning coordinates and through a direct interface with DIII-D equilibrium data via the EFIT and ONETWO codes, as well as Holger Saint John's PLOTEQ code for the (R, Z) position of each flux surface. The effect of real geometry is being elucidated with CYCLONE shot by comparing results for growth rates and diffusivities from PGEQ_NC to those of its circular counterpart. The PG3EQ_NC module is also being used to model ion channel turbulence in DIII-D discharges 118561 and 120327. Linear results will be compared to growth rate calculations with the GKS code. Nonlinear results will also be compared with scattering measurements of turbulence, as well as with accessible measurements of fluctuation amplitudes and spectra from other diagnostics.
NASA Astrophysics Data System (ADS)
Leboeuf, Jean-Noel; Decyk, Viktor; Rhodes, Terry; Dimits, Andris; Shumaker, Dan
2006-04-01
The PG3EQ_/NC module within the SUMMIT Gyrokinetic PIC FORTRAN90 Framework makes possible 3D nonlinear toroidal computations of ion turbulence in the real geometry of DIII-D discharges. This is accomplished with the use of local, field line following, quasi-ballooning coordinates and through a direct interface with DIII-D equilibrium data via the EFIT and ONETWO codes, as well as Holger Saint John's PLOTEQ code for the (R, Z) position of each flux surface. The effect of real geometry is being elucidated with CYCLONE shot 81499 by comparing results from PGEQ_/NC to those of its circular counterpart. The PG3EQ_/NC module is also being used to model ion channel turbulence in advanced tokamak discharges 118561 and 120327. Linear results will be compared to growth rate calculations with the GKS code. Nonlinear results will also be compared with scattering measurements of turbulence, as well as with accessible measurements of fluctuation amplitudes and spectra from other diagnostics.
NASA Astrophysics Data System (ADS)
Oberparleiter, M.; Jenko, F.; Told, D.; Doerk, H.; Görler, T.
2016-04-01
Neoclassical and turbulent transport in tokamaks has been studied extensively over the past decades, but their possible interaction remains largely an open question. The two are only truly independent if the length scales governing each of them are sufficiently separate, i.e., if the ratio ρ* between ion gyroradius and the pressure gradient scale length is small. This is not the case in particularly interesting regions such as transport barriers. Global simulations of a collisional ion-temperature-gradient-driven microturbulence performed with the nonlinear global gyrokinetic code Gene are presented. In particular, comparisons are made between systems with and without neoclassical effects. In fixed-gradient simulations, the modified radial electric field is shown to alter the zonal flow pattern such that a significant increase in turbulent transport is observed for ρ*≳1 /300 . Furthermore, the dependency of the flux on the collisionality changes. In simulations with fixed power input, we find that the presence of neoclassical effects decreases the frequency and amplitude of intermittent turbulent transport bursts (avalanches) and thus plays an important role for the self-organisation behaviour.
NASA Astrophysics Data System (ADS)
Viljamaa, Panu; Jacobs, J. Richard; Chris; JamesHyman; Halma, Matthew; EricNolan; Coxon, Paul
2014-07-01
In reply to a Physics World infographic (part of which is given above) about a study showing that Euler's equation was deemed most beautiful by a group of mathematicians who had been hooked up to a functional magnetic-resonance image (fMRI) machine while viewing mathematical expressions (14 May, http://ow.ly/xHUFi).
Schekochihin, A. A.; Cowley, S. C.; Dorland, W.; Hammett, G. W.; Howes, G. G.; Quataert, E.; Tatsuno, T.
2009-04-23
This paper presents a theoretical framework for understanding plasma turbulence in astrophysical plasmas. It is motivated by observations of electromagnetic and density fluctuations in the solar wind, interstellar medium and galaxy clusters, as well as by models of particle heating in accretion disks. All of these plasmas and many others have turbulentmotions at weakly collisional and collisionless scales. The paper focuses on turbulence in a strong mean magnetic field. The key assumptions are that the turbulent fluctuations are small compared to the mean field, spatially anisotropic with respect to it and that their frequency is low compared to the ion cyclotron frequency. The turbulence is assumed to be forced at some system-specific outer scale. The energy injected at this scale has to be dissipated into heat, which ultimately cannot be accomplished without collisions. A kinetic cascade develops that brings the energy to collisional scales both in space and velocity. The nature of the kinetic cascade in various scale ranges depends on the physics of plasma fluctuations that exist there. There are four special scales that separate physically distinct regimes: the electron and ion gyroscales, the mean free path and the electron diffusion scale. In each of the scale ranges separated by these scales, the fully kinetic problem is systematically reduced to a more physically transparent and computationally tractable system of equations, which are derived in a rigorous way. In the "inertial range" above the ion gyroscale, the kinetic cascade separates into two parts: a cascade of Alfvenic fluctuations and a passive cascade of density and magnetic-fieldstrength fluctuations. The former are governed by the Reduced Magnetohydrodynamic (RMHD) equations at both the collisional and collisionless scales; the latter obey a linear kinetic equation along the (moving) field lines associated with the Alfvenic component (in the collisional limit, these compressive fluctuations
Wang, W X; Hahm, T S; Ethier, S; Rewoldt, G; Lee, W W; Tang, W M; Kaye, S M; Diamond, P H
2009-01-23
A significant inward flux of toroidal momentum is found in global gyrokinetic simulations of ion temperature gradient turbulence, leading to core plasma rotation spin-up. The underlying mechanism is identified to be the generation of residual stress due to the k parallel symmetry breaking induced by global quasistationary zonal flow shear. Simulations also show a significant off-diagonal element associated with the ion temperature gradient in the neoclassical momentum flux, while the overall neoclassical flux is small. In addition, the residual turbulence found in the presence of strong E x B flow shear may account for neoclassical-level ion heat and anomalous momentum transport widely observed in experiments.
Byers, J.A.; Williams, T.J.; Cohen, B.I.; Dimits, A.M.
1994-04-27
One of the programs of the Magnetic fusion Energy (MFE) Theory and computations Program is studying the anomalous transport of thermal energy across the field lines in the core of a tokamak. We use the method of gyrokinetic particle-in-cell simulation in this study. For this LDRD project we employed massively parallel processing, new algorithms, and new algorithms, and new formal techniques to improve this research. Specifically, we sought to take steps toward: researching experimentally-relevant parameters in our simulations, learning parallel computing to have as a resource for our group, and achieving a 100 {times} speedup over our starting-point Cray2 simulation code`s performance.
Wang, Zhenyu; Lin, Yu; Wang, Xueyi; Tummel, Kurt; Chen, Liu
2016-07-07
The eigenmode stability properties of three-dimensional lower-hybrid-drift-instabilities (LHDI) in a Harris current sheet with a small but finite guide magnetic field have been systematically studied by employing the gyrokinetic electron and fully kinetic ion (GeFi) particle-in-cell (PIC) simulation model with a realistic ion-to-electron mass ratio m_{i}/m_{e}. In contrast to the fully kinetic PIC simulation scheme, the fast electron cyclotron motion and plasma oscillations are systematically removed in the GeFi model, and hence one can employ the realistic m_{i}/m_{e}. The GeFi simulations are benchmarked against and show excellent agreement with both the fully kinetic PIC simulation and the analytical eigenmode theory. Our studies indicate that, for small wavenumbers, ky, along the current direction, the most unstable eigenmodes are peaked at the location where $\\vec{k}$• $\\vec{B}$ =0, consistent with previous analytical and simulation studies. Here, $\\vec{B}$ is the equilibrium magnetic field and $\\vec{k}$ is the wavevector perpendicular to the nonuniformity direction. As ky increases, however, the most unstable eigenmodes are found to be peaked at $\\vec{k}$ •$\\vec{B}$ ≠0. Additionally, the simulation results indicate that varying m_{i}/m_{e}, the current sheet width, and the guide magnetic field can affect the stability of LHDI. Simulations with the varying mass ratio confirm the lower hybrid frequency and wave number scalings.
Benchmark studies of the gyro-Landau-fluid code and gyro-kinetic codes on kinetic ballooning modes
NASA Astrophysics Data System (ADS)
Tang, T. F.; Xu, X. Q.; Ma, C. H.; Bass, E. M.; Holland, C.; Candy, J.
2016-03-01
A Gyro-Landau-Fluid (GLF) 3 + 1 model has been recently implemented in BOUT++ framework, which contains full Finite-Larmor-Radius effects, Landau damping, and toroidal resonance [Ma et al., Phys. Plasmas 22, 055903 (2015)]. A linear global beta scan has been conducted using the JET-like circular equilibria (cbm18 series), showing that the unstable modes are kinetic ballooning modes (KBMs). In this work, we use the GYRO code, which is a gyrokinetic continuum code widely used for simulation of the plasma microturbulence, to benchmark with GLF 3 + 1 code on KBMs. To verify our code on the KBM case, we first perform the beta scan based on "Cyclone base case parameter set." We find that the growth rate is almost the same for two codes, and the KBM mode is further destabilized as beta increases. For JET-like global circular equilibria, as the modes localize in peak pressure gradient region, a linear local beta scan using the same set of equilibria has been performed at this position for comparison. With the drift kinetic electron module in the GYRO code by including small electron-electron collision to damp electron modes, GYRO generated mode structures and parity suggest that they are kinetic ballooning modes, and the growth rate is comparable to the GLF results. However, a radial scan of the pedestal for a particular set of cbm18 equilibria, using GYRO code, shows different trends for the low-n and high-n modes. The low-n modes show that the linear growth rate peaks at peak pressure gradient position as GLF results. However, for high-n modes, the growth rate of the most unstable mode shifts outward to the bottom of pedestal and the real frequency of what was originally the KBMs in ion diamagnetic drift direction steadily approaches and crosses over to the electron diamagnetic drift direction.
Muñoz, P. A. Kilian, P.; Büchner, J.; Told, D.; Jenko, F.
2015-08-15
In this work, we compare gyrokinetic (GK) with fully kinetic Particle-in-Cell (PIC) simulations of magnetic reconnection in the limit of strong guide field. In particular, we analyze the limits of applicability of the GK plasma model compared to a fully kinetic description of force free current sheets for finite guide fields (b{sub g}). Here, we report the first part of an extended comparison, focusing on the macroscopic effects of the electron flows. For a low beta plasma (β{sub i} = 0.01), it is shown that both plasma models develop magnetic reconnection with similar features in the secondary magnetic islands if a sufficiently high guide field (b{sub g} ≳ 30) is imposed in the kinetic PIC simulations. Outside of these regions, in the separatrices close to the X points, the convergence between both plasma descriptions is less restrictive (b{sub g} ≳ 5). Kinetic PIC simulations using guide fields b{sub g} ≲ 30 reveal secondary magnetic islands with a core magnetic field and less energetic flows inside of them in comparison to the GK or kinetic PIC runs with stronger guide fields. We find that these processes are mostly due to an initial shear flow absent in the GK initialization and negligible in the kinetic PIC high guide field regime, in addition to fast outflows on the order of the ion thermal speed that violate the GK ordering. Since secondary magnetic islands appear after the reconnection peak time, a kinetic PIC/GK comparison is more accurate in the linear phase of magnetic reconnection. For a high beta plasma (β{sub i} = 1.0) where reconnection rates and fluctuations levels are reduced, similar processes happen in the secondary magnetic islands in the fully kinetic description, but requiring much lower guide fields (b{sub g} ≲ 3)
DOE R&D Accomplishments Database
1998-09-21
In the late 1950s to early 1960s Rudolph A. Marcus developed a theory for treating the rates of outer-sphere electron-transfer reactions. Outer-sphere reactions are reactions in which an electron is transferred from a donor to an acceptor without any chemical bonds being made or broken. (Electron-transfer reactions in which bonds are made or broken are referred to as inner-sphere reactions.) Marcus derived several very useful expressions, one of which has come to be known as the Marcus cross-relation or, more simply, as the Marcus equation. It is widely used for correlating and predicting electron-transfer rates. For his contributions to the understanding of electron-transfer reactions, Marcus received the 1992 Nobel Prize in Chemistry. This paper discusses the development and use of the Marcus equation. Topics include self-exchange reactions; net electron-transfer reactions; Marcus cross-relation; and proton, hydride, atom and group transfers.
Beyond single-stream with the Schrödinger method
NASA Astrophysics Data System (ADS)
Uhlemann, Cora; Kopp, Michael
2016-10-01
We investigate large scale structure formation of collisionless dark matter in the phase space description based on the Vlasov-Poisson equation. We present the Schrödinger method, originally proposed by \\cite{WK93} as numerical technique based on the Schrödinger Poisson equation, as an analytical tool which is superior to the common standard pressureless fluid model. Whereas the dust model fails and develops singularities at shell crossing the Schrödinger method encompasses multi-streaming and even virialization.
Micah Beck
2008-09-14
This project focused on the use of Logistical Networking technology to address the challenges involved in rapid sharing of data from the the Center's gyrokinetic particle simulations, which can be on the order of terabytes per time step, among researchers at a number of geographically distributed locations. There is a great need to manage data on this scale in a flexible manner, with simulation code, file system, database and visualization functions requiring access. The project used distributed data management infrastructure based on Logistical Networking technology to address these issues in a way that maximized interoperability and achieved the levels of performance the required by the Center's application community. The work focused on the development and deployment of software tools and infrastructure for the storage and distribution of terascale datasets generated by simulations running at the National Center for Computational Science at Oak Ridge National Laboratory.
NASA Astrophysics Data System (ADS)
Bass, E. M.; Waltz, R. E.
2013-01-01
The unstable spectrum of Alfvén eigenmodes (AEs) driven by neutral beam-sourced energetic particles (EPs) in a benchmark DIII-D discharge (142111) is calculated in a fully gyrokinetic model using the GYRO code's massively parallel linear eigenvalue solver. One cycle of the slow (equilibrium scale) frequency sweep of the reverse shear Alfvén eigenmode (RSAE) at toroidal mode number n =3 is mapped. The RSAE second harmonic and an unstable beta-induced Alfvén eigenmode (BAE) are simultaneously tracked alongside the primary RSAE. An observed twist in the eigenmode pattern, caused mostly by shear in the driving EP profile, is shown through artificially varying the E ×B rotational velocity shear to depend generally on shear in the local wave phase velocity. Coupling to the BAE and to the toroidal Alfvén eigenmode limit the RSAE frequency sweeps at the lower and upper end, respectively. While the present fully gyrokinetic model (including thermal ions and electrons) constitutes the best treatment of compressibility physics available, the BAE frequency is overpredicted by about 20% against experiment here and is found to be sensitive to energetic beam ion pressure. The RSAE frequency is more accurately matched except when it is limited by the BAE. Simulations suggest that the experiment is very close to marginal AE stability at points of RSAE-BAE coupling. A recipe for comparing the radial profile of quasilinear transport flux from local modes to that from global modes paves the way for the development of a stiff (critical gradient) local AE transport model based on local mode stability thresholds.
A splitting algorithm for Vlasov simulation with filamentation filtration
NASA Technical Reports Server (NTRS)
Klimas, A. J.; Farrell, W. M.
1994-01-01
A Fourier-Fourier transformed version of the splitting algorithm for simulating solutions of the Vlasov-Poisson system of equations is introduced. It is shown that with the inclusion of filamentation filtration in this transformed algorithm it is both faster and more stable than the standard splitting algorithm. It is further shown that in a scalar computer environment this new algorithm is approximately equal in speed and far less noisy than its particle-in-cell counterpart. It is conjectured that in a multiprocessor environment the filtered splitting algorithm would be faster while producing more precise results.
Electromagnetic Gyrokinetic Simulations
Wan, W
2003-11-19
A new electromagnetic kinetic electron {delta} particle simulation model has been demonstrated to work well at large values of plasma {beta} times the ion-to-electron mass ratio. The simulation is three-dimensional using toroidal flux-tube geometry and includes electron-ion collisions. The model shows accurate shear Alfven wave damping and microtearing physics. Zonal flows with kinetic electrons are found to be turbulent with the spectrum peaking at zero and having a width in the frequency range of the driving turbulence. This is in contrast with adiabatic electron cases where the zonal flows are near stationary, even though the linear behavior of the zonal flow is not significantly affected by kinetic electrons. zonal fields are found to be very weak, consistent with theoretical predictions for {beta} below the kinetic ballooning limit. Detailed spectral analysis of the turbulence data is presented in the various limits.
NASA Technical Reports Server (NTRS)
Hamrock, B. J.; Dowson, D.
1981-01-01
Lubricants, usually Newtonian fluids, are assumed to experience laminar flow. The basic equations used to describe the flow are the Navier-Stokes equation of motion. The study of hydrodynamic lubrication is, from a mathematical standpoint, the application of a reduced form of these Navier-Stokes equations in association with the continuity equation. The Reynolds equation can also be derived from first principles, provided of course that the same basic assumptions are adopted in each case. Both methods are used in deriving the Reynolds equation, and the assumptions inherent in reducing the Navier-Stokes equations are specified. Because the Reynolds equation contains viscosity and density terms and these properties depend on temperature and pressure, it is often necessary to couple the Reynolds with energy equation. The lubricant properties and the energy equation are presented. Film thickness, a parameter of the Reynolds equation, is a function of the elastic behavior of the bearing surface. The governing elasticity equation is therefore presented.
Variational Derivation of Dissipative Equations
NASA Astrophysics Data System (ADS)
Sogo, Kiyoshi
2017-03-01
A new variational principle is formulated to derive various dissipative equations. Model equations considered are the damping equation, Bloch equation, diffusion equation, Fokker-Planck equation, Kramers equation and Smoluchowski equation. Each equation and its time reversal equation are simultaneously obtained in our variational principle.
NASA Astrophysics Data System (ADS)
Parker, Scott
2012-10-01
Global electromagnetic gyrokinetic simulations show the existence of near threshold conditions, for both a high-n Kinetic Ballooning Mode (KBM) and an intermediate-n kinetic version of Peeling-Ballooning Mode (PBM). The KBM and the PBM have been used to constrain the EPED model [1]. Global gyrokinetic simulations show that the H-mode pedestal, just prior to the onset of the Edge Localized Mode (ELM), is very near the KBM threshold. Two DIII-D experimental discharges are studied, one reporting KBM features in fluctuation measurements [2]. Simulations find that in addition to the high-n KBM, an intermediate-n electromagnetic mode is unstable. This kinetic version of the PBM has phase velocity in the electron diamagnetic direction, but otherwise has features similar to the MHD PBM. When the magnetic shear is reduced in a narrow region near the steep pressure gradient, the intermediate-n ``kinetic PBM'' is stabilized, while the high-n KBM becomes the most unstable mode. Global simulation results of the KBM compare favorably with flux tube simulations. The KBM transitions to an unstable electrostatic ion mode as the plasma beta is reduced. The intermediate-n ``kinetic peeling ballooning mode'' is sensitive to the q-profile and only seen in global electromagnetic simulations. Collisions increase the KBM critical beta and growth rate. These results indicate that an improved pedestal model should include, in detail, any corrections to the bootstrap current, and any other equilibrium effects that might reduce the local magnetic shear. It is known that the bootstrap current may flatten the q-profile in the steep gradient region [3]. Simulations are carried out using the global electromagnetic GEM code, including kinetic electrons, electron-ion collisions and the effects of realistic magnetic geometry. In addition to global linear analysis, nonlinear simulations will be reported showing that, while the equilibrium radial electric field has a weak effect on the linear growth
ERIC Educational Resources Information Center
Blakley, G. R.
1982-01-01
Reviews mathematical techniques for solving systems of homogeneous linear equations and demonstrates that the algebraic method of balancing chemical equations is a matter of solving a system of homogeneous linear equations. FORTRAN programs using this matrix method to chemical equation balancing are available from the author. (JN)
NASA Astrophysics Data System (ADS)
Field, A. R.; Dunai, D.; Ghim, Y.-c.; Hill, P.; McMillan, B.; Roach, C. M.; Saarelma, S.; Schekochihin, A. A.; Zoletnik, S.; the MAST Team
2014-02-01
Observations of ion-scale (kyρi ⩽ 1) density turbulence of relative amplitude ≳0.2% are available on the Mega Amp Spherical Tokamak (MAST) using a 2D (8 radial × 4 poloidal channel) imaging beam emission spectroscopy diagnostic. Spatial and temporal characteristics of this turbulence, i.e., amplitudes, correlation times, radial and perpendicular correlation lengths and apparent phase velocities of the density contours, are determined by means of correlation analysis. For a low-density, L-mode discharge with strong equilibrium flow shear exhibiting an internal transport barrier in the ion channel, the observed turbulence characteristics are compared with synthetic density turbulence data generated from global, non-linear, gyro-kinetic simulations using the particle-in-cell code NEMORB. This validation exercise highlights the need to include increasingly sophisticated physics, e.g., kinetic treatment of trapped electrons, equilibrium flow shear and collisions, to reproduce most of the characteristics of the observed turbulence. Even so, significant discrepancies remain: an underprediction by the simulations of the turbulence amplitude and heat flux at plasma periphery and the finding that the correlation times of the numerically simulated turbulence are typically two orders of magnitude longer than those measured in MAST. Comparison of these correlation times with various linear timescales suggests that, while the measured turbulence is strong and may be ‘critically balanced’, the simulated turbulence is weak.
NASA Astrophysics Data System (ADS)
Cheng, Junyi; Chen, Yang; Zhang, Wenlu; Lin, Zhihong; Chen, Wei; Yu, Limin; Ding, Xuantong
2015-11-01
A verification and validation study is carried out for a sequence of fast electron driven beta-induced Alfven eigenmodes in HL-2A tokamak plasma. The fast electron driven beta induced Alfvén eigenmode (e-BAE) in toroidal plasmas is investigated for the first time using global gyrokinetic particle simulations, where the fast electrons are described by the drift kinetic model. The phase space structure shows that only the processional resonance is responsible for the e-BAE excitations while fast-ion driven BAE can be excited through all the channels such as transit, drift-bounce, and processional resonance. For weakly nonlinear driven case, frequency is observed to be in phase with the particle energy flux, and mode structure is almost the same as linear stage . While in the strongly driven nonlinear case, BAAE is excited after the BAE mode saturated. The simulation of e-BAE with the HL-2A tokamak parameters is taken. Three modes n/m = 1/2 , 2/5 ,1/3 in HL-2A can be excited in our simulations.
NASA Astrophysics Data System (ADS)
Leboeuf, Jean-Noel; Decyk, Viktor; Newman, David; Sanchez, Raul
2013-10-01
The massively parallel, 2D domain-decomposed, nonlinear, 3D, toroidal, electrostatic, gyrokinetic, Particle in Cell (PIC), Cartesian geometry UCAN2 code, with particle ions and adiabatic electrons, has been ported to two emerging mainframes. These two computers, one at NERSC in the US built by Cray named Edison and the other at the Barcelona Supercomputer Center (BSC) in Spain built by IBM named MareNostrum III (MNIII) just happen to share the same Intel ``Sandy Bridge'' processors. The successful port of UCAN2 to MNIII which came online first has enabled us to be up and running efficiently in record time on Edison. Overall, the performance of UCAN2 on Edison is superior to that on MNIII, particularly at large numbers of processors (>1024) for the same Intel IFORT compiler. This appears to be due to different MPI modules (OpenMPI on MNIII and MPICH2 on Edison) and different interconnection networks (Infiniband on MNIII and Cray's Aries on Edison) on the two mainframes. Details of these ports and comparative benchmarks are presented. Work supported by OFES, USDOE, under contract no. DE-FG02-04ER54741 with the University of Alaska at Fairbanks.
Single wall penetration equations
NASA Technical Reports Server (NTRS)
Hayashida, K. B.; Robinson, J. H.
1991-01-01
Five single plate penetration equations are compared for accuracy and effectiveness. These five equations are two well-known equations (Fish-Summers and Schmidt-Holsapple), two equations developed by the Apollo project (Rockwell and Johnson Space Center (JSC), and one recently revised from JSC (Cour-Palais). They were derived from test results, with velocities ranging up to 8 km/s. Microsoft Excel software was used to construct a spreadsheet to calculate the diameters and masses of projectiles for various velocities, varying the material properties of both projectile and target for the five single plate penetration equations. The results were plotted on diameter versus velocity graphs for ballistic and spallation limits using Cricket Graph software, for velocities ranging from 2 to 15 km/s defined for the orbital debris. First, these equations were compared to each other, then each equation was compared with various aluminum projectile densities. Finally, these equations were compared with test results performed at JSC for the Marshall Space Flight Center. These equations predict a wide variety of projectile diameters at a given velocity. Thus, it is very difficult to choose the 'right' prediction equation. The thickness of a single plate could have a large variation by choosing a different penetration equation. Even though all five equations are empirically developed with various materials, especially for aluminum alloys, one cannot be confident in the shield design with the predictions obtained by the penetration equations without verifying by tests.
Reflections on Chemical Equations.
ERIC Educational Resources Information Center
Gorman, Mel
1981-01-01
The issue of how much emphasis balancing chemical equations should have in an introductory chemistry course is discussed. The current heavy emphasis on finishing such equations is viewed as misplaced. (MP)
Parametrically defined differential equations
NASA Astrophysics Data System (ADS)
Polyanin, A. D.; Zhurov, A. I.
2017-01-01
The paper deals with nonlinear ordinary differential equations defined parametrically by two relations. It proposes techniques to reduce such equations, of the first or second order, to standard systems of ordinary differential equations. It obtains the general solution to some classes of nonlinear parametrically defined ODEs dependent on arbitrary functions. It outlines procedures for the numerical solution of the Cauchy problem for parametrically defined differential equations.
ERIC Educational Resources Information Center
Fay, Temple H.
2002-01-01
We investigate the pendulum equation [theta] + [lambda][squared] sin [theta] = 0 and two approximations for it. On the one hand, we suggest that the third and fifth-order Taylor series approximations for sin [theta] do not yield very good differential equations to approximate the solution of the pendulum equation unless the initial conditions are…
NASA Astrophysics Data System (ADS)
Kostov, Ivan; Serban, Didina; Volin, Dmytro
2008-08-01
We give a realization of the Beisert, Eden and Staudacher equation for the planar Script N = 4 supersymetric gauge theory which seems to be particularly useful to study the strong coupling limit. We are using a linearized version of the BES equation as two coupled equations involving an auxiliary density function. We write these equations in terms of the resolvents and we transform them into a system of functional, instead of integral, equations. We solve the functional equations perturbatively in the strong coupling limit and reproduce the recursive solution obtained by Basso, Korchemsky and Kotański. The coefficients of the strong coupling expansion are fixed by the analyticity properties obeyed by the resolvents.
Fractional chemotaxis diffusion equations.
Langlands, T A M; Henry, B I
2010-05-01
We introduce mesoscopic and macroscopic model equations of chemotaxis with anomalous subdiffusion for modeling chemically directed transport of biological organisms in changing chemical environments with diffusion hindered by traps or macromolecular crowding. The mesoscopic models are formulated using continuous time random walk equations and the macroscopic models are formulated with fractional order differential equations. Different models are proposed depending on the timing of the chemotactic forcing. Generalizations of the models to include linear reaction dynamics are also derived. Finally a Monte Carlo method for simulating anomalous subdiffusion with chemotaxis is introduced and simulation results are compared with numerical solutions of the model equations. The model equations developed here could be used to replace Keller-Segel type equations in biological systems with transport hindered by traps, macromolecular crowding or other obstacles.
Solving Ordinary Differential Equations
NASA Technical Reports Server (NTRS)
Krogh, F. T.
1987-01-01
Initial-value ordinary differential equation solution via variable order Adams method (SIVA/DIVA) package is collection of subroutines for solution of nonstiff ordinary differential equations. There are versions for single-precision and double-precision arithmetic. Requires fewer evaluations of derivatives than other variable-order Adams predictor/ corrector methods. Option for direct integration of second-order equations makes integration of trajectory problems significantly more efficient. Written in FORTRAN 77.
Chang, Z.; Callen, J.D. )
1992-05-01
Unified fluid/kinetic equations for the plasma perturbed density ({ital {tilde n}}), parallel flow velocity ({ital {tilde u}}{sub {parallel}}) and temperature ({ital {tilde T}}) are developed in a sheared slab geometry by calculating the fluid moment closure relations kinetically. At first, a set of (unclosed) nonlinear perturbed fluid equations for {ital {tilde n}}, {ital {tilde u}}{sub {parallel}} and {ital {tilde T}} is developed using a drift ordering analysis and a new gyroviscous force ((spec. char. missing){center dot}{Pi}{sub {ital g}}). Thereafter, to develop linear closure relations for {bold b}{center dot}{del}{center dot}{tilde {Pi}}{sub {parallel}} and {ital {tilde q}}{sub {parallel}}, a drift-kinetic version of a new Chapman--Enskog-like (CEL) equation is developed and solved by using a moment approach and a physically realistic collision operator (Lorentz scattering operator plus the momentum restoring terms). The resultant closure relations for {bold b}{center dot}(spec. char. missing){center dot}{tilde {Pi}}{sub {parallel}} and {ital {tilde q}}{sub {parallel}} unify the fluid and kinetic approaches. In the collisional fluid limit the equations reduce to the well-known Braginskii equations. In the adiabatic limit they reproduce the usual kinetic results, including Landau damping. It is shown that this new CEL approach is more compatible with a fluidlike description of plasmas than the usual drift/gyrokinetic approach. Remarkably simplified forms of the closure relations are presented. The results are compared with other Landau damping models and shown to be more accurate, complete, and physically realistic. Applications of this set of equations to various microinstabilities in tokamak plasmas are presented in a separate paper (Part II) (Phys. Fluids B {bold 4}, 1182 (1992)).
Uniqueness of Maxwell's Equations.
ERIC Educational Resources Information Center
Cohn, Jack
1978-01-01
Shows that, as a consequence of two feasible assumptions and when due attention is given to the definition of charge and the fields E and B, the lowest-order equations that these two fields must satisfy are Maxwell's equations. (Author/GA)
Linear Equations: Equivalence = Success
ERIC Educational Resources Information Center
Baratta, Wendy
2011-01-01
The ability to solve linear equations sets students up for success in many areas of mathematics and other disciplines requiring formula manipulations. There are many reasons why solving linear equations is a challenging skill for students to master. One major barrier for students is the inability to interpret the equals sign as anything other than…
Yagi, M.; Horton, W. )
1994-07-01
A set of reduced Braginskii equations is derived without assuming flute ordering and the Boussinesq approximation. These model equations conserve the physical energy. It is crucial at finite [beta] that the perpendicular component of Ohm's law be solved to ensure [del][center dot][bold j]=0 for energy conservation.
NASA Astrophysics Data System (ADS)
Shabat, A. B.
2016-12-01
We consider the class of entire functions of exponential type in relation to the scattering theory for the Schrödinger equation with a finite potential that is a finite Borel measure. These functions have a special self-similarity and satisfy q-difference functional equations. We study their asymptotic behavior and the distribution of zeros.
NASA Astrophysics Data System (ADS)
White, A. E.
2012-10-01
New experiments at Alcator C-Mod are challenging and expanding our understanding of electron, impurity particle, and momentum transport. Understanding these coupled transport channels is important, since alpha particles will slow down mostly on electrons in ITER intrinsically rotating, alpha-heated plasmas. Core density fluctuations are reduced in high performance (H98˜1) I-mode plasmas by up to 30% compared to L-mode. At the L-I transition, the core turbulence changes lead the edge turbulence changes. This is in contrast to observations at L-H transitions, where the edge turbulence changes lead the core turbulence changes. It is also found that L-mode and I-mode plasmas are both ITG dominant, with lower linear ITG growth rates in I-mode. TRANSP analysis indicates that core electron heat transport is reduced in I-mode compared to L-mode, with little change in ion heat transport. This indicates that significant electron heat transport is driven by ITG turbulence in L-mode. However, impurity particle transport is similar in L-mode and I-mode, suggesting that tracking changes in ITG drive alone does not describe as well the transport in this channel. Additionally, in L-mode plasmas, small < 20% changes in line averaged density lead to dramatic differences in the on-axis rotation, due to a hollowing of the radial profile at marginally lower density. This change in the shape of the rotation profile in L-mode plasmas does not appear to occur at the ITG/TEM boundary, contrary to similar phenomena seen in Ohmic plasmas. For the first time, local, long-wavelength density and electron temperature fluctuations in the core plasma (0.4 < r/a < 0.9) at C-Mod have been measured with reflectometry and correlation ECE. These new fluctuation measurements at C-Mod will allow for better testing and validation of transport models. Gyrokinetic simulations are in progress to interpret the new multichannel and multifield turbulence/transport results from C-Mod.
NASA Astrophysics Data System (ADS)
Kuksin, Sergei; Maiocchi, Alberto
In this chapter we present a general method of constructing the effective equation which describes the behavior of small-amplitude solutions for a nonlinear PDE in finite volume, provided that the linear part of the equation is a hamiltonian system with a pure imaginary discrete spectrum. The effective equation is obtained by retaining only the resonant terms of the nonlinearity (which may be hamiltonian, or may be not); the assertion that it describes the limiting behavior of small-amplitude solutions is a rigorous mathematical theorem. In particular, the method applies to the three- and four-wave systems. We demonstrate that different possible types of energy transport are covered by this method, depending on whether the set of resonances splits into finite clusters (this happens, e.g. in case of the Charney-Hasegawa-Mima equation), or is connected (this happens, e.g. in the case of the NLS equation if the space-dimension is at least two). For equations of the first type the energy transition to high frequencies does not hold, while for equations of the second type it may take place. Our method applies to various weakly nonlinear wave systems, appearing in plasma, meteorology and oceanography.
The Quadrature Master Equations
NASA Astrophysics Data System (ADS)
Hassan, N. J.; Pourdarvish, A.; Sadeghi, J.; Olaomi, J. O.
2017-04-01
In this paper, we derive the non-Markovian stochastic equation of motion (SEM) and master equations (MEs) for the open quantum system by using the non-Markovian stochastic Schrödinger equations (SSEs) for the quadrature unraveling in linear and nonlinear cases. The SSEs for quadrature unraveling arise as a special case of a quantum system. Also we derive the Markovian SEM and ME by using linear and nonlinear Itô SSEs for the measurement probabilities. In linear non-Markovian case, we calculate the convolutionless linear quadrature non-Markovian SEM and ME. We take advantage from example and show that corresponding theory.
Nonlinear ordinary difference equations
NASA Technical Reports Server (NTRS)
Caughey, T. K.
1979-01-01
Future space vehicles will be relatively large and flexible, and active control will be necessary to maintain geometrical configuration. While the stresses and strains in these space vehicles are not expected to be excessively large, their cumulative effects will cause significant geometrical nonlinearities to appear in the equations of motion, in addition to the nonlinearities caused by material properties. Since the only effective tool for the analysis of such large complex structures is the digital computer, it will be necessary to gain a better understanding of the nonlinear ordinary difference equations which result from the time discretization of the semidiscrete equations of motion for such structures.
Regularized Structural Equation Modeling.
Jacobucci, Ross; Grimm, Kevin J; McArdle, John J
A new method is proposed that extends the use of regularization in both lasso and ridge regression to structural equation models. The method is termed regularized structural equation modeling (RegSEM). RegSEM penalizes specific parameters in structural equation models, with the goal of creating easier to understand and simpler models. Although regularization has gained wide adoption in regression, very little has transferred to models with latent variables. By adding penalties to specific parameters in a structural equation model, researchers have a high level of flexibility in reducing model complexity, overcoming poor fitting models, and the creation of models that are more likely to generalize to new samples. The proposed method was evaluated through a simulation study, two illustrative examples involving a measurement model, and one empirical example involving the structural part of the model to demonstrate RegSEM's utility.
Equations For Rotary Transformers
NASA Technical Reports Server (NTRS)
Salomon, Phil M.; Wiktor, Peter J.; Marchetto, Carl A.
1988-01-01
Equations derived for input impedance, input power, and ratio of secondary current to primary current of rotary transformer. Used for quick analysis of transformer designs. Circuit model commonly used in textbooks on theory of ac circuits.
ERIC Educational Resources Information Center
Shumway, Richard J.
1989-01-01
Illustrated is the problem of solving equations and some different strategies students might employ when using available technology. Gives illustrations for: exact solutions, approximate solutions, and approximate solutions which are graphically generated. (RT)
Nonlinear differential equations
Dresner, L.
1988-01-01
This report is the text of a graduate course on nonlinear differential equations given by the author at the University of Wisconsin-Madison during the summer of 1987. The topics covered are: direction fields of first-order differential equations; the Lie (group) theory of ordinary differential equations; similarity solutions of second-order partial differential equations; maximum principles and differential inequalities; monotone operators and iteration; complementary variational principles; and stability of numerical methods. The report should be of interest to graduate students, faculty, and practicing scientists and engineers. No prior knowledge is required beyond a good working knowledge of the calculus. The emphasis is on practical results. Most of the illustrative examples are taken from the fields of nonlinear diffusion, heat and mass transfer, applied superconductivity, and helium cryogenics.
Discrete wave equation upscaling
NASA Astrophysics Data System (ADS)
Fichtner, Andreas; Hanasoge, Shravan M.
2017-01-01
We present homogenisation technique for the uniformly discretised wave equation, based on the derivation of an effective equation for the low-wavenumber component of the solution. The method produces a down-sampled, effective medium, thus making the solution of the effective equation less computationally expensive. Advantages of the method include its conceptual simplicity and ease of implementation, the applicability to any uniformly discretised wave equation in one, two or three dimensions, and the absence of any constraints on the medium properties. We illustrate our method with a numerical example of wave propagation through a one-dimensional multiscale medium, and demonstrate the accurate reproduction of the original wavefield for sufficiently low frequencies.
Relativistic Guiding Center Equations
White, R. B.; Gobbin, M.
2014-10-01
In toroidal fusion devices it is relatively easy that electrons achieve relativistic velocities, so to simulate runaway electrons and other high energy phenomena a nonrelativistic guiding center formalism is not sufficient. Relativistic guiding center equations including flute mode time dependent field perturbations are derived. The same variables as used in a previous nonrelativistic guiding center code are adopted, so that a straightforward modifications of those equations can produce a relativistic version.
SIMULTANEOUS DIFFERENTIAL EQUATION COMPUTER
Collier, D.M.; Meeks, L.A.; Palmer, J.P.
1960-05-10
A description is given for an electronic simulator for a system of simultaneous differential equations, including nonlinear equations. As a specific example, a homogeneous nuclear reactor system including a reactor fluid, heat exchanger, and a steam boiler may be simulated, with the nonlinearity resulting from a consideration of temperature effects taken into account. The simulator includes three operational amplifiers, a multiplier, appropriate potential sources, and interconnecting R-C networks.
The Bernoulli-Poiseuille Equation.
ERIC Educational Resources Information Center
Badeer, Henry S.; Synolakis, Costas E.
1989-01-01
Describes Bernoulli's equation and Poiseuille's equation for fluid dynamics. Discusses the application of the combined Bernoulli-Poiseuille equation in real flows, such as viscous flows under gravity and acceleration. (YP)
Spatial equation for water waves
NASA Astrophysics Data System (ADS)
Dyachenko, A. I.; Zakharov, V. E.
2016-02-01
A compact spatial Hamiltonian equation for gravity waves on deep water has been derived. The equation is dynamical and can describe extreme waves. The equation for the envelope of a wave train has also been obtained.
Introducing Chemical Formulae and Equations.
ERIC Educational Resources Information Center
Dawson, Chris; Rowell, Jack
1979-01-01
Discusses when the writing of chemical formula and equations can be introduced in the school science curriculum. Also presents ways in which formulae and equations learning can be aided and some examples for balancing and interpreting equations. (HM)
Budini, Adrian A.
2006-11-15
In this paper we derive an extra class of non-Markovian master equations where the system state is written as a sum of auxiliary matrixes whose evolution involve Lindblad contributions with local coupling between all of them, resembling the structure of a classical rate equation. The system dynamics may develop strong nonlocal effects such as the dependence of the stationary properties with the system initialization. These equations are derived from alternative microscopic interactions, such as complex environments described in a generalized Born-Markov approximation and tripartite system-environment interactions, where extra unobserved degrees of freedom mediates the entanglement between the system and a Markovian reservoir. Conditions that guarantee the completely positive condition of the solution map are found. Quantum stochastic processes that recover the system dynamics in average are formulated. We exemplify our results by analyzing the dynamical action of nontrivial structured dephasing and depolarizing reservoirs over a single qubit.
Nonlocal electrical diffusion equation
NASA Astrophysics Data System (ADS)
Gómez-Aguilar, J. F.; Escobar-Jiménez, R. F.; Olivares-Peregrino, V. H.; Benavides-Cruz, M.; Calderón-Ramón, C.
2016-07-01
In this paper, we present an analysis and modeling of the electrical diffusion equation using the fractional calculus approach. This alternative representation for the current density is expressed in terms of the Caputo derivatives, the order for the space domain is 0<β≤1 and for the time domain is 0<γ≤2. We present solutions for the full fractional equation involving space and time fractional derivatives using numerical methods based on Fourier variable separation. The case with spatial fractional derivatives leads to Levy flight type phenomena, while the time fractional equation is related to sub- or super diffusion. We show that the mathematical concept of fractional derivatives can be useful to understand the behavior of semiconductors, the design of solar panels, electrochemical phenomena and the description of anomalous complex processes.
Stochastic differential equations
Sobczyk, K. )
1990-01-01
This book provides a unified treatment of both regular (or random) and Ito stochastic differential equations. It focuses on solution methods, including some developed only recently. Applications are discussed, in particular an insight is given into both the mathematical structure, and the most efficient solution methods (analytical as well as numerical). Starting from basic notions and results of the theory of stochastic processes and stochastic calculus (including Ito's stochastic integral), many principal mathematical problems and results related to stochastic differential equations are expounded here for the first time. Applications treated include those relating to road vehicles, earthquake excitations and offshore structures.
NASA Technical Reports Server (NTRS)
Markley, F. Landis
1995-01-01
Kepler's Equation is solved over the entire range of elliptic motion by a fifth-order refinement of the solution of a cubic equation. This method is not iterative, and requires only four transcendental function evaluations: a square root, a cube root, and two trigonometric functions. The maximum relative error of the algorithm is less than one part in 10(exp 18), exceeding the capability of double-precision computer arithmetic. Roundoff errors in double-precision implementation of the algorithm are addressed, and procedures to avoid them are developed.
Obtaining Maxwell's equations heuristically
NASA Astrophysics Data System (ADS)
Diener, Gerhard; Weissbarth, Jürgen; Grossmann, Frank; Schmidt, Rüdiger
2013-02-01
Starting from the experimental fact that a moving charge experiences the Lorentz force and applying the fundamental principles of simplicity (first order derivatives only) and linearity (superposition principle), we show that the structure of the microscopic Maxwell equations for the electromagnetic fields can be deduced heuristically by using the transformation properties of the fields under space inversion and time reversal. Using the experimental facts of charge conservation and that electromagnetic waves propagate with the speed of light, together with Galilean invariance of the Lorentz force, allows us to finalize Maxwell's equations and to introduce arbitrary electrodynamics units naturally.
Comparison of Kernel Equating and Item Response Theory Equating Methods
ERIC Educational Resources Information Center
Meng, Yu
2012-01-01
The kernel method of test equating is a unified approach to test equating with some advantages over traditional equating methods. Therefore, it is important to evaluate in a comprehensive way the usefulness and appropriateness of the Kernel equating (KE) method, as well as its advantages and disadvantages compared with several popular item…
Accumulative Equating Error after a Chain of Linear Equatings
ERIC Educational Resources Information Center
Guo, Hongwen
2010-01-01
After many equatings have been conducted in a testing program, equating errors can accumulate to a degree that is not negligible compared to the standard error of measurement. In this paper, the author investigates the asymptotic accumulative standard error of equating (ASEE) for linear equating methods, including chained linear, Tucker, and…
The Statistical Drake Equation
NASA Astrophysics Data System (ADS)
Maccone, Claudio
2010-12-01
We provide the statistical generalization of the Drake equation. From a simple product of seven positive numbers, the Drake equation is now turned into the product of seven positive random variables. We call this "the Statistical Drake Equation". The mathematical consequences of this transformation are then derived. The proof of our results is based on the Central Limit Theorem (CLT) of Statistics. In loose terms, the CLT states that the sum of any number of independent random variables, each of which may be ARBITRARILY distributed, approaches a Gaussian (i.e. normal) random variable. This is called the Lyapunov Form of the CLT, or the Lindeberg Form of the CLT, depending on the mathematical constraints assumed on the third moments of the various probability distributions. In conclusion, we show that: The new random variable N, yielding the number of communicating civilizations in the Galaxy, follows the LOGNORMAL distribution. Then, as a consequence, the mean value of this lognormal distribution is the ordinary N in the Drake equation. The standard deviation, mode, and all the moments of this lognormal N are also found. The seven factors in the ordinary Drake equation now become seven positive random variables. The probability distribution of each random variable may be ARBITRARY. The CLT in the so-called Lyapunov or Lindeberg forms (that both do not assume the factors to be identically distributed) allows for that. In other words, the CLT "translates" into our statistical Drake equation by allowing an arbitrary probability distribution for each factor. This is both physically realistic and practically very useful, of course. An application of our statistical Drake equation then follows. The (average) DISTANCE between any two neighboring and communicating civilizations in the Galaxy may be shown to be inversely proportional to the cubic root of N. Then, in our approach, this distance becomes a new random variable. We derive the relevant probability density
Do Differential Equations Swing?
ERIC Educational Resources Information Center
Maruszewski, Richard F., Jr.
2006-01-01
One of the units of in a standard differential equations course is a discussion of the oscillatory motion of a spring and the associated material on forcing functions and resonance. During the presentation on practical resonance, the instructor may tell students that it is similar to when they take their siblings to the playground and help them on…
Modelling by Differential Equations
ERIC Educational Resources Information Center
Chaachoua, Hamid; Saglam, Ayse
2006-01-01
This paper aims to show the close relation between physics and mathematics taking into account especially the theory of differential equations. By analysing the problems posed by scientists in the seventeenth century, we note that physics is very important for the emergence of this theory. Taking into account this analysis, we show the…
NASA Astrophysics Data System (ADS)
Mejjaoli, Hatem
2008-12-01
We introduce and study the Dunkl symmetric systems. We prove the well-posedness results for the Cauchy problem for these systems. Eventually we describe the finite speed of it. Next the semi-linear Dunkl-wave equations are also studied.
Structural Equation Model Trees
ERIC Educational Resources Information Center
Brandmaier, Andreas M.; von Oertzen, Timo; McArdle, John J.; Lindenberger, Ulman
2013-01-01
In the behavioral and social sciences, structural equation models (SEMs) have become widely accepted as a modeling tool for the relation between latent and observed variables. SEMs can be seen as a unification of several multivariate analysis techniques. SEM Trees combine the strengths of SEMs and the decision tree paradigm by building tree…
ERIC Educational Resources Information Center
Fay, Temple H.
2010-01-01
Through numerical investigations, we study examples of the forced quadratic spring equation [image omitted]. By performing trial-and-error numerical experiments, we demonstrate the existence of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions, investigate the resonance boundary in the [omega]…
Parallel Multigrid Equation Solver
Adams, Mark
2001-09-07
Prometheus is a fully parallel multigrid equation solver for matrices that arise in unstructured grid finite element applications. It includes a geometric and an algebraic multigrid method and has solved problems of up to 76 mullion degrees of feedom, problems in linear elasticity on the ASCI blue pacific and ASCI red machines.
Quenching equation for scintillation
NASA Astrophysics Data System (ADS)
Kato, Takahisa
1980-06-01
A mathematical expression is postulated showing the relationship between counting rate and quenching agent concentration in a liquid scintillation solution. The expression is more suited to a wider range of quenching agent concentrations than the Stern-Volmer equation. An estimation of the quenched correction is demonstrated using the expression.
Nonlinear equations of 'variable type'
NASA Astrophysics Data System (ADS)
Larkin, N. A.; Novikov, V. A.; Ianenko, N. N.
In this monograph, new scientific results related to the theory of equations of 'variable type' are presented. Equations of 'variable type' are equations for which the original type is not preserved within the entire domain of coefficient definition. This part of the theory of differential equations with partial derivatives has been developed intensively in connection with the requirements of mechanics. The relations between equations of the considered type and the problems of mathematical physics are explored, taking into account quasi-linear equations, and models of mathematical physics which lead to equations of 'variable type'. Such models are related to transonic flows, problems involving a separation of the boundary layer, gasdynamics and the van der Waals equation, shock wave phenomena, and a combustion model with a turbulent diffusion flame. Attention is also given to nonlinear parabolic equations, and nonlinear partial differential equations of the third order.
Methods for Equating Mental Tests.
1984-11-01
1983) compared conventional and IRT methods for equating the Test of English as a Foreign Language ( TOEFL ) after chaining. Three conventional and...three IRT equating methods were examined in this study; two sections of TOEFL were each (separately) equated. The IRT methods included the following: (a...group. A separate base form was established for each of the six equating methods. Instead of equating the base-form TOEFL to itself, the last (eighth
Fluid and kinetic models of negative ion sheaths
Cavenago, M.
2011-09-26
Due to the presence of a large transverse magnetic field (B{sub x} and B{sub y} where z is the extraction axis), the extraction of electrons from a negative ion source is likely to happen with a large angle with respect to z axis. The negative ion and electron sheaths are here studied both with kinetic and with fluid models. First, Vlasov-Poisson models are reduced to one dimensional integrodifferential equations, discussing also trapped orbits. The integrodifferential equations for electron transport are analytically solved for a variety of extraction potentials (in 1D). Collision frequency dependency from electron flow speed and temperature is discussed. Then both ion and electron space charge and fluid motion are solved, using electron densities expression consistent with kinetic model. Results for the sheath charge profile and extraction field as a function of B{sub x} are shown.
Averaged variational principle for autoresonant Bernstein-Greene-Kruskal modes
Khain, P.; Friedland, L.
2010-10-15
Whitham's averaged variational principle is applied in studying dynamics of formation of autoresonant (continuously phase-locked) Bernstein-Greene-Kruskal (BGK) modes in a plasma driven by a chirped frequency ponderomotive wave. A flat-top electron velocity distribution is used as a model allowing a variational formulation within the water bag theory. The corresponding Lagrangian, averaged over the fast phase variable yields evolution equations for the slow field variables, allows uniform description of all stages of excitation of driven-chirped BGK modes, and predicts modulational stability of these nonlinear phase-space structures. Numerical solutions of the system of slow variational equations are in good agreement with Vlasov-Poisson simulations.
A New Class of Non-Linear, Finite-Volume Methods for Vlasov Simulation
Banks, J W; Hittinger, J A
2009-11-24
Methods for the numerical discretization of the Vlasov equation should efficiently use the phase space discretization and should introduce only enough numerical dissipation to promote stability and control oscillations. A new high-order, non-linear, finite-volume algorithm for the Vlasov equation that discretely conserves particle number and controls oscillations is presented. The method is fourth-order in space and time in well-resolved regions, but smoothly reduces to a third-order upwind scheme as features become poorly resolved. The new scheme is applied to several standard problems for the Vlasov-Poisson system, and the results are compared with those from other finite-volume approaches, including an artificial viscosity scheme and the Piecewise Parabolic Method. It is shown that the new scheme is able to control oscillations while preserving a higher degree of fidelity of the solution than the other approaches.
Simulations of plasma sheaths using continuum kinetic models
NASA Astrophysics Data System (ADS)
Srinivasan, Bhuvana; Hakim, Ammar
2015-11-01
Understanding plasma sheath physics is important for the performance of devices such as Hall thrusters due to the effect of energetic particles on electrode erosion. Plasma sheath physics is studied using kinetic and multi-fluid models with relevance to secondary electron emissions and plasma-surface interactions. Continuum kinetic models are developed to directly solve the Vlasov-Poisson equation using the discontinuous Galerkin method for each of the ion and electron species. A steady-state sheath is simulated by including a simple model for a neutral fluid. Multi-fluid simulations for the plasma sheath are also performed using the discontinuous Galerkin method to solve a complete set of fluid equations for each of the ion and electron species. The kinetic plasma sheath is compared to a multi-fluid plasma sheath. Supported by Air Force Office of Scientific Research.
Flavored quantum Boltzmann equations
Cirigliano, Vincenzo; Lee, Christopher; Ramsey-Musolf, Michael J.; Tulin, Sean
2010-05-15
We derive from first principles, using nonequilibrium field theory, the quantum Boltzmann equations that describe the dynamics of flavor oscillations, collisions, and a time-dependent mass matrix in the early universe. Working to leading nontrivial order in ratios of relevant time scales, we study in detail a toy model for weak-scale baryogenesis: two scalar species that mix through a slowly varying time-dependent and CP-violating mass matrix, and interact with a thermal bath. This model clearly illustrates how the CP asymmetry arises through coherent flavor oscillations in a nontrivial background. We solve the Boltzmann equations numerically for the density matrices, investigating the impact of collisions in various regimes.
NASA Astrophysics Data System (ADS)
Trzetrzelewski, Maciej
2016-11-01
Starting with a Nambu-Goto action, a Dirac-like equation can be constructed by taking the square-root of the momentum constraint. The eigenvalues of the resulting Hamiltonian are real and correspond to masses of the excited string. In particular there are no tachyons. A special case of radial oscillations of a closed string in Minkowski space-time admits exact solutions in terms of wave functions of the harmonic oscillator.
Perturbed nonlinear differential equations
NASA Technical Reports Server (NTRS)
Proctor, T. G.
1974-01-01
For perturbed nonlinear systems, a norm, other than the supremum norm, is introduced on some spaces of continuous functions. This makes possible the study of new types of behavior. A study is presented on a perturbed nonlinear differential equation defined on a half line, and the existence of a family of solutions with special boundedness properties is established. The ideas developed are applied to the study of integral manifolds, and examples are given.
Quantum molecular master equations
NASA Astrophysics Data System (ADS)
Brechet, Sylvain D.; Reuse, Francois A.; Maschke, Klaus; Ansermet, Jean-Philippe
2016-10-01
We present the quantum master equations for midsize molecules in the presence of an external magnetic field. The Hamiltonian describing the dynamics of a molecule accounts for the molecular deformation and orientation properties, as well as for the electronic properties. In order to establish the master equations governing the relaxation of free-standing molecules, we have to split the molecule into two weakly interacting parts, a bath and a bathed system. The adequate choice of these systems depends on the specific physical system under consideration. Here we consider a first system consisting of the molecular deformation and orientation properties and the electronic spin properties and a second system composed of the remaining electronic spatial properties. If the characteristic time scale associated with the second system is small with respect to that of the first, the second may be considered as a bath for the first. Assuming that both systems are weakly coupled and initially weakly correlated, we obtain the corresponding master equations. They describe notably the relaxation of magnetic properties of midsize molecules, where the change of the statistical properties of the electronic orbitals is expected to be slow with respect to the evolution time scale of the bathed system.
Double-Plate Penetration Equations
NASA Technical Reports Server (NTRS)
Hayashida, K. B.; Robinson, J. H.
2000-01-01
This report compares seven double-plate penetration predictor equations for accuracy and effectiveness of a shield design. Three of the seven are the Johnson Space Center original, modified, and new Cour-Palais equations. The other four are the Nysmith, Lundeberg-Stern-Bristow, Burch, and Wilkinson equations. These equations, except the Wilkinson equation, were derived from test results, with the velocities ranging up to 8 km/sec. Spreadsheet software calculated the projectile diameters for various velocities for the different equations. The results were plotted on projectile diameter versus velocity graphs for the expected orbital debris impact velocities ranging from 2 to 15 km/sec. The new Cour-Palais double-plate penetration equation was compared to the modified Cour-Palais single-plate penetration equation. Then the predictions from each of the seven double-plate penetration equations were compared to each other for a chosen shield design. Finally, these results from the equations were compared with test results performed at the NASA Marshall Space Flight Center. Because the different equations predict a wide range of projectile diameters at any given velocity, it is very difficult to choose the "right" prediction equation for shield configurations other than those exactly used in the equations' development. Although developed for various materials, the penetration equations alone cannot be relied upon to accurately predict the effectiveness of a shield without using hypervelocity impact tests to verify the design.
Computing generalized Langevin equations and generalized Fokker-Planck equations.
Darve, Eric; Solomon, Jose; Kia, Amirali
2009-07-07
The Mori-Zwanzig formalism is an effective tool to derive differential equations describing the evolution of a small number of resolved variables. In this paper we present its application to the derivation of generalized Langevin equations and generalized non-Markovian Fokker-Planck equations. We show how long time scales rates and metastable basins can be extracted from these equations. Numerical algorithms are proposed to discretize these equations. An important aspect is the numerical solution of the orthogonal dynamics equation which is a partial differential equation in a high dimensional space. We propose efficient numerical methods to solve this orthogonal dynamics equation. In addition, we present a projection formalism of the Mori-Zwanzig type that is applicable to discrete maps. Numerical applications are presented from the field of Hamiltonian systems.
Reduction operators of Burgers equation.
Pocheketa, Oleksandr A; Popovych, Roman O
2013-02-01
The solution of the problem on reduction operators and nonclassical reductions of the Burgers equation is systematically treated and completed. A new proof of the theorem on the special "no-go" case of regular reduction operators is presented, and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case. All possible nonclassical reductions of the Burgers equation to single ordinary differential equations are exhaustively described. Any Lie reduction of the Burgers equation proves to be equivalent via the Hopf-Cole transformation to a parameterized family of Lie reductions of the linear heat equation.
Reduction operators of Burgers equation
Pocheketa, Oleksandr A.; Popovych, Roman O.
2013-01-01
The solution of the problem on reduction operators and nonclassical reductions of the Burgers equation is systematically treated and completed. A new proof of the theorem on the special “no-go” case of regular reduction operators is presented, and the representation of the coefficients of operators in terms of solutions of the initial equation is constructed for this case. All possible nonclassical reductions of the Burgers equation to single ordinary differential equations are exhaustively described. Any Lie reduction of the Burgers equation proves to be equivalent via the Hopf–Cole transformation to a parameterized family of Lie reductions of the linear heat equation. PMID:23576819
Evaluating Equating Results: Percent Relative Error for Chained Kernel Equating
ERIC Educational Resources Information Center
Jiang, Yanlin; von Davier, Alina A.; Chen, Haiwen
2012-01-01
This article presents a method for evaluating equating results. Within the kernel equating framework, the percent relative error (PRE) for chained equipercentile equating was computed under the nonequivalent groups with anchor test (NEAT) design. The method was applied to two data sets to obtain the PRE, which can be used to measure equating…
Differential Equations Compatible with Boundary Rational qKZ Equation
NASA Astrophysics Data System (ADS)
Takeyama, Yoshihiro
2011-10-01
We give diffierential equations compatible with the rational qKZ equation with boundary reflection. The total system contains the trigonometric degeneration of the bispectral qKZ equation of type (Cěen, Cn) which in the case of type GLn was studied by van Meer and Stokman. We construct an integral formula for solutions to our compatible system in a special case.
Problems, Perspectives, and Practical Issues in Equating.
ERIC Educational Resources Information Center
Weiss, David J., Ed.
1987-01-01
Issues concerning equating test scores are discussed in an introduction, four papers, and two commentaries. Equating methods research, sampling errors, linear equating, population differences, sources of equating errors, and a circular equating paradigm are considered. (SLD)
Perturbed nonlinear differential equations
NASA Technical Reports Server (NTRS)
Proctor, T. G.
1972-01-01
The existence of a solution defined for all t and possessing a type of boundedness property is established for the perturbed nonlinear system y = f(t,y) + F(t,y). The unperturbed system x = f(t,x) has a dichotomy in which some solutions exist and are well behaved as t increases to infinity, and some solution exists and are well behaved as t decreases to minus infinity. A similar study is made for a perturbed nonlinear differential equation defined on a half line, R+, and the existence of a family of solutions with special boundedness properties is established. The ideas are applied to integral manifolds.
Noncommutativity and the Friedmann Equations
Sabido, M.; Socorro, J.; Guzman, W.
2010-07-12
In this paper we study noncommutative scalar field cosmology, we find the noncommutative Friedmann equations as well as the noncommutative Klein-Gordon equation, interestingly the noncommutative contributions are only present up to second order in the noncommutitive parameter.
Conservational PDF Equations of Turbulence
NASA Technical Reports Server (NTRS)
Shih, Tsan-Hsing; Liu, Nan-Suey
2010-01-01
Recently we have revisited the traditional probability density function (PDF) equations for the velocity and species in turbulent incompressible flows. They are all unclosed due to the appearance of various conditional means which are modeled empirically. However, we have observed that it is possible to establish a closed velocity PDF equation and a closed joint velocity and species PDF equation through conditions derived from the integral form of the Navier-Stokes equations. Although, in theory, the resulted PDF equations are neither general nor unique, they nevertheless lead to the exact transport equations for the first moment as well as all higher order moments. We refer these PDF equations as the conservational PDF equations. This observation is worth further exploration for its validity and CFD application
``Riemann equations'' in bidifferential calculus
NASA Astrophysics Data System (ADS)
Chvartatskyi, O.; Müller-Hoissen, F.; Stoilov, N.
2015-10-01
We consider equations that formally resemble a matrix Riemann (or Hopf) equation in the framework of bidifferential calculus. With different choices of a first-order bidifferential calculus, we obtain a variety of equations, including a semi-discrete and a fully discrete version of the matrix Riemann equation. A corresponding universal solution-generating method then either yields a (continuous or discrete) Cole-Hopf transformation, or leaves us with the problem of solving Riemann equations (hence an application of the hodograph method). If the bidifferential calculus extends to second order, solutions of a system of "Riemann equations" are also solutions of an equation that arises, on the universal level of bidifferential calculus, as an integrability condition. Depending on the choice of bidifferential calculus, the latter can represent a number of prominent integrable equations, like self-dual Yang-Mills, as well as matrix versions of the two-dimensional Toda lattice, Hirota's bilinear difference equation, (2+1)-dimensional Nonlinear Schrödinger (NLS), Kadomtsev-Petviashvili (KP) equation, and Davey-Stewartson equations. For all of them, a recent (non-isospectral) binary Darboux transformation result in bidifferential calculus applies, which can be specialized to generate solutions of the associated "Riemann equations." For the latter, we clarify the relation between these specialized binary Darboux transformations and the aforementioned solution-generating method. From (arbitrary size) matrix versions of the "Riemann equations" associated with an integrable equation, possessing a bidifferential calculus formulation, multi-soliton-type solutions of the latter can be generated. This includes "breaking" multi-soliton-type solutions of the self-dual Yang-Mills and the (2+1)-dimensional NLS equation, which are parametrized by solutions of Riemann equations.
The Forced Hard Spring Equation
ERIC Educational Resources Information Center
Fay, Temple H.
2006-01-01
Through numerical investigations, various examples of the Duffing type forced spring equation with epsilon positive, are studied. Since [epsilon] is positive, all solutions to the associated homogeneous equation are periodic and the same is true with the forcing applied. The damped equation exhibits steady state trajectories with the interesting…
Successfully Transitioning to Linear Equations
ERIC Educational Resources Information Center
Colton, Connie; Smith, Wendy M.
2014-01-01
The Common Core State Standards for Mathematics (CCSSI 2010) asks students in as early as fourth grade to solve word problems using equations with variables. Equations studied at this level generate a single solution, such as the equation x + 10 = 25. For students in fifth grade, the Common Core standard for algebraic thinking expects them to…
Solving Nonlinear Coupled Differential Equations
NASA Technical Reports Server (NTRS)
Mitchell, L.; David, J.
1986-01-01
Harmonic balance method developed to obtain approximate steady-state solutions for nonlinear coupled ordinary differential equations. Method usable with transfer matrices commonly used to analyze shaft systems. Solution to nonlinear equation, with periodic forcing function represented as sum of series similar to Fourier series but with form of terms suggested by equation itself.
Transient growth in stable linearized Vlasov-Maxwell plasmas
Podesta, J. J.
2010-12-15
Large amplitude transient growth of kinetic scale perturbations in stable collisionless magnetized plasmas has recently been demonstrated using a linearized Landau fluid model. Initial perturbations with lengthscales of the order of the ion gyroradius were shown to have transient timescales that in some cases were long compared to the ion gyroperiod, {Omega}{sub i}t>>1. Moreover, it was suggested that such perturbations are not rare but instead form a large class within the set of all possible initial conditions. For collisionless plasmas, the Vlasov-Maxwell equations provide a more complete description of kinetic physics and the existence of transient growth of solutions for the linearized Vlasov-Maxwell system is an interesting question. The existence of transient growth of solutions is demonstrated here for a special case of the Vlasov-Maxwell equations, namely, the one dimensional Vlasov-Poisson system. The analysis is different from the standard approach of nonmodal analysis since the initial value problem is described by a Volterra integral equation of the second kind, reflecting the fact that the time evolution of the system depends on the memory of the state from time zero through time t. For the case of a thermal equilibrium plasma, it is shown how initial conditions may be constructed to obtain solutions that grow linearly in time; the duration of this growth is the time required for a thermal electron to traverse the wavelength of the initial perturbation, a timescale that can last for many plasma periods 2{pi}/{omega}{sub pe}, thus demonstrating the existence of transient growth of solutions for the linearized Vlasov-Poisson system. The results suggest that the phenomenon of transient growth may be a common feature of the linearized Vlasov-Maxwell system as well as for Landau fluid models.
Generalized Klein-Kramers equations.
Fa, Kwok Sau
2012-12-21
A generalized Klein-Kramers equation for a particle interacting with an external field is proposed. The equation generalizes the fractional Klein-Kramers equation introduced by Barkai and Silbey [J. Phys. Chem. B 104, 3866 (2000)]. Besides, the generalized Klein-Kramers equation can also recover the integro-differential Klein-Kramers equation for continuous-time random walk; this means that it can describe the subdiffusive and superdiffusive regimes in the long-time limit. Moreover, analytic solutions for first two moments both in velocity and displacement (for force-free case) are obtained, and their dynamic behaviors are investigated.
Large amplitude relativistic plasma waves
Coffey, Timothy
2010-05-15
Relativistic, longitudinal plasma oscillations are studied for the case of a simple water bag distribution of electrons having cylindrical symmetry in momentum space with the axis of the cylinder parallel to the velocity of wave propagation. The plasma is required to obey the relativistic Vlasov-Poisson equations, and solutions are sought in the wave frame. An exact solution for the plasma density as a function of the electrostatic field is derived. The maximum electric field is presented in terms of an integral over the known density. It is shown that when the perpendicular momentum is neglected, the maximum electric field approaches infinity as the wave phase velocity approaches the speed of light. It is also shown that for any nonzero perpendicular momentum, the maximum electric field will remain finite as the wave phase velocity approaches the speed of light. The relationship to previously published solutions is discussed as is some recent controversy regarding the proper modeling of large amplitude relativistic plasma waves.
The polarization response function and the dielectric permittivity of a plasma
Gnavi, G.; Gratton, F.
1984-09-01
We give a simple direct derivation of the polarization response function h for linear electrostatic excitations of a plasma (without magnetic field) considering the effect of a percussion on the electrons. The physical meaning of the procedure is discussed, thus bringing into light basic facts of the plasma dielectric behavior. The result h = S/sub p/ fo(x/t) (where f/sub o/ is the electron distribution function in velocity space and /sub p/ the plasma frequency) is obtained without passing through the Vlasov-Poisson equations as in the standard theory. We show that the equivalence between the present method and the classic Landau analysis rests on properties of the Fourier transform applied on velocity space.
NASA Technical Reports Server (NTRS)
Klimas, A. J.
1983-01-01
A numerical method is presented for studying one-dimensional electron plasma evolution under typical interplanetary conditions. The method applies the Fourier-Fourier transform approach to a plasma model that is a generalization of the electrostatic Vlasov-Poisson system of equations. Conservation laws that are modified to include the plasma model generalization and also the boundary effects of nonperiodic solutions are given. A new conservation law for entropy in the transformed space is then introduced. These conservation laws are used to verify the numerical solutions. A discretization error analysis is presented. Two numerical instabilities and the methods used for their suppression are treated. It is shown that in interplanetary plasma conditions, the bump-on-tail instability produces significant excitation of plasma oscillations at the Bohm-Gross frequency and its second harmonic. An explanation of the second harmonic excitation is given in terms of wave-wave coupling during the growth phase of the instability.
Kinetic study of ion acoustic twisted waves with kappa distributed electrons
NASA Astrophysics Data System (ADS)
Arshad, Kashif; Aman-ur-Rehman, Mahmood, Shahzad
2016-05-01
The kinetic theory of Landau damping of ion acoustic twisted modes is developed in the presence of orbital angular momentum of the helical (twisted) electric field in plasmas with kappa distributed electrons and Maxwellian ions. The perturbed distribution function and helical electric field are considered to be decomposed by Laguerre-Gaussian mode function defined in cylindrical geometry. The Vlasov-Poisson equation is obtained and solved analytically to obtain the weak damping rates of the ion acoustic twisted waves in a non-thermal plasma. The strong damping effects of ion acoustic twisted waves at low values of temperature ratio of electrons and ions are also obtained by using exact numerical method and illustrated graphically, where the weak damping wave theory fails to explain the phenomenon properly. The obtained results of Landau damping rates of the twisted ion acoustic wave are discussed at different values of azimuthal wave number and non-thermal parameter kappa for electrons.
Landau damping of Langmuir twisted waves with kappa distributed electrons
Arshad, Kashif Aman-ur-Rehman; Mahmood, Shahzad
2015-11-15
The kinetic theory of Landau damping of Langmuir twisted modes is investigated in the presence of orbital angular momentum of the helical (twisted) electric field in plasmas with kappa distributed electrons. The perturbed distribution function and helical electric field are considered to be decomposed by Laguerre-Gaussian mode function defined in cylindrical geometry. The Vlasov-Poisson equation is obtained and solved analytically to obtain the weak damping rates of the Langmuir twisted waves in a nonthermal plasma. The strong damping effects of the Langmuir twisted waves at wavelengths approaching Debye length are also obtained by using an exact numerical method and are illustrated graphically. The damping rates of the planar Langmuir waves are found to be larger than the twisted Langmuir waves in plasmas which shows opposite behavior as depicted in Fig. 3 by J. T. Mendoça [Phys. Plasmas 19, 112113 (2012)].
Langmuir oscillations in a nonthermal nonextensive electron-positron plasma
NASA Astrophysics Data System (ADS)
El-Taibany, W. F.; Zedan, N. A.
2017-02-01
The high-frequency Langmuir-type oscillations in a pure pair plasma are studied using Vlasov-Poisson's equations in the presence of hybrid nonthermal nonextensive distributed species. The characteristics of the Langmuir oscillations, Landau damping, and growing unstable modes in a nonthermal nonextensive electron-positron (EP) plasma are remarkably modified. It is found that the phase velocity of the Langmuir waves increases by decreasing (increasing) the value of nonextensive (nonthermal) parameter, q ( α). In particular, depending on the degree of nonthermality and nonextensivity, both damping and growing oscillations are predicted in the proposed EP plasma. It is seen that the Langmuir waves suffer from Landau damping in two different q regions. Furthermore, the mechanism that leads to unstable modes is established in the context of the nonthermal nonextensive formalism, yet the damping mechanism is the same developed by Landau. The present study is useful in the regions where such mixed distributions in space or laboratory plasmas exist.
Thermodynamic Bounds on Nonlinear Electrostatic Perturbations in Intense Charged Particle Beams
Nikolas C. Logan and Ronald C. Davidson
2012-07-18
This paper places a lowest upper bound on the field energy in electrostatic perturbations in single-species charged particle beams with initial temperature anisotropy (TllT⊥ < 1). The result applies to all electrostatic perturbations driven by the natural anisotropies that develop in accelerated particle beams, including Harris-type electrostatic instabilities, known to limit the luminosity and minimum spot size attainable in experiments. The thermodynamic bound on the field perturbation energy of the instabilities is obtained from the nonlinear Vlasov-Poisson equations for an arbitrary initial distribution function, including the effects of intense self-fields, finite geometry and nonlinear processes. This paper also includes analytical estimates of the nonlinear bounds for space-charge-dominated and emittance-dominated anisotropic bi-Maxwellian distributions.
Shukla-Spatschek diffusion effects on surface plasma waves in astrophysical turbulent plasmas
NASA Astrophysics Data System (ADS)
Lee, Myoung-Jae; Jung, Young-Dae
2017-02-01
The effects of Shukla-Spatschek turbulent diffusion on a temporal mode of surface waves propagating at the interface of an astrophysical turbulent plasma are investigated. The damping rates for high and low modes of surface wave are kinetically derived by employing the Vlasov-Poisson equation and the specular reflection boundary condition. We found that the diffusion caused by the fluctuating electric fields leads to damping for both high and low modes of surface waves. The high-mode damping is enhanced with an increase of the wavenumber and the diffusion coefficient, but suppressed by an increase of electron thermal energy. By contrast, the low-mode damping is suppressed as the wavenumber and the thermal energy increase although it is enhanced as the diffusion increases. The variation of the damping rate due to the Shukla-Spatschek turbulent diffusion is also discussed.
NASA Astrophysics Data System (ADS)
Coulette, David; Hirstoaga, Sever A.; Manfredi, Giovanni
2016-08-01
We develop a hybrid model to describe the parallel transport in a tokamak scrape-off layer following an edge-localized mode (ELM) event. The parallel dynamics is treated with a kinetic Vlasov-Poisson model, while the evolution of the perpendicular temperature {{T}\\bot} is governed by a fluid equation. The coupling is ensured by isotropising collisions. The model generalises an earlier approach where {{T}\\bot} was constant in space and time (Manfredi et al 2011 Plasma Phys. Control. Fusion 53 015012). Numerical results show that the main effect comes from electron-electron collisions, which limit the decrease of the parallel electron temperature and increase the potential drop in the Debye sheath in front of the surface. Ion-ion collisions have an almost negligible impact. The net effect is an increased peak power load on the target plates.
ERIC Educational Resources Information Center
Powers, Sonya Jean
2010-01-01
When test forms are administered to examinee groups that differ in proficiency, equating procedures are used to disentangle group differences from form differences. This dissertation investigates the extent to which equating results are population invariant, the impact of group differences on equating results, the impact of group differences on…
On nonautonomous Dirac equation
Hovhannisyan, Gro; Liu Wen
2009-12-15
We construct the fundamental solution of time dependent linear ordinary Dirac system in terms of unknown phase functions. This construction gives approximate representation of solutions which is useful for the study of asymptotic behavior. Introducing analog of Rayleigh quotient for differential equations we generalize Hartman-Wintner asymptotic integration theorems with the error estimates for applications to the Dirac system. We also introduce the adiabatic invariants for the Dirac system, which are similar to the adiabatic invariant of Lorentz's pendulum. Using a small parameter method it is shown that the change in the adiabatic invariants approaches zero with the power speed as a small parameter approaches zero. As another application we calculate the transition probabilities for the Dirac system. We show that for the special choice of electromagnetic field, the only transition of an electron to the positron with the opposite spin orientation is possible.
NASA Astrophysics Data System (ADS)
Gomez, Humberto
2016-06-01
The CHY representation of scattering amplitudes is based on integrals over the moduli space of a punctured sphere. We replace the punctured sphere by a double-cover version. The resulting scattering equations depend on a parameter Λ controlling the opening of a branch cut. The new representation of scattering amplitudes possesses an enhanced redundancy which can be used to fix, modulo branches, the location of four punctures while promoting Λ to a variable. Via residue theorems we show how CHY formulas break up into sums of products of smaller (off-shell) ones times a propagator. This leads to a powerful way of evaluating CHY integrals of generic rational functions, which we call the Λ algorithm.
NASA Astrophysics Data System (ADS)
Frimmer, Martin; Novotny, Lukas
2014-10-01
Coherent control of a quantum mechanical two-level system is at the heart of magnetic resonance imaging, quantum information processing, and quantum optics. Among the most prominent phenomena in quantum coherent control are Rabi oscillations, Ramsey fringes, and Hahn echoes. We demonstrate that these phenomena can be derived classically by use of a simple coupled-harmonic-oscillator model. The classical problem can be cast in a form that is formally equivalent to the quantum mechanical Bloch equations with the exception that the longitudinal and the transverse relaxation times (T1 and T2) are equal. The classical analysis is intuitive and well suited for familiarizing students with the basic concepts of quantum coherent control, while at the same time highlighting the fundamental differences between classical and quantum theories.
Solitary Wave Solutions of KP equation, Cylindrical KP Equation and Spherical KP Equation
NASA Astrophysics Data System (ADS)
Li, Xiang-Zheng; Zhang, Jin-Liang; Wang, Ming-Liang
2017-02-01
Three (2+1)-dimensional equations–KP equation, cylindrical KP equation and spherical KP equation, have been reduced to the same KdV equation by different transformation of variables respectively. Since the single solitary wave solution and 2-solitary wave solution of the KdV equation have been known already, substituting the solutions of the KdV equation into the corresponding transformation of variables respectively, the single and 2-solitary wave solutions of the three (2+1)-dimensional equations can be obtained successfully. Supported by the National Natural Science Foundation of China under Grant No. 11301153 and the Doctoral Foundation of Henan University of Science and Technology under Grant No. 09001562, and the Science and Technology Innovation Platform of Henan University of Science and Technology under Grant No. 2015XPT001
Mode decomposition evolution equations
Wang, Yang; Wei, Guo-Wei; Yang, Siyang
2011-01-01
Partial differential equation (PDE) based methods have become some of the most powerful tools for exploring the fundamental problems in signal processing, image processing, computer vision, machine vision and artificial intelligence in the past two decades. The advantages of PDE based approaches are that they can be made fully automatic, robust for the analysis of images, videos and high dimensional data. A fundamental question is whether one can use PDEs to perform all the basic tasks in the image processing. If one can devise PDEs to perform full-scale mode decomposition for signals and images, the modes thus generated would be very useful for secondary processing to meet the needs in various types of signal and image processing. Despite of great progress in PDE based image analysis in the past two decades, the basic roles of PDEs in image/signal analysis are only limited to PDE based low-pass filters, and their applications to noise removal, edge detection, segmentation, etc. At present, it is not clear how to construct PDE based methods for full-scale mode decomposition. The above-mentioned limitation of most current PDE based image/signal processing methods is addressed in the proposed work, in which we introduce a family of mode decomposition evolution equations (MoDEEs) for a vast variety of applications. The MoDEEs are constructed as an extension of a PDE based high-pass filter (Europhys. Lett., 59(6): 814, 2002) by using arbitrarily high order PDE based low-pass filters introduced by Wei (IEEE Signal Process. Lett., 6(7): 165, 1999). The use of arbitrarily high order PDEs is essential to the frequency localization in the mode decomposition. Similar to the wavelet transform, the present MoDEEs have a controllable time-frequency localization and allow a perfect reconstruction of the original function. Therefore, the MoDEE operation is also called a PDE transform. However, modes generated from the present approach are in the spatial or time domain and can be
Menikoff, Ralph
2015-12-15
The JWL equation of state (EOS) is frequently used for the products (and sometimes reactants) of a high explosive (HE). Here we review and systematically derive important properties. The JWL EOS is of the Mie-Grueneisen form with a constant Grueneisen coefficient and a constants specific heat. It is thermodynamically consistent to specify the temperature at a reference state. However, increasing the reference state temperature restricts the EOS domain in the (V, e)-plane of phase space. The restrictions are due to the conditions that P ≥ 0, T ≥ 0, and the isothermal bulk modulus is positive. Typically, this limits the low temperature regime in expansion. The domain restrictions can result in the P-T equilibrium EOS of a partly burned HE failing to have a solution in some cases. For application to HE, the heat of detonation is discussed. Example JWL parameters for an HE, both products and reactions, are used to illustrate the restrictions on the domain of the EOS.
NASA Astrophysics Data System (ADS)
Sivron, Ran
2006-12-01
With the introduction of "Ranking Tests" some quantitative ideas were added to a large body of successful techniques for teaching conceptual astronomy. We incorporated those methods into our classes, and added a new ingredient: On a biweekly basis we included a quantitative excercise: Students working in groups of 2-3 draw geometrical figures, say: a circle, and use some trivial geometry equations, such as circumference = 2 x pi x r, in solving astronomy problems on 3'x4' white boards. A few examples included: Finding the distance to the moon with the Aristarchus method, finding the Solar Constant with the inverse square law, etc. Our methodolgy was similar to problem solving techniques in introductory physics. We were therefore worried that the students may be intimidated. To our surprize, not only did most students succeed in solving the problems, but they were not intimidated at all (that is: after the first class...) As a matter of fact, their test results improved, and the students interviewed expressed great enthusiasm for the new method. Warning: Our classes were relatively small <40 studets). For larger classes TA help is needed.
A note on "Kepler's equation".
NASA Astrophysics Data System (ADS)
Dutka, J.
1997-07-01
This note briefly points out the formal similarity between Kepler's equation and equations developed in Hindu and Islamic astronomy for describing the lunar parallax. Specifically, an iterative method for calculating the lunar parallax has been developed by the astronomer Habash al-Hasib al-Marwazi (about 850 A.D., Turkestan), which is surprisingly similar to the iterative method for solving Kepler's equation invented by Leonhard Euler (1707 - 1783).
Electronic representation of wave equation
NASA Astrophysics Data System (ADS)
Veigend, Petr; Kunovský, Jiří; Kocina, Filip; Nečasová, Gabriela; Šátek, Václav; Valenta, Václav
2016-06-01
The Taylor series method for solving differential equations represents a non-traditional way of a numerical solution. Even though this method is not much preferred in the literature, experimental calculations done at the Department of Intelligent Systems of the Faculty of Information Technology of TU Brno have verified that the accuracy and stability of the Taylor series method exceeds the currently used algorithms for numerically solving differential equations. This paper deals with solution of Telegraph equation using modelling of a series small pieces of the wire. Corresponding differential equations are solved by the Modern Taylor Series Method.
Delay equations and radiation damping
NASA Astrophysics Data System (ADS)
Chicone, C.; Kopeikin, S. M.; Mashhoon, B.; Retzloff, D. G.
2001-06-01
Starting from delay equations that model field retardation effects, we study the origin of runaway modes that appear in the solutions of the classical equations of motion involving the radiation reaction force. When retardation effects are small, we argue that the physically significant solutions belong to the so-called slow manifold of the system and we identify this invariant manifold with the attractor in the state space of the delay equation. We demonstrate via an example that when retardation effects are no longer small, the motion could exhibit bifurcation phenomena that are not contained in the local equations of motion.
Complete solution of Boolean equations
NASA Technical Reports Server (NTRS)
Tapia, M. A.; Tucker, J. H.
1980-01-01
A method is presented for generating a single formula involving arbitary Boolean parameters, which includes in it each and every possible solution of a system of Boolean equations. An alternate condition equivalent to a known necessary and sufficient condition for solving a system of Boolean equations is given.
Uncertainty of empirical correlation equations
NASA Astrophysics Data System (ADS)
Feistel, R.; Lovell-Smith, J. W.; Saunders, P.; Seitz, S.
2016-08-01
The International Association for the Properties of Water and Steam (IAPWS) has published a set of empirical reference equations of state, forming the basis of the 2010 Thermodynamic Equation of Seawater (TEOS-10), from which all thermodynamic properties of seawater, ice, and humid air can be derived in a thermodynamically consistent manner. For each of the equations of state, the parameters have been found by simultaneously fitting equations for a range of different derived quantities using large sets of measurements of these quantities. In some cases, uncertainties in these fitted equations have been assigned based on the uncertainties of the measurement results. However, because uncertainties in the parameter values have not been determined, it is not possible to estimate the uncertainty in many of the useful quantities that can be calculated using the parameters. In this paper we demonstrate how the method of generalised least squares (GLS), in which the covariance of the input data is propagated into the values calculated by the fitted equation, and in particular into the covariance matrix of the fitted parameters, can be applied to one of the TEOS-10 equations of state, namely IAPWS-95 for fluid pure water. Using the calculated parameter covariance matrix, we provide some preliminary estimates of the uncertainties in derived quantities, namely the second and third virial coefficients for water. We recommend further investigation of the GLS method for use as a standard method for calculating and propagating the uncertainties of values computed from empirical equations.
Graphical Solution of Polynomial Equations
ERIC Educational Resources Information Center
Grishin, Anatole
2009-01-01
Graphing utilities, such as the ubiquitous graphing calculator, are often used in finding the approximate real roots of polynomial equations. In this paper the author offers a simple graphing technique that allows one to find all solutions of a polynomial equation (1) of arbitrary degree; (2) with real or complex coefficients; and (3) possessing…
The report describes a program for computing equation of state parameters for a material which undergoes a phase transition, either rate-dependent or...obtaining explicit temperature dependence if measurements are made at three temperatures. It is applied to data from calcite. Finally a theoretical equation of state is described for solid iron. (Author)
Homotopy Solutions of Kepler's Equations
NASA Technical Reports Server (NTRS)
Fitz-Coy, Norman; Jang, Jiann-Woei
1996-01-01
Kepler's Equation is solved using an integrative algorithm developed using homotropy theory. The solution approach is applicable to both elliptic and hyperbolic forms of Kepler's Equation. The results from the proposed algorithm compare quite favorably with those from existing iterative schemes.
Drug Levels and Difference Equations
ERIC Educational Resources Information Center
Acker, Kathleen A.
2004-01-01
American university offers a course in finite mathematics whose focus is difference equation with emphasis on real world applications. The conclusion states that students learned to look for growth and decay patterns in raw data, to recognize both arithmetic and geometric growth, and to model both scenarios with graphs and difference equations.
Students' Understanding of Quadratic Equations
ERIC Educational Resources Information Center
López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael
2016-01-01
Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help…
How Students Understand Physics Equations.
ERIC Educational Resources Information Center
Sherin, Bruce L.
2001-01-01
Analyzed a corpus of videotapes in which university students solved physics problems to determine how students learn to understand a physics equation. Found that students learn to understand physics equations in terms of a vocabulary of elements called symbolic forms, each associating a simple conceptual schema with a pattern of symbols. Findings…
Generalized Multilevel Structural Equation Modeling
ERIC Educational Resources Information Center
Rabe-Hesketh, Sophia; Skrondal, Anders; Pickles, Andrew
2004-01-01
A unifying framework for generalized multilevel structural equation modeling is introduced. The models in the framework, called generalized linear latent and mixed models (GLLAMM), combine features of generalized linear mixed models (GLMM) and structural equation models (SEM) and consist of a response model and a structural model for the latent…
The Bessel Equation and Dissipation
NASA Astrophysics Data System (ADS)
Alfinito, Eleonora; Vitiello, Giuseppe
The Bessel equation can be cast, by means of suitable transformations, into a system of two damped/amplified parametric oscillator equations. The role of group contraction and the breakdown of loop-antiloop symmetry is discussed. The relation between the Virasoro algebra and the Euclidean algebras e(2) and e(3) is also presented.
Nonlinear analysis of the cold fluid-Poisson plasma by using the characteristic method
NASA Astrophysics Data System (ADS)
Lee, Hee J.
2016-10-01
We show that the Vlasov and Euler equations can be transformed into each other along the same characteristics on the ( x, t) plane. Therefore, the Vlasov-Poisson plasma may have common features that are contained in the cold fluid equations: the Euler equation, the continuity equation, and the Poisson equation. Here, the cold fluid equation does not mean the moment equation of the Boltzmann equation. We show that the compensated electron fluid equations can be solved linearly along the characteristics. We address an ion plasma with Boltzmann-distributed electrons as a Cauchy initial-boundary value problem for which initial data are provided by compatible solutions of the Poisson equation. In this plasma, the set of nonlinear cold fluid equations can be approached by arranging them in Riemann invariant equations or via a hodograph transform. The result of this arrangement is linear equations, thus suggesting a way to investigate a cold fluid nonlinear plasma without directly engaging the nonlinearity. The Poisson equation corresponds to the entropy equation in the gas dynamic equations. Analogously, a power law similar to the polytropic gas law in gas dynamics is assumed between the electric potential and the density.
NASA Astrophysics Data System (ADS)
Newman, D. L.; Goldman, M. V.
2007-05-01
The bipolar signatures of weak (eφmax/Te ≪ 1) electron phase-space holes have now been observed in numerous near-Earth space-plasma environments such as the polar cusp region1 and the solar wind2 at 1 AU. While families of stationary solutions of the Vlasov-Poisson equations consistent with these observations have been found,3 the question of how shallow phase-space density depressions supporting these bipolar fields form remains an open one. While strong bipolar fields associated with deep phase-space holes in Earth auroral downward-current region are consistent with saturated two-stream instabilities resulting from double-layer electron acceleration,4 the weak bipolar fields observed in other space environments may require an alternative generation mechanism. One such mechanism involves the formation of narrow minima in the electron velocity-space distribution resulting from stretching due to velocity dispersion of phase-space density minima that are initially localized in physical space (e.g., constant-density regions with temperatures greater than their surroundings). These velocity-space minima, which become narrower as they are dispersively stretched, eventually cross the threshold condition for a "notch" instability, which saturates by forming an expanding series of shallow phase-space holes and their associated weak bipolar fields. 1-D Vlasov-Poisson simulations show that this process can be a robust mechanism for generating a large ensemble of shallow holes. Simulations with different background electron distributions show that the properties of the holes that form depend sensitively on the characteristics of the embedding plasma environment. * Research supported by NSF, NASA, and DOE. 1 J. R. Franz, et al., JGR, 110, doi:10.1029/2005JA011095 (2005). 2 A. Mangeney, private communication. 3 M. V. Goldman, et al., this meeting. 4 R. E. Ergun, et al., PRL, 87, 045003 (2001); D. L. Newman, et al. PRL, 87, 255001 (2001).
The Equations of Oceanic Motions
NASA Astrophysics Data System (ADS)
Müller, Peter
2006-10-01
Modeling and prediction of oceanographic phenomena and climate is based on the integration of dynamic equations. The Equations of Oceanic Motions derives and systematically classifies the most common dynamic equations used in physical oceanography, from large scale thermohaline circulations to those governing small scale motions and turbulence. After establishing the basic dynamical equations that describe all oceanic motions, M|ller then derives approximate equations, emphasizing the assumptions made and physical processes eliminated. He distinguishes between geometric, thermodynamic and dynamic approximations and between the acoustic, gravity, vortical and temperature-salinity modes of motion. Basic concepts and formulae of equilibrium thermodynamics, vector and tensor calculus, curvilinear coordinate systems, and the kinematics of fluid motion and wave propagation are covered in appendices. Providing the basic theoretical background for graduate students and researchers of physical oceanography and climate science, this book will serve as both a comprehensive text and an essential reference.
Upper bounds for parabolic equations and the Landau equation
NASA Astrophysics Data System (ADS)
Silvestre, Luis
2017-02-01
We consider a parabolic equation in nondivergence form, defined in the full space [ 0 , ∞) ×Rd, with a power nonlinearity as the right-hand side. We obtain an upper bound for the solution in terms of a weighted control in Lp. This upper bound is applied to the homogeneous Landau equation with moderately soft potentials. We obtain an estimate in L∞ (Rd) for the solution of the Landau equation, for positive time, which depends only on the mass, energy and entropy of the initial data.
Higher derivative gravity: Field equation as the equation of state
NASA Astrophysics Data System (ADS)
Dey, Ramit; Liberati, Stefano; Mohd, Arif
2016-08-01
One of the striking features of general relativity is that the Einstein equation is implied by the Clausius relation imposed on a small patch of locally constructed causal horizon. The extension of this thermodynamic derivation of the field equation to more general theories of gravity has been attempted many times in the last two decades. In particular, equations of motion for minimally coupled higher-curvature theories of gravity, but without the derivatives of curvature, have previously been derived using a thermodynamic reasoning. In that derivation the horizon slices were endowed with an entropy density whose form resembles that of the Noether charge for diffeomorphisms, and was dubbed the Noetheresque entropy. In this paper, we propose a new entropy density, closely related to the Noetheresque form, such that the field equation of any diffeomorphism-invariant metric theory of gravity can be derived by imposing the Clausius relation on a small patch of local causal horizon.
Langevin equations from time series.
Racca, E; Porporato, A
2005-02-01
We discuss the link between the approach to obtain the drift and diffusion of one-dimensional Langevin equations from time series, and Pope and Ching's relationship for stationary signals. The two approaches are based on different interpretations of conditional averages of the time derivatives of the time series at given levels. The analysis provides a useful indication for the correct application of Pope and Ching's relationship to obtain stochastic differential equations from time series and shows its validity, in a generalized sense, for nondifferentiable processes originating from Langevin equations.
Accuracy of perturbative master equations.
Fleming, C H; Cummings, N I
2011-03-01
We consider open quantum systems with dynamics described by master equations that have perturbative expansions in the system-environment interaction. We show that, contrary to intuition, full-time solutions of order-2n accuracy require an order-(2n+2) master equation. We give two examples of such inaccuracies in the solutions to an order-2n master equation: order-2n inaccuracies in the steady state of the system and order-2n positivity violations. We show how these arise in a specific example for which exact solutions are available. This result has a wide-ranging impact on the validity of coupling (or friction) sensitive results derived from second-order convolutionless, Nakajima-Zwanzig, Redfield, and Born-Markov master equations.
On systems of Boolean equations
NASA Astrophysics Data System (ADS)
Leont'ev, V. K.; Tonoyan, G. P.
2013-05-01
Systems of Boolean equations are considered. The order of maximal consistent subsystems is estimated in the general and "typical" (in a probability sense) cases. Applications for several well-known discrete problems are given.
Comment on "Quantum Raychaudhuri equation"
NASA Astrophysics Data System (ADS)
Lashin, E. I.; Dou, Djamel
2017-03-01
We address the validity of the formalism and results presented in S. Das, Phys. Rev. D 89, 084068 (2014), 10.1103/PhysRevD.89.084068 with regard to the quantum Raychaudhuri equation. The author obtained the so-called quantum Raychaudhuri equation by replacing classical geodesics with quantal trajectories arising from Bhommian mechanics. The resulting modified equation was used to draw some conclusions about the inevitability of focusing and the formation of conjugate points and therefore singularity. We show that the whole procedure is full of problematic points, on both physical relevancy and mathematical correctness. In particular, we illustrate the problems associated with the technical derivation of the so-called quantum Raychaudhuri equation, as well as its invalid physical implications.
Taxis equations for amoeboid cells.
Erban, Radek; Othmer, Hans G
2007-06-01
The classical macroscopic chemotaxis equations have previously been derived from an individual-based description of the tactic response of cells that use a "run-and-tumble" strategy in response to environmental cues [17,18]. Here we derive macroscopic equations for the more complex type of behavioral response characteristic of crawling cells, which detect a signal, extract directional information from a scalar concentration field, and change their motile behavior accordingly. We present several models of increasing complexity for which the derivation of population-level equations is possible, and we show how experimentally measured statistics can be obtained from the transport equation formalism. We also show that amoeboid cells that do not adapt to constant signals can still aggregate in steady gradients, but not in response to periodic waves. This is in contrast to the case of cells that use a "run-and-tumble" strategy, where adaptation is essential.
Parametric Equations, Maple, and Tubeplots.
ERIC Educational Resources Information Center
Feicht, Louis
1997-01-01
Presents an activity that establishes a graphical foundation for parametric equations by using a graphing output form called tubeplots from the computer program Maple. Provides a comprehensive review and exploration of many previously learned topics. (ASK)
NASA Technical Reports Server (NTRS)
Shebalin, John V.
1987-01-01
The Boussinesq approximation is extended so as to explicitly account for the transfer of fluid energy through viscous action into thermal energy. Ideal and dissipative integral invariants are discussed, in addition to the general equations for thermal-fluid motion.
Friedmann equation with quantum potential
Siong, Ch'ng Han; Radiman, Shahidan; Nikouravan, Bijan
2013-11-27
Friedmann equations are used to describe the evolution of the universe. Solving Friedmann equations for the scale factor indicates that the universe starts from an initial singularity where all the physical laws break down. However, the Friedmann equations are well describing the late-time or large scale universe. Hence now, many physicists try to find an alternative theory to avoid this initial singularity. In this paper, we generate a version of first Friedmann equation which is added with an additional term. This additional term contains the quantum potential energy which is believed to play an important role at small scale. However, it will gradually become negligible when the universe evolves to large scale.
Derivation of the Simon equation
NASA Astrophysics Data System (ADS)
Fedorov, P. P.
2016-09-01
The form of the empirical Simon equation describing the dependence of the phase-transition temperature on pressure is shown to be asymptotically strict when the system tends to absolute zero of temperature, and then only for crystalline phases.
Hidden Statistics of Schroedinger Equation
NASA Technical Reports Server (NTRS)
Zak, Michail
2011-01-01
Work was carried out in determination of the mathematical origin of randomness in quantum mechanics and creating a hidden statistics of Schr dinger equation; i.e., to expose the transitional stochastic process as a "bridge" to the quantum world. The governing equations of hidden statistics would preserve such properties of quantum physics as superposition, entanglement, and direct-product decomposability while allowing one to measure its state variables using classical methods.
Program solves line flow equation
McCaslin, K.P.
1981-01-19
A program written for the TI-59 programmable calculator solves the Panhandle Eastern A equation - an industry-accepted equation for calculating pressure losses in high-pressure gas-transmission pipelines. The input variables include the specific gravity of the gas, the flowing temperature, the pipeline efficiency, the base temperature and pressure, the inlet pressure, the pipeline's length and inside diameter, and the flow rate (SCF/day); the program solves for the discharge pressure.
Wave equations for pulse propagation
Shore, B.W.
1987-06-24
Theoretical discussions of the propagation of pulses of laser radiation through atomic or molecular vapor rely on a number of traditional approximations for idealizing the radiation and the molecules, and for quantifying their mutual interaction by various equations of propagation (for the radiation) and excitation (for the molecules). In treating short-pulse phenomena it is essential to consider coherent excitation phenomena of the sort that is manifest in Rabi oscillations of atomic or molecular populations. Such processes are not adequately treated by rate equations for excitation nor by rate equations for radiation. As part of a more comprehensive treatment of the coupled equations that describe propagation of short pulses, this memo presents background discussion of the equations that describe the field. This memo discusses the origin, in Maxwell's equations, of the wave equation used in the description of pulse propagation. It notes the separation into lamellar and solenoidal (or longitudinal and transverse) and positive and negative frequency parts. It mentions the possibility of separating the polarization field into linear and nonlinear parts, in order to define a susceptibility or index of refraction and, from these, a phase and group velocity. The memo discusses various ways of characterizing the polarization characteristics of plane waves, that is, of parameterizing a transverse unit vector, such as the Jones vector, the Stokes vector, and the Poincare sphere. It discusses the connection between macroscopically defined quantities, such as the intensity or, more generally, the Stokes parameters, and microscopic field amplitudes. The material presented here is a portion of a more extensive treatment of propagation to be presented separately. The equations presented here have been described in various books and articles. They are collected here as a summary and review of theory needed when treating pulse propagation.
An Exact Mapping from Navier-Stokes Equation to Schr"odinger Equation via Riccati Equation
NASA Astrophysics Data System (ADS)
Christianto, Vic; Smarandache, Florentin
2010-03-01
In the present article we argue that it is possible to write down Schr"odinger representation of Navier-Stokes equation via Riccati equation. The proposed approach, while differs appreciably from other method such as what is proposed by R. M. Kiehn, has an advantage, i.e. it enables us extend further to quaternionic and biquaternionic version of Navier-Stokes equation, for instance via Kravchenko's and Gibbon's route. Further observation is of course recommended in order to refute or verify this proposition.
Turbulent fluid motion 3: Basic continuum equations
NASA Technical Reports Server (NTRS)
Deissler, Robert G.
1991-01-01
A derivation of the continuum equations used for the analysis of turbulence is given. These equations include the continuity equation, the Navier-Stokes equations, and the heat transfer or energy equation. An experimental justification for using a continuum approach for the study of turbulence is given.
Optimization of one-way wave equations.
Lee, M.W.; Suh, S.Y.
1985-01-01
The theory of wave extrapolation is based on the square-root equation or one-way equation. The full wave equation represents waves which propagate in both directions. On the contrary, the square-root equation represents waves propagating in one direction only. A new optimization method presented here improves the dispersion relation of the one-way wave equation. -from Authors
Linear determining equations for differential constraints
Kaptsov, O V
1998-12-31
A construction of differential constraints compatible with partial differential equations is considered. Certain linear determining equations with parameters are used to find such differential constraints. They generalize the classical determining equations used in the search for admissible Lie operators. As applications of this approach equations of an ideal incompressible fluid and non-linear heat equations are discussed.
Solving Parker's transport equation with stochastic differential equations on GPUs
NASA Astrophysics Data System (ADS)
Dunzlaff, P.; Strauss, R. D.; Potgieter, M. S.
2015-07-01
The numerical solution of transport equations for energetic charged particles in space is generally very costly in terms of time. Besides the use of multi-core CPUs and computer clusters in order to decrease the computation times, high performance calculations on graphics processing units (GPUs) have become available during the last years. In this work we introduce and describe a GPU-accelerated implementation of Parker's equation using Stochastic Differential Equations (SDEs) for the simulation of the transport of energetic charged particles with the CUDA toolkit, which is the focus of this work. We briefly discuss the set of SDEs arising from Parker's transport equation and their application to boundary value problems such as that of the Jovian magnetosphere. We compare the runtimes of the GPU code with a CPU version of the same algorithm. Compared to the CPU implementation (using OpenMP and eight threads) we find a performance increase of about a factor of 10-60, depending on the assumed set of parameters. Furthermore, we benchmark our simulation using the results of an existing SDE implementation of Parker's transport equation.
Cable equation for general geometry.
López-Sánchez, Erick J; Romero, Juan M
2017-02-01
The cable equation describes the voltage in a straight cylindrical cable, and this model has been employed to model electrical potential in dendrites and axons. However, sometimes this equation might give incorrect predictions for some realistic geometries, in particular when the radius of the cable changes significantly. Cables with a nonconstant radius are important for some phenomena, for example, discrete swellings along the axons appear in neurodegenerative diseases such as Alzheimers, Parkinsons, human immunodeficiency virus associated dementia, and multiple sclerosis. In this paper, using the Frenet-Serret frame, we propose a generalized cable equation for a general cable geometry. This generalized equation depends on geometric quantities such as the curvature and torsion of the cable. We show that when the cable has a constant circular cross section, the first fundamental form of the cable can be simplified and the generalized cable equation depends on neither the curvature nor the torsion of the cable. Additionally, we find an exact solution for an ideal cable which has a particular variable circular cross section and zero curvature. For this case we show that when the cross section of the cable increases the voltage decreases. Inspired by this ideal case, we rewrite the generalized cable equation as a diffusion equation with a source term generated by the cable geometry. This source term depends on the cable cross-sectional area and its derivates. In addition, we study different cables with swelling and provide their numerical solutions. The numerical solutions show that when the cross section of the cable has abrupt changes, its voltage is smaller than the voltage in the cylindrical cable. Furthermore, these numerical solutions show that the voltage can be affected by geometrical inhomogeneities on the cable.
Cable equation for general geometry
NASA Astrophysics Data System (ADS)
López-Sánchez, Erick J.; Romero, Juan M.
2017-02-01
The cable equation describes the voltage in a straight cylindrical cable, and this model has been employed to model electrical potential in dendrites and axons. However, sometimes this equation might give incorrect predictions for some realistic geometries, in particular when the radius of the cable changes significantly. Cables with a nonconstant radius are important for some phenomena, for example, discrete swellings along the axons appear in neurodegenerative diseases such as Alzheimers, Parkinsons, human immunodeficiency virus associated dementia, and multiple sclerosis. In this paper, using the Frenet-Serret frame, we propose a generalized cable equation for a general cable geometry. This generalized equation depends on geometric quantities such as the curvature and torsion of the cable. We show that when the cable has a constant circular cross section, the first fundamental form of the cable can be simplified and the generalized cable equation depends on neither the curvature nor the torsion of the cable. Additionally, we find an exact solution for an ideal cable which has a particular variable circular cross section and zero curvature. For this case we show that when the cross section of the cable increases the voltage decreases. Inspired by this ideal case, we rewrite the generalized cable equation as a diffusion equation with a source term generated by the cable geometry. This source term depends on the cable cross-sectional area and its derivates. In addition, we study different cables with swelling and provide their numerical solutions. The numerical solutions show that when the cross section of the cable has abrupt changes, its voltage is smaller than the voltage in the cylindrical cable. Furthermore, these numerical solutions show that the voltage can be affected by geometrical inhomogeneities on the cable.
Fredholm's equations for subwavelength focusing
NASA Astrophysics Data System (ADS)
Velázquez-Arcos, J. M.
2012-10-01
Subwavelength focusing (SF) is a very useful tool that can be carried out with the use of left hand materials for optics that involve the range of the microwaves. Many recent works have described a successful alternative procedure using time reversal methods. The advantage is that we do not need devices which require the complicated manufacture of left-hand materials; nevertheless, the theoretical mathematical bases are far from complete because before now we lacked an adequate easy-to-apply frame. In this work we give, for a broad class of discrete systems, a solid support for the theory of electromagnetic SF that can be applied to communications and nanotechnology. The very central procedure is the development of vector-matrix formalism (VMF) based on exploiting both the inhomogeneous and homogeneous Fredholm's integral equations in cases where the last two kinds of integral equations are applied to some selected discrete systems. To this end, we first establish a generalized Newmann series for the Fourier transform of the Green's function in the inhomogeneous Fredholm's equation of the problem. Then we go from an integral operator equation to a vector-matrix algebraic one. In this way we explore the inhomogeneous case and later on also the very interesting one about the homogeneous equation. Thus, on the one hand we can relate in a simple manner the arriving electromagnetic signals with those at their sources and we can use them to perform a SF. On the other hand, we analyze the homogeneous version of the equations, finding resonant solutions that have analogous properties to their counterparts in quantum mechanical scattering, that can be used in a proposed very powerful way in communications. Also we recover quantum mechanical operator relations that are identical for classical electromagnetics. Finally, we prove two theorems that formalize the relation between the theory of Fredholm's integral equations and the VMF we present here.
Equation of State of Simple Metals.
1982-05-10
This is the final report of A. L. Ruoff and N. W. Ashcroft on Equation of State of Simple Metals. It includes experimental equation of state results for potassium and theoretical calculations of its equation of state . (Author)
How to Obtain the Covariant Form of Maxwell's Equations from the Continuity Equation
ERIC Educational Resources Information Center
Heras, Jose A.
2009-01-01
The covariant Maxwell equations are derived from the continuity equation for the electric charge. This result provides an axiomatic approach to Maxwell's equations in which charge conservation is emphasized as the fundamental axiom underlying these equations.
An Equation of State for Fluid Ethylene.
an equation of state , vapor pressure equation, and equation for the ideal gas heat capacity. The coefficients were determined by a least squares fit...of selected experimental data. Comparisons of property values calculated using the equation of state with measured values are given. The equation of state is...vapor phases for temperatures from the freezing line of 450 K with pressures to 40 MPa are presented. The equation of state and its derivative and
Spectral Models Based on Boussinesq Equations
2006-10-03
equations assume periodic solutions apriori. This, however, also forces the question of which extended Boussinesq model to use. Various one- equation ... equations of Nwogu (1993), without the traditional reduction to a one- equation model. Optimal numerical techniques to solve this system of equations are...A. and Madsen, P. A. (2004). "Boussinesq evolution equations : numerical efficiency, breaking and amplitude dispersion," Coastal Engineering, 51, 1117
Integration rules for scattering equations
NASA Astrophysics Data System (ADS)
Baadsgaard, Christian; Bjerrum-Bohr, N. E. J.; Bourjaily, Jacob L.; Damgaard, Poul H.
2015-09-01
As described by Cachazo, He and Yuan, scattering amplitudes in many quantum field theories can be represented as integrals that are fully localized on solutions to the so-called scattering equations. Because the number of solutions to the scattering equations grows quite rapidly, the contour of integration involves contributions from many isolated components. In this paper, we provide a simple, combinatorial rule that immediately provides the result of integration against the scattering equation constraints fo any Möbius-invariant integrand involving only simple poles. These rules have a simple diagrammatic interpretation that makes the evaluation of any such integrand immediate. Finally, we explain how these rules are related to the computation of amplitudes in the field theory limit of string theory.
Students' understanding of quadratic equations
NASA Astrophysics Data System (ADS)
López, Jonathan; Robles, Izraim; Martínez-Planell, Rafael
2016-05-01
Action-Process-Object-Schema theory (APOS) was applied to study student understanding of quadratic equations in one variable. This required proposing a detailed conjecture (called a genetic decomposition) of mental constructions students may do to understand quadratic equations. The genetic decomposition which was proposed can contribute to help students achieve an understanding of quadratic equations with improved interrelation of ideas and more flexible application of solution methods. Semi-structured interviews with eight beginning undergraduate students explored which of the mental constructions conjectured in the genetic decomposition students could do, and which they had difficulty doing. Two of the mental constructions that form part of the genetic decomposition are highlighted and corresponding further data were obtained from the written work of 121 undergraduate science and engineering students taking a multivariable calculus course. The results suggest the importance of explicitly considering these two highlighted mental constructions.
Special solutions to Chazy equation
NASA Astrophysics Data System (ADS)
Varin, V. P.
2017-02-01
We consider the classical Chazy equation, which is known to be integrable in hypergeometric functions. But this solution has remained purely existential and was never used numerically. We give explicit formulas for hypergeometric solutions in terms of initial data. A special solution was found in the upper half plane H with the same tessellation of H as that of the modular group. This allowed us to derive some new identities for the Eisenstein series. We constructed a special solution in the unit disk and gave an explicit description of singularities on its natural boundary. A global solution to Chazy equation in elliptic and theta functions was found that allows parametrization of an arbitrary solution to Chazy equation. The results have applications to analytic number theory.
Numerical optimization using flow equations.
Punk, Matthias
2014-12-01
We develop a method for multidimensional optimization using flow equations. This method is based on homotopy continuation in combination with a maximum entropy approach. Extrema of the optimizing functional correspond to fixed points of the flow equation. While ideas based on Bayesian inference such as the maximum entropy method always depend on a prior probability, the additional step in our approach is to perform a continuous update of the prior during the homotopy flow. The prior probability thus enters the flow equation only as an initial condition. We demonstrate the applicability of this optimization method for two paradigmatic problems in theoretical condensed matter physics: numerical analytic continuation from imaginary to real frequencies and finding (variational) ground states of frustrated (quantum) Ising models with random or long-range antiferromagnetic interactions.
The room acoustic rendering equation.
Siltanen, Samuel; Lokki, Tapio; Kiminki, Sami; Savioja, Lauri
2007-09-01
An integral equation generalizing a variety of known geometrical room acoustics modeling algorithms is presented. The formulation of the room acoustic rendering equation is adopted from computer graphics. Based on the room acoustic rendering equation, an acoustic radiance transfer method, which can handle both diffuse and nondiffuse reflections, is derived. In a case study, the method is used to predict several acoustic parameters of a room model. The results are compared to measured data of the actual room and to the results given by other acoustics prediction software. It is concluded that the method can predict most acoustic parameters reliably and provides results as accurate as current commercial room acoustic prediction software. Although the presented acoustic radiance transfer method relies on geometrical acoustics, it can be extended to model diffraction and transmission through materials in future.
Numerical optimization using flow equations
NASA Astrophysics Data System (ADS)
Punk, Matthias
2014-12-01
We develop a method for multidimensional optimization using flow equations. This method is based on homotopy continuation in combination with a maximum entropy approach. Extrema of the optimizing functional correspond to fixed points of the flow equation. While ideas based on Bayesian inference such as the maximum entropy method always depend on a prior probability, the additional step in our approach is to perform a continuous update of the prior during the homotopy flow. The prior probability thus enters the flow equation only as an initial condition. We demonstrate the applicability of this optimization method for two paradigmatic problems in theoretical condensed matter physics: numerical analytic continuation from imaginary to real frequencies and finding (variational) ground states of frustrated (quantum) Ising models with random or long-range antiferromagnetic interactions.
Explicit integration of Friedmann's equation with nonlinear equations of state
Chen, Shouxin; Gibbons, Gary W.; Yang, Yisong E-mail: gwg1@damtp.cam.ac.uk
2015-05-01
In this paper we study the integrability of the Friedmann equations, when the equation of state for the perfect-fluid universe is nonlinear, in the light of the Chebyshev theorem. A series of important, yet not previously touched, problems will be worked out which include the generalized Chaplygin gas, two-term energy density, trinomial Friedmann, Born-Infeld, two-fluid models, and Chern-Simons modified gravity theory models. With the explicit integration, we are able to understand exactly the roles of the physical parameters in various models play in the cosmological evolution which may also offer clues to a profound understanding of the problems in general settings. For example, in the Chaplygin gas universe, a few integrable cases lead us to derive a universal formula for the asymptotic exponential growth rate of the scale factor, of an explicit form, whether the Friedmann equation is integrable or not, which reveals the coupled roles played by various physical sectors and it is seen that, as far as there is a tiny presence of nonlinear matter, conventional linear matter makes contribution to the dark matter, which becomes significant near the phantom divide line. The Friedmann equations also arise in areas of physics not directly related to cosmology. We provide some examples ranging from geometric optics and central orbits to soap films and the shape of glaciated valleys to which our results may be applied.
Transport Equations In Tokamak Plasmas
NASA Astrophysics Data System (ADS)
Callen, J. D.
2009-11-01
Tokamak plasma transport equations are usually obtained by flux surface averaging the collisional Braginskii equations. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for: neoclassical effects on the parallel Ohm's law (trapped particle effects on resistivity, bootstrap current); fluctuation-induced transport; heating, current-drive and flow sources and sinks; small B field non-axisymmetries; magnetic field transients etc. A set of self-consistent second order in gyroradius fluid-moment-based transport equations for nearly axisymmetric tokamak plasmas has been developed recently using a kinetic-based framework. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales (and constraints they impose) are considered sequentially: compressional Alfv'en waves (Grad-Shafranov equilibrium, ion radial force balance); sound waves (pressure constant along field lines, incompressible flows within a flux surface); and ion collisions (damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on the plasma fluid: 7 ambipolar collision-based ones (classical, neoclassical, etc.) and 8 non-ambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients etc.). The plasma toroidal rotation equation [1] results from setting to zero the net radial current induced by the non-ambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the non-ambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport equations do not involve an ambipolar constraint and hence are more directly obtained. The resultant transport equations will be presented and contrasted with the usual ones. [4pt] [1] J.D. Callen, A.J. Cole, C.C. Hegna, ``Toroidal Rotation In
Transport equations in tokamak plasmas
Callen, J. D.; Hegna, C. C.; Cole, A. J.
2010-05-15
Tokamak plasma transport equations are usually obtained by flux surface averaging the collisional Braginskii equations. However, tokamak plasmas are not in collisional regimes. Also, ad hoc terms are added for neoclassical effects on the parallel Ohm's law, fluctuation-induced transport, heating, current-drive and flow sources and sinks, small magnetic field nonaxisymmetries, magnetic field transients, etc. A set of self-consistent second order in gyroradius fluid-moment-based transport equations for nearly axisymmetric tokamak plasmas has been developed using a kinetic-based approach. The derivation uses neoclassical-based parallel viscous force closures, and includes all the effects noted above. Plasma processes on successive time scales and constraints they impose are considered sequentially: compressional Alfven waves (Grad-Shafranov equilibrium, ion radial force balance), sound waves (pressure constant along field lines, incompressible flows within a flux surface), and collisions (electrons, parallel Ohm's law; ions, damping of poloidal flow). Radial particle fluxes are driven by the many second order in gyroradius toroidal angular torques on a plasma species: seven ambipolar collision-based ones (classical, neoclassical, etc.) and eight nonambipolar ones (fluctuation-induced, polarization flows from toroidal rotation transients, etc.). The plasma toroidal rotation equation results from setting to zero the net radial current induced by the nonambipolar fluxes. The radial particle flux consists of the collision-based intrinsically ambipolar fluxes plus the nonambipolar fluxes evaluated at the ambipolarity-enforcing toroidal plasma rotation (radial electric field). The energy transport equations do not involve an ambipolar constraint and hence are more directly obtained. The 'mean field' effects of microturbulence on the parallel Ohm's law, poloidal ion flow, particle fluxes, and toroidal momentum and energy transport are all included self-consistently. The
Young's Equation at the Nanoscale
NASA Astrophysics Data System (ADS)
Seveno, David; Blake, Terence D.; De Coninck, Joël
2013-08-01
In 1805, Thomas Young was the first to propose an equation to predict the value of the equilibrium contact angle of a liquid on a solid. Today, the force exerted by a liquid on a solid, such as a flat plate or fiber, is routinely used to assess this angle. Moreover, it has recently become possible to study wetting at the nanoscale using an atomic force microscope. Here, we report the use of molecular-dynamics simulations to investigate the force distribution along a 15 nm fiber dipped into a liquid meniscus. We find very good agreement between the measured force and that predicted by Young’s equation.
Investigation of the kinetic model equations.
Liu, Sha; Zhong, Chengwen
2014-03-01
Currently the Boltzmann equation and its model equations are widely used in numerical predictions for dilute gas flows. The nonlinear integro-differential Boltzmann equation is the fundamental equation in the kinetic theory of dilute monatomic gases. By replacing the nonlinear fivefold collision integral term by a nonlinear relaxation term, its model equations such as the famous Bhatnagar-Gross-Krook (BGK) equation are mathematically simple. Since the computational cost of solving model equations is much less than that of solving the full Boltzmann equation, the model equations are widely used in predicting rarefied flows, multiphase flows, chemical flows, and turbulent flows although their predictions are only qualitatively right for highly nonequilibrium flows in transitional regime. In this paper the differences between the Boltzmann equation and its model equations are investigated aiming at giving guidelines for the further development of kinetic models. By comparing the Boltzmann equation and its model equations using test cases with different nonequilibrium types, two factors (the information held by nonequilibrium moments and the different relaxation rates of high- and low-speed molecules) are found useful for adjusting the behaviors of modeled collision terms in kinetic regime. The usefulness of these two factors are confirmed by a generalized model collision term derived from a mathematical relation between the Boltzmann equation and BGK equation that is also derived in this paper. After the analysis of the difference between the Boltzmann equation and the BGK equation, an attempt at approximating the collision term is proposed.
The Forced Soft Spring Equation
ERIC Educational Resources Information Center
Fay, T. H.
2006-01-01
Through numerical investigations, this paper studies examples of the forced Duffing type spring equation with [epsilon] negative. By performing trial-and-error numerical experiments, the existence is demonstrated of stability boundaries in the phase plane indicating initial conditions yielding bounded solutions. Subharmonic boundaries are…
Sonar equations for planetary exploration.
Ainslie, Michael A; Leighton, Timothy G
2016-08-01
The set of formulations commonly known as "the sonar equations" have for many decades been used to quantify the performance of sonar systems in terms of their ability to detect and localize objects submerged in seawater. The efficacy of the sonar equations, with individual terms evaluated in decibels, is well established in Earth's oceans. The sonar equations have been used in the past for missions to other planets and moons in the solar system, for which they are shown to be less suitable. While it would be preferable to undertake high-fidelity acoustical calculations to support planning, execution, and interpretation of acoustic data from planetary probes, to avoid possible errors for planned missions to such extraterrestrial bodies in future, doing so requires awareness of the pitfalls pointed out in this paper. There is a need to reexamine the assumptions, practices, and calibrations that work well for Earth to ensure that the sonar equations can be accurately applied in combination with the decibel to extraterrestrial scenarios. Examples are given for icy oceans such as exist on Europa and Ganymede, Titan's hydrocarbon lakes, and for the gaseous atmospheres of (for example) Jupiter and Venus.
The Symbolism Of Chemical Equations
ERIC Educational Resources Information Center
Jensen, William B.
2005-01-01
A question about the historical origin of equal sign and double arrow symbolism in balanced chemical equation is raised. The study shows that Marshall proposed the symbolism in 1902, which includes the use of currently favored double barb for equilibrium reactions.
Renaissance Learning Equating Study. Report
ERIC Educational Resources Information Center
Sewell, Julie; Sainsbury, Marian; Pyle, Katie; Keogh, Nikki; Styles, Ben
2007-01-01
An equating study was carried out in autumn 2006 by the National Foundation for Educational Research (NFER) on behalf of Renaissance Learning, to provide validation evidence for the use of the Renaissance Star Reading and Star Mathematics tests in English schools. The study investigated the correlation between the Star tests and established tests.…
Wave-equation dispersion inversion
NASA Astrophysics Data System (ADS)
Li, Jing; Feng, Zongcai; Schuster, Gerard
2017-03-01
We present the theory for wave-equation inversion of dispersion curves, where the misfit function is the sum of the squared differences between the wavenumbers along the predicted and observed dispersion curves. The dispersion curves are obtained from Rayleigh waves recorded by vertical-component geophones. Similar to wave-equation traveltime tomography, the complicated surface wave arrivals in traces are skeletonized as simpler data, namely the picked dispersion curves in the phase-velocity and frequency domains. Solutions to the elastic wave equation and an iterative optimization method are then used to invert these curves for 2-D or 3-D S-wave velocity models. This procedure, denoted as wave-equation dispersion inversion (WD), does not require the assumption of a layered model and is significantly less prone to the cycle-skipping problems of full waveform inversion. The synthetic and field data examples demonstrate that WD can approximately reconstruct the S-wave velocity distributions in laterally heterogeneous media if the dispersion curves can be identified and picked. The WD method is easily extended to anisotropic data and the inversion of dispersion curves associated with Love waves.
Optimized solution of Kepler's equation
NASA Technical Reports Server (NTRS)
Kohout, J. M.; Layton, L.
1972-01-01
A detailed description is presented of KEPLER, an IBM 360 computer program used for the solution of Kepler's equation for eccentric anomaly. The program KEPLER employs a second-order Newton-Raphson differential correction process, and it is faster than previously developed programs by an order of magnitude.
Blink, J.A.
1983-09-01
In 1977, Dave Young published an equation-of-state (EOS) for lithium. This EOS was used by Lew Glenn in his AFTON calculations of the HYLIFE inertial-fusion-reactor hydrodynamics. In this paper, I summarize Young's development of the EOS and demonstrate a computer program (MATHSY) that plots isotherms, isentropes and constant energy lines on a P-V diagram.
The solution of transcendental equations
NASA Technical Reports Server (NTRS)
Agrawal, K. M.; Outlaw, R.
1973-01-01
Some of the existing methods to globally approximate the roots of transcendental equations namely, Graeffe's method, are studied. Summation of the reciprocated roots, Whittaker-Bernoulli method, and the extension of Bernoulli's method via Koenig's theorem are presented. The Aitken's delta squared process is used to accelerate the convergence. Finally, the suitability of these methods is discussed in various cases.
Empirical equation estimates geothermal gradients
Kutasov, I.M. )
1995-01-02
An empirical equation can estimate geothermal (natural) temperature profiles in new exploration areas. These gradients are useful for cement slurry and mud design and for improving electrical and temperature log interpretation. Downhole circulating temperature logs and surface outlet temperatures are used for predicting the geothermal gradients.
Pendulum Motion and Differential Equations
ERIC Educational Resources Information Center
Reid, Thomas F.; King, Stephen C.
2009-01-01
A common example of real-world motion that can be modeled by a differential equation, and one easily understood by the student, is the simple pendulum. Simplifying assumptions are necessary for closed-form solutions to exist, and frequently there is little discussion of the impact if those assumptions are not met. This article presents a…
Statistical Equating of Direct Writing Assessment.
ERIC Educational Resources Information Center
Phillips, Gary W.
This paper provides empirical data on two approaches to statistically equate scores derived from the direct assessment of writing. These methods are linear equating and equating based on the general polychotomous form of the Rasch model. Data from the Maryland Functional Writing Test are used to equate scores obtained from two prompts given in…
The Complexity of One-Step Equations
ERIC Educational Resources Information Center
Ngu, Bing
2014-01-01
An analysis of one-step equations from a cognitive load theory perspective uncovers variation within one-step equations. The complexity of one-step equations arises from the element interactivity across the operational and relational lines. The higher the number of operational and relational lines, the greater the complexity of the equations.…
A Bayesian Nonparametric Approach to Test Equating
ERIC Educational Resources Information Center
Karabatsos, George; Walker, Stephen G.
2009-01-01
A Bayesian nonparametric model is introduced for score equating. It is applicable to all major equating designs, and has advantages over previous equating models. Unlike the previous models, the Bayesian model accounts for positive dependence between distributions of scores from two tests. The Bayesian model and the previous equating models are…
Relativistic equations with fractional and pseudodifferential operators
Babusci, D.; Dattoli, G.; Quattromini, M.
2011-06-15
In this paper we use different techniques from the fractional and pseudo-operators calculus to solve partial differential equations involving operators with noninteger exponents. We apply the method to equations resembling generalizations of the heat equations and discuss the possibility of extending the procedure to the relativistic Schroedinger and Dirac equations.
Simple Derivation of the Lindblad Equation
ERIC Educational Resources Information Center
Pearle, Philip
2012-01-01
The Lindblad equation is an evolution equation for the density matrix in quantum theory. It is the general linear, Markovian, form which ensures that the density matrix is Hermitian, trace 1, positive and completely positive. Some elementary examples of the Lindblad equation are given. The derivation of the Lindblad equation presented here is…
Interplays between Harper and Mathieu equations.
Papp, E; Micu, C
2001-11-01
This paper deals with the application of relationships between Harper and Mathieu equations to the derivation of energy formulas. Establishing suitable matching conditions, one proceeds by inserting a concrete solution to the Mathieu equation into the Harper equation. For this purpose, one resorts to the nonlinear oscillations characterizing the Mathieu equation. This leads to the derivation of two kinds of energy formulas working in terms of cubic and quadratic algebraic equations, respectively. Combining such results yields quadratic equations to the energy description of the Harper equation, incorporating all parameters needed.
Lattice Boltzmann equation method for the Cahn-Hilliard equation
NASA Astrophysics Data System (ADS)
Zheng, Lin; Zheng, Song; Zhai, Qinglan
2015-01-01
In this paper a lattice Boltzmann equation (LBE) method is designed that is different from the previous LBE for the Cahn-Hilliard equation (CHE). The starting point of the present CHE LBE model is from the kinetic theory and the work of Lee and Liu [T. Lee and L. Liu, J. Comput. Phys. 229, 8045 (2010), 10.1016/j.jcp.2010.07.007]; however, because the CHE does not conserve the mass locally, a modified equilibrium density distribution function is introduced to treat the diffusion term in the CHE. Numerical simulations including layered Poiseuille flow, static droplet, and Rayleigh-Taylor instability have been conducted to validate the model. The results show that the predictions of the present LBE agree well with the analytical solution and other numerical results.
Isothermal Equation Of State For Compressed Solids
NASA Technical Reports Server (NTRS)
Vinet, Pascal; Ferrante, John
1989-01-01
Same equation with three adjustable parameters applies to different materials. Improved equation of state describes pressure on solid as function of relative volume at constant temperature. Even though types of interatomic interactions differ from one substance to another, form of equation determined primarily by overlap of electron wave functions during compression. Consequently, equation universal in sense it applies to variety of substances, including ionic, metallic, covalent, and rare-gas solids. Only three parameters needed to describe equation for given material.
Applications of film thickness equations
NASA Technical Reports Server (NTRS)
Hamrock, B. J.; Dowson, D.
1983-01-01
A number of applications of elastohydrodynamic film thickness expressions were considered. The motion of a steel ball over steel surfaces presenting varying degrees of conformity was examined. The equation for minimum film thickness in elliptical conjunctions under elastohydrodynamic conditions was applied to roller and ball bearings. An involute gear was also introduced, it was again found that the elliptical conjunction expression yielded a conservative estimate of the minimum film thickness. Continuously variable-speed drives like the Perbury gear, which present truly elliptical elastohydrodynamic conjunctions, are favored increasingly in mobile and static machinery. A representative elastohydrodynamic condition for this class of machinery is considered for power transmission equipment. The possibility of elastohydrodynamic films of water or oil forming between locomotive wheels and rails is examined. The important subject of traction on the railways is attracting considerable attention in various countries at the present time. The final example of a synovial joint introduced the equation developed for isoviscous-elastic regimes of lubrication.