Abstract
A set of key properties for an ideal dissipation scheme in gyrokinetic simulations is proposed, and implementation of a modelcollision operator satisfying these properties is described. This operator is based on the exact linearized testparticle collision operator, with approximations to the fieldparticle terms that preserve conservation laws and an theorem. It includes energy diffusion, pitchangle scattering, and finite Larmor radius effects corresponding to classical (realspace) diffusion. The numerical implementation in the continuum gyrokinetic code GS2 [Kotschenreuther et al., Comput. Phys. Comm.88, 128 (1995)] is fully implicit and guarantees exact satisfaction of conservation properties. Numerical results are presented showing that the correct physics is captured over the entire range of collisionalities, from the collisionless to the strongly collisional regimes, without recourse to artificial dissipation.
We thank S. C. Cowley, R. Numata, and I. Broemstrup for useful discussions. M.B., W.D., and P.R. were supported by the US DOE Center for Multiscale Plasma Dynamics. I.G.A. was supported by a CASE EPSRC studentship in association with UKAEA Fusion (Culham). A.A.S. was supported by an STFC (UK) Advanced Fellowship and STFC Grant ST/F002505/1. M.B., G.W.H., and W.D. would also like to thank the Leverhulme Trust (UK) International Network for Magnetized Plasma Turbulence for travel support.
I. INTRODUCTION
II. PROPERTIES OF THE GYROAVERAGED COLLISION OPERATOR
A. Collision operator amplitude
B. Local moment conservation
C. theorem
III. NUMERICAL IMPLEMENTATION
A. Conserving terms
B. Discretization in energy and pitch angle
IV. NUMERICAL TESTS
A. Homogeneous plasma slab
B. Resistive damping
C. Slow mode damping
D. Electrostatic turbulence
V. SUMMARY
Key Topics
 Plasma gyrokinetics
 34.0
 Diffusion
 11.0
 Collision theories
 9.0
 Integration
 9.0
 Entropy
 8.0
Figures
(Left) Solid line indicates the scaling of the leading order error, averaged over all grid points, of the conservative finite difference scheme for a Gauss–Legendre grid (the grid used in GS2). The slope of the dotted line corresponds to a first order scheme. (Right) Factor by which the conservative finite difference scheme of Eq. (43) amplifies the true collision operator amplitude at the boundaries of the Gauss–Legendre grid.
(Left) Solid line indicates the scaling of the leading order error, averaged over all grid points, of the conservative finite difference scheme for a Gauss–Legendre grid (the grid used in GS2). The slope of the dotted line corresponds to a first order scheme. (Right) Factor by which the conservative finite difference scheme of Eq. (43) amplifies the true collision operator amplitude at the boundaries of the Gauss–Legendre grid.
Plots showing evolution of the perturbed local density, parallel momentum, and energy over fifty collision times. Without the conserving terms (9)–(11), both parallel momentum and energy decay significantly over a few collision times (long dashed lines). Inclusion of conserving terms with the conservative scheme detailed in Sec. III leads to exact moment conservation (solid lines). Use of a nonconservative scheme leads to inexact conservation that depends on grid spacing (short dashed lines).
Plots showing evolution of the perturbed local density, parallel momentum, and energy over fifty collision times. Without the conserving terms (9)–(11), both parallel momentum and energy decay significantly over a few collision times (long dashed lines). Inclusion of conserving terms with the conservative scheme detailed in Sec. III leads to exact moment conservation (solid lines). Use of a nonconservative scheme leads to inexact conservation that depends on grid spacing (short dashed lines).
Plot of the evolution of entropy generation for the homogeneous plasma slab over 20 collision times. Our initial distribution in velocity space is random noise, and we use a grid with 16 pitch angles and 8 energies. The entropy generation rate is always nonnegative and approaches zero in the longtime limit.
Plot of the evolution of entropy generation for the homogeneous plasma slab over 20 collision times. Our initial distribution in velocity space is random noise, and we use a grid with 16 pitch angles and 8 energies. The entropy generation rate is always nonnegative and approaches zero in the longtime limit.
Evolution of for the electromagnetic plasma slab with , , and . Inclusion of the ion drag term in the electronion collision operator leads to the theoretically predicted damping rate for the parallel current given in Eq. (50). Without the ion drag term, the parallel current decays past zero (at ) and converges to a negative value as the electron flow damps to zero.
Evolution of for the electromagnetic plasma slab with , , and . Inclusion of the ion drag term in the electronion collision operator leads to the theoretically predicted damping rate for the parallel current given in Eq. (50). Without the ion drag term, the parallel current decays past zero (at ) and converges to a negative value as the electron flow damps to zero.
Evolution of perturbed parallel flow for the electromagnetic plasma slab with , , and . Without inclusion of the ion drag term in Eq. (12), the electron flow is erroneously damped to zero (instead of to the ion flow).
Evolution of perturbed parallel flow for the electromagnetic plasma slab with , , and . Without inclusion of the ion drag term in Eq. (12), the electron flow is erroneously damped to zero (instead of to the ion flow).
Damping rate of the slow mode for a range of collisionalities spanning the collisionless to strongly collisional regimes. Dashed lines correspond to the theoretical prediction for the damping rate in the collisional and collisionless limits. The solid line is the result obtained numerically with GS2. Vertical dotted lines denote approximate regions (collisional and collisionless) for which the analytic theory is valid.
Damping rate of the slow mode for a range of collisionalities spanning the collisionless to strongly collisional regimes. Dashed lines correspond to the theoretical prediction for the damping rate in the collisional and collisionless limits. The solid line is the result obtained numerically with GS2. Vertical dotted lines denote approximate regions (collisional and collisionless) for which the analytic theory is valid.
Evolution of ion particle and heat fluxes for an electrostatic, twospecies Zpinch simulation. We are considering and . The particle flux is indicated by the solid line and is given in units of . The heat flux is indicated by the dashed line and is given in units of . We see that a steady state is achieved for both fluxes without artificial dissipation.
Evolution of ion particle and heat fluxes for an electrostatic, twospecies Zpinch simulation. We are considering and . The particle flux is indicated by the solid line and is given in units of . The heat flux is indicated by the dashed line and is given in units of . We see that a steady state is achieved for both fluxes without artificial dissipation.
Linear growth rate spectrum of the entropy mode in a Z pinch for , where is major radius and is density gradient scale length. The solid line is the collisionless result, and the other lines represent the result of including collisions. The short dashed line corresponds to using only the Lorentz operator; the dotted line corresponds to the full model collision operator without conserving terms; and the long dashed line corresponds to the full model collision operator with conserving terms. All collisional cases were carried out with .
Linear growth rate spectrum of the entropy mode in a Z pinch for , where is major radius and is density gradient scale length. The solid line is the collisionless result, and the other lines represent the result of including collisions. The short dashed line corresponds to using only the Lorentz operator; the dotted line corresponds to the full model collision operator without conserving terms; and the long dashed line corresponds to the full model collision operator with conserving terms. All collisional cases were carried out with .
Tables
Sherman–Morrison variable definitions for Lorentz and energy diffusion operator equations
Sherman–Morrison variable definitions for Lorentz and energy diffusion operator equations
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