Solving gyrokinetic systems with higherorder time dependence
Abstract
We discuss theoretical and numerical aspects of gyrokinetics as a Lagrangian field theory when the field perturbation is introduced into the symplectic part. A consequence is that the field equations and particle equations of motion in general depend on the time derivatives of the field. The most wellknown example is when the parallel vector potential is introduced as a perturbation, where a time derivative of the field arises only in the equations of motion, so an explicit equation for the fields may still be written. We will consider the conceptually more problematic case where the timedependent fields appear in both the field equations and equations of motion, but where the additional term in the field equations is formally small. The conceptual issues were described by Burby (J. Plasma Phys., vol. 82 (3), 2016, 905820304): these terms lead to apparent additional degrees of freedom to the problem, so that the electric field now requires an initial condition, which is not required in lowfrequency (Darwin) Vlasov–Maxwell equations. Also, the small terms in the Euler–Lagrange equations are a singular perturbation, and these two issues are interlinked. For wellbehaved problems the apparent additional degrees of freedom are spurious, and the physically relevant solution may bemore »
 Authors:

 Univ. of Warick (United Kingdom)
 Publication Date:
 Research Org.:
 Princeton Plasma Physics Lab. (PPPL), Princeton, NJ (United States)
 Sponsoring Org.:
 Euratom research and training programme; USDOE
 OSTI Identifier:
 1668078
 Grant/Contract Number:
 AC0209CH11466
 Resource Type:
 Accepted Manuscript
 Journal Name:
 Journal of Plasma Physics
 Additional Journal Information:
 Journal Volume: 86; Journal Issue: 4; Journal ID: ISSN 00223778
 Publisher:
 Cambridge University Press
 Country of Publication:
 United States
 Language:
 English
 Subject:
 70 PLASMA PHYSICS AND FUSION TECHNOLOGY
Citation Formats
Sharma, A. Y., and McMillan, B. F. Solving gyrokinetic systems with higherorder time dependence. United States: N. p., 2020.
Web. doi:10.1017/s0022377820000653.
Sharma, A. Y., & McMillan, B. F. Solving gyrokinetic systems with higherorder time dependence. United States. doi:10.1017/s0022377820000653.
Sharma, A. Y., and McMillan, B. F. Thu .
"Solving gyrokinetic systems with higherorder time dependence". United States. doi:10.1017/s0022377820000653. https://www.osti.gov/servlets/purl/1668078.
@article{osti_1668078,
title = {Solving gyrokinetic systems with higherorder time dependence},
author = {Sharma, A. Y. and McMillan, B. F.},
abstractNote = {We discuss theoretical and numerical aspects of gyrokinetics as a Lagrangian field theory when the field perturbation is introduced into the symplectic part. A consequence is that the field equations and particle equations of motion in general depend on the time derivatives of the field. The most wellknown example is when the parallel vector potential is introduced as a perturbation, where a time derivative of the field arises only in the equations of motion, so an explicit equation for the fields may still be written. We will consider the conceptually more problematic case where the timedependent fields appear in both the field equations and equations of motion, but where the additional term in the field equations is formally small. The conceptual issues were described by Burby (J. Plasma Phys., vol. 82 (3), 2016, 905820304): these terms lead to apparent additional degrees of freedom to the problem, so that the electric field now requires an initial condition, which is not required in lowfrequency (Darwin) Vlasov–Maxwell equations. Also, the small terms in the Euler–Lagrange equations are a singular perturbation, and these two issues are interlinked. For wellbehaved problems the apparent additional degrees of freedom are spurious, and the physically relevant solution may be directly identified. Because we needed to assume that the system is well behaved for small perturbations when deriving gyrokinetic theory, we must continue to assume that when solving it, and the physical solutions are thus the regular ones. The spurious nature of the singular degrees of freedom may also be seen by changing coordinate systems so the varying field appears only in the Hamiltonian. We then describe how methods appropriate for singular perturbation theory may be used to solve these asymptotic equations numerically. We then describe a proofofprinciple implementation of these methods for an electrostatic strongflow gyrokinetic system; two basic test cases are presented to illustrate code functionality.},
doi = {10.1017/s0022377820000653},
journal = {Journal of Plasma Physics},
number = 4,
volume = 86,
place = {United States},
year = {2020},
month = {7}
}
Works referenced in this record:
Orb5: A global electromagnetic gyrokinetic code using the PIC approach in toroidal geometry
journal, June 2020
 Lanti, E.; Ohana, N.; Tronko, N.
 Computer Physics Communications, Vol. 251
Noncurvaturedriven modes in a transport barrier
journal, June 2005
 Rogers, B. N.; Dorland, W.
 Physics of Plasmas, Vol. 12, Issue 6
On scrape off layer plasma transport
journal, May 2001
 Krasheninnikov, S. I.
 Physics Letters A, Vol. 283, Issue 56
Mitigation of the cancellation problem in the gyrokinetic particleincell simulations of global electromagnetic modes
journal, August 2017
 Mishchenko, Alexey; Bottino, Alberto; Hatzky, Roman
 Physics of Plasmas, Vol. 24, Issue 8
Effect of finite ion Larmor radius on the Kelvin–Helmholtz instability
journal, October 1978
 Nagano, Hirosh
 Journal of Plasma Physics, Vol. 20, Issue 2
Difference approximations for singular perturbations of systems of ordinary differential equations
journal, October 1974
 Abrahamsson, L. R.; Keller, H. B.; Kreiss, H. O.
 Numerische Mathematik, Vol. 22, Issue 5
Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks
journal, February 1995
 Brizard, Alain J.
 Physics of Plasmas, Vol. 2, Issue 2
The global version of the gyrokinetic turbulence code GENE
journal, August 2011
 Görler, T.; Lapillonne, X.; Brunner, S.
 Journal of Computational Physics, Vol. 230, Issue 18
Radial convection of finite ion temperature, high amplitude plasma blobs
journal, September 2014
 Wiesenberger, M.; Madsen, J.; Kendl, A.
 Physics of Plasmas, Vol. 21, Issue 9
A Modification of the GuidingCentre Fundamental 1Form with Strong E × B Flow
journal, October 2009
 Miyato, Naoaki; D. Scott, Bruce; Strintzi, Dafni
 Journal of the Physical Society of Japan, Vol. 78, Issue 10
Transport and evolution of ion gyroscale plasma blobs in perpendicular magnetic fields
journal, May 2012
 Gingell, P. W.; Chapman, S. C.; Dendy, R. O.
 Plasma Physics and Controlled Fusion, Vol. 54, Issue 6
Full f gyrokinetic method for particle simulation of tokamak transport
journal, May 2008
 Heikkinen, J. A.; Janhunen, S. J.; Kiviniemi, T. P.
 Journal of Computational Physics, Vol. 227, Issue 11
A very general electromagnetic gyrokinetic formalism
journal, September 2016
 McMillan, B. F.; Sharma, A.
 Physics of Plasmas, Vol. 23, Issue 9
A reanalysis of a strongflow gyrokinetic formalism
journal, March 2015
 Sharma, A. Y.; McMillan, B. F.
 Physics of Plasmas, Vol. 22, Issue 3
The influence of finite Larmor radius effects on the radial interchange motions of plasma filaments
journal, November 2011
 Madsen, Jens; Garcia, Odd E.; Stærk Larsen, Jeppe
 Physics of Plasmas, Vol. 18, Issue 11
Particle simulations with a generalized gyrokinetic solver
journal, June 2005
 Mishchenko, Alexey; Könies, Axel; Hatzky, Roman
 Physics of Plasmas, Vol. 12, Issue 6
Limitations of gyrokinetics on transport time scales
journal, April 2008
 Parra, Felix I.; Catto, Peter J.
 Plasma Physics and Controlled Fusion, Vol. 50, Issue 6
The initial value problem in Lagrangian drift kinetic theory
journal, May 2016
 Burby, J. W.
 Journal of Plasma Physics, Vol. 82, Issue 3
The splitweight particle simulation scheme for plasmas
journal, May 2000
 Manuilskiy, Igor; Lee, W. W.
 Physics of Plasmas, Vol. 7, Issue 5
Stationary Spectrum of Strong Turbulence in Magnetized Nonuniform Plasma
journal, July 1977
 Hasegawa, Akira; Mima, Kunioki
 Physical Review Letters, Vol. 39, Issue 4
Quasi‐two‐dimensional dynamics of plasmas and fluids
journal, June 1994
 Horton, Wendell; Hasegawa, Akira
 Chaos: An Interdisciplinary Journal of Nonlinear Science, Vol. 4, Issue 2
Electromagnetic full gyrokinetics in the tokamak edge with discontinuous Galerkin methods
journal, February 2020
 Mandell, N. R.; Hakim, A.; Hammett, G. W.
 Journal of Plasma Physics, Vol. 86, Issue 1
A numerical instability in an ADI algorithm for gyrokinetics
journal, November 2005
 Belli, E. A.; Hammett, G. W.
 Computer Physics Communications, Vol. 172, Issue 2
Foundations of nonlinear gyrokinetic theory
journal, April 2007
 Brizard, A. J.; Hahm, T. S.
 Reviews of Modern Physics, Vol. 79, Issue 2