Perturbative variational formulation of the Vlasov-Maxwell equations

Brizard, Alain J.

doi:10.1063/1.5049570

Title: Perturbative variational formulation of the Vlasov-Maxwell equations

Journal Article · Mon Nov 26 00:00:00 EST 2018 · Physics of Plasmas

DOI:https://doi.org/10.1063/1.5049570· OSTI ID:1612062

^[1]

Saint Michael's College, Colchester, VT (United States)

The perturbative variational formulation of the Vlasov-Maxwell equations is presented up to the third order in the perturbation analysis. From the second and third-order Lagrangian densities, the first-order and second-order Vlasov-Maxwell equations are expressed in gauge-invariant and gauge-independent forms, respectively. Upon deriving the reduced second-order Vlasov-Maxwell Lagrangian for the linear nonadiabatic gyrokinetic Vlasov-Maxwell equations, the reduced Lagrangian densities for the linear drift-wave equation and the linear hybrid kinetic-magnetohydrodynamic (MHD) equations are derived, with their associated wave-action conservation laws obtained by the Noether method. The exact wave-action conservation law for the linear hybrid kinetic-MHD equations is written explicitly. To conclude, a new form of the third-order Vlasov-Maxwell Lagrangian is derived in which ponderomotive effects play a crucial role.

View Accepted Manuscript (DOE)

Cite

Export

Save

Research Organization:: Saint Michael's College, Colchester, VT (United States)

Sponsoring Organization:: USDOE Office of Science (SC); National Science Foundation (NSF)

Grant/Contract Number:: SC0014032; PHY-1805164

OSTI ID:: 1612062

Journal Information:: Physics of Plasmas, Vol. 25, Issue 11; ISSN 1070-664X

Publisher:: American Institute of Physics (AIP)Copyright Statement

Country of Publication:: United States

Language:: English

Citation Metrics:

Cited by: 2 works

Citation information provided by
Web of Science

References (42)

Double-Cross Instability: An Absolute Instability Caused by Counter-Propagating Positive- and Negative-Energy Waves Brizard, A. J.; Morehead, J. J.; Kaufman, A. N. Physical Review Letters, Vol. 77, Issue 8 https://doi.org/10.1103/PhysRevLett.77.1500	journal	August 1996
Physics of Alfvén waves and energetic particles in burning plasmas Chen, Liu; Zonca, Fulvio Reviews of Modern Physics, Vol. 88, Issue 1 https://doi.org/10.1103/RevModPhys.88.015008	journal	March 2016
How a Wave Flips Its Energy Sign by Linear Conversion Brizard, A. J.; Kaufman, A. N. Physical Review Letters, Vol. 76, Issue 10 https://doi.org/10.1103/PhysRevLett.76.1639	journal	March 1996
Invariant variation problems Noether, Emmy Transport Theory and Statistical Physics, Vol. 1, Issue 3 https://doi.org/10.1080/00411457108231446	journal	January 1971
Kinetic theory of a weakly unstable plasma Frieman, E.; Rutherford, P. Annals of Physics, Vol. 28, Issue 1 https://doi.org/10.1016/0003-4916(64)90145-9	journal	June 1964
Variational principles of guiding centre motion Littlejohn, Robert G. Journal of Plasma Physics, Vol. 29, Issue 1 https://doi.org/10.1017/S002237780000060X	journal	February 1983
Hamiltonian theory of guiding-center motion Cary, John R.; Brizard, Alain J. Reviews of Modern Physics, Vol. 81, Issue 2 https://doi.org/10.1103/RevModPhys.81.693	journal	May 2009
On wave-action and its relatives Andrews, D. G.; Mcintyre, M. E. Journal of Fluid Mechanics, Vol. 89, Issue 4 https://doi.org/10.1017/S0022112078002785	journal	December 1978
The Gyrokinetic Description of Microturbulence in Magnetized Plasmas Krommes, John A. Annual Review of Fluid Mechanics, Vol. 44, Issue 1 https://doi.org/10.1146/annurev-fluid-120710-101223	journal	January 2012
Gyrokinetic simulations of turbulent transport Garbet, X.; Idomura, Y.; Villard, L. Nuclear Fusion, Vol. 50, Issue 4 https://doi.org/10.1088/0029-5515/50/4/043002	journal	March 2010
Excitation of Internal Kink Modes by Trapped Energetic Beam Ions Chen, Liu; White, R. B.; Rosenbluth, M. N. Physical Review Letters, Vol. 52, Issue 13 https://doi.org/10.1103/PhysRevLett.52.1122	journal	March 1984
Hermitian structure for linearized ideal MHD equations with equilibrium flows Brizard, Alain Physics Letters A, Vol. 168, Issue 5-6 https://doi.org/10.1016/0375-9601(92)90518-Q	journal	September 1992
Nonlinear interactions of positive and negative energy modes in rarefied plasmas (I) Coppi, B.; Rosenbluth, M. N.; Sudan, R. N. Annals of Physics, Vol. 55, Issue 2 https://doi.org/10.1016/0003-4916(69)90178-X	journal	November 1969
Double-crossing mode conversion in nonuniform media Brizard, A. J.; Morehead, J. J.; Kaufman, A. N. Physics of Plasmas, Vol. 5, Issue 1 https://doi.org/10.1063/1.872674	journal	January 1998
Lagrangian studies of plasma wave interactions. I Boyd, T. J. M.; Turner, J. G. Journal of Physics A: General Physics, Vol. 5, Issue 6 https://doi.org/10.1088/0305-4470/5/6/013	journal	June 1972
Variational principles for reduced plasma physics Brizard, Alain J. Journal of Physics: Conference Series, Vol. 169 https://doi.org/10.1088/1742-6596/169/1/012003	journal	May 2009
Generalized action invariants for drift waves-zonal flow systems Smolyakov, A. I.; Diamond, P. H. Physics of Plasmas, Vol. 6, Issue 12 https://doi.org/10.1063/1.873725	journal	December 1999
Variational structure for dissipationless linear drift‐wave equations Brizard, Alain J. Physics of Plasmas, Vol. 3, Issue 3 https://doi.org/10.1063/1.871808	journal	March 1996
Foundations of nonlinear gyrokinetic theory Brizard, A. J.; Hahm, T. S. Reviews of Modern Physics, Vol. 79, Issue 2 https://doi.org/10.1103/RevModPhys.79.421	journal	April 2007
Local Manley-Rowe Relations for Noneikonal Wave Fields Brizard, A. J.; Kaufman, A. N. Physical Review Letters, Vol. 74, Issue 23 https://doi.org/10.1103/PhysRevLett.74.4567	journal	June 1995
Linear‐conversion theory of energetic minority‐ion Bernstein‐wave propagation across gyroresonance in nonuniform magnetic field Brizard, A. J.; Kaufman, A. N. Physics of Plasmas, Vol. 3, Issue 1 https://doi.org/10.1063/1.871824	journal	January 1996
Nonlinear ideal magnetohydrodynamics instabilities Pfirsch, D.; Sudan, R. N. Physics of Fluids B: Plasma Physics, Vol. 5, Issue 7 https://doi.org/10.1063/1.860792	journal	July 1993
Beyond linear gyrocenter polarization in gyrokinetic theory Brizard, Alain J. Physics of Plasmas, Vol. 20, Issue 9 https://doi.org/10.1063/1.4823716	journal	September 2013
On the dynamical reduction of the Vlasov equation Brizard, Alain J. Communications in Nonlinear Science and Numerical Simulation, Vol. 13, Issue 1 https://doi.org/10.1016/j.cnsns.2007.05.006	journal	February 2008
Quadratic free energy for the linearized gyrokinetic Vlasov–Maxwell equations Brizard, Alain Physics of Plasmas, Vol. 1, Issue 8 https://doi.org/10.1063/1.870575	journal	August 1994
Eulerian action principles for linearized reduced dynamical equations Brizard, Alain Physics of Plasmas, Vol. 1, Issue 8 https://doi.org/10.1063/1.870574	journal	August 1994
Hamiltonian perturbation theory in noncanonical coordinates Littlejohn, Robert G. Journal of Mathematical Physics, Vol. 23, Issue 5 https://doi.org/10.1063/1.525429	journal	May 1982
Ponderomotive effects in collisionless plasma: A Lie transform approach Cary, John R. Physics of Fluids, Vol. 24, Issue 7 https://doi.org/10.1063/1.863527	journal	January 1981
The free energy of Maxwell–Vlasov equilibria Morrison, P. J.; Pfirsch, D. Physics of Fluids B: Plasma Physics, Vol. 2, Issue 6 https://doi.org/10.1063/1.859246	journal	June 1990
Drift wave propagation as a source of plasma edge turbulence: Slab theory Mattor, Nathan; Diamond, Patrick H. Physics of Plasmas, Vol. 1, Issue 12 https://doi.org/10.1063/1.870870	journal	December 1994
Lie transform perturbation theory for Hamiltonian systems Cary, J. Physics Reports, Vol. 79, Issue 2 https://doi.org/10.1016/0370-1573(81)90175-7	journal	December 1981
Regular and Stochastic Motion Lichtenberg, A. J.; Lieberman, M. A. Applied Mathematical Sciences https://doi.org/10.1007/978-1-4757-4257-2	book	January 1983
Variational formulation for weakly nonlinear perturbations of ideal magnetohydrodynamics Hirota, M. Journal of Plasma Physics, Vol. 77, Issue 5 https://doi.org/10.1017/S0022377810000760	journal	January 2011
Nonlinear gyrokinetic Maxwell-Vlasov equations using magnetic co-ordinates Brizard, A. Journal of Plasma Physics, Vol. 41, Issue 3 https://doi.org/10.1017/S0022377800014070	journal	June 1989
Excitation of zonal flow by drift waves in toroidal plasmas Chen, Liu; Lin, Zhihong; White, Roscoe Physics of Plasmas, Vol. 7, Issue 8 https://doi.org/10.1063/1.874222	journal	August 2000
On Hydromagnetic Stability of Stationary Equilibria Frieman, E.; Rotenberg, Manuel Reviews of Modern Physics, Vol. 32, Issue 4 https://doi.org/10.1103/RevModPhys.32.898	journal	October 1960
Ray Tracing and Beyond Tracy, E. R.; Brizard, A. J.; Richardson, A. S. Cambridge University Press https://doi.org/10.1017/CBO9780511667565	book	January 2014
A geometric view of Hamiltonian perturbation theory Brizard, Alain J. Physics Letters A, Vol. 291, Issue 2-3 https://doi.org/10.1016/S0375-9601(01)00722-8	journal	December 2001
Wave-action conservation for pseudo-Hermitian fields Brizard, Alain J.; Cook, Daniel R.; Kaufman, Allan N. Physical Review Letters, Vol. 70, Issue 5 https://doi.org/10.1103/PhysRevLett.70.521	journal	February 1993
Variational Principles for Reduced Plasma Physics Brizard, Alain J. arXiv https://doi.org/10.48550/arxiv.0812.1737	text	January 2008
Beyond linear gyrocenter polarization in gyrokinetic theory Brizard, Alain J. arXiv https://doi.org/10.48550/arxiv.1306.6942	text	January 2013
Generalized Action Invariants for Drift Waves-Zonal Flow Systems Smolyakov, A. I.; Diamond, P. H. arXiv https://doi.org/10.48550/arxiv.physics/9910018	text	January 1999

Similar Records

Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems

Journal Article · Tue May 19 00:00:00 EDT 2020 · Journal of Physics. A, Mathematical and Theoretical · OSTI ID:1612062

Hirvijoki, Eero; Burby, Joshua W.; Pfefferlé, David; +1 more

Variational principle for the parallel-symplectic representation of electromagnetic gyrokinetic theory

Journal Article · Fri Aug 04 00:00:00 EDT 2017 · Physics of Plasmas · OSTI ID:1612062

Brizard, Alain J.

Variational principle for nonlinear gyrokinetic Vlasov--Maxwell equations

Journal Article · Fri Dec 01 00:00:00 EST 2000 · Physics of Plasmas · OSTI ID:1612062

Brizard, Alain J

Related Subjects

71 CLASSICAL AND QUANTUM MECHANICS
GENERAL PHYSICS
physics

Title: Perturbative variational formulation of the Vlasov-Maxwell equations

Citation Formats

References (42)

Similar Records

Related Subjects