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Exact conservation laws for gauge-free electromagnetic gyrokinetic equations

Part of: Focus on Fusion

Published online by Cambridge University Press:  25 May 2021

Alain J. Brizard Open the ORCID record for Alain J. Brizard [Opens in a new window]
Alain J. Brizard*
Affiliation:
Department of Physics, Saint Michael's College, Colchester, VT05439, USA
*
†Email address for correspondence: [email protected]
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Abstract

The exact energy and angular momentum conservation laws are derived by the Noether method for the Hamiltonian and symplectic representations of the gauge-free electromagnetic gyrokinetic Vlasov–Maxwell equations. These gyrokinetic equations, which are solely expressed in terms of electromagnetic fields, describe the low-frequency turbulent fluctuations that perturb a time-independent toroidally-axisymmetric magnetized plasma. The explicit proofs presented here provide a complete picture of the transfer of energy and angular momentum between the gyrocentres and the perturbed electromagnetic fields, in which the crucial roles played by gyrocentre polarization and magnetization effects are highlighted. In addition to yielding an exact angular momentum conservation law, the gyrokinetic Noether equation yields an exact momentum transport equation, which might be useful in more general equilibrium magnetic geometries.

Keywords

plasma nonlinear phenomenaplasma dynamics
Type
Research Article
Information
Journal of Plasma Physics , Volume 87 , Issue 3 , June 2021 , 905870307
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Copyright
Copyright © The Author(s), 2021. Published by Cambridge University Press

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References

REFERENCES

Abiteboul, J., Garbet, X., Grandgirard, V., Allfrey, S.J., Ghendrih, P., Latu, G., Sarazin, Y. & Strugarek, A. 2011 Conservation equations and calculation of mean flows in gyrokinetics. Phys. Plasmas 18, 082503.CrossRefGoogle Scholar
Belova, E.V., Gorlenkov, N.N. & Cheng, C.Z. 2003 Self-consistent equilibrium model of low aspect-ratio toroidal plasma with energetic beam ions. Phys. Plasmas 10, 3240–3251.CrossRefGoogle Scholar
Brizard, A. 1989 a Gyrokinetic energy conservation and Poisson-bracket formulation. Phys. Fluids B 1, 1381–1384.CrossRefGoogle Scholar
Brizard, A.J. 1989 b Nonlinear gyrokinetic Maxwell–Vlasov equations using magnetic coordinates. J. Plasma Phys. 41, 541–559.CrossRefGoogle Scholar
Brizard, A.J. 1995 Nonlinear gyrokinetic vlasov equation for toroidally rotating axisymmetric tokamaks. Phys. Plasmas 2, 459–471.CrossRefGoogle Scholar
Brizard, A.J. 2000 a A new variational principle for Vlasov–Maxwell equations. Phys. Rev. Lett. 84, 5768–5771.CrossRefGoogle ScholarPubMed
Brizard, A.J. 2000 b Variational principle for nonlinear gyrokinetic Vlasov–Maxwell equations. Phys. Plasmas 7, 4816–4822.CrossRefGoogle Scholar
Brizard, A.J. 2005 a Energy-conserving finite-beta electromagnetic drift-fluid equations. Phys. Plasmas 12, 092302.CrossRefGoogle Scholar
Brizard, A.J. 2005 b Noether methods for fluids and plasmas. J. Plasma Phys. 71, 225–236.CrossRefGoogle Scholar
Brizard, A.J. 2008 On the dynamical reduction of the Vlasov equation. Commun. Nonlinear Sci. Numer. Simul. 13, 24–33.CrossRefGoogle Scholar
Brizard, A.J. 2009 Variational principles for reduced plasma physics. J. Phys.: Conf. Ser. 169, 012003.Google Scholar
Brizard, A.J. 2010 a Exact energy conservation laws for full and truncated nonlinear gyrokinetic equations. Phys. Plasmas 17, 042303.CrossRefGoogle Scholar
Brizard, A.J. 2010 b Noether derivation of exact conservation laws for dissipationless reduced fluid models. Phys. Plasmas 17, 112503.CrossRefGoogle Scholar
Brizard, A.J. 2017 Variational principle for the parallel-symplectic representation of electromagnetic gyrokinetic theory. Phys. Plasmas 24, 081201.CrossRefGoogle Scholar
Brizard, A.J. 2020 Symplectic gyrokinetic Vlasov–Maxwell theory. arXiv:1907.11204v2.Google Scholar
Brizard, A.J. & Chan, A.A. 1999 Nonlinear relativistic gyrokinetic Vlasov–Maxwell equations. Phys. Plasmas 6, 4548–4558.CrossRefGoogle Scholar
Brizard, A.J., Denton, R.E., Rogers, B. & Lotko, W. 2008 Nonlinear finite-Larmor-radius effects in reduced fluid models. Phys. Plasmas 15, 082302.CrossRefGoogle Scholar
Brizard, A.J. & Hahm, T.S. 2007 Foundations of nonlinear gyrokinetic theory. Rev. Mod. Phys. 79, 421–468.CrossRefGoogle Scholar
Brizard, A.J. & Tronci, C. 2016 Variational formulations of guiding-center Vlasov–Maxwell theory. Phys. Plasmas 23, 062107.CrossRefGoogle Scholar
Brizard, A.J. & Tronko, N. 2011 Exact momentum conservation laws for the gyrokinetic Vlasov–Poisson equations. Phys. Plasmas 18, 082307.CrossRefGoogle Scholar
Burby, J.W. & Brizard, A.J. 2019 Gauge-free electromagnetic gyrokinetic theory. Phys. Lett. A 383, 2172–2175.CrossRefGoogle Scholar
Burby, J.W., Brizard, A.J., Morrison, P.J. & Qin, H. 2015 Hamiltonian gyrokinetic Vlasov–Maxwell system. Phys. Lett. A 379, 2073–2077.CrossRefGoogle Scholar
Cary, J.R. & Kaufman, A.N. 1981 Ponderomotive effects in collisionless plasmas: a Lie transform approach. Phys. Fluids 24, 1238–1250.CrossRefGoogle Scholar
Chen, L., Lin, Y., Wang, X.Y. & Bao, J. 2019 A new particle simulation scheme using electromagnetic fields. Plasma Phys. Control. Fusion 61, 035004.CrossRefGoogle Scholar
Chen, L., Zonca, F. & Chen, H. 2020 Unexpanded nonlinear electromagnetic gyrokinetic equations for magnetized plasmas. Plasma Sci. Tech. 22, 102001.CrossRefGoogle Scholar
Chen, Y. & Parker, S.E. 2009 Particle-in-cell simulation with Vlasov ions and drift kinetic electrons. Phys. Plasmas 16, 052305.CrossRefGoogle Scholar
Correa-Restrepo, D. & Pfirsch, D. 2004 New method of deriving local energy- and momentum-conserving Maxwell-collisionless drift-kinetic and gyrokinetic theories: conservation laws and their structures. J. Plasma Phys. 70, 757–797.CrossRefGoogle Scholar
Dubin, D.H.E., Krommes, J.A., Oberman, C. & Lee, W.W. 1983 Nonlinear gyrokinetic equations. Phys. Fluids 26, 3524–3535.CrossRefGoogle Scholar
Duthoit, F.-X., Hahm, T.S. & Wang, L. 2014 Electromagnetic nonlinear gyrokinetics with polarization drift. Phys. Plasmas 21, 082301.CrossRefGoogle Scholar
Fan, P., Qin, H. & Xiao, J. 2020 Discovering exact local energy-momentum conservation laws for electromagnetic gyrokinetic system by high-order field theory on heterogeneous manifolds. arXiv:2006.11039v2.Google Scholar
Frieman, E.A. & Chen, L. 1982 Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria. Phys. Fluids 25, 502–508.CrossRefGoogle Scholar
Garbet, X., Idomura, Y., Villard, L. & Watanabe, T.H. 2010 Gyrokinetic simulations of turbulent transport. Nucl. Fusion 50, 043002.CrossRefGoogle Scholar
Goldstein, H., Poole, C. & Safko, J. 2002 Classical Mechanics, 3rd edn. Addison Wesley.Google Scholar
Hahm, T.S., Diamond, P.H., Gürcan, Ö. D. & Rewoldt, G. 2007 Nonlinear gyrokinetic theory of toroidal momentum pinch. Phys. Plasmas 14, 072302.CrossRefGoogle Scholar
Hahm, T.S., Lee, W.W. & Brizard, A.J. 1988 Nonlinear gyrokinetic theory for finite-beta plasmas. Phys. Fluids 31, 1940–1948.CrossRefGoogle Scholar
Hirvijoki, E., Burby, J.W., Pfefferlé, D. & Brizard, A.J. 2020 Energy and momentum conservation in the Euler–Poincaré formulation of local Vlasov–Maxwell-type systems. J. Phys. A: Theor. Math. 53, 235204.CrossRefGoogle Scholar
Landau, L.D. & Lifshitz, E.M. 1971 The Classical Theory of Fields, 3rd edn. Pergamon.Google Scholar
Leerink, S., Parra, F.I. & Heikkinen, J.A. 2010 Comment on ‘Nonlinear gyrokinetic theory with polarization drift’. Phys. Plasmas 17, 124701.CrossRefGoogle Scholar
McDevitt, C.J., Diamond, P.H., Gürcan, Ö.D. & Hahm, T.S. 2009 Toroidal rotation driven by the polarization drift. Phys. Rev. Lett. 103, 205003.CrossRefGoogle ScholarPubMed
Parra, F.I. & Catto, P.J. 2010 a Transport of momentum in full f gyrokinetics. Phys. Plasmas 17, 056106.CrossRefGoogle Scholar
Parra, F.I. & Catto, P.J. 2010 b Turbulent transport of toroidal angular momentum in low flow gyrokinetics. Plasma Phys. Control. Fusion 52, 045004.CrossRefGoogle Scholar
Peeters, A.G., Angioni, C., Bortolon, A., Camenen, Y., Casson, F.J., Duval, B., Fiederspiel, L., Hornsby, W.A., Idomura, Y., Hein, T., et al. 2011 Overview of toroidal momentum transport. Nucl. Fusion 51, 094027.CrossRefGoogle Scholar
Pfirsch, D. & Morrison, P.J. 1985 Local conservation laws for the Maxwell–Vlasov and collisionless guiding-center theories. Phys. Rev. A 32, 1714–1721.CrossRefGoogle ScholarPubMed
Porazik, P. & Lin, Z. 2011 Gyrokinetic simulation of magnetic compressional modes in general geometry. Commun. Comput. Phys. 10, 899–911.CrossRefGoogle Scholar
Scott, B. & Smirnov, J. 2010 Energetic consistency and momentum conservation in the gyrokinetic description of tokamak plasmas. Phys. Plasmas 17, 112302.CrossRefGoogle Scholar
Similon, P.L. 1985 Conservation laws for relativistic guiding-center plasmas. Phys. Lett. A 112, 33–37.CrossRefGoogle Scholar
Squire, J., Qin, H., Tang, W.M. & Chandre, C. 2013 The Hamiltonian structure and Euler–Poincaré formulation of the Vlasov–Maxwell and gyrokinetic systems. Phys. Plasmas 20, 022501.CrossRefGoogle Scholar
Stoltzfus-Dueck, T. 2019 Intrinsic rotation in axisymmetric devices. Plasma Phys. Control. Fusion 61, 124003.CrossRefGoogle Scholar
Strintzi, D., Scott, B.D. & Brizard, A.J. 2005 Nonlocal nonlinear electrostatic gyrofluid equations: a four-moment model. Phys. Plasmas 12, 052517.CrossRefGoogle Scholar
Sugama, H. 2000 Gykokinetic field theory. Phys. Plasmas 7, 466–480.CrossRefGoogle Scholar
Sugama, H., Matsuoka, S., Nunami, M. & Satake, S. 2021 The eulerian variational formulation of the gyrokinetic system in general spatial coordinates. Phys. Plasmas 28, 022312.CrossRefGoogle Scholar
Sugama, H., Matsuoka, S., Satake, S. & Kanno, R. 2016 Radially local approximation of the drift kinetic equation. Phys. Plasmas 23, 042502.CrossRefGoogle Scholar
Tronko, N. & Brizard, A.J. 2015 Lagrangian and Hamiltonian constraints for guiding-center Hamiltonian theories. Phys. Plasmas 22, 112507.CrossRefGoogle Scholar
Waltz, R.E., Staebler, G.M., Candy, J. & Hinton, F.L. 2007 Gyrokinetic theory and simulation of angular momentum transport. Phys. Plasmas 14, 122507.CrossRefGoogle Scholar
Wang, L. & Hahm, T.S. 2010 a Nonlinear gyrokinetic theory with polarization drift. Phys. Plasmas 17, 082304.CrossRefGoogle Scholar
Wang, L. & Hahm, T.S. 2010 b Response to “Comment on ‘Nonlinear gyrokinetic theory with polarization drift’ ”. Phys. Plasmas 17, 124702.CrossRefGoogle Scholar
Wang, L., Peng, S. & Diamond, P.H. 2018 Gyrokinetic theory of turbulent acceleration and momentum conservation in tokamak plasmas. Plasma Sci. Technol. 20, 074004.CrossRefGoogle Scholar