Long time gyrokinetic equations
Authors:
Christophe Cheverry and Shahnaz Farhat
Journal:
Quart. Appl. Math.
MSC (2020):
Primary 35Q60, 34E05, 34E20, 78A35
DOI:
https://doi.org/10.1090/qam/1666
Published electronically:
June 15, 2023
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Abstract: The aim of this text is to elucidate the oscillating patterns (see C. Cheverry [Res. Rep. Math. (2018)]) which are generated in a toroidal plasma by a strong external magnetic field and a nonzero electric field. It is also to justify and then study new modulation equations which are valid for longer times than before. Oscillating coherent structures are induced by the collective motions of charged particles which satisfy a system of ODEs implying a large parameter, the gyrofrequency $\varepsilon ^{-1} \gg 1$. By exploiting the properties of underlying integrable systems, we can complement the KAM picture (see G. Benettin and P. Sempio [Nonlinearity 7 (1994), pp. 281–303]; M. Braun [SIAM Rev. 23 (1981), pp. 61–93]) and go beyond the classical results about gyrokinetics (see M. Bostan [Multiscale Model. Simul. 8 (2010), pp. 1923–1957]; A. J. Brizard and T. S. Hahm [Rev. Modern Phys. 79 (2007), pp. 421–468]). The purely magnetic situation was addressed by C. Cheverry [Comm. Math. Phys. 338 (2015), pp. 641–703; J. Differential Equations 262 (2017), pp. 2987–3033]. We are concerned here with the numerous additional difficulties due to the influence of a nonzero electric field.
References
- Giancarlo Benettin and Paolo Sempio, Adiabatic invariants and trapping of a point charge in a strong nonuniform magnetic field, Nonlinearity 7 (1994), no. 1, 281–303. MR 1260143
- Mihai Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, Multiscale Model. Simul. 8 (2010), no. 5, 1923–1957. MR 2769087, DOI 10.1137/090777621
- Martin Braun, Mathematical remarks on the Van Allen radiation belt: a survey of old and new results, SIAM Rev. 23 (1981), no. 1, 61–93. MR 605441, DOI 10.1137/1023005
- Alain J. Brizard, Jacobi zeta function and action-angle coordinates for the pendulum, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), no. 3, 511–518. MR 2990693, DOI 10.1016/j.cnsns.2012.08.023
- A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys. 79 (2007), no. 2, 421–468. MR 2336960, DOI 10.1103/RevModPhys.79.421
- P. J. Catto, A. N. Simakov, and Los Alamos National Laboratory, Evaluation of the neoclassical radial electric field in a collisional tokamak, Physics of Plasmas 12 (2005), no. 1.
- Christophe Cheverry, Can one hear whistler waves?, Comm. Math. Phys. 338 (2015), no. 2, 641–703. MR 3351054, DOI 10.1007/s00220-015-2389-6
- Christophe Cheverry, Anomalous transport, J. Differential Equations 262 (2017), no. 3, 2987–3033. MR 3582249, DOI 10.1016/j.jde.2016.11.012
- C. Cheverry, Mathematical perspectives in plasma turbulence, Research and Reports on Mathematics, 2018.
- Rémi Carles and Christophe Cheverry, Constructive and destructive interferences in nonlinear hyperbolic equations, Mém. Soc. Math. Fr. (N.S.) 174 (2022), 105 (English, with English and French summaries). MR 4491902
- Christophe Cheverry and Shahnaz Farhat, Paradigm for the creation of scales and phases in nonlinear evolution equations, Electron. J. Differential Equations (2023), Paper No. 9, 59. MR 4541718
- V. I. Davydenko, A. A. Ivanov, A. N. Karpushov, R. Pozzoli, M. Rome, and D. D. Ryutov, Radial electric field measurement in a tokamak by the injection of a pulsed neutral beam, Plasma Physics and Controlled Fusion 36 (1994), no. 11, 1805–1817.
- P. Donnel, X. Garbet, Y. Sarazin, Y. Asahi, F. Wilczynski, E. Caschera, G. Dif-Pradalier, P. Ghendrih, and C. Gillot. Turbulent generation of poloidal asymmetries of the electric potential in a tokamak, Plasma Physics and Controlled Fusion 61 (2018), no. 1, 014003.
- Emmanuel Frénod and Frédérique Watbled, The Vlasov equation with strong magnetic field and oscillating electric field as a model for isotop resonant separation, Electron. J. Differential Equations (2002), No. 06, 20. MR 1872801
- Youjun Hu, Notes on tokamak equilibrium, Plasma Physics and Controlled Fusion.
- John David Jackson, Classical electrodynamics, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1975. MR 0436782
- Claudia Negulescu, Kinetic modelling of strongly magnetized tokamak plasmas with mass disparate particles. The electron Boltzmann relation, Multiscale Model. Simul. 16 (2018), no. 4, 1732–1755. MR 3874010, DOI 10.1137/17M113109X
- Ian Percival and Derek Richards, Introduction to dynamics, Cambridge University Press, Cambridge-New York, 1982. MR 680581
- J. Wesson and D. J. Campbell, Tokamaks, International Series of Monographs on Physics, OUP, Oxford, 2011.
- H. P. Zehrfeld, G. Fussmann, and B. J. Green, Electric field effects on relativistic charged particle motion in tokamaks, Plasma Physics 23 (1981), no. 5, 473–489.
References
- Giancarlo Benettin and Paolo Sempio, Adiabatic invariants and trapping of a point charge in a strong nonuniform magnetic field, Nonlinearity 7 (1994), no. 1, 281–303. MR 1260143
- Mihai Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, Multiscale Model. Simul. 8 (2010), no. 5, 1923–1957. MR 2769087, DOI 10.1137/090777621
- Martin Braun, Mathematical remarks on the Van Allen radiation belt: a survey of old and new results, SIAM Rev. 23 (1981), no. 1, 61–93. MR 605441, DOI 10.1137/1023005
- Alain J. Brizard, Jacobi zeta function and action-angle coordinates for the pendulum, Commun. Nonlinear Sci. Numer. Simul. 18 (2013), no. 3, 511–518. MR 2990693, DOI 10.1016/j.cnsns.2012.08.023
- A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys. 79 (2007), no. 2, 421–468. MR 2336960, DOI 10.1103/RevModPhys.79.421
- P. J. Catto, A. N. Simakov, and Los Alamos National Laboratory, Evaluation of the neoclassical radial electric field in a collisional tokamak, Physics of Plasmas 12 (2005), no. 1.
- Christophe Cheverry, Can one hear whistler waves?, Comm. Math. Phys. 338 (2015), no. 2, 641–703. MR 3351054, DOI 10.1007/s00220-015-2389-6
- Christophe Cheverry, Anomalous transport, J. Differential Equations 262 (2017), no. 3, 2987–3033. MR 3582249, DOI 10.1016/j.jde.2016.11.012
- C. Cheverry, Mathematical perspectives in plasma turbulence, Research and Reports on Mathematics, 2018.
- Rémi Carles and Christophe Cheverry, Constructive and destructive interferences in nonlinear hyperbolic equations, Mém. Soc. Math. Fr. (N.S.) 174 (2022), 105 (English, with English and French summaries). MR 4491902
- Christophe Cheverry and Shahnaz Farhat, Paradigm for the creation of scales and phases in nonlinear evolution equations, Electron. J. Differential Equations (2023), Paper No. 9, 59. MR 4541718
- V. I. Davydenko, A. A. Ivanov, A. N. Karpushov, R. Pozzoli, M. Rome, and D. D. Ryutov, Radial electric field measurement in a tokamak by the injection of a pulsed neutral beam, Plasma Physics and Controlled Fusion 36 (1994), no. 11, 1805–1817.
- P. Donnel, X. Garbet, Y. Sarazin, Y. Asahi, F. Wilczynski, E. Caschera, G. Dif-Pradalier, P. Ghendrih, and C. Gillot. Turbulent generation of poloidal asymmetries of the electric potential in a tokamak, Plasma Physics and Controlled Fusion 61 (2018), no. 1, 014003.
- Emmanuel Frénod and Frédérique Watbled, The Vlasov equation with strong magnetic field and oscillating electric field as a model for isotop resonant separation, Electron. J. Differential Equations (2002), No. 06, 20. MR 1872801
- Youjun Hu, Notes on tokamak equilibrium, Plasma Physics and Controlled Fusion.
- John David Jackson, Classical electrodynamics, 2nd ed., John Wiley & Sons, Inc., New York-London-Sydney, 1975. MR 0436782
- Claudia Negulescu, Kinetic modelling of strongly magnetized tokamak plasmas with mass disparate particles. The electron Boltzmann relation, Multiscale Model. Simul. 16 (2018), no. 4, 1732–1755. MR 3874010, DOI 10.1137/17M113109X
- Ian Percival and Derek Richards, Introduction to dynamics, Cambridge University Press, Cambridge-New York, 1982. MR 680581
- J. Wesson and D. J. Campbell, Tokamaks, International Series of Monographs on Physics, OUP, Oxford, 2011.
- H. P. Zehrfeld, G. Fussmann, and B. J. Green, Electric field effects on relativistic charged particle motion in tokamaks, Plasma Physics 23 (1981), no. 5, 473–489.
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Additional Information
Christophe Cheverry
Affiliation:
Université de Rennes, CNRS, IRMAR – UMR 6625, F-35000 Rennes, France
Email:
christophe.cheverry@univ-rennes1.fr
Shahnaz Farhat
Affiliation:
Université de Rennes, CNRS, IRMAR – UMR 6625, F-35000 Rennes, France
Email:
shahnaz.farhat@univ-rennes1.fr
Keywords:
Nonlinear differential equations,
nearly integrable systems,
WKB analysis,
toroidal magnetized plasmas,
coherent structures,
gyrokinetic equations
Received by editor(s):
December 28, 2022
Received by editor(s) in revised form:
February 21, 2023
Published electronically:
June 15, 2023
Article copyright:
© Copyright 2023
Brown University
