On the Geometrical Gyro-Kinetic theory

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December 2014, 7(4): 621-659. doi: 10.3934/krm.2014.7.621

On the Geometrical Gyro-Kinetic theory

Emmanuel Frénod ^1, and Mathieu Lutz ^2,

1.	LMBA (UMR 6205) Université de Bretagne-Sud, F-56017 Vannes, France
2.	IRMA (UMR 7501) Université de Strasbourg, F-67094 Strasbourg Cedex, France

Received July 2014 Revised July 2014 Published November 2014

Abstract
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Considering a Hamiltonian Dynamical System describing the motion of charged particle in a Tokamak or a Stellarator, we build a change of coordinates to reduce its dimension. This change of coordinates is in fact an intricate succession of mappings that are built using Hyperbolic Partial Differential Equations, Differential Geometry, Hamiltonian Dynamical System Theory and Symplectic Geometry, Lie Transforms and a new tool which is here introduced : Partial Lie Sums.

Keywords: Hyperbolic Partial Differential Equations, Differential Geometry, Stellarator, Partial Lie Sums., Lie Transforms, Tokamak, Gyro-Kinetic Approximation, Hamiltonian Dynamical System Theory, Symplectic Geometry.

Mathematics Subject Classification: Primary: 58Z05, 58J37, 58J45; Secondary: 82D1.

Citation: Emmanuel Frénod, Mathieu Lutz. On the Geometrical Gyro-Kinetic theory. Kinetic & Related Models, 2014, 7 (4) : 621-659. doi: 10.3934/krm.2014.7.621

References:

[1]	J. A. Brizard, Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks,, Physics of Plasmas, 2 (1995), 459. doi: 10.1063/1.871465.
[2]	L Brouwer, Über abbildung von mannigfaltigkeiten,, Mathematische Annalen, 71 (1912), 97.
[3]	D. H. E. Dubin, J. A. Krommes, C. Oberman and W. W. Lee, Nonlinear gyrokinetic equations,, Physics of Fluids, 26 (1983), 3524. doi: 10.1063/1.864113.
[4]	E. Frénod, P. A. Raviart and E. Sonnendrücker, Asymptotic expansion of the Vlasov equation in a large external magnetic field,, J. Math. Pures et Appl., 80 (2001), 815. doi: 10.1016/S0021-7824(01)01215-6.
[5]	E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field,, Asymp. Anal., 18 (1998), 193.
[6]	E. Frénod and E. Sonnendrücker, Long time behavior of the two dimensionnal Vlasov equation with a strong external magnetic field,, Math. Models Methods Appl. Sci., 10 (2000), 539. doi: 10.1142/S021820250000029X.
[7]	E. Frénod and E. Sonnendrücker, The Finite Larmor Radius Approximation,, SIAM J. Math. Anal., 32 (2001), 1227. doi: 10.1137/S0036141099364243.
[8]	E. A. Frieman and L. Chen, Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria,, Physics of Fluids, 25 (1982), 502. doi: 10.1063/1.863762.
[9]	X. Garbet, Y. Idomura, L. Villard and T. H. Watanabe, Gyrokinetic simulations of turbulent transport,, Nuclear Fusion, 50 (2010). doi: 10.1088/0029-5515/50/4/043002.
[10]	C. S. Gardner, Adiabatic invariants of periodic classical systems,, Physical Rieview, 115 (1959), 791. doi: 10.1103/PhysRev.115.791.
[11]	P. Ghendrih, M. Hauray and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solutions,, Kinet. Relat. Models, 2 (2009), 707. doi: 10.3934/krm.2009.2.707.
[12]	F. Golse and L. Saint Raymond, The Vlasov-Poisson system with strong magnetic field,, J. Math. Pures. Appl., 78 (1999), 791. doi: 10.1016/S0021-7824(99)00021-5.
[13]	V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Ghendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendrücker, J. Vaclavik and L. Villard, A drift-kinetic semi-lagrangian 4d code for ion turbulence simulation,, Journal of Computational Physics, 217 (2006), 395. doi: 10.1016/j.jcp.2006.01.023.
[14]	V. Grandgirard, Y. Sarazin, P Angelino, A. Bottino, N. Crouseilles, G. Darmet, G. Dif-Pradalier, X. Garbet, Ph. Ghendrih, S. Jolliet, G. Latu, E. Sonnendrücker and L. Villard, Global full-$f$ gyrokinetic simulations of plasma turbulence,, Plasma Physics and Controlled Fusion, 49 (2007).
[15]	T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence,, Physics of Fluids, 31 (1988), 2670. doi: 10.1063/1.866544.
[16]	T. S. Hahm, Nonlinear gyrokinetic equations for turbulence in core transport barriers,, Physics of Plasmas, 3 (1996), 4658. doi: 10.1063/1.872034.
[17]	T. S. Hahm, W. W. Lee and A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas,, Physics of Fluids, 31 (1988), 1940. doi: 10.1063/1.866641.
[18]	T. S. Hahm, Lu Wang and J. Madsen, Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence,, Physics of Plasmas, 16 (2009). doi: 10.2172/938981.
[19]	V. I. Istratescu, Fixed Point Theory an Introduction,, Dordrecht-Boston, (1981).
[20]	G. Kawamura and A. Fukuyama, Refinement of the gyrokinetic equations for edge plasmas with large flow shears,, Physics of Plasmas, 15 (2008). doi: 10.1063/1.2902016.
[21]	M. D. Kruskal, Plasma Physics, chapter Elementary Orbit and Drift Theory,, International Atomic Energy Agency, (1965).
[22]	R. G. Littlejohn, A guiding center Hamiltonian: A new approach,, Journal of Mathematical Physics, 20 (1979), 2445. doi: 10.1063/1.524053.
[23]	R. G. Littlejohn, Hamiltonian formulation of guiding center motion,, Physics of Fluids, 24 (1981), 1730. doi: 10.1063/1.863594.
[24]	R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates,, Journal of Mathematical Physics, 23 (1982), 742. doi: 10.1063/1.525429.
[25]	T. G. Northrop, The guiding center approximation to charged particle motion,, Annals of Physics, 15 (1961), 79. doi: 10.1016/0003-4916(61)90167-1.
[26]	T. G. Northrop and J. A. Rome, Extensions of guiding center motion to higher order,, Physics of Fluids, 21 (1978), 384. doi: 10.1063/1.862226.
[27]	P. J. Olver, Applications of Lie Groups to Differential Equations,, Second edition. Graduate Texts in Mathematics, (1993). doi: 10.1007/978-1-4612-4350-2.
[28]	F. I. Parra and P. J. Catto, Limitations of gyrokinetics on transport time scales,, Plasma Physics and Controlled Fusion, 50 (2008). doi: 10.1088/0741-3335/50/6/065014.
[29]	F. I. Parra and P. J. Catto, Gyrokinetic equivalence,, Plasma Physics and Controlled Fusion, 51 (2009). doi: 10.1088/0741-3335/51/6/065002.
[30]	F. I. Parra and P. J. Catto, Turbulent transport of toroidal angular momentum in low flow gyrokinetics,, Plasma Physics and Controlled Fusion, 52 (2010). doi: 10.1088/0741-3335/52/4/045004.
[31]	H. Qin, R. H. Cohen, W. M. Nevins and X. Q. Xu, General gyrokinetic equations for edge plasmas,, Contributions to Plasma Physics, 46 (2006), 477. doi: 10.1002/ctpp.200610034.
[32]	H. Qin, R. H. Cohen, W. M. Nevins and X. Q. Xu, Geometric gyrokinetic theory for edge plasmas,, Physics of Plasmas, 14 (2007). doi: 10.1063/1.2472596.

show all references

References:

[1]	J. A. Brizard, Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks,, Physics of Plasmas, 2 (1995), 459. doi: 10.1063/1.871465.
[2]	L Brouwer, Über abbildung von mannigfaltigkeiten,, Mathematische Annalen, 71 (1912), 97.
[3]	D. H. E. Dubin, J. A. Krommes, C. Oberman and W. W. Lee, Nonlinear gyrokinetic equations,, Physics of Fluids, 26 (1983), 3524. doi: 10.1063/1.864113.
[4]	E. Frénod, P. A. Raviart and E. Sonnendrücker, Asymptotic expansion of the Vlasov equation in a large external magnetic field,, J. Math. Pures et Appl., 80 (2001), 815. doi: 10.1016/S0021-7824(01)01215-6.
[5]	E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field,, Asymp. Anal., 18 (1998), 193.
[6]	E. Frénod and E. Sonnendrücker, Long time behavior of the two dimensionnal Vlasov equation with a strong external magnetic field,, Math. Models Methods Appl. Sci., 10 (2000), 539. doi: 10.1142/S021820250000029X.
[7]	E. Frénod and E. Sonnendrücker, The Finite Larmor Radius Approximation,, SIAM J. Math. Anal., 32 (2001), 1227. doi: 10.1137/S0036141099364243.
[8]	E. A. Frieman and L. Chen, Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria,, Physics of Fluids, 25 (1982), 502. doi: 10.1063/1.863762.
[9]	X. Garbet, Y. Idomura, L. Villard and T. H. Watanabe, Gyrokinetic simulations of turbulent transport,, Nuclear Fusion, 50 (2010). doi: 10.1088/0029-5515/50/4/043002.
[10]	C. S. Gardner, Adiabatic invariants of periodic classical systems,, Physical Rieview, 115 (1959), 791. doi: 10.1103/PhysRev.115.791.
[11]	P. Ghendrih, M. Hauray and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solutions,, Kinet. Relat. Models, 2 (2009), 707. doi: 10.3934/krm.2009.2.707.
[12]	F. Golse and L. Saint Raymond, The Vlasov-Poisson system with strong magnetic field,, J. Math. Pures. Appl., 78 (1999), 791. doi: 10.1016/S0021-7824(99)00021-5.
[13]	V. Grandgirard, M. Brunetti, P. Bertrand, N. Besse, X. Garbet, P. Ghendrih, G. Manfredi, Y. Sarazin, O. Sauter, E. Sonnendrücker, J. Vaclavik and L. Villard, A drift-kinetic semi-lagrangian 4d code for ion turbulence simulation,, Journal of Computational Physics, 217 (2006), 395. doi: 10.1016/j.jcp.2006.01.023.
[14]	V. Grandgirard, Y. Sarazin, P Angelino, A. Bottino, N. Crouseilles, G. Darmet, G. Dif-Pradalier, X. Garbet, Ph. Ghendrih, S. Jolliet, G. Latu, E. Sonnendrücker and L. Villard, Global full-$f$ gyrokinetic simulations of plasma turbulence,, Plasma Physics and Controlled Fusion, 49 (2007).
[15]	T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence,, Physics of Fluids, 31 (1988), 2670. doi: 10.1063/1.866544.
[16]	T. S. Hahm, Nonlinear gyrokinetic equations for turbulence in core transport barriers,, Physics of Plasmas, 3 (1996), 4658. doi: 10.1063/1.872034.
[17]	T. S. Hahm, W. W. Lee and A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas,, Physics of Fluids, 31 (1988), 1940. doi: 10.1063/1.866641.
[18]	T. S. Hahm, Lu Wang and J. Madsen, Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence,, Physics of Plasmas, 16 (2009). doi: 10.2172/938981.
[19]	V. I. Istratescu, Fixed Point Theory an Introduction,, Dordrecht-Boston, (1981).
[20]	G. Kawamura and A. Fukuyama, Refinement of the gyrokinetic equations for edge plasmas with large flow shears,, Physics of Plasmas, 15 (2008). doi: 10.1063/1.2902016.
[21]	M. D. Kruskal, Plasma Physics, chapter Elementary Orbit and Drift Theory,, International Atomic Energy Agency, (1965).
[22]	R. G. Littlejohn, A guiding center Hamiltonian: A new approach,, Journal of Mathematical Physics, 20 (1979), 2445. doi: 10.1063/1.524053.
[23]	R. G. Littlejohn, Hamiltonian formulation of guiding center motion,, Physics of Fluids, 24 (1981), 1730. doi: 10.1063/1.863594.
[24]	R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates,, Journal of Mathematical Physics, 23 (1982), 742. doi: 10.1063/1.525429.
[25]	T. G. Northrop, The guiding center approximation to charged particle motion,, Annals of Physics, 15 (1961), 79. doi: 10.1016/0003-4916(61)90167-1.
[26]	T. G. Northrop and J. A. Rome, Extensions of guiding center motion to higher order,, Physics of Fluids, 21 (1978), 384. doi: 10.1063/1.862226.
[27]	P. J. Olver, Applications of Lie Groups to Differential Equations,, Second edition. Graduate Texts in Mathematics, (1993). doi: 10.1007/978-1-4612-4350-2.
[28]	F. I. Parra and P. J. Catto, Limitations of gyrokinetics on transport time scales,, Plasma Physics and Controlled Fusion, 50 (2008). doi: 10.1088/0741-3335/50/6/065014.
[29]	F. I. Parra and P. J. Catto, Gyrokinetic equivalence,, Plasma Physics and Controlled Fusion, 51 (2009). doi: 10.1088/0741-3335/51/6/065002.
[30]	F. I. Parra and P. J. Catto, Turbulent transport of toroidal angular momentum in low flow gyrokinetics,, Plasma Physics and Controlled Fusion, 52 (2010). doi: 10.1088/0741-3335/52/4/045004.
[31]	H. Qin, R. H. Cohen, W. M. Nevins and X. Q. Xu, General gyrokinetic equations for edge plasmas,, Contributions to Plasma Physics, 46 (2006), 477. doi: 10.1002/ctpp.200610034.
[32]	H. Qin, R. H. Cohen, W. M. Nevins and X. Q. Xu, Geometric gyrokinetic theory for edge plasmas,, Physics of Plasmas, 14 (2007). doi: 10.1063/1.2472596.

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