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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

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), 459-471. doi: 10.1063/1.871465.  Google Scholar

[2]

L Brouwer, Über abbildung von mannigfaltigkeiten, Mathematische Annalen, 71 (1912), 97-115. Google Scholar

[3]

D. H. E. Dubin, J. A. Krommes, C. Oberman and W. W. Lee, Nonlinear gyrokinetic equations, Physics of Fluids, 26 (1983), 3524-3535. doi: 10.1063/1.864113.  Google Scholar

[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), 815-843. doi: 10.1016/S0021-7824(01)01215-6.  Google Scholar

[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), 193-214.  Google Scholar

[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), 539-553. doi: 10.1142/S021820250000029X.  Google Scholar

[7]

E. Frénod and E. Sonnendrücker, The Finite Larmor Radius Approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243.  Google Scholar

[8]

E. A. Frieman and L. Chen, Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria, Physics of Fluids, 25 (1982), 502-508. doi: 10.1063/1.863762.  Google Scholar

[9]

X. Garbet, Y. Idomura, L. Villard and T. H. Watanabe, Gyrokinetic simulations of turbulent transport, Nuclear Fusion, 50 (2010), 043002. doi: 10.1088/0029-5515/50/4/043002.  Google Scholar

[10]

C. S. Gardner, Adiabatic invariants of periodic classical systems, Physical Rieview, 115 (1959), 791-794. doi: 10.1103/PhysRev.115.791.  Google Scholar

[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), 707-725. doi: 10.3934/krm.2009.2.707.  Google Scholar

[12]

F. Golse and L. Saint Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures. Appl., 78 (1999), 791-817. doi: 10.1016/S0021-7824(99)00021-5.  Google Scholar

[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), 395-423. doi: 10.1016/j.jcp.2006.01.023.  Google Scholar

[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), B173. Google Scholar

[15]

T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence, Physics of Fluids, 31 (1988), 2670-2673. doi: 10.1063/1.866544.  Google Scholar

[16]

T. S. Hahm, Nonlinear gyrokinetic equations for turbulence in core transport barriers, Physics of Plasmas, 3 (1996), 4658-4664. doi: 10.1063/1.872034.  Google Scholar

[17]

T. S. Hahm, W. W. Lee and A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas, Physics of Fluids, 31 (1988), 1940-1948. doi: 10.1063/1.866641.  Google Scholar

[18]

T. S. Hahm, Lu Wang and J. Madsen, Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence, Physics of Plasmas, 16 (2009), 022305. doi: 10.2172/938981.  Google Scholar

[19]

V. I. Istratescu, Fixed Point Theory an Introduction, Dordrecht-Boston, Mass., 1981.  Google Scholar

[20]

G. Kawamura and A. Fukuyama, Refinement of the gyrokinetic equations for edge plasmas with large flow shears, Physics of Plasmas, 15 (2008), 042304. doi: 10.1063/1.2902016.  Google Scholar

[21]

M. D. Kruskal, Plasma Physics, chapter Elementary Orbit and Drift Theory, International Atomic Energy Agency, Vienna, 1965. Google Scholar

[22]

R. G. Littlejohn, A guiding center Hamiltonian: A new approach, Journal of Mathematical Physics, 20 (1979), 2445-2458. doi: 10.1063/1.524053.  Google Scholar

[23]

R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Physics of Fluids, 24 (1981), 1730-1749. doi: 10.1063/1.863594.  Google Scholar

[24]

R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates, Journal of Mathematical Physics, 23 (1982), 742-747. doi: 10.1063/1.525429.  Google Scholar

[25]

T. G. Northrop, The guiding center approximation to charged particle motion, Annals of Physics, 15 (1961), 79-101. doi: 10.1016/0003-4916(61)90167-1.  Google Scholar

[26]

T. G. Northrop and J. A. Rome, Extensions of guiding center motion to higher order, Physics of Fluids, 21 (1978), 384-389. doi: 10.1063/1.862226.  Google Scholar

[27]

P. J. Olver, Applications of Lie Groups to Differential Equations, Second edition. Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.  Google Scholar

[28]

F. I. Parra and P. J. Catto, Limitations of gyrokinetics on transport time scales, Plasma Physics and Controlled Fusion, 50 (2008), 065014. doi: 10.1088/0741-3335/50/6/065014.  Google Scholar

[29]

F. I. Parra and P. J. Catto, Gyrokinetic equivalence, Plasma Physics and Controlled Fusion, 51 (2009), 065002. doi: 10.1088/0741-3335/51/6/065002.  Google Scholar

[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), 045004. doi: 10.1088/0741-3335/52/4/045004.  Google Scholar

[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), 477-489. doi: 10.1002/ctpp.200610034.  Google Scholar

[32]

H. Qin, R. H. Cohen, W. M. Nevins and X. Q. Xu, Geometric gyrokinetic theory for edge plasmas, Physics of Plasmas, 14 (2007), 056110. doi: 10.1063/1.2472596.  Google Scholar

show all references

References:
[1]

J. A. Brizard, Nonlinear gyrokinetic Vlasov equation for toroidally rotating axisymmetric tokamaks, Physics of Plasmas, 2 (1995), 459-471. doi: 10.1063/1.871465.  Google Scholar

[2]

L Brouwer, Über abbildung von mannigfaltigkeiten, Mathematische Annalen, 71 (1912), 97-115. Google Scholar

[3]

D. H. E. Dubin, J. A. Krommes, C. Oberman and W. W. Lee, Nonlinear gyrokinetic equations, Physics of Fluids, 26 (1983), 3524-3535. doi: 10.1063/1.864113.  Google Scholar

[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), 815-843. doi: 10.1016/S0021-7824(01)01215-6.  Google Scholar

[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), 193-214.  Google Scholar

[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), 539-553. doi: 10.1142/S021820250000029X.  Google Scholar

[7]

E. Frénod and E. Sonnendrücker, The Finite Larmor Radius Approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243.  Google Scholar

[8]

E. A. Frieman and L. Chen, Nonlinear gyrokinetic equations for low-frequency electromagnetic waves in general plasma equilibria, Physics of Fluids, 25 (1982), 502-508. doi: 10.1063/1.863762.  Google Scholar

[9]

X. Garbet, Y. Idomura, L. Villard and T. H. Watanabe, Gyrokinetic simulations of turbulent transport, Nuclear Fusion, 50 (2010), 043002. doi: 10.1088/0029-5515/50/4/043002.  Google Scholar

[10]

C. S. Gardner, Adiabatic invariants of periodic classical systems, Physical Rieview, 115 (1959), 791-794. doi: 10.1103/PhysRev.115.791.  Google Scholar

[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), 707-725. doi: 10.3934/krm.2009.2.707.  Google Scholar

[12]

F. Golse and L. Saint Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures. Appl., 78 (1999), 791-817. doi: 10.1016/S0021-7824(99)00021-5.  Google Scholar

[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), 395-423. doi: 10.1016/j.jcp.2006.01.023.  Google Scholar

[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), B173. Google Scholar

[15]

T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence, Physics of Fluids, 31 (1988), 2670-2673. doi: 10.1063/1.866544.  Google Scholar

[16]

T. S. Hahm, Nonlinear gyrokinetic equations for turbulence in core transport barriers, Physics of Plasmas, 3 (1996), 4658-4664. doi: 10.1063/1.872034.  Google Scholar

[17]

T. S. Hahm, W. W. Lee and A. Brizard, Nonlinear gyrokinetic theory for finite-beta plasmas, Physics of Fluids, 31 (1988), 1940-1948. doi: 10.1063/1.866641.  Google Scholar

[18]

T. S. Hahm, Lu Wang and J. Madsen, Fully electromagnetic nonlinear gyrokinetic equations for tokamak edge turbulence, Physics of Plasmas, 16 (2009), 022305. doi: 10.2172/938981.  Google Scholar

[19]

V. I. Istratescu, Fixed Point Theory an Introduction, Dordrecht-Boston, Mass., 1981.  Google Scholar

[20]

G. Kawamura and A. Fukuyama, Refinement of the gyrokinetic equations for edge plasmas with large flow shears, Physics of Plasmas, 15 (2008), 042304. doi: 10.1063/1.2902016.  Google Scholar

[21]

M. D. Kruskal, Plasma Physics, chapter Elementary Orbit and Drift Theory, International Atomic Energy Agency, Vienna, 1965. Google Scholar

[22]

R. G. Littlejohn, A guiding center Hamiltonian: A new approach, Journal of Mathematical Physics, 20 (1979), 2445-2458. doi: 10.1063/1.524053.  Google Scholar

[23]

R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Physics of Fluids, 24 (1981), 1730-1749. doi: 10.1063/1.863594.  Google Scholar

[24]

R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates, Journal of Mathematical Physics, 23 (1982), 742-747. doi: 10.1063/1.525429.  Google Scholar

[25]

T. G. Northrop, The guiding center approximation to charged particle motion, Annals of Physics, 15 (1961), 79-101. doi: 10.1016/0003-4916(61)90167-1.  Google Scholar

[26]

T. G. Northrop and J. A. Rome, Extensions of guiding center motion to higher order, Physics of Fluids, 21 (1978), 384-389. doi: 10.1063/1.862226.  Google Scholar

[27]

P. J. Olver, Applications of Lie Groups to Differential Equations, Second edition. Graduate Texts in Mathematics, 107. Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4350-2.  Google Scholar

[28]

F. I. Parra and P. J. Catto, Limitations of gyrokinetics on transport time scales, Plasma Physics and Controlled Fusion, 50 (2008), 065014. doi: 10.1088/0741-3335/50/6/065014.  Google Scholar

[29]

F. I. Parra and P. J. Catto, Gyrokinetic equivalence, Plasma Physics and Controlled Fusion, 51 (2009), 065002. doi: 10.1088/0741-3335/51/6/065002.  Google Scholar

[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), 045004. doi: 10.1088/0741-3335/52/4/045004.  Google Scholar

[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), 477-489. doi: 10.1002/ctpp.200610034.  Google Scholar

[32]

H. Qin, R. H. Cohen, W. M. Nevins and X. Q. Xu, Geometric gyrokinetic theory for edge plasmas, Physics of Plasmas, 14 (2007), 056110. doi: 10.1063/1.2472596.  Google Scholar

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