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

On the Geometrical Gyro-Kinetic theory

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.
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.  Google Scholar

[2]

L Brouwer, Über abbildung von mannigfaltigkeiten,, Mathematische Annalen, 71 (1912), 97.   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.  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.  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.   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.  doi: 10.1142/S021820250000029X.  Google Scholar

[7]

E. Frénod and E. Sonnendrücker, The Finite Larmor Radius Approximation,, SIAM J. Math. Anal., 32 (2001), 1227.  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.  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).  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.  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.  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.  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.  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).   Google Scholar

[15]

T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence,, Physics of Fluids, 31 (1988), 2670.  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.  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.  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).  doi: 10.2172/938981.  Google Scholar

[19]

V. I. Istratescu, Fixed Point Theory an Introduction,, Dordrecht-Boston, (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).  doi: 10.1063/1.2902016.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates,, Journal of Mathematical Physics, 23 (1982), 742.  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.  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.  doi: 10.1063/1.862226.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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).  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.  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).  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.  doi: 10.1063/1.871465.  Google Scholar

[2]

L Brouwer, Über abbildung von mannigfaltigkeiten,, Mathematische Annalen, 71 (1912), 97.   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.  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.  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.   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.  doi: 10.1142/S021820250000029X.  Google Scholar

[7]

E. Frénod and E. Sonnendrücker, The Finite Larmor Radius Approximation,, SIAM J. Math. Anal., 32 (2001), 1227.  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.  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).  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.  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.  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.  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.  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).   Google Scholar

[15]

T. S. Hahm, Nonlinear gyrokinetic equations for tokamak microturbulence,, Physics of Fluids, 31 (1988), 2670.  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.  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.  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).  doi: 10.2172/938981.  Google Scholar

[19]

V. I. Istratescu, Fixed Point Theory an Introduction,, Dordrecht-Boston, (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).  doi: 10.1063/1.2902016.  Google Scholar

[21]

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

[22]

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

[23]

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

[24]

R. G. Littlejohn, Hamiltonian perturbation theory in noncanonical coordinates,, Journal of Mathematical Physics, 23 (1982), 742.  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.  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.  doi: 10.1063/1.862226.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  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).  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.  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).  doi: 10.1063/1.2472596.  Google Scholar

[1]

Oǧul Esen, Hasan Gümral. Geometry of plasma dynamics II: Lie algebra of Hamiltonian vector fields. Journal of Geometric Mechanics, 2012, 4 (3) : 239-269. doi: 10.3934/jgm.2012.4.239

[2]

Wilhelm Schlag. Spectral theory and nonlinear partial differential equations: A survey. Discrete & Continuous Dynamical Systems - A, 2006, 15 (3) : 703-723. doi: 10.3934/dcds.2006.15.703

[3]

Rongmei Cao, Jiangong You. The existence of integrable invariant manifolds of Hamiltonian partial differential equations. Discrete & Continuous Dynamical Systems - A, 2006, 16 (1) : 227-234. doi: 10.3934/dcds.2006.16.227

[4]

Yuhki Hosoya. First-order partial differential equations and consumer theory. Discrete & Continuous Dynamical Systems - S, 2018, 11 (6) : 1143-1167. doi: 10.3934/dcdss.2018065

[5]

Mogtaba Mohammed, Mamadou Sango. Homogenization of nonlinear hyperbolic stochastic partial differential equations with nonlinear damping and forcing. Networks & Heterogeneous Media, 2019, 14 (2) : 341-369. doi: 10.3934/nhm.2019014

[6]

David Blázquez-Sanz, Juan J. Morales-Ruiz. Lie's reduction method and differential Galois theory in the complex analytic context. Discrete & Continuous Dynamical Systems - A, 2012, 32 (2) : 353-379. doi: 10.3934/dcds.2012.32.353

[7]

Michela Eleuteri, Pavel Krejčí. An asymptotic convergence result for a system of partial differential equations with hysteresis. Communications on Pure & Applied Analysis, 2007, 6 (4) : 1131-1143. doi: 10.3934/cpaa.2007.6.1131

[8]

Yves Achdou, Mathieu Laurière. On the system of partial differential equations arising in mean field type control. Discrete & Continuous Dynamical Systems - A, 2015, 35 (9) : 3879-3900. doi: 10.3934/dcds.2015.35.3879

[9]

Răzvan M. Tudoran, Anania Gîrban. On the Hamiltonian dynamics and geometry of the Rabinovich system. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 789-823. doi: 10.3934/dcdsb.2011.15.789

[10]

Fernando Casas, Cristina Chiralt. A Lie--Deprit perturbation algorithm for linear differential equations with periodic coefficients. Discrete & Continuous Dynamical Systems - A, 2014, 34 (3) : 959-975. doi: 10.3934/dcds.2014.34.959

[11]

Isaac A. García, Jaume Giné, Jaume Llibre. Liénard and Riccati differential equations related via Lie Algebras. Discrete & Continuous Dynamical Systems - B, 2008, 10 (2&3, September) : 485-494. doi: 10.3934/dcdsb.2008.10.485

[12]

José F. Cariñena, Irina Gheorghiu, Eduardo Martínez. Jacobi fields for second-order differential equations on Lie algebroids. Conference Publications, 2015, 2015 (special) : 213-222. doi: 10.3934/proc.2015.0213

[13]

Xin Chen, Ana Bela Cruzeiro. Stochastic geodesics and forward-backward stochastic differential equations on Lie groups. Conference Publications, 2013, 2013 (special) : 115-121. doi: 10.3934/proc.2013.2013.115

[14]

Herbert Koch. Partial differential equations with non-Euclidean geometries. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 481-504. doi: 10.3934/dcdss.2008.1.481

[15]

Eugenia N. Petropoulou, Panayiotis D. Siafarikas. Polynomial solutions of linear partial differential equations. Communications on Pure & Applied Analysis, 2009, 8 (3) : 1053-1065. doi: 10.3934/cpaa.2009.8.1053

[16]

Arnulf Jentzen. Taylor expansions of solutions of stochastic partial differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 515-557. doi: 10.3934/dcdsb.2010.14.515

[17]

Nguyen Thieu Huy, Ngo Quy Dang. Dichotomy and periodic solutions to partial functional differential equations. Discrete & Continuous Dynamical Systems - B, 2017, 22 (8) : 3127-3144. doi: 10.3934/dcdsb.2017167

[18]

Barbara Abraham-Shrauner. Exact solutions of nonlinear partial differential equations. Discrete & Continuous Dynamical Systems - S, 2018, 11 (4) : 577-582. doi: 10.3934/dcdss.2018032

[19]

Masaki Hibino. Gevrey asymptotic theory for singular first order linear partial differential equations of nilpotent type — Part I —. Communications on Pure & Applied Analysis, 2003, 2 (2) : 211-231. doi: 10.3934/cpaa.2003.2.211

[20]

Jiahui Zhu, Zdzisław Brzeźniak. Nonlinear stochastic partial differential equations of hyperbolic type driven by Lévy-type noises. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3269-3299. doi: 10.3934/dcdsb.2016097

2018 Impact Factor: 1.38

Metrics

  • PDF downloads (10)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]