April  2012, 5(2): 257-269. doi: 10.3934/dcdss.2012.5.257

Gyrokinetic models for strongly magnetized plasmas with general magnetic shape

1. 

Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex, France

Received  July 2009 Revised  February 2010 Published  September 2011

One of the main applications in plasma physics concerns the energy production through thermo-nuclear fusion. The controlled fusion requires the confinement of the plasma into a bounded domain and for this, we appeal to the magnetic confinement. Several models exist for describing the evolution of strongly magnetized plasmas. The subject matter of this paper is to provide a rigorous derivation of the guiding-center approximation in the general three dimensional setting, under the action of large stationary inhomogeneous magnetic fields.
Citation: Mihai Bostan. Gyrokinetic models for strongly magnetized plasmas with general magnetic shape. Discrete & Continuous Dynamical Systems - S, 2012, 5 (2) : 257-269. doi: 10.3934/dcdss.2012.5.257
References:
[1]

N. N. Bogoliubov and Y. A. Mitropolsky, "Asymptotic Methods in the Theory of Nonlinear Oscillations," Translated from the second Russian edition, International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, New York, 1961.  Google Scholar

[2]

M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123.  Google Scholar

[3]

M. Bostan, The Vlasov-Maxwell system with strong initial magnetic field: Guiding-center approximation, Multiscale Model. Simul., 6 (2007), 1026-1058. doi: 10.1137/070689383.  Google Scholar

[4]

M. Bostan, Transport equations with singular coefficients. Application to the gyrokinetic models in plasma physics,, research report INRIA., ().   Google Scholar

[5]

M. Bostan and T. Goudon, High-electric-field limit for the Vlasov-Maxwell-Fokker-Planck system, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 25 (2008), 1221-1251.  Google Scholar

[6]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754.  Google Scholar

[7]

Y. Brenier, N. Mauser and M. Puel, Incompressible Euler and e-MHD as scaling limits of the Vlasov-Maxwell system, Commun. Math. Sci., 1 (2003), 437-447.  Google Scholar

[8]

A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys., 79 (2007), 421-468. doi: 10.1103/RevModPhys.79.421.  Google Scholar

[9]

E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with strong external magnetic field, Asymptotic Anal., 18 (1998), 193-213.  Google Scholar

[10]

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

[11]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9), 78 (1999), 791-817. doi: 10.1016/S0021-7824(99)00021-5.  Google Scholar

[12]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci., 13 (2003), 661-714. doi: 10.1142/S0218202503002647.  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, J. Comput. Phys., 217 (2006), 395-423. doi: 10.1016/j.jcp.2006.01.023.  Google Scholar

[14]

R. D. Hazeltine and J. D. Meiss, "Plasma Confinement," Dover Publications, Inc., Mineola, New York, 2003. Google Scholar

[15]

P. Morel, E. Gravier, N. Besse, A. Ghizzo and P. Bertrand, The water bag model and gyrokinetic applications, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 11-17. doi: 10.1016/j.cnsns.2007.03.016.  Google Scholar

[16]

J.-M. Rax, "Physique des Plasmas, Cours et Applications," Dunod, 2007. Google Scholar

[17]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. I. Functional Analysis," Second edition, Academic Press, New York, 1980.  Google Scholar

show all references

References:
[1]

N. N. Bogoliubov and Y. A. Mitropolsky, "Asymptotic Methods in the Theory of Nonlinear Oscillations," Translated from the second Russian edition, International Monographs on Advanced Mathematics and Physics, Hindustan Publishing Corp., Delhi, Gordon and Breach Science Publishers, New York, 1961.  Google Scholar

[2]

M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123.  Google Scholar

[3]

M. Bostan, The Vlasov-Maxwell system with strong initial magnetic field: Guiding-center approximation, Multiscale Model. Simul., 6 (2007), 1026-1058. doi: 10.1137/070689383.  Google Scholar

[4]

M. Bostan, Transport equations with singular coefficients. Application to the gyrokinetic models in plasma physics,, research report INRIA., ().   Google Scholar

[5]

M. Bostan and T. Goudon, High-electric-field limit for the Vlasov-Maxwell-Fokker-Planck system, Ann. Inst. H. Poincaré, Anal. Non Linéaire, 25 (2008), 1221-1251.  Google Scholar

[6]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754.  Google Scholar

[7]

Y. Brenier, N. Mauser and M. Puel, Incompressible Euler and e-MHD as scaling limits of the Vlasov-Maxwell system, Commun. Math. Sci., 1 (2003), 437-447.  Google Scholar

[8]

A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys., 79 (2007), 421-468. doi: 10.1103/RevModPhys.79.421.  Google Scholar

[9]

E. Frénod and E. Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with strong external magnetic field, Asymptotic Anal., 18 (1998), 193-213.  Google Scholar

[10]

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

[11]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9), 78 (1999), 791-817. doi: 10.1016/S0021-7824(99)00021-5.  Google Scholar

[12]

F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field in quasineutral regime, Math. Models Methods Appl. Sci., 13 (2003), 661-714. doi: 10.1142/S0218202503002647.  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, J. Comput. Phys., 217 (2006), 395-423. doi: 10.1016/j.jcp.2006.01.023.  Google Scholar

[14]

R. D. Hazeltine and J. D. Meiss, "Plasma Confinement," Dover Publications, Inc., Mineola, New York, 2003. Google Scholar

[15]

P. Morel, E. Gravier, N. Besse, A. Ghizzo and P. Bertrand, The water bag model and gyrokinetic applications, Commun. Nonlinear Sci. Numer. Simul., 13 (2008), 11-17. doi: 10.1016/j.cnsns.2007.03.016.  Google Scholar

[16]

J.-M. Rax, "Physique des Plasmas, Cours et Applications," Dunod, 2007. Google Scholar

[17]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics. I. Functional Analysis," Second edition, Academic Press, New York, 1980.  Google Scholar

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