# American Institute of Mathematical Sciences

December  2011, 15(3): 513-544. doi: 10.3934/dcdsb.2011.15.513

## Mathematical models for strongly magnetized plasmas with mass disparate particles

 1 Laboratoire de Mathématiques de Besançon, Université de Franche-Comté, 16 route de Gray, Besançon, 25030 Cedex 2 CMI/LATP (UMR 6632), Université de Provence, 39, rue Joliot Curie, 13453 Marseille Cedex 13

Received  January 2010 Revised  May 2010 Published  February 2011

The controlled fusion is achieved by magnetic confinement : the plasma is confined into toroidal devices called tokamaks, under the action of strong magnetic fields. The particle motion reduces to advection along the magnetic lines combined to rotation around the magnetic lines. The rotation around the magnetic lines is much faster than the parallel motion and efficient numerical resolution requires homogenization procedures. Moreover the rotation period, being proportional to the particle mass, introduces very different time scales in the case when the plasma contains disparate particles; the electrons turn much faster than the ions, the ratio between their cyclotronic periods being the mass ratio of the electrons with respect to the ions. The subject matter of this paper concerns the mathematical study of such plasmas, under the action of strong magnetic fields. In particular, we are interested in the limit models when the small parameter, representing the mass ratio as we ll as the fast cyclotronic motion, tends to zero.
Citation: Mihai Bostan, Claudia Negulescu. Mathematical models for strongly magnetized plasmas with mass disparate particles. Discrete and Continuous Dynamical Systems - B, 2011, 15 (3) : 513-544. doi: 10.3934/dcdsb.2011.15.513
##### References:
 [1] H. D. I. Abarbanel, Hamiltonian description of almost geostrophic flow, Geophysical and Astrophysical Fluid Dynamics, 33 (1985), 145-171. doi: 10.1080/03091928508245427. [2] J. S. Allen and D. Holm, Extended-geostrophic Hamiltonian models for rotating shallow water motion, Physica D, 98 (1996), 229-248. doi: 10.1016/0167-2789(96)00120-0. [3] N. N. Bogoliubov and Y. A. Mitropolsky, "Asymptotic Methods in the Theory of Nonlinear Oscillations," Gordon and Breach Sciences Publishers, New York, 1961. [4] M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123. [5] M. Bostan, Transport equations with singular coefficients. Application to the gyrokinetic models in plasma physics, Research Report INRIA, hal:inria-00232800, submitted 2009. [6] M. Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, hal-00431289, submitted 2009. [7] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754. doi: 10.1080/03605300008821529. [8] A. H. Boozer, Physics of magnetically confined plasmas, Rev. Modern Phys., 76 (2004), 1071-1141. doi: 10.1103/RevModPhys.76.1071. [9] 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. [10] F. Charles and L. Desvillettes, Small mass ratio limit of Boltzmann equations in the context of the study of evolution of dust particles in a rarefied atmosphere, J. Stat. Phys., 137 (2009), 539-567. doi: 10.1007/s10955-009-9858-2. [11] J.-Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier, "Mathematical Geophysics: An Introduction to Rotating Fluids and the Navier-Stokes Equations," Oxford Lecture Series in Mathematics and Its Applications 32, The Clarendon Press, Oxford University Press, 2006. [12] P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases, Transport Theory Statist. Phys, 25 (1996), 595-633. doi: 10.1080/00411459608222915. [13] 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. [14] E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243. [15] 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. [16] T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime, Indiana Univ. Math. J., 53 (2004), 1495-1515. doi: 10.1512/iumj.2004.53.2508. [17] 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. [18] R. D. Hazeltine and J. D. Meiss, "Plasma Confinement," Dover Publications, Inc. Mineola, New York, 2003. [19] R. G. Littlejohn, A guiding center Hamiltonian : A new approach, J. Math. Phys. 20 (1979) 2445-2458. doi: 10.1063/1.524053. [20] R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Phys. Fluids, 24 (1981), 1730-1749. doi: 10.1063/1.863594. [21] J.-M. Rax, "Physique des plasmas, Cours et applications," Dunod, 2007. [22] M. Reed and B. Simon, "Methods of Modern Mathematical Physics," Vol. I, Functional Analysis, Academic Press, 1980. [23] R. Salmon, New equations for nearly geostrophic flow, J. Fluid Mechanics, 153 (1985), 461-477. doi: 10.1017/S0022112085001343. [24] J. Vanneste and O. Bokhove, Dirac-bracket approach to nearly geostrophic Hamiltonian balanced models, Physica D Nonlinear Phenomena, 164 (2002), 152-167. doi: 10.1016/S0167-2789(02)00375-5.

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##### References:
 [1] H. D. I. Abarbanel, Hamiltonian description of almost geostrophic flow, Geophysical and Astrophysical Fluid Dynamics, 33 (1985), 145-171. doi: 10.1080/03091928508245427. [2] J. S. Allen and D. Holm, Extended-geostrophic Hamiltonian models for rotating shallow water motion, Physica D, 98 (1996), 229-248. doi: 10.1016/0167-2789(96)00120-0. [3] N. N. Bogoliubov and Y. A. Mitropolsky, "Asymptotic Methods in the Theory of Nonlinear Oscillations," Gordon and Breach Sciences Publishers, New York, 1961. [4] M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal., 61 (2009), 91-123. [5] M. Bostan, Transport equations with singular coefficients. Application to the gyrokinetic models in plasma physics, Research Report INRIA, hal:inria-00232800, submitted 2009. [6] M. Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, hal-00431289, submitted 2009. [7] Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations, Comm. Partial Differential Equations, 25 (2000), 737-754. doi: 10.1080/03605300008821529. [8] A. H. Boozer, Physics of magnetically confined plasmas, Rev. Modern Phys., 76 (2004), 1071-1141. doi: 10.1103/RevModPhys.76.1071. [9] 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. [10] F. Charles and L. Desvillettes, Small mass ratio limit of Boltzmann equations in the context of the study of evolution of dust particles in a rarefied atmosphere, J. Stat. Phys., 137 (2009), 539-567. doi: 10.1007/s10955-009-9858-2. [11] J.-Y. Chemin, B. Desjardins, I. Gallagher, E. Grenier, "Mathematical Geophysics: An Introduction to Rotating Fluids and the Navier-Stokes Equations," Oxford Lecture Series in Mathematics and Its Applications 32, The Clarendon Press, Oxford University Press, 2006. [12] P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases, Transport Theory Statist. Phys, 25 (1996), 595-633. doi: 10.1080/00411459608222915. [13] 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. [14] E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal., 32 (2001), 1227-1247. doi: 10.1137/S0036141099364243. [15] 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. [16] T. Goudon, P.-E. Jabin and A. Vasseur, Hydrodynamic limit for the Vlasov-Navier-Stokes equations. I. Light particles regime, Indiana Univ. Math. J., 53 (2004), 1495-1515. doi: 10.1512/iumj.2004.53.2508. [17] 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. [18] R. D. Hazeltine and J. D. Meiss, "Plasma Confinement," Dover Publications, Inc. Mineola, New York, 2003. [19] R. G. Littlejohn, A guiding center Hamiltonian : A new approach, J. Math. Phys. 20 (1979) 2445-2458. doi: 10.1063/1.524053. [20] R. G. Littlejohn, Hamiltonian formulation of guiding center motion, Phys. Fluids, 24 (1981), 1730-1749. doi: 10.1063/1.863594. [21] J.-M. Rax, "Physique des plasmas, Cours et applications," Dunod, 2007. [22] M. Reed and B. Simon, "Methods of Modern Mathematical Physics," Vol. I, Functional Analysis, Academic Press, 1980. [23] R. Salmon, New equations for nearly geostrophic flow, J. Fluid Mechanics, 153 (1985), 461-477. doi: 10.1017/S0022112085001343. [24] J. Vanneste and O. Bokhove, Dirac-bracket approach to nearly geostrophic Hamiltonian balanced models, Physica D Nonlinear Phenomena, 164 (2002), 152-167. doi: 10.1016/S0167-2789(02)00375-5.
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