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 & 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.  doi: 10.1080/03091928508245427.  Google Scholar

[2]

J. S. Allen and D. Holm, Extended-geostrophic Hamiltonian models for rotating shallow water motion,, Physica D, 98 (1996), 229.  doi: 10.1016/0167-2789(96)00120-0.  Google Scholar

[3]

N. N. Bogoliubov and Y. A. Mitropolsky, "Asymptotic Methods in the Theory of Nonlinear Oscillations,", Gordon and Breach Sciences Publishers, (1961).   Google Scholar

[4]

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

[5]

M. Bostan, Transport equations with singular coefficients. Application to the gyrokinetic models in plasma physics,, Research Report INRIA, (2009).   Google Scholar

[6]

M. Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation,, hal-00431289, (2009).   Google Scholar

[7]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, Comm. Partial Differential Equations, 25 (2000), 737.  doi: 10.1080/03605300008821529.  Google Scholar

[8]

A. H. Boozer, Physics of magnetically confined plasmas,, Rev. Modern Phys., 76 (2004), 1071.  doi: 10.1103/RevModPhys.76.1071.  Google Scholar

[9]

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

[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.  doi: 10.1007/s10955-009-9858-2.  Google Scholar

[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, 32 (2006).   Google Scholar

[12]

P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases,, Transport Theory Statist. Phys, 25 (1996), 595.  doi: 10.1080/00411459608222915.  Google Scholar

[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.   Google Scholar

[14]

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

[15]

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

[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.  doi: 10.1512/iumj.2004.53.2508.  Google Scholar

[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.  doi: 10.1016/j.jcp.2006.01.023.  Google Scholar

[18]

R. D. Hazeltine and J. D. Meiss, "Plasma Confinement,", Dover Publications, (2003).   Google Scholar

[19]

R. G. Littlejohn, A guiding center Hamiltonian : A new approach,, J. Math. Phys. 20 (1979) 2445-2458., 20 (1979), 2445.  doi: 10.1063/1.524053.  Google Scholar

[20]

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

[21]

J.-M. Rax, "Physique des plasmas, Cours et applications,", Dunod, (2007).   Google Scholar

[22]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics,", Vol. I, I (1980).   Google Scholar

[23]

R. Salmon, New equations for nearly geostrophic flow,, J. Fluid Mechanics, 153 (1985), 461.  doi: 10.1017/S0022112085001343.  Google Scholar

[24]

J. Vanneste and O. Bokhove, Dirac-bracket approach to nearly geostrophic Hamiltonian balanced models,, Physica D Nonlinear Phenomena, 164 (2002), 152.  doi: 10.1016/S0167-2789(02)00375-5.  Google Scholar

show all references

References:
[1]

H. D. I. Abarbanel, Hamiltonian description of almost geostrophic flow,, Geophysical and Astrophysical Fluid Dynamics, 33 (1985), 145.  doi: 10.1080/03091928508245427.  Google Scholar

[2]

J. S. Allen and D. Holm, Extended-geostrophic Hamiltonian models for rotating shallow water motion,, Physica D, 98 (1996), 229.  doi: 10.1016/0167-2789(96)00120-0.  Google Scholar

[3]

N. N. Bogoliubov and Y. A. Mitropolsky, "Asymptotic Methods in the Theory of Nonlinear Oscillations,", Gordon and Breach Sciences Publishers, (1961).   Google Scholar

[4]

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

[5]

M. Bostan, Transport equations with singular coefficients. Application to the gyrokinetic models in plasma physics,, Research Report INRIA, (2009).   Google Scholar

[6]

M. Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation,, hal-00431289, (2009).   Google Scholar

[7]

Y. Brenier, Convergence of the Vlasov-Poisson system to the incompressible Euler equations,, Comm. Partial Differential Equations, 25 (2000), 737.  doi: 10.1080/03605300008821529.  Google Scholar

[8]

A. H. Boozer, Physics of magnetically confined plasmas,, Rev. Modern Phys., 76 (2004), 1071.  doi: 10.1103/RevModPhys.76.1071.  Google Scholar

[9]

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

[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.  doi: 10.1007/s10955-009-9858-2.  Google Scholar

[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, 32 (2006).   Google Scholar

[12]

P. Degond and B. Lucquin-Desreux, Transport coefficients of plasmas and disparate mass binary gases,, Transport Theory Statist. Phys, 25 (1996), 595.  doi: 10.1080/00411459608222915.  Google Scholar

[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.   Google Scholar

[14]

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

[15]

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

[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.  doi: 10.1512/iumj.2004.53.2508.  Google Scholar

[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.  doi: 10.1016/j.jcp.2006.01.023.  Google Scholar

[18]

R. D. Hazeltine and J. D. Meiss, "Plasma Confinement,", Dover Publications, (2003).   Google Scholar

[19]

R. G. Littlejohn, A guiding center Hamiltonian : A new approach,, J. Math. Phys. 20 (1979) 2445-2458., 20 (1979), 2445.  doi: 10.1063/1.524053.  Google Scholar

[20]

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

[21]

J.-M. Rax, "Physique des plasmas, Cours et applications,", Dunod, (2007).   Google Scholar

[22]

M. Reed and B. Simon, "Methods of Modern Mathematical Physics,", Vol. I, I (1980).   Google Scholar

[23]

R. Salmon, New equations for nearly geostrophic flow,, J. Fluid Mechanics, 153 (1985), 461.  doi: 10.1017/S0022112085001343.  Google Scholar

[24]

J. Vanneste and O. Bokhove, Dirac-bracket approach to nearly geostrophic Hamiltonian balanced models,, Physica D Nonlinear Phenomena, 164 (2002), 152.  doi: 10.1016/S0167-2789(02)00375-5.  Google Scholar

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