Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



Finite Larmor radius approximation for collisional magnetic confinement. Part I: The linear Boltzmann equation

Authors: Mihai Bostan and Céline Caldini-Queiros
Journal: Quart. Appl. Math. 72 (2014), 323-345
MSC (2010): Primary 35Q75, 78A35, 82D10
DOI: https://doi.org/10.1090/S0033-569X-2014-01356-1
Published electronically: March 28, 2014
MathSciNet review: 3186240
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Abstract | References | Similar Articles | Additional Information

Abstract: The subject matter of this paper concerns the derivation of the finite Larmor radius approximation, when collisions are taken into account. Several studies are performed, corresponding to different collision kernels : the relaxation and the Fokker-Planck operators. Gyroaveraging the relaxation operator leads to a position-velocity integral operator, whereas gyroaveraging the linear Fokker-Planck operator leads to diffusion in velocity but also with respect to the perpendicular position coordinates.

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  • [1] Mihai Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime, Asymptot. Anal. 61 (2009), no. 2, 91-123. MR 2499194 (2009k:82122)
  • [2] Mihai Bostan, Transport equations with disparate advection fields. Application to the gyrokinetic models in plasma physics, J. Differential Equations 249 (2010), no. 7, 1620-1663. MR 2677810 (2011i:82074), https://doi.org/10.1016/j.jde.2010.07.010
  • [3] Mihai Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, Multiscale Model. Simul. 8 (2010), no. 5, 1923-1957. MR 2769087 (2012b:35342), https://doi.org/10.1137/090777621
  • [4] M. Bostan, Transport of charged particles under fast oscillating magnetic fields, SIAM J. Math. Anal. 44(2012) 1415-1447.
  • [5] Mihaï Bostan and Céline Caldini-Queiros, Approximation de rayon de Larmor fini pour les plasmas magnétisés collisionnels, C. R. Math. Acad. Sci. Paris 350 (2012), no. 19-20, 879-884. MR 2990896, https://doi.org/10.1016/j.crma.2012.09.019
  • [6] M. Bostan, C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part II : The Fokker-Planck-Landau equation, to appear in Quart. Appl. Math.
  • [7] Alain J. Brizard, Variational principle for nonlinear gyrokinetic Vlasov-Maxwell equations, Phys. Plasmas 7 (2000), no. 12, 4816-4822. MR 1800518 (2001g:76065), https://doi.org/10.1063/1.1322063
  • [8] Alain J. Brizard, A guiding-center Fokker-Planck collision operator for nonuniform magnetic fields, Phys. Plasmas 11 (2004), no. 9, 4429-4438. MR 2095562 (2005e:82094), https://doi.org/10.1063/1.1780532
  • [9] A. J. Brizard and T. S. Hahm, Foundations of nonlinear gyrokinetic theory, Rev. Modern Phys. 79 (2007), no. 2, 421-468. MR 2336960 (2008e:76188), https://doi.org/10.1103/RevModPhys.79.421
  • [10] E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptot. Anal. 46 (2006), no. 1, 1-28. MR 2196630 (2007a:34090)
  • [11] E. Frénod and A. Mouton, Two-dimensional finite Larmor radius approximation in canonical gyrokinetic coordinates, J. Pure Appl. Math. Adv. Appl. 4 (2010), no. 2, 135-169. MR 2816864 (2012d:76146)
  • [12] Emmanuel Frénod and Eric Sonnendrücker, Homogenization of the Vlasov equation and of the Vlasov-Poisson system with a strong external magnetic field, Asymptot. Anal. 18 (1998), no. 3-4, 193-213. MR 1668938 (99m:82046)
  • [13] Emmanuel Frénod and Eric Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal. 32 (2001), no. 6, 1227-1247 (electronic). MR 1856246 (2002g:82049), https://doi.org/10.1137/S0036141099364243
  • [14] X. Garbet, G. Dif-Pradalier, C. Nguyen, Y. Sarazin, V. Grandgirard, Ph. Ghendrih, Neoclassical equilibrium in gyrokinetic simulations, Phys. Plasmas 16(2009).
  • [15] François Golse and Laure Saint-Raymond, The Vlasov-Poisson system with strong magnetic field, J. Math. Pures Appl. (9) 78 (1999), no. 8, 791-817. MR 1715342 (2000g:35209), https://doi.org/10.1016/S0021-7824(99)00021-5
  • [16] P. A. Markowich, C. A. Ringhofer, and C. Schmeiser, Semiconductor equations, Springer-Verlag, Vienna, 1990. MR 1063852 (91j:78011)
  • [17] Frédéric Poupaud and Christian Schmeiser, Charge transport in semiconductors with degeneracy effects, Math. Methods Appl. Sci. 14 (1991), no. 5, 301-318. MR 1113606 (92c:82102), https://doi.org/10.1002/mma.1670140503
  • [18] F. Poupaud, Runaway phenomena and fluid approximation under high fields in semiconductor kinetic theory, Z. Angew. Math. Mech. 72 (1992), no. 8, 359-372 (English, with English, German and Russian summaries). MR 1178932 (93h:82074), https://doi.org/10.1002/zamm.19920720813
  • [19] Laure Saint-Raymond, Control of large velocities in the two-dimensional gyrokinetic approximation, J. Math. Pures Appl. (9) 81 (2002), no. 4, 379-399. MR 1967354 (2004b:76166), https://doi.org/10.1016/S0021-7824(01)01245-4
  • [20] X. Q. Xu, M. N. Rosenbluth, Numerical simulation of ion-temperature-gradient-driven modes, Phys. Fluids, B 3(1991) 627-643.

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Additional Information

Mihai Bostan
Affiliation: Laboratoire d’Analyse, Topologie, Probabilités LATP, Centre de Mathématiques et Informatique CMI, UMR CNRS 7353, 39 rue Frédéric Joliot Curie, 13453 Marseille Cedex 13 France
Email: bostan@cmi.univ-mrs.fr

Céline Caldini-Queiros
Affiliation: Laboratoire de Mathématiques de Besançon, UMR CNRS 6623, Université de Franche-Comté, 16 route de Gray, 25030 Besançon Cedex France
Email: celine.caldini-queiros@univ-fcomte.fr

DOI: https://doi.org/10.1090/S0033-569X-2014-01356-1
Received by editor(s): July 4, 2012
Received by editor(s) in revised form: December 15, 2012
Published electronically: March 28, 2014
Article copyright: © Copyright 2014 Brown University

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