Finite Larmor radius approximation for collisional magnetic confinement. Part II: the Fokker-Planck-Landau equation

Bostan, Mihai; Caldini-Queiros, Céline

doi:10.1090/S0033-569X-2014-01357-4

Authors: Mihai Bostan and Céline Caldini-Queiros
Journal: Quart. Appl. Math. 72 (2014), 513-548
MSC (2010): Primary 35Q75, 78A35, 82D10.
DOI: https://doi.org/10.1090/S0033-569X-2014-01357-4
Published electronically: June 9, 2014
MathSciNet review: 3237562
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: This paper is devoted to the finite Larmor radius approximation of the Fokker-Planck-Landau equation, which plays a major role in plasma physics. We obtain a completely explicit form for the gyroaverage of the Fokker-Planck-Landau kernel, accounting for diffusion and convolution with respect to both velocity and (perpendicular) position coordinates. We show that the new collision operator enjoys the usual physical properties; the averaged kernel balances the mass, momentum, and kinetic energy and dissipates the entropy, globally in velocity and perpendicular position coordinates.

References

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
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, DOI https://doi.org/10.1016/j.jde.2010.07.010
Mihai Bostan, Gyrokinetic Vlasov equation in three dimensional setting. Second order approximation, Multiscale Model. Simul. 8 (2010), no. 5, 1923–1957. MR 2769087, DOI https://doi.org/10.1137/090777621
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 (French, with English and French summaries). MR 2990896, DOI https://doi.org/10.1016/j.crma.2012.09.019

M. Bostan, C. Caldini-Queiros, Finite Larmor radius approximation for collisional magnetic confinement. Part I : The linear Boltzmann equation, Quart. Appl. Math. 72 (2014), no. 2, 323–345.

Carlo Cercignani, Reinhard Illner, and Mario Pulvirenti, The mathematical theory of dilute gases, Applied Mathematical Sciences, vol. 106, Springer-Verlag, New York, 1994. MR 1307620

Laurent Desvillettes and Cédric Villani, On the spatially homogeneous Landau equation for hard potentials. I. Existence, uniqueness and smoothness, Comm. Partial Differential Equations 25 (2000), no. 1-2, 179–259. MR 1737547, DOI https://doi.org/10.1080/03605300008821512

Laurent Desvillettes and Cédric Villani, On the spatially homogeneous Landau equation for hard potentials. II. $H$-theorem and applications, Comm. Partial Differential Equations 25 (2000), no. 1-2, 261–298. MR 1737548, DOI https://doi.org/10.1080/03605300008821513

Laurent Desvillettes, Plasma kinetic models: the Fokker-Planck-Landau equation, Modeling and computational methods for kinetic equations, Model. Simul. Sci. Eng. Technol., Birkhäuser Boston, Boston, MA, 2004, pp. 171–193. MR 2068584

E. Frénod, Application of the averaging method to the gyrokinetic plasma, Asymptot. Anal. 46 (2006), no. 1, 1–28. MR 2196630

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

Emmanuel Frénod and Eric Sonnendrücker, The finite Larmor radius approximation, SIAM J. Math. Anal. 32 (2001), no. 6, 1227–1247. MR 1856246, DOI https://doi.org/10.1137/S0036141099364243

X. Garbet, G. Dif-Pradalier, C. Nguyen, Y. Sarazin, V. Grandgirard, Ph. Ghendrih, Neoclassical equilibrium in gyrokinetic simulations, Phys. Plasmas 16(2009).

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, DOI https://doi.org/10.1016/S0021-7824%2899%2900021-5

R. D. Hazeltine, J. D. Meiss, Plasma confinement, Dover Publications, Inc., Mineola, New York, 2003.

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, DOI https://doi.org/10.1016/S0021-7824%2801%2901245-4

X. Q. Xu, M. N. Rosenbluth, Numerical simulation of ion-temperature-gradient-driven modes, Phys. Fluids, B 3(1991) 627-643.