Effect of the polarization drift in a strongly magnetized plasma
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 4, p. 929-947

We consider a strongly magnetized plasma described by a Vlasov-Poisson system with a large external magnetic field. The finite Larmor radius scaling allows to describe its behaviour at very fine scales. We give a new interpretation of the asymptotic equations obtained by Frénod and Sonnendrücker [SIAM J. Math. Anal. 32 (2001) 1227-1247] when the intensity of the magnetic field goes to infinity. We introduce the so-called polarization drift and show that its contribution is not negligible in the limit, contrary to what is usually said. This is due to the non linear coupling between the Vlasov and Poisson equations.

DOI : https://doi.org/10.1051/m2an/2011068
Classification:  35Q83,  76X05,  82D10
Keywords: Vlasov-Poisson equation, strong magnetic field regime, finite larmor radius scaling, electric drift, polarization drift, oscillations in time
@article{M2AN_2012__46_4_929_0,
author = {Han-Kwan, Daniel},
title = {Effect of the polarization drift in a strongly magnetized plasma},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
publisher = {EDP-Sciences},
volume = {46},
number = {4},
year = {2012},
pages = {929-947},
doi = {10.1051/m2an/2011068},
mrnumber = {2891475},
language = {en},
url = {http://www.numdam.org/item/M2AN_2012__46_4_929_0}
}

Han-Kwan, Daniel. Effect of the polarization drift in a strongly magnetized plasma. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Volume 46 (2012) no. 4, pp. 929-947. doi : 10.1051/m2an/2011068. http://www.numdam.org/item/M2AN_2012__46_4_929_0/

[1] G. Allaire, Homogenization and two-scale convergence. SIAM J. Math. Anal. XXIII (1992) 1482-1518. | MR 1185639 | Zbl 0770.35005

[2] A.A. Arsenev, Existence in the large of a weak solution of Vlasov's system of equations. Z. Vychisl. Mat. Mat. Fiz. 15 (1975) 136-147. | MR 371322 | Zbl 0345.35083

[3] M. Bostan, The Vlasov-Poisson system with strong external magnetic field. Finite Larmor radius regime. Asymptot. Anal. 61 (2009) 91-123. | MR 2499194 | Zbl 1180.35501

[4] P. Degond, Global existence of smooth solutions for the Vlasov-Fokker-Planck equations in 1 and 2 space dimensions. Ann. Sci. École Norm. Sup. 19 (1986) 519-542. | Numdam | MR 875086 | Zbl 0619.35087

[5] E. Frénod and A. Mouton, Two-dimensional finite Larmor radius approximation in canonical gyrokinetic coordinates. J. Pure Appl. Math. : Adv. Appl. 4 (2010) 135-166. | MR 2816864 | Zbl 1225.35016

[6] E. Frénod and E. Sonnendrücker, The finite Larmor radius approximation. SIAM J. Math. Anal. 32 (2001) 1227-1247. | MR 1856246 | Zbl 0980.82030

[7] E. Frénod, A. Mouton and E. Sonnendrücker, Two-scale numerical simulation of the weakly compressible 1D isentropic Euler equations. Numer. Math. 108 (2007) 263-293. | MR 2358005 | Zbl 1127.76034

[8] E. Frénod, F. Salvarani and E. Sonnendrücker, Long time simulation of a beam in a periodic focusing channel via a two-scale PIC-method. Math. Models Methods Appl. Sci. 19 (2009) 175-197. | MR 2498432 | Zbl 1168.82026

[9] P. Ghendrih, M. Hauray and A. Nouri, Derivation of a gyrokinetic model. Existence and uniqueness of specific stationary solutions. KRM 2 (2009) 707-725. | MR 2556718 | Zbl 1195.82087

[10] F. Golse and L. Saint-Raymond, The Vlasov-Poisson system with strong magnetic field. J. Math. Pures Appl. 78 (1999) 791-817. | MR 1715342 | Zbl 0977.35108

[11] V. Grandgirard et al., Global full-f gyrokinetic simulations of plasma turbulence. Plasma Phys. Control. Fusion 49 (2007) 173-182.

[12] D. Han-Kwan, The three-dimensional finite Larmor radius approximation. Asymptot. Anal. 66 (2010) 9-33. | MR 2582446 | Zbl 1191.35267

[13] D. Han-Kwan, On the three-dimensional finite Larmor radius approximation : the case of electrons in a fixed background of ions. Submitted (2010). | MR 2582446 | Zbl 1191.35267

[14] Z. Lin, S. Ethier, T.S. Hahm and W.M. Tang, Size scaling of turbulent transport in magnetically confined plasmas. Phys. Rev. Lett. 88 (2002) 195004-1-195004-4.

[15] P.L. Lions and B. Perthame, Propagation of moments and regularity for the three-dimensional Vlasov-Poisson system. Invent. Math. 105 (1991) 415-430. | MR 1115549 | Zbl 0741.35061

[16] A. Mouton, Two-scale semi-Lagrangian simulation of a charged particle beam in a periodic focusing channel. KRM 2 (2009) 251-274. | MR 2507448 | Zbl 1191.35040

[17] G. Nguetseng, A general convergence result for a functional related to the theory of homogenization. SIAM J. Math. Anal. 20 (1989) 608-623. | MR 990867 | Zbl 0688.35007

[18] S. Ukai and T. Okabe, On classical solutions in the large in time of two-dimensional Vlasov's equation. Osaka J. Math. 15 (1978) 245-261. | MR 504289 | Zbl 0405.35002

[19] J. Wesson, Tokamaks.Clarendon Press-Oxford (2004). | Zbl 1111.82054