no. 4
Effect of the polarization drift in a strongly magnetized plasma
ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 4, pp. 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 : 10.1051/m2an/2011068
Classification : 35Q83, 76X05, 82D10
Mots clés : 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 },
     pages = {929--947},
     publisher = {EDP-Sciences},
     volume = {46},
     number = {4},
     year = {2012},
     doi = {10.1051/m2an/2011068},
     mrnumber = {2891475},
     language = {en},
     url = {http://www.numdam.org/articles/10.1051/m2an/2011068/}
}
TY  - JOUR
AU  - Han-Kwan, Daniel
TI  - Effect of the polarization drift in a strongly magnetized plasma
JO  - ESAIM: Mathematical Modelling and Numerical Analysis 
PY  - 2012
SP  - 929
EP  - 947
VL  - 46
IS  - 4
PB  - EDP-Sciences
UR  - http://www.numdam.org/articles/10.1051/m2an/2011068/
DO  - 10.1051/m2an/2011068
LA  - en
ID  - M2AN_2012__46_4_929_0
ER  - 
%0 Journal Article
%A Han-Kwan, Daniel
%T Effect of the polarization drift in a strongly magnetized plasma
%J ESAIM: Mathematical Modelling and Numerical Analysis 
%D 2012
%P 929-947
%V 46
%N 4
%I EDP-Sciences
%U http://www.numdam.org/articles/10.1051/m2an/2011068/
%R 10.1051/m2an/2011068
%G en
%F 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 , Tome 46 (2012) no. 4, pp. 929-947. doi : 10.1051/m2an/2011068. http://www.numdam.org/articles/10.1051/m2an/2011068/

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

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

[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 | Zbl

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

[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 | Zbl

[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 | Zbl

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

Cité par Sources :