Reduced resistive MHD in Tokamaks with general density
ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) no. 5, pp. 1081-1106.

The aim of this paper is to derive a general model for reduced viscous and resistive Magnetohydrodynamics (MHD) and to study its mathematical structure. The model is established for arbitrary density profiles in the poloidal section of the toroidal geometry of Tokamaks. The existence of global weak solutions, on the one hand, and the stability of the fundamental mode around initial data, on the other hand, are investigated.

DOI : https://doi.org/10.1051/m2an/2011078
Classification : 93A30,  35Q35,  76E25,  82D10
Mots clés : tokamaks, reduced magnetohydrodynamics
@article{M2AN_2012__46_5_1081_0,
author = {Despr\'es, Bruno and Sart, R\'emy},
title = {Reduced resistive MHD in Tokamaks with general density},
journal = {ESAIM: Mathematical Modelling and Numerical Analysis - Mod\'elisation Math\'ematique et Analyse Num\'erique},
pages = {1081--1106},
publisher = {EDP-Sciences},
volume = {46},
number = {5},
year = {2012},
doi = {10.1051/m2an/2011078},
zbl = {1267.76034},
mrnumber = {2916373},
language = {en},
url = {http://www.numdam.org/item/M2AN_2012__46_5_1081_0/}
}
Després, Bruno; Sart, Rémy. Reduced resistive MHD in Tokamaks with general density. ESAIM: Mathematical Modelling and Numerical Analysis - Modélisation Mathématique et Analyse Numérique, Tome 46 (2012) no. 5, pp. 1081-1106. doi : 10.1051/m2an/2011078. http://www.numdam.org/item/M2AN_2012__46_5_1081_0/

[1] G. Allaire, Numerical Analysis and Optimization : An Introduction to Mathematical Modelling and Numerical Simulation in Numerical Mathematics and Scientific Computation series. Oxford University Press (2007). | MR 2326223 | Zbl 1120.65001

[2] D. Biskamp, Nonlinear Magnetohydrodynamics. Cambridge University Press (1992). | MR 1250152

[3] J. Blum, Numerical simulation and optimal control in plasma physics, with application to Tokamaks. Series in Modern Applied Mathematics. Wiley/Gauthier-Villard (1989). | MR 996236 | Zbl 0717.76009

[4] J. Blum, Numerical identification of the plasma current density in a Tokamak fusion reactor : the determination of a non-linear source in an elliptic pde, invited conference, in Proceedings of PICOF02. Carthage, Tunisie (2002).

[5] J. Blum, T. Gallouet and J. Simon, Existence and control of plasma equilibirum in a Tokamak. SIAM J. Math. Anal. 17 (1986) 1158-1177. | MR 853522 | Zbl 0614.35082

[6] J. Blum, C. Boulbe and B. Faugeras, Real time reconstruction of plasma equilibrium in a Tokamak, International conference on burning plasma diagnostics. Villa Manoastero, Varenna (2007).

[7] H. Brezis and H. Berestycki, On a free boundary problem arising in plasma physics. Nonlinear Anal. 4 (1980) 415-436. | MR 574364 | Zbl 0437.35032

[8] S. Briguglio, G. Wad, F. Zonca and C. Kar, Hybrid magnetohydrodynamic-gyrokinetic simulation of toroidal Alfven modes. Phys. Plasmas 2 (1995) 3711-3723.

[9] S. Briguglio, F. Zonca and C. Kar, Hybrid magnetohydrodynamic-particle simulation of linear and nonlinear evolution of Alfven modes in tokamaks. Phys. Plasmas 5 (1998) 3287-3301.

[10] L.A. Caffarelli and S. Salsa, A geometric approach to free boundary problems, Graduate Studies in Mathematics. AMS, Providence, RI 68 (2005). | Zbl 1083.35001

[11] F. Chen, Introduction to plasma physics and controlled fusion. Springer, New York (1984).

[12] O. Czarny and G. Huysmans, MHD stability in X-point geometry : simulation of ELMs. Nucl. Fusion 47 (2007) 659-666.

[13] O. Czarny and G. Huysmans, Bézier surfaces and finite elements for MHD simulations. J. Comput. Phys. 227 (2008) 7423-7445. | MR 2437577 | Zbl 1141.76035

[14] E. Deriaz, B. Després, G. Faccanoni, K.P. Gostaf, L.-M. Imbert-Gérard, G. Sadaka and R. Sart, Magnetic equations with FreeFem++, The Grad-Shafranov equation and the Current Hole. ESAIM Proc. 32 (2011) 76-94. | MR 2862440 | Zbl 1235.76064

[15] J.I. Diaz and J.F. Padial, On a free-boundary problem modeling the action of a limiter on a plasma. Discrete Contin. Dyn. Syst. Suppl. (2007) 313-322. | MR 2409226 | Zbl 1163.35400

[16] J.I. Diaz and J.-M. Rakotoson, On a two-dimensional stationary free boundary problem arising in the confinement of a plasma in a Stellarator. C. R. Acad. Sci. Paris, Sér. I 317 (1993) 353-359. | MR 1235448 | Zbl 0783.76106

[17] E. Feireisl, Dynamics of viscous compressible fluids. Oxford University Press (2004). | MR 2040667 | Zbl 1080.76001

[18] J. Freidberg, Plasma physics and fusion energy. Cambridge (2007).

[19] A. Friedman, Variational principles and free-boundary problems. Wiley-interscience publication, Wiley, New York (1982). | MR 679313 | Zbl 0564.49002

[20] T. Fujita, Tokamak equilibria with nearly zero central current : the current hole (review article). Nucl. Fusion 50 (2010).

[21] T. Fujita, T. Oikawa, T. Suzuki, S. Ide, Y. Sakamoto, Y. Koide, T. Hatae, O. Naito, A. Isayama, N. Hayashi and H. Shirai, Plasma equilibrium and confinement in a Tokamak with nearly zero central current density in JT-60U. Phys. Rev. Lett. 87 (2001) 245001-245005.

[22] J.F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical methods for the magnetohydrodynamics of liquid metals. Oxford University Press, USA (2006). | MR 2289481 | Zbl 1107.76001

[23] G. Huysmans, T.C. Hender, N.C. Hawkes and X. Litaudon, MHD stability of advanced Tokamak scenarios with reversed central current : an explanation of the “Current Hole”. Phys. Rev. Lett. 87 (2001) 245002-245006.

[24] G.T.A. Huysmans, S. Pamela, E. Van Der Plas and P. Ramet, Non-linear MHD simulations of edge localized modes (ELMs). Plasma Phys. Control. Fusion 51 (2009) 124012.

[25] B.B. Kadomtsev and O.P. Pogutse, Non linear helical perturbations of a plasma in a Tokamak. Sov. Phys.-JETP 38 (1974) 283-290.

[26] S.-E. Kruger, C.C. Hegna and J.D. Callen, Generalized reduced magnetohydrodynamic equations. Phys. Plasmas 5 (1998) 4169-4183. | MR 1656032

[27] J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Études Mathématiques. Dunod (1969). | Zbl 0189.40603

[28] P.-L. Lions, Mathematical topics in fluid mechanics. Incompressible models, edited by Oxford Science Publication 1 (1996). | MR 1422251 | Zbl 0866.76002

[29] P.-L. Lions, Mathematical topics in fluid mechanics. Compressible models, edited by Oxford Science Publication 2 (1998). | MR 1637634 | Zbl 0908.76004

[30] H. Lütjens and J.-F. Luciani, The XTOR code for nonlinear 3D simulations of MHD instabilities in tokamak plasmas. J. Comput. Phys. 227 (2008) 6944-6966. | MR 2435437

[31] H. Lütjens and J.-F. Luciani, XTOR-2F : A fully implicit NewtonKrylov solver applied to nonlinear 3D extended MHD in tokamaks. J. Comput. Phys. 229 (2010) 8130-8143. | MR 2719164 | Zbl 1220.76055

[32] K. Miyamoto, Plasma physics and controlled nuclear fusion. Springer (2005). | Zbl 1276.81126

[33] B. Nkonga, Private communication (2010).

[34] M.N. Rosenbluth, D.A. Monticello, H.R. Strauss and R.B. White, Dynamics of high β plasmas. Phys. Fluids 19 (1976) 1987.

[35] R. Smaltz, Reduced, three-dimensional, nonlinear equations for high-β plasmas including toroidal effects. Phys. Lett. A 82 (1981) 14-17.

[36] H.R. Strauss, Nonlinear three-dimensional magnetohydrodynamics of noncircular Tokamaks. Phys. Fluids 19 (1976) 134-140.

[37] H.R. Strauss, Dynamics of high β plasmas. Phys. Fluids 20 (1977) 1354-1360.

[38] R. Temam, Remarks on a free boundary value problem arising in plasma physics. Commun. Partial Differ. Equ. 2 (1977) 563-585. | MR 602544 | Zbl 0355.35023

[39] R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland (1979). | MR 603444 | Zbl 0426.35003

[40] Z. Yoshida, S.M. Mahajan, S. Ohsaki, M. Iqbal and N. Shatashvili, Beltrami fields in plasmas : High-confinement mode boundary layers and high beta equilibria. Phys. Plasmas 8 (2001) 2125.

[41] Z. Yoshida et al., Potential Control and Flow Generation in a Toroidal Internal-Coil System - a New Approach to High-beta Equilibrium, in 20th IAEA Fusion Energy Conference. Online at http://www-naweb.iaea.org/napc/physics/fec/fec2004/papers/icp6-16.pdf (2004).