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.
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 },
pages = {1081--1106},
publisher = {EDP-Sciences},
volume = {46},
number = {5},
year = {2012},
doi = {10.1051/m2an/2011078},
mrnumber = {2916373},
zbl = {1267.76034},
language = {en},
url = {http://www.numdam.org/articles/10.1051/m2an/2011078/}
}
TY - JOUR AU - Després, Bruno AU - Sart, Rémy TI - Reduced resistive MHD in Tokamaks with general density JO - ESAIM: Mathematical Modelling and Numerical Analysis PY - 2012 SP - 1081 EP - 1106 VL - 46 IS - 5 PB - EDP-Sciences UR - http://www.numdam.org/articles/10.1051/m2an/2011078/ DO - 10.1051/m2an/2011078 LA - en ID - M2AN_2012__46_5_1081_0 ER -
%0 Journal Article %A Després, Bruno %A Sart, Rémy %T Reduced resistive MHD in Tokamaks with general density %J ESAIM: Mathematical Modelling and Numerical Analysis %D 2012 %P 1081-1106 %V 46 %N 5 %I EDP-Sciences %U http://www.numdam.org/articles/10.1051/m2an/2011078/ %R 10.1051/m2an/2011078 %G en %F M2AN_2012__46_5_1081_0
Després, Bruno; Sart, Rémy. Reduced resistive MHD in Tokamaks with general density. ESAIM: Mathematical Modelling and Numerical Analysis , Tome 46 (2012) no. 5, pp. 1081-1106. doi : 10.1051/m2an/2011078. http://www.numdam.org/articles/10.1051/m2an/2011078/
[1] , Numerical Analysis and Optimization : An Introduction to Mathematical Modelling and Numerical Simulation in Numerical Mathematics and Scientific Computation series. Oxford University Press (2007). | MR | Zbl
[2] , Nonlinear Magnetohydrodynamics. Cambridge University Press (1992). | MR
[3] , Numerical simulation and optimal control in plasma physics, with application to Tokamaks. Series in Modern Applied Mathematics. Wiley/Gauthier-Villard (1989). | MR | Zbl
[4] , 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] , and , Existence and control of plasma equilibirum in a Tokamak. SIAM J. Math. Anal. 17 (1986) 1158-1177. | MR | Zbl
[6] , and , Real time reconstruction of plasma equilibrium in a Tokamak, International conference on burning plasma diagnostics. Villa Manoastero, Varenna (2007).
[7] and , On a free boundary problem arising in plasma physics. Nonlinear Anal. 4 (1980) 415-436. | MR | Zbl
[8] , , and , Hybrid magnetohydrodynamic-gyrokinetic simulation of toroidal Alfven modes. Phys. Plasmas 2 (1995) 3711-3723.
[9] , and , Hybrid magnetohydrodynamic-particle simulation of linear and nonlinear evolution of Alfven modes in tokamaks. Phys. Plasmas 5 (1998) 3287-3301.
[10] and , A geometric approach to free boundary problems, Graduate Studies in Mathematics. AMS, Providence, RI 68 (2005). | Zbl
[11] , Introduction to plasma physics and controlled fusion. Springer, New York (1984).
[12] and , MHD stability in X-point geometry : simulation of ELMs. Nucl. Fusion 47 (2007) 659-666.
[13] and , Bézier surfaces and finite elements for MHD simulations. J. Comput. Phys. 227 (2008) 7423-7445. | MR | Zbl
[14] , , , , , and , Magnetic equations with FreeFem++, The Grad-Shafranov equation and the Current Hole. ESAIM Proc. 32 (2011) 76-94. | MR | Zbl
[15] and , On a free-boundary problem modeling the action of a limiter on a plasma. Discrete Contin. Dyn. Syst. Suppl. (2007) 313-322. | MR | Zbl
[16] and , 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 | Zbl
[17] , Dynamics of viscous compressible fluids. Oxford University Press (2004). | MR | Zbl
[18] , Plasma physics and fusion energy. Cambridge (2007).
[19] , Variational principles and free-boundary problems. Wiley-interscience publication, Wiley, New York (1982). | MR | Zbl
[20] , Tokamak equilibria with nearly zero central current : the current hole (review article). Nucl. Fusion 50 (2010).
[21] , , , , , , , , , and , Plasma equilibrium and confinement in a Tokamak with nearly zero central current density in JT-60U. Phys. Rev. Lett. 87 (2001) 245001-245005.
[22] , and , Mathematical methods for the magnetohydrodynamics of liquid metals. Oxford University Press, USA (2006). | MR | Zbl
[23] , , and , MHD stability of advanced Tokamak scenarios with reversed central current : an explanation of the “Current Hole”. Phys. Rev. Lett. 87 (2001) 245002-245006.
[24] , , and , Non-linear MHD simulations of edge localized modes (ELMs). Plasma Phys. Control. Fusion 51 (2009) 124012.
[25] and , Non linear helical perturbations of a plasma in a Tokamak. Sov. Phys.-JETP 38 (1974) 283-290.
[26] , and , Generalized reduced magnetohydrodynamic equations. Phys. Plasmas 5 (1998) 4169-4183. | MR
[27] , Quelques méthodes de résolution des problèmes aux limites non linéaires, Études Mathématiques. Dunod (1969). | Zbl
[28] , Mathematical topics in fluid mechanics. Incompressible models, edited by Oxford Science Publication 1 (1996). | MR | Zbl
[29] , Mathematical topics in fluid mechanics. Compressible models, edited by Oxford Science Publication 2 (1998). | MR | Zbl
[30] and , The XTOR code for nonlinear 3D simulations of MHD instabilities in tokamak plasmas. J. Comput. Phys. 227 (2008) 6944-6966. | MR
[31] and , XTOR-2F : A fully implicit NewtonKrylov solver applied to nonlinear 3D extended MHD in tokamaks. J. Comput. Phys. 229 (2010) 8130-8143. | MR | Zbl
[32] , Plasma physics and controlled nuclear fusion. Springer (2005). | Zbl
[33] , Private communication (2010).
[34] , , and , Dynamics of high β plasmas. Phys. Fluids 19 (1976) 1987.
[35] , Reduced, three-dimensional, nonlinear equations for high-β plasmas including toroidal effects. Phys. Lett. A 82 (1981) 14-17.
[36] , Nonlinear three-dimensional magnetohydrodynamics of noncircular Tokamaks. Phys. Fluids 19 (1976) 134-140.
[37] , Dynamics of high β plasmas. Phys. Fluids 20 (1977) 1354-1360.
[38] , Remarks on a free boundary value problem arising in plasma physics. Commun. Partial Differ. Equ. 2 (1977) 563-585. | MR | Zbl
[39] , Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland (1979). | MR | Zbl
[40] , , , and , Beltrami fields in plasmas : High-confinement mode boundary layers and high beta equilibria. Phys. Plasmas 8 (2001) 2125.
[41] 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).
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