# Reduced resistive MHD in Tokamaks with general density

ESAIM: Mathematical Modelling and Numerical Analysis (2012)

- Volume: 46, Issue: 5, page 1081-1106
- ISSN: 0764-583X

## Access Full Article

top## Abstract

top## How to cite

topDesprés, Bruno, and Sart, Rémy. "Reduced resistive MHD in Tokamaks with general density." ESAIM: Mathematical Modelling and Numerical Analysis 46.5 (2012): 1081-1106. <http://eudml.org/doc/222141>.

@article{Després2012,

abstract = {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.},

author = {Després, Bruno, Sart, Rémy},

journal = {ESAIM: Mathematical Modelling and Numerical Analysis},

keywords = {Tokamaks; reduced Magnetohydrodynamics; tokamaks; reduced magnetohydrodynamics},

language = {eng},

month = {2},

number = {5},

pages = {1081-1106},

publisher = {EDP Sciences},

title = {Reduced resistive MHD in Tokamaks with general density},

url = {http://eudml.org/doc/222141},

volume = {46},

year = {2012},

}

TY - JOUR

AU - Després, Bruno

AU - Sart, Rémy

TI - Reduced resistive MHD in Tokamaks with general density

JO - ESAIM: Mathematical Modelling and Numerical Analysis

DA - 2012/2//

PB - EDP Sciences

VL - 46

IS - 5

SP - 1081

EP - 1106

AB - 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.

LA - eng

KW - Tokamaks; reduced Magnetohydrodynamics; tokamaks; reduced magnetohydrodynamics

UR - http://eudml.org/doc/222141

ER -

## References

top- 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). Zbl1120.65001
- D. Biskamp, Nonlinear Magnetohydrodynamics. Cambridge University Press (1992).
- J. Blum, Numerical simulation and optimal control in plasma physics, with application to Tokamaks. Series in Modern Applied Mathematics. Wiley/Gauthier-Villard (1989). Zbl0717.76009
- 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).
- J. Blum, T. Gallouet and J. Simon, Existence and control of plasma equilibirum in a Tokamak. SIAM J. Math. Anal.17 (1986) 1158–1177. Zbl0614.35082
- 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). Zbl06039412
- H. Brezis and H. Berestycki, On a free boundary problem arising in plasma physics. Nonlinear Anal.4 (1980) 415–436. Zbl0437.35032
- S. Briguglio, G. Wad, F. Zonca and C. Kar, Hybrid magnetohydrodynamic-gyrokinetic simulation of toroidal Alfven modes. Phys. Plasmas2 (1995) 3711–3723.
- S. Briguglio, F. Zonca and C. Kar, Hybrid magnetohydrodynamic-particle simulation of linear and nonlinear evolution of Alfven modes in tokamaks. Phys. Plasmas5 (1998) 3287–3301.
- L.A. Caffarelli and S. Salsa, A geometric approach to free boundary problems, Graduate Studies in Mathematics. AMS, Providence, RI 68 (2005). Zbl1083.35001
- F. Chen, Introduction to plasma physics and controlled fusion. Springer, New York (1984).
- O. Czarny and G. Huysmans, MHD stability in X-point geometry : simulation of ELMs. Nucl. Fusion47 (2007) 659–666.
- O. Czarny and G. Huysmans, Bézier surfaces and finite elements for MHD simulations. J. Comput. Phys.227 (2008) 7423–7445. Zbl1141.76035
- 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. Zbl1235.76064
- 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. Zbl1163.35400
- 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. I317 (1993) 353–359. Zbl0783.76106
- E. Feireisl, Dynamics of viscous compressible fluids. Oxford University Press (2004). Zbl1080.76001
- J. Freidberg, Plasma physics and fusion energy. Cambridge (2007).
- A. Friedman, Variational principles and free-boundary problems. Wiley-interscience publication, Wiley, New York (1982).
- T. Fujita, Tokamak equilibria with nearly zero central current : the current hole (review article). Nucl. Fusion50 (2010).
- 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.
- J.F. Gerbeau, C. Le Bris and T. Lelièvre, Mathematical methods for the magnetohydrodynamics of liquid metals. Oxford University Press, USA (2006).
- 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.
- 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. Fusion51 (2009) 124012.
- B.B. Kadomtsev and O.P. Pogutse, Non linear helical perturbations of a plasma in a Tokamak. Sov. Phys.-JETP38 (1974) 283–290.
- S.-E. Kruger, C.C. Hegna and J.D. Callen, Generalized reduced magnetohydrodynamic equations. Phys. Plasmas5 (1998) 4169–4183.
- J.-L. Lions, Quelques méthodes de résolution des problèmes aux limites non linéaires, Études Mathématiques. Dunod (1969). Zbl0189.40603
- P.-L. Lions, Mathematical topics in fluid mechanics. Incompressible models, edited by Oxford Science Publication 1 (1996).
- P.-L. Lions, Mathematical topics in fluid mechanics. Compressible models, edited by Oxford Science Publication 2 (1998).
- 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. Zbl1338.76143
- 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. Zbl1220.76055
- K. Miyamoto, Plasma physics and controlled nuclear fusion. Springer (2005). Zbl1276.81126
- B. Nkonga, Private communication (2010).
- M.N. Rosenbluth, D.A. Monticello, H.R. Strauss and R.B. White, Dynamics of high β plasmas. Phys. Fluids19 (1976) 1987.
- R. Smaltz, Reduced, three-dimensional, nonlinear equations for high-β plasmas including toroidal effects. Phys. Lett. A82 (1981) 14–17.
- H.R. Strauss, Nonlinear three-dimensional magnetohydrodynamics of noncircular Tokamaks. Phys. Fluids19 (1976) 134–140.
- H.R. Strauss, Dynamics of high β plasmas. Phys. Fluids20 (1977) 1354–1360.
- R. Temam, Remarks on a free boundary value problem arising in plasma physics. Commun. Partial Differ. Equ.2 (1977) 563–585. Zbl0355.35023
- R. Temam, Navier-Stokes Equations, Theory and Numerical Analysis. North-Holland (1979). Zbl0426.35003
- 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. Plasmas8 (2001) 2125.
- 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 (2004). URIhttp://www-naweb.iaea.org/napc/physics/fec/fec2004/papers/icp6-16.pdf

## NotesEmbed ?

topTo embed these notes on your page include the following JavaScript code on your page where you want the notes to appear.