Spectral approximation for the linearized MHD operator in cylindrical region

Abstract

We propose a numerical algorithm to calculate the spectrum of linearized roagnetohydrodynamic operators in the cylindrical region. The method is based on the Ritz-Galerkin approximation with special finite element basis functions. The detailed analysis is made in the presence of the magnetic axis. Ir is proved that there is no spectral pollution in this case. Some numerical examples ate included.w IntroductionThe study of the magnetohydrodynamic stability of an ideal plasma leads to the investigation of the spectrum of a linear operator. Both for theoretical and practical aspects, two major difficulties arise: a) the resolvent of the operator is not compact, b) the coefficients of the operator possess complicated singularity in the neighborhood of the magnetic axis.A mathematically rigorous characterization of the continuous spectrum of the operator has been obtained in References [4], [11] for the toroidal geometry; however it supposes that the plasma is isolated from the magnetic axis. The cylinder case, in the presence of the magnetic axis, is treated in [13].On the other hand, References [5], [8], [9], [10], [171 contain mathematical justifications of some numerical algorithms used by physicists [1], [2], [19] in cylindrical and toroidal geometries; it is proved that the approximations have good spectral properties, in particular "non poUution'; however these analyses do not take account of the difficulty of the singularity at the magnetic axis.The purpose of this paper is to propose and to discuss a numerical method for computing the spectrum of the operator for the cylinder case in the presence of the magnetic axis. The algorithm and its analysis rely strongly on the formulation of the operator given in [13]; in particular, in order to resolve the seeming singularity at the magnetic axis, the radial variable r is replaced by s = logr so that the independent variable varies over the in¡ interval (-c~, so]. In Section 2, we recall some relevant results of [13 I. In Section 3, we define the algorithm and state the main result of spectral convergence which will be proved in Section 4. Section 5 contains remarks on numerical integration and some numerical examples. The

Essential Spectrum Due to Singularity

“…Similar phenomena in connection with problems of magnetohydrodynamics have been studied by V Adamyan, J Descloux, G Geymonat, G Grubb, T Kako, H Langer, A E Lifchitz, R Mennicken, M Möller, A Shkalikov, and others [1,3,4,5,6,7,8,10,12,14,15,16,17,21,22,24]. Problems of magnetohydrodynamics are described by 3 × 3 matrix differential operators [14,12].…”

Eigenvalue enclosures and convergence for the linearized MHD operator

“…Thus ( 7) is satisfied and we may obtain upper and lower bounds on the eigenvalue λ − 1 from the left side of (10).…”

“…A considerable amount of research has been devoted to the formulation of a rigorous operator theoretic framework for K and to the structure of the spectrum, [17]. Particular attention has been payed to the plane slab (plasma layer) and the cylindrical (plasma pinch) configurations [9,10,21,1] where K is reduced to a block ordinary differential operator matrix. A systematic description (analytical or numerical) of properties of the eigensolutions turns out to be difficult even for these, the simplest configurations.…”

“…If a sequence of subspaces is guaranteed not to produce spectral pollution, then the standard Galerkin method can be used. A prescribed recipe for avoiding spurious modes when these subspaces are generated by the finite element method dates back to [22,10]. In this classical approach convergence is guaranteed, however, it is never clear whether a computed eigenvalue is on the left or on the right of the exact eigenvalue.…”

Essential spectrum of non-self-adjoint singular matrix differential operators

“…The appearance of the branch of essential spectrum due to the singularity at the boundary was first predicted by Descloux and Geymonat [4] in connection with a physical model describing the oscillations of plasma in an equilibrium configuration in a cylindrical domain and proven much later by Faierman, Mennicken and Möller [8]. Similar phenomena in connection with problems of theoretical physics have been studied by many authors including Kako [14,15,16], Descloux and Kako [17], Raikov [29,30], Beyer [3], Atkinson, H. Langer, Mennicken and Shkalikov [2], H. Langer and Möller [21], Faierman, Mennicken and Möller [6,7], Hardt, Mennicken and Naboko [10], Konstantinov [18], Mennicken, Naboko and Tretter [24], Kurasov and Naboko [20], Möller [26], Marletta and Tretter [23], Kurasov, Lelyavin and Naboko [19], Qi and Chen [27,28].…”