A WKB analysis of the Alfvén spectrum of the linearized magnetohydrodynamics equations. (English) Zbl 0778.76100

Summary: Small perturbations of an equilibrium plasma satisfy the linearized magnetohydrodynamics equations. These form a mixed elliptic-hyperbolic system that in a straight-field geometry and for a fixed time frequency may be reduced to a single scalar equation \(div(A_ 1\bigtriangledown u)+A_ 2u=0\), where \(A_ 1\) may have singularities in the domain \(U\) of definition. We study the case when \(U\) is a half-plane and \(u\) possesses high Fourier components, analyzing the changes brought about by the singularity \(A_ 1=\infty\). We show that absorption of energy takes place precisely at this singularity, that the solutions have a near harmonic character, and the integrability characteristics of the boundary data are kept throughout \(U\).


76W05 Magnetohydrodynamics and electrohydrodynamics
35Q60 PDEs in connection with optics and electromagnetic theory
34E05 Asymptotic expansions of solutions to ordinary differential equations
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