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Keywords:
magnetohydrodynamics; Alfvén waves; Fourier analysis; singularity; small perturbations; equilibrium plasma; mixed elliptic-hyperbolic system
magnetohydrodynamics; Alfvén waves; Fourier analysis; singularity; small perturbations; equilibrium plasma; mixed elliptic-hyperbolic system
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$\left(A_1\Delta_u\right) + A_2u =0$, where $A_1$ may have singularities in the domaind $U$ 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 absorptions 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 $U$.
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