On the stability of Alfvén discontinuity

Abstract

The stability of Alfvén discontinuities for the equations of ideal compressible magneto-hydrodynamics (MHD) is studied. The Alfvén discontinuity is a characteristic discontinuity for the hyperbolic system of MHD equations but, as in the case of shock waves, there is a mass flux through its front. The Lopatinskii condition for a planar Alfvén discontinuity is tested numerically, and the domain in the space of parameters of the discontinuity where it is unstable is determined. In fact, in this domain the Alfvén discontinuity is not only unstable, but the initial-boundary-value problem for corresponding linearized equations is ill-posed in the sense of Hadamard.

Shock waves and characteristic discontinuities in ideal compressible two-fluid MHD

“…In classical MHD, there are still no nonlinear (structural stability) results for them. At the same time, the domains of neutral stability and violent instability of planar Alfvén discontinuities were numerically found in [7]. It is still unclear whether neutrally stable nonplanar Alfvén discontinuities do exist locally in time.…”

“…It is still unclear whether neutrally stable nonplanar Alfvén discontinuities do exist locally in time. However, the linear results in [7] are automatically carried over two-fluid MHD because the free boundary problem (20), (21) has absolutely the same form as that for Alfvén discontinuities in classical MHD.…”

“…Stability/existence results for the multidimensional case (2D and/or 3D) for shock waves and characteristic discontinuities (current-vortex sheets, contact and Alfvén discontinuities) in ideal compressible one-fluid (classical) MHD can be found in [1,3,4,7,18,19,22,23,24,27]. For vortex sheets for a compressible inviscid liquid-gas two-phase model (without magnetic field)…”

On stability of Alfvén discontinuities

“…Moreover, using the boundary conditions (8), from the first equation in system (11) one gets (we omit calculations)…”

“…Problem (13), (14) is the genuine linearization of (7), (8) in the sense that we keep all the lower-order terms in (13). It should be noted that the differential operator in system (13) is a first-order operator in f .…”