Stability of magnetohydrodynamic shear flows with and without bounding walls

Abstract

We have solved the shear flow stability problem of compressible electrically conducting fluid (plasma) in a magnetic field. A comparative analysis is made of the influence of different boundary conditions on shear flow stability. We consider the problems for flows in an unbounded medium and a flow between fixed walls, as well as in the presence of a fixed wall on one side of the shear flow only. Shear flow velocity profiles are treated both in the form of a tangential discontinuity and as being described by a hyperbolic tangent function. For the case of an unbounded flow with a velocity vortex sheet, analytical solutions are found; for all other cases, the solutions are found numerically. Problems are solved for two different positions of the tangential wave vector of oscillations and magnetic field ${\bf k}_t\,{\perp}\, {\bf B}_0$ and ${\bf k}_t\,{\parallel}\, {\bf B}_0$. For shear flows bounded by a fixed wall we found an unstable mode of oscillations produced by the wave, reflected from the wall and transmitted through the shear layer.