Nonlocal stability analysis of the MHD Kelvin-Helmholtz instability in a compressible plasma

A general stability analysis is performed for the Kelvin-Helmholtz instability is sheared magnetohydrodynamic flow of finite thickness in a compressible plasma. The analysis allows for arbitrary orientation of the magnetic field B₀, velocity flow v₀, and wave vector k in the plane perpendicular to the velocity gradient, and no restrictions are imposed on the sound or Alfvén Mach numbers. The stability problem is reduced to the solution of a single-order differential equation, which includes a gravitational term to represent coupling between the Kelvin-Helmholtz mode and the interchange mode. In the incompressible limit it is shown that the Kelvin-Helmholtz mode is completely stabilized for any velocity profile as long as the condition V₀<2v_A (k.B₀)/(k.v₀) is satisfied, where V₀ is the total velocity jump across the shear layer. Numerical results are obtained for a hyperbolic tangent velocity profile for the transverse (B₀⊥v₀) and parallel (B₀∥v₀) flow configurations. Only modes with k∆<2 are unstable, where ∆ is the scale length of the shear layer. The fastest growing modes occur for k∆~0.5-1.0. Compressibility and a magnetic field component parallel to the flow are found to be stabilizing effects. For the transverse case, only the fast magnetosonic mode is destabilized, but if the k.B₀≠0, the instability contains Alfvén-mode and slow-mode components as well. The Alfvén component gives rise to a field-aligned current inside the shear layer. In the parallel case, both Alfvén and slow magnetosonic components are present, with the Alfvén mode confined inside the shear layer. The results of the analysis are used to discuss the stability of sheared plasma flow at the magnetopause boundary and in the solar wind. At the magnetopause boundary, the fastest growing Kelvin-Helmholtz mode has a frequency of 0 (V₀/2∆), which overlaps with the frequency range of geomagnetic pulsations (Pc 3-5). It is suggested that the MHD Kelvin-Helmholtz instability could serve as a dynamo process driving small-scale field-aligned currents in the presence of the sheared plasma flow in the magnetosphere.