  Three eigenvalue bounds are derived for the instability of ideal compressible
stratified magnetohydrodynamic shear flows in which the base velocity, density,
and magnetic field vary in two directions. The first bound can be obtained by
combining the Howard semi-circle theorem with the energy principle of the
Lagrangian displacement. Remarkably, no special conditions are needed to use
this bound, and for some cases, we can establish the stability of the flow. The
second and third bounds come out from a generalisation of the Miles-Howard
theory and have some similarity to the semi-ellipse theorem by Kochar & Jain
(J. Fluid Mech., vol. 91, 1979, 489) and the bound found by Cally (Astrophys.
Fluid Dyn., vol. 31,1983, 43), respectively. An important byproduct of this
investigation is that the Miles-Howard stability condition holds only when
there is no applied magnetic field and, in addition, the directions of the
shear and the stratification are aligned everywhere.

Fluid Dynamics