On the instability of magnetohydrodynamic shear flows

We consider the linear stability of an ideal, plane–parallel magnetohydrodynamic shear flow with velocity u = (U(z),0, 0) and magnetic field B = (B(z), 0,0). We show how Howard's semicircle theorem is modified by the presence of a magnetic field and how this leads to new sufficient conditions for the stability of the flow. The issue of how tight these stability bounds actually are is addressed, by numerical computation, for one particular shear flow.