A Sufficient Condition for Instability in a Sheared Incompressible Magnetofluid

It is shown that a sufficient condition for the stability of an incompressible sheared gravitationally stratified ideal magnetofluid with flow-aligned horizontal magnetic field is that there exists a Galilean frame in which the flow is nowhere super-Alfvénic (similarly, stability is assured in a compressible shear flow without gravity if there exists a frame in which the flow nowhere exceeds the cusp speed). Complex eigenvalue bounds are presented for unstable flows. The stability condition is applied to the solar tachocline; it suggests that any shear instabilities associated with radial gradients in flow speed should be stabilized by fields of above about 7 kG.