Stability of magnetohydrodynamic stratified shear flows

Summary The linear stability of a stratified shear flow of a perfectly conducting bounded fluid in the presence of a magnetic field aligned with the flow and buoyancy forces has been studied under Boussinesq approximation. A new upper bound has been obtained for the range of real and imaginary parts of the complex wave velocity for growing perturbations. The upper bound depends on minimum Richardson number, wave number, Alfvén velocity and basic flow velocity. H?iland's necessary criterion for instability of hydrodynamic stratified homogeneous shear flow is modified and its analog for nonhomogeneous magnetohydrodynamic cases is derived. Finally the upper bound for the growth rate ofKC _i and its variants, whereK is the wave number andC _i the imaginary part of complex wave velocity, is derived as the necessary condition of instability. All estimates remain valid even when the minimum richardson numberJ ₁, for some practical problems, exceeds 1/4 for growing perturbations. The authors of this paper have agreed to not receive the proofs for correction.