Instability of stratified flows as a result of resonance

Abstract

The stability of stratified flows consisting of a middle layer of homogeneous fluid in linear shear flow and contiguous upper and lower stratified layers of constant velocities is considered. The density and velocity are continuous throughout. It is shown that there are infinitely many pairs of neutral modes with the phase velocity cr in the range of the velocity which, in general, are not represented by the stability boundary, but which merge as the parameter N (the inverse of the Froude number squared) increases. As N increases further the coalesced neutral modes become modes with complex c, one of which is unstable. The instability may be considered to be caused by the resonance of the original pair of separate modes. In addition, instability of stratified flows is considered for general density and velocity distributions without a layer of constant density and linear velocity, and it is concluded that in that case the modes with complex c on the unstable side of a stability boundary do not continue into any neutral normal modes on the stable side