NONAXISYMMETRIC INSTABILITIES IN ROTATING SHEAR FLOWS - INTERNAL GRAVITY MODES IN STRATIFIED MEDIA AND ANALOGIES WITH PLANE FLOWS

We study the instability of stratified, incompressible, differentially rotating fluid cylinders to global, nonaxisymmetric modes. We find new unstable branches of the dispersion relation mediated by internal gravity modes of the system (analogous to g-modes of stars), in addition to a modified version of the well-known principal branch, which is mediated by surface modes of the system (analogous to f-modes of stars). We show that the modified principal branch has interesting scaling properties, and that the maximum growth rate on one of the new branches can sometimes equal or exceed that on the principal one. We explore the close analogy between the stratified, incompressible, rotating shear flow described above and a similarly stratified, incompressible, plane-parallel shear flow. We comment on the role of rotation and show that, in the limit where f-modes are suppressed, the new branches of the dispersion relation for the plane flow reduce to a single one which was originally found by Taylor in 1931. We show that the regions of instability increase as the Richardson number is increased from zero, because the g-modes introduced by (statically) stable stratifications in fact further destabilize the shear flow. Thus, the tapping of the shear energy required for driving these instabilities occurs through resonant interactions between normal modes of the system that are similar to the g-, p-, and f-modes of stars.