Instability Associated Baroclinic Critical Layers in Rotating Stratified Shear Flow

The notion of over-reflection has been used to rationalize the linear instability of shear flows that are directed and sheared in a horizontal plane but stratified and subject to rotation in the vertical. In the linear stability analysis of these types of flows, two types of critical levels may appear that translate to singular points in the corresponding differential eigenvalue problem. The first type corresponds to the classical kind of critical level encountered for unstratified flows whereat the phase speed of a wave matches the local flow speed. The second type are more novel, arising where the Döppler-shifted wavespeed matches the natural gravity wavespeed. Such “baroclinic” critical levels are chacterized by unusual properties of reflection and transmission; we explore how this impacts linear over-reflectional instability using a combination of WKB theory and numerical solution of the linear eigenvalue problem. In particular, we examine how Kelvin waves riding on the wall of a channel become unstably coupled to gravity waves propagating on the other side of a baroclinic critical level.