A buoyancy-vorticity wave interaction approach to stratified shear flow

Motivated by the success of potential vorticity (PV) thinking for Rossby waves and related shear flow phenomena, this work develops a buoyancy-vorticity formulation of gravity waves in stratified shear How, for which the nonlocality enters in the same way as it does for barotropic/baroclinic shear flows. This formulation provides a time integration scheme that is analogous to the time integration of the quasigeostrophic equations with two, rather than one, prognostic equations, and a diagnostic equation forestream-function through a vorticity inversion. The invertibility of vorticity allows the development of a gravity wave kernel view, which provides a mechanistic rationalization of many aspects of the linear dynamics of stratified shear flow. The resulting kernel formulation is similar to the Rossby-based one obtained for barotropic and baroclinic instability; however, since there are two independent variables - vorticity and buoyancy - there are also two independent kernels at each level. Though having two kernels complicates the picture, the kernels are constructed so that they do not interact with each other at a given level.