On the stratorotational instability in the quasi-hydrostatic semi-geostrophic limit

The linear normal-mode stratorotational instability (SRI) is analytically reexamined in the inviscid limit where the length scales of horizontal disturbances are large compared their vertical and radial counterparts. Boundary conditions different than channel walls are also considered. This quasi-hydrostatic, semi-geostrophic (QHSG) approximation allows one to examine the effect of a vertically varying Brunt-Vaisaila frequency, $N^2$. It is found that the normal-mode instability persists when $N^2$ increases quadratically with respect to the disc vertical coordinate. However we also find that the SRI seems to exist in this inviscid QHSG extreme only for channel wall conditions: when one or both of the reflecting walls are removed there is no instability in the asymptotic limit explored here. It is also found that only exponential-type SRI modes (as defined by Dubrulle et al. 2005) exist under these conditions. These equations also admit non-normal mode behaviour. Fixed Lagrangian pressure conditions on both radial boundaries predicts there to be no normal mode behaviour in the QHSG limit. The mathematical relationship between the results obtained here and that of the classic Eady (1949) problem for baroclinic instability is drawn. We conjecture as to the mathematical/physical nature of the SRI. The general linear problem, analyzed without approximation in the context of the Boussinesq equations, admits a potential vorticity-like quantity that is advectively conserved by the shear. Its existence means that a continuous spectrum \emph{is a generic feature of this system}. It also implies that in places where the Brunt-Vaisaila frequency becomes dominant the linearized flow may two-dimensionalize by advectively conserving its vertical vorticity.