The Importance of Divergence and Vertical Extent of the Jet to the Growth Rates of Barotropic Instability in Realistic Mean Flows on a Sphere

The effects of divergence and of the vertical extent of a barotropically unstable layer on the growth rates of barotropic instabilities are studied by examining the stability of strong polar jets and comparing the growth rates obtained from the Shallow Water Equations (SWEs) with those from the non-divergent barotropic vorticity equation. Two mean zonal wind profiles are considered: (i) An idealized profile identical to the one used in Hartmann (1983). (ii) A profile that approximates the observed SH polar stratospheric jet. The main result of the present study is that the depth over which a layer is barotropically unstable is a crucial parameter in controlling the growth rate of eddies, and this dependence is only evident in the SWEs. For shallow water depths of 30km or more the growth rates predicted by the non-divergent model provide a good approximation to those of the SWEs.
However, for smaller shallow water depths, the growth rates predicted by the SWEs become smaller than those of the non-divergent model, and for shallow water depths of between 5 and 10km they can be more than 50% smaller. For shallow water depths below 5km the mean height in geostrophic balance with the strong zonal jets becomes negative, and hence the barotropic instability problem is ill-defined. Furthermore, the solutions of the non-divergent vorticity equation violate the ``balance equation'', which they require to satisfy in a consistent first-order approximation of the non-divergent model.