Non-divergent 2d vorticity dynamics and the shallow water equations on the rotating earth

From a physical viewpoint the assumption of flow's non-divergence, which greatly simplifies the Shallow Water Equations, is justified by the addition of a virtual "rigid lid" that overlies the surface of the fluid and which supplies the pressure gradient forces that drive the (non-divergent) velocity field. In the presence of rotation any initial vorticity field generates divergence by the Coriolis force in the same way that any initial horizontal velocity component generates the other component in finite time, which implies that an initial non-divergent flow is bound to become divergent at later times. Using a particular scaling of the Shallow Water Equations it can be shown that non-divergent flows are regular limits of the Shallow Water Equations when the layer of fluid is sufficiently thick (high) even though the required surface pressure is not determined by the height of the fluid. These analytical considerations are supported by numerical calculations of the instability of a shear flow on the f-plane that show how the non-divergent instability is the limit of the divergent instability when the mean layer thickness becomes large.