The roll-up of vorticity strips on the surface of a sphere

We derive the conditions for the stability of strips or filaments of vorticity on the surface of a sphere. We find that the spherical results are surprisingly different from the planar ones, owing to the nature of the spherical geometry. Strips of vorticity on the surface of a sphere show a greater tendency to roll-up into vortices than do strips on a planar surface. The results are obtained by performing a linear stability analysis of the simplest, piecewise-constant vorticity configuration, namely a zonal band of uniform vorticity located in equilibrium between two latitudes. The presence of polar vortices is also considered, this having the effect of introducing adverse shear, a known stabilizing mechanism for planar flows. In several representative examples, the fully developed stages of the instabilities are illustrated by direct numerical simulation. The implication for planetary atmospheres is that barotropic flows on the sphere have a more pronounced tendency to produce small, long-lived vortices, especially in equatorial and mid-latitude regions, than was previously anticipated from the theoretical results for planar flows. Essentially, the curvature of the sphere's surface weakens the interaction between different parts of the flow, enabling these parts to behave in relative isolation.