A Potential Vorticity Dynamics on the Sphere

Mesoscale dynamics in the midlatitudes are well-described by quasigeostrophy (QG), in which the slow, balanced motions are described through the advection of potential vorticity (PV). The mathematics of QG is often justified by a limit of small Rossby number, and hence its validity does not extend across the equator where the Coriolis effects vanish.
A model based upon the dynamics of PV is developed for rotating shallow water on the sphere. Specifically, a PV-streamfunction relationship is defined which allows for the inversion of PV over the entire sphere. At midlatitudes, these dynamics are equivalent to beta-plane QG, in the usual small Rossby number sense. In the equatorial regions, wave propagation at short-scales mimics the dispersion relation for equatorial beta waves. These dynamics compare favorably with computations of the equatorial crossing of topographic waves by Grose & Hoskins (1979).
Despite that this PV model is not obtained in the classical manner of small Rossby number asymptotic analysis, the propagation of mesoscale waves across the equatorial region retains QG-like accuracy. Preliminary computations, in this single-layer shallow water system, illustrate that the upscale cascade tends to favor large-scale waves over vortices, as controlled by the presence of the planetary PV gradient.