**David J. Muraki**

Simon Fraser University

in collaboration with Andrea Blazenko and Chris Snyder

### 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.