Stuart vortices on a sphere

Abstract

The exact steady two-dimensional solutions of the Euler equations due to Stuart (1967) are generalized to the surface of a sphere. The solutions are parametrized by three parameters $N$,$\theta_0$ and $\Omega_{\hbox{\scriptsize\it max}}$; the integer $N > 1$ denotes the number of smooth vorticity extrema, with extremal vorticity $\Omega_{\hbox{\scriptsize\it max}}$, that are equally spaced in longitudinal angle around a latitude circle at latitudinal spherical polar angle $\theta_0$. The solutions have two equal point-vortex singularities at the north and south spherical poles. Like Stuart's, the solutions are exact and explicit. The solutions are expected to be useful in geophysical and astrophysical applications where curvature effects are important.