Geophysics and Stuart vortices on a sphere meet differential geometry

TL;DR

This paper establishes new existence criteria for a nonlinear elliptic PDE on a sphere, with applications to modeling ocean surface currents and connecting geophysical phenomena with differential geometry.

Contribution

It introduces novel existence criteria for solutions of a specific nonlinear PDE on the sphere, extending previous results and linking geophysical models with differential geometry.

Findings

New existence criteria for PDE solutions on $S^2$ for $2 \,\leq C < 4$

Application of criteria to ocean surface current models

Extension of known results for $C<2$ to the regime $2 \leq C < 4$

Abstract

We prove new existence criteria relevant for the non-linear elliptic PDE of the form $\Delta_{S^2} u=C-he^{u}$, considered on a two dimensional sphere $S^2$, in the parameter regime $2\leq C<4$. We apply this result, as well as several previously known results valid when $C<2$, to discuss existence of solutions of a particular PDE modelling ocean surface currents.