Geophysics and Stuart vortices on a sphere meet differential geometry

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