Euler equations on a fast rotating sphere --- time-averages and zonal flows

Motivated by recent studies in geophysical and planetary sciences, we investigate the PDE-analytical aspects of time-averages for barotropic, inviscid flows on a fast rotating sphere $S^2$. Of particular interests are the incompressible Euler equations. We prove that the finite-time-average of the solution stays close to a subspace of \emph{longitude-independent zonal flows}. The intial data can be arbitrarily far away from this subspace. Meridional variation of the Coriolis parameter underlies this phenomenon. Our proofs use Riemannian geometric tools, in particular the Hodge Theory.