Exact solutions of the vorticity equation on the sphere as a manifold

Abstract

The purpose of this paper is to represent the exact solutions of the barotropic vorticity equations (BVE) on the rotating unit sphere S² as a manifold, which are zonal flows, Rossby-Haurwitz waves and generalized solutions named modons. Modern methods of the function theory are connected to the sphere defined as a compact differentiable manifold. When the differentiable manifold S² is well understood, the abstract notion of local chart, change of chart, and atlases becomes evident. One of the aims of this paper is to better understand the solution of the barotropic vorticity equation on the manifold S² and its usefulness to identify the properties of the solutions on the Riemannian manifold (S², g). Therefore, a more general type of space will be available, which can also contain substantial geometric and analytic information about solutions for the barotropic vorticity equation.