Introduction

This book is about some mathematical problems of the dynamics of an incompressible two-dimensional (viscous and ideal) fluid on a rotating sphere. The fluid motion is governed by the barotropic vorticity equation (BVE). The large-scale vortex motions of the atmosphere can also be adequately described by the barotropic vorticity equation. Therefore, since the mid-twentieth century, this equation has played an important role in the study of hydrodynamic and meteorological processes. Although the vorticity equation is relatively simple, it takes into account such important dynamic processes as the nonlinear interaction and dispersion of waves, the dissipation and external forcing, and the solid-body rotation of a fluid about the polar axis. In the case of an ileal and unforced fluid, there are four sets of exact solutions of this equation: the zonal flows, meteorologically significant Rossby-Haurwitz waves, modons, and Wu-Verkley waves. In 1950, the BVE was chosen as the first approximate model of the atmosphere in the general plan of attacking the problem of numerical weather prediction. Also, in the case of an ideal fluid, the conservation laws for a BVE solution allowed studying the changes in time of the spectral distribution of the kinetic energy. In 1933, a very useful simplification of the hydrodynamic equations for three-dimensional perturbations was obtained by Squire who showed that each unstable three-dimensional perturbation corresponds to a more unstable two-dimensional perturbation. This result stimulated research on the stability of barotropic fluid flows. Also, the BVE proved a convenient model for studying the asymptotic behavior and attractive sets of solutions. In general, the study of the stability of the BVE solutions and the structure of their stable and unstable manifolds can be considered as the first step in understanding the complex mechanism of low-frequency variability of the atmosphere. Unlike the studies in which the β-plane approximation is used, the vortex dynamics is analyzed here on a sphere. This approach is more natural from the viewpoint of meteorological applications. Besides, it removes the problem of artificial boundary conditions. At the same time, the use of spherical geometry enables us to apply modern methods of the theory of functions on a sphere. Special attention is paid to such problems as the existence, uniqueness, and asymptotic behavior of BVE solutions. These topics are very important for better understanding of the structure of the phase space of solutions, and also for choosing correct norms in the study of stability of solutions.