Evolution of Energy of Perturbations in Barotropic Atmosphere

Abstract

The barotropic vorticity equation describing the vortex dynamics of viscous and forced incompressible fluid on a rotating sphere is considered. This equation is also used for studying the large-scale dynamics of barotropic atmosphere. Operators of orthogonal projection on the subspaces of homogeneous spherical polynomials and derivatives of real order for functions are introduced. A family of Hilbert spaces of generalized functions having fractional derivatives of real order s is introduced, and a few embedding theorems are given. An equation for the evolution of kinetic energy of perturbations to a basic flow is analyzed. A relationship between the rate of generation of kinetic energy perturbations and the eigenfunctions of the symmetric part of the operator linearized about the basic flow is shown. As an illustrative example, the numerical solution of the spectral problem for such operator is discussed in the case when the basic flow is the climatic January circulation.