TABLE OF CONTENTS Parallel to the flat geometry case, conserved quantities and special exact solutions as well as their non-linear stabilities are central to our understanding of the basic dynamics and statistics of the barotropic quasi-geostrophic flows. The purpose of this section is to study conserved quantities, special exact solutions and their non-linear stabilities to the barotropic quasi-geostrophic equations on the sphere (16.4). The techniques and most of the results are similar to the flat geometry case. However, we also have distinctive features associated with the spherical geometry as we shall demonstrate below.
We first derive several exact solutions for the barotropic quasi-geostrophic equations (16.4) on the sphere. Parallel to the flat geometry case (see Section 1.2 from Chapter 1), we have simple steady state solutions, where the potential vorticity is a function of the stream function, and exact solutions with generalized Kolmogorov forcing. Due to the special spherical geometry, we also have special exact solutions that have no obvious analogy in the flat geometry case. For instance, the dynamics of the ground state modes will be independent of the dynamics of the higher modes, if the topography lives on the ground energy shell. This also has implications for the conserved quantities and non-linear stability to be discussed later in the section and statistical theories to be discussed later in the chapter. We will also discover that the space spanned by the ground energy shell and another arbitrary energy shell...
