TABLE OF CONTENTS Recall that in the flat geometry case without topographic effects ( h ? 0), the long time behavior of the solutions of the barotropic quasi-geostrophic equations with damping and large-scale (ground state modes) forcing is the unique exact solution on the ground state modes determined by the external large-scale forcing (see Chapter 2). This remains true in the spherical geometry case. Once again due to the special geometry we are able to relax the assumption on the external forcing a little bit: instead of requiring the external forcing acting on the largest scale (ground state modes) only, we can allow external forcing acting on both the ground energy shell and the second energy shell. A key step in our proof is to utilize the fact that the dynamics on the ground energy shell is independent of motions on other modes (see Subsection 16.2.1).
We consider a force on the first two energy shells, i.e.
where
(See Subsections 16.1.2 and 16.2.1 for notation).
It follows from our discussion on exact dynamics on the first two energy shells for the quasi-geostrophic equation in the absence of topography that the space W 1 + W 2 is an invariant subspace of the dynamical system generated by this damped driven equation. See equation (16.51) for more details.
Our goal here is to prove that the long time dynamics is that of the one on the first two energy shells (first eight eigenfunctions) with the initial...
