TABLE OF CONTENTS In this section, we apply statistical theories with two conserved quantities, energy and potential enstrophy, both the empirical theory introduced in Chapter 6 and the complete statistical mechanics theory in Chapter 8, to the inviscid unforced barotropic quasi-geostrophic equation on the sphere (16.62). The mean field of the most probable state turns out to be those exact steady state solutions of the equation that we introduced in Subsection 16.2.1 and hence their stability is discussed in Subsection 16.2.3. The results here are parallel to the flat geometry case and the argument here is very much the same as in Chapters 6 and 8, since they rely mostly on conserved quantities. Nevertheless, there is an important special case of ground state topography. In this special case, the ground state modes have their independent linear dynamics and thus no sufficient mixing on the ground energy shell. Thus, we need to consider the statistics of the high modes (motion off the ground energy shell) in the case of h ? W 1.
For empirical statistics theory we will postulate the one-point statistics, define the Shannon entropy and then apply the maximum entropy principle to derive the most probable state. For complete statistical mechanics theory, we introduce various truncations, verify the Liouville property as well as the conservation of the truncated energy and potential enstrophy, apply the equilibrium statistical mechanics theory for ODEs from Chapter 7 to the truncated system, compute the mean state,...
