9.1 Introduction

In this chapter we continue the study of statistical theories for the most probable state of the barotropic quasi-geostrophic equations

where q = ? ? + h + ?y is the potential vorticity, ? = ? V( t)y + ? ? is the stream function, V is the large-scale mean flow, h is the topography, ? is the beta-plane constant, ? ? is the small-scale stream function, ? = ? ? ? is the relative vorticity, and q ? = ? ? ? + h = ? + h is the small-scale potential vorticity. The bar in the integral sign indicates that the space integral has been normalized by the area of the domain ?. Here we assume either periodic geometry, where all the functions are 2 ?-periodic, so that ? = [0, 2 ?] [0, 2 ?] and the normalization constant is ? = 4 ? ² or channel geometry ( ? = [0, 2 ?] [0, ?]) as described in Section 1.4. We will also consider the case of disk domain with radius R and centered at the origin, or the entire plane for the point-vortex theory without mean flow.

In Chapters 6 and 8 we discussed the classical statistical theory with two conserved quantities both from the non-traditional point of view of...