- Statistical mechanics studies the probability that a system is in a certain state given one or more constraints which are usually fixed conserved quantities. It is a particularly useful and powerful approach for problems with a large number of degrees of freedom where a complete knowledge of the system is not practical or even possible. By allowing to reduce the complexity of the system to a few parameters, statistical mechanics allows avoiding the question of 'what is the state of a system?' by asking instead 'what is the most likely state of the system given some known constraints?'.
Holloway (1986) has a review of successful applications of statistical mechanics to a variety of geophysical fluid dynamics (GFD) problems, including geostrophic turbulence over topography, two-dimensional turbulence on a plane and on a sphere, closed-basin circulation and Western intensification, the shape of a thermocline, baroclinic turbulence, eddy heat transport, predictability (i.e. sensitivity of flow evolution to perturbations in the initial conditions), stirring of tracer fields, internal gravity waves and buoyant turbulence among others. More recently, statistical mechanics has been successfully used to understand aspects of large-scale GFD. For example the Robert-Sommeria-Miller (RSM) equilibrium statistical mechanics has been used to interpret rings and jets as statistical equilibria (Bouchet & Venaille 2012).
Statistical Mechanics has also been successfully applied to numerical GFD, a subject of great interest for humanity not only for purely scientific reasons, but also for the large number of applications to real life. A sharp increase in the ability to numerically simulate oceans and atmospheres, as well as in the interest of projecting current states into the future have fueled important developments in numerical GFD.
The rest of this paper is organized into a section 2 where the equations of geophysical fluid dynamics and some simplifications are introduced; some details of the simplified equations like non-linear stability of the steady state solution are also presented. Section 3 develops the theory of statistichal mechanics. We obtain a probability function with which we can predict the most-likely state of the system describing a generalized flow. In section 4 we predict the most likely state of a basic geophysical flow. The first step applies the theory of section 3 to a Galerkin approximation of the simplified equations from section 2. The final step extends the result to a continuum by taking the limit to infinity of the truncated system. An appendix includes material for some of the concepts we use.