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We are not only interested in modelling fluid dynamics problems on the sphere. The unbounded plane is of obvious interest, and we want to generate meshes on it. We cannot hope to do more than cover a finite region of the plane. And yet if we run a Monte Carlo algorithm with vortices all of the same strength initially placed randomly over any region of the plane and with a positive β we can get a reasonably uniform mesh – but it never settles to a statistical equilibrium. This is obvious in hindsight: whatever the arrangement of points z, z, z, et cetera, one can always reduce the energy of the system by moving points farther away from the center of vorticity.

We develop a more realistic model of fluid flow based on the horizontal and restricted vertical motions of the atmosphere and oceans of a planet [442]. We will continue to study the vorticity field of an inviscid fluid, but will consider the atmosphere as having two (or, in principle, more) layers to it. The actual atmosphere is a complex of many layers, with interactions and behaviors on multiple time-scales. We also add to consideration the rotation of the earth; this is the “strophic” part of the term “geostrophic models.” This model can be represented as a vorticity, dependent on latitude superimposed on the rest of the fluid motion. Several terms must be introduced before we can construct these models.

In this chapter we apply the methods developed already to a simple geophysical flow with rotation with the aim of deriving and solving a crude statistical mechanics model for atmospheric super-rotation. Consider the system consisting of a rotating high density rigid sphere of radius , enveloped by a thin shell of barotropic (non-divergent) fluid. The barotropic flow is assumed to be inviscid, apart from an ability to exchange angular momentum and energy with the heavy solid sphere. In addition we assume that the fluid is in radiation balance and there is no net energy gain or loss from insolation.