Phase transitions of barotropic flow coupled to a massive rotating sphere—Derivation of a fixed point equation by the Bragg method

Abstract

The kinetic energy of barotropic flow coupled to an infinitely massive rotating sphere by an unresolved complex torque mechanism is approximated by a discrete spin–lattice model of fluid vorticity on a rotating sphere, analogous to a one-step renormalized Ising model on a sphere with global interactions. The constrained energy functional is a function of spin–spin coupling and spin coupling with the rotation of the sphere. A mean field approximation similar to the Curie–Weiss theory, modeled after that used by Bragg and Williams to treat a 2D Ising model of ferromagnetism, is used to find the barotropic vorticity states at thermal equilibrium for a given temperature and rotational frequency of the sphere. A fixed point equation for the most probable barotropic flow state is one of the main results. This provides a crude model of super- and sub-rotating planetary atmospheres in which the barotropic flow can be considered to be the vertically averaged rotating stratified atmosphere and where a key order parameter is the changeable amount of angular momentum in the barotropic fluid. Using the crudest two domains partition of the resulting fixed point equation, we find that in positive temperatures associated with low-energy flows, for fixed planetary spin larger than Ω c > 0 there is a continuous transition from a disordered state in higher temperatures to a counter-rotating solid-body flow state in lower positive temperatures. The most probable state is a weakly counter-rotating mixed state for all positive temperatures when planetary spin is smaller than Ω c . For sufficiently large spins Ω > 2 Ω c , there is a single smooth change from slightly pro-rotating mixed states to a strongly pro-rotating ordered state as the negative value of T increases (or decreases in absolute value). But for smaller spins Ω < 2 Ω c there is a transition from a predominantly mixed state (for T ⪡ - 1 ) to a pro-rotating state at β - ( Ω ) < 0 plus a second β c Ω , for which the fixed point equation has three fixed points when β < β c Ω instead of just the pro-rotating one when β c Ω < β . An argument based on comparing free energy shows to that the pro-rotating state is preferred when there are three fixed points because it has the highest free energy—at negative temperatures the thermodynamically stable state is the one with the maximum free energy. In the non-rotating case ( Ω = 0 ) the most probable state changes from a mixed state for all positive and large absolute-valued negative temperatures to an ordered state of solid-body flow at small absolute-valued negative temperatures through a standard symmetry-breaking second-order phase transition. The predictions of this model for the non-rotating problem and the rotating problem agree with the predictions of the simple mean field model and the spherical model. This model differs from previous mean field theories for quasi-2D turbulence in not fixing angular momentum nor relative enstrophy—a property which increases its applicability to coupled fluid–sphere systems and by extension to 2D turbulent flows in complex domains such as no-slip square boundaries where only the total circulation is fixed—as opposed to classical statistical equilibrium models such as the vortex gas model and Miller–Robert theories that fix all the vorticity moments. Furthermore, this Bragg mean field theory is well-defined for all positive and negative temperatures unlike the classical energy–enstrophy theories.