Phase transition to super-rotating atmospheres in a simple planetary model for a nonrotating massive planet: exact solution

• Published in: Physical review. E, Statistical, nonlinear, and soft matter physics Statistical, nonlinear, and soft matter physics, 2012-12, Vol.86 (6 Pt 2), p.066304-066304, Article 066304
• Main Author: Lim, Chjan C
• Format: Article
• Language: English
• ISSN: 1539-3755; 1550-2376
• DOI: 10.1103/PhysRevE.86.066304

An energy-enstrophy model for the equilibrium statistical mechanics of barotropic flow on a massive nonrotating sphere is introduced and solved exactly for phase transitions to rotating solid-body atmospheres when the kinetic energy level is high. Unlike the Kraichnan theory which is a Gaussian model, we substitute a microcanonical enstrophy constraint for the usual canonical one, a step which is based on sound physical principles. This yields a spherical model with zero total circulation, microcanonical enstrophy constraint, and canonical constraint on energy, leaving angular momentum free as is required for any model whose objective is to predict super-rotation in planetary atmospheres. A closed-form solution of this spherical model, obtained by the Kac-Berlin method of steepest descent, provides critical temperatures and amplitudes of the symmetry-breaking rotating solid-body flows. The critical values depend linearly on the relative enstrophy, with proportionality constant derived from the spectrum of the Laplace-Beltrami operator on the sphere, as expected within an energy-enstrophy theory for macroscopic turbulent flows. This model and its results differ from previous solvable models for related phenomena in the sense that the model is not based on a mean-field assumption.