Phase transitions of the energy-relative enstrophy theory for a coupled barotropic fluid–rotating sphere system

Abstract

The statistical equilibrium of a coupled barotropic fluid–rotating solid sphere system is simulated using a energy-relative enstrophy spherical model in a wide range of parameter space by Monte Carlo (MC) methods [J.M. Hammersley, D.C. Handscomb, Monte Carlo Methods, Methuen & Co, London, Wiley, New York City, 1964; C.C. Lim, J. Nebus, Vorticity, Statistical Mechanics and Simulations, Springer, Berlin, 2006]. The energy-relative enstrophy model does not have the low temperature defect of the classical energy–enstrophy theory [R.H. Kraichnan, Statistical dynamics of two-dimensional flows, J. Fluid Mech. 67 (1975) 155–175] because of its microcanonical constraint on relative enstrophy. This model also differs from previous work in not fixing the angular momentum. A family of spin–lattice models are derived as convergent finite dimensional approximations to the total kinetic energy. MC simulations are used to calculate the mean nearest neighbor parity as order parameter or indicator of phase transitions in the system. We find that at extremely high energy levels or small negative temperatures, the preferred state is a super-rotational equilibrium state aligned with the planetary spin. This phenomenon has been observed in the Venusian middle atmosphere for many decades but remains difficult to explain [ 〈 http://www-atm.physics.ox.ac.uk/project/virtis/venus-super.html 〉 ]. For large values of the planetary spin compared with the relative enstrophy there appears to be two phase transitions when the temperature varies from a numerically large negative value to a large positive value. The latter arises as constrained energy minimizers.