Dynamics and Statistical Mechanics of Rotating and non-Rotating Vortical Flows

Three projects were analyzed with the overall aim of developing a computational/analytical model for estimating values of the energy, angular momentum, enstrophy and total variation of fluid height at phase transitions between disordered and self-organized flow states in planetary atmospheres. It is believed that these transitions in equilibrium statistical mechanics models play a role in the construction of large-scale, stable structures including super-rotation in the Venusian atmosphere and the formation of the Great Red Spot on Jupiter. Exact solutions of the spherical energy-enstrophy models for rotating planetary atmospheres by Kac's method of steepest descent predicted phase transitions to super-rotating solid-body flows at high energy to enstrophy ratio for all planetary spins and to sub-rotating modes if the planetary spin is large enough. These canonical statistical ensembles are well-defined for the long-range energy interactions that arise from 2D fluid flows on compact oriented manifolds such as the surface of the sphere and torus. This is because in Fourier space available through Hodge theory, the energy terms are exactly diagonalizable and hence has zero range, leading to well-defined heat baths