Probabilistic and numerical techniques in the study of statistical theories of turbulence. Final Technical Report

Abstract

In this research project we made fundamental advances in a number of problems arising in statistical equilibrium theories of turbulence. Here are the highlights. In most cases the mathematical analysis was supplemented by numerical calculations. (a) Maximum entropy principles. We analyzed in a unified framework the Miller-Robert continuum model of equilibrium states in an ideal fluid and a modification of that model due to Turkington. (b) Equivalence and nonequivalence of ensembles. We gave a complete analysis of the equivalence and nonequivalence of the microcanonical, canonical, and mixed ensembles at the level of equilibrium macrostates for a large class of models of turbulence. (c) Nonlinear stability of flows. We refined the well known Arnold stability theorems by proving the nonlinear stability of steady mean flows for the quasi-geostrophic potential vorticity equation in the case when the ensembles are nonequivalent. (d) Geophysical application. The theories developed in items (a), (b), and (c) were applied to predict the emergence and persistence of coherent structures in the active weather layer of the Jovian atmosphere. This is the first work in which sophisticated statistical theories are synthesized with actual observations data from the Voyager and Galileo space missions. (e) Nonlinear dispersive waves. For a class of nonlinear Schroedinger equations we demonstrated that the self-organization of solutions into a ground-state solitary wave immersed in fine-scale fluctuations is a relaxation into statistical equilibrium.