Review of Non-Equilibrium Thermodynamics And Statistical Mechanics of Vortex Gases in Tornado Theory

Abstract

This work puts into mathematical, statistical mechanical, and thermodynamical context the initial stages of the genesis of tornado-like vortices with the aim to be consistent with the current state of knowledge of the process of tornadogenesis. In particular, it discusses a mathematical foundation of the formation of coherent structures such as ``cusps'' and ``hairpins'' using variants of the nonlinear Schr\"odinger equation that arise via the Hasimoto transform of a vortex filament model. The behavior of such structures is then analyzed within a quasi-two-dimensional boundary layer model using the statistical mechanics of vortex gases to explain the rearrangement of cusps and other vertical vortex filaments into patches and possibly supercritical vortices. Non-equilibrium thermodynamics is used to obtain the entropic balance and the internal entropy production rate, and connect them to the turbulent heat flux. A formula for the non-equilibrium turbulent heat supply and formulas for the entropy supply and entropy production in the boundary layer are also provided. A relationship between the vorticity and the entropy gradient based on macroscopic fluctuations is given with implications to stretching and tilting of vorticity in the vertical direction. We conclude with some remarks on equivalence of Schr\"odinger and Gross-Pitaevskii equations in describing vortex filaments.