Abstract
A recent numerical study of tornadogenesis suggests that processes and structures in the turbulent boundary layer below a supercell thunderstorm's mesocyclone may play an important role in tornado formation. A theoretical explanation using concepts from turbulence theory would be desirable. This work puts into mathematical, statistical mechanics, 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 Schrödinger 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. An example involving vorticity, potential vorticity, and the gradient of potential temperature is given in a thermodynamic non-equilibrium context with its implications for stretching and tilting of vorticity in the vertical direction. We conclude with some remarks on the equivalence of the Schrödinger and Gross-Pitaevskii equations in describing vortex filaments.