On the Axisymmetric Steady Incompressible Beltrami Flows

In this paper, Beltrami vector fields in several orthogonal coordinate systems are obtained analytically and numerically. Specifically, axisymmetric incompressible inviscid steady state Beltrami (Trkalian) fluid flows are obtained with the motivation to model flows that have been hypothesized to occur in tornadic flows. The studied coordinate systems include those that appear amenable to modeling such flows: the cylindrical, spherical, paraboloidal, and prolate and oblate spheroidal systems. The usual Euler equations are reformulated using the Bragg-Hawthorne equation for the stream function of the flow, which is solved analytically or numerically in each coordinate system under the assumption of separability of variables. Many of the obtained flows are visualized via contour plots of their stream functions in the rz-plane. Finally, the results are combined to provide a qualitative quasi-static model for a progression of tornado-like flows that develop as swirl increases. The results in this paper are equally applicable in electromagnetics, where the equivalent concept is that of a force-free magnetic field.

Equilibrium Energy and Entropy of Vortex Filaments in the Context of Tornadogenesis and Tornadic Flows

In this work, we study approximations of supercritical or suction vortices in tornadic flows and their contribution to tornadogenesis and tornado maintenance using self-avoiding walks on a cubic lattice. We extend the previous work on turbulence by A. Chorin and collaborators to approximate the statistical equilibrium quantities of vortex filaments on a cubic lattice when both an energy and a statistical temperature are involved. Our results confirm that supercritical (smooth, “straight”) vortices have the highest average energy and correspond to negative temperatures in this model. The lowest-energy configurations are folded up and “balled up” to a great extent. The results support A. Chorin’s findings that, in the context of supercritical vortices in a tornadic flow, when such high-energy vortices stretch, they need to fold and transfer energy to the surrounding flow, contributing to tornado maintenance or leading to its genesis. The computations are performed using a Markov Chain Monte Carlo approach with a simple sampling algorithm using local transformations that allow the results to be reliable over a wide range of statistical temperatures, unlike the originally used pivot algorithm that only performs well near infinite temperatures. Efficient ways to compute entropy are discussed and show that a system with supercritical vortices will increase entropy by having these vortices fold and transfer their energy to the surrounding flow.

Applications of Vortex Gas Models to Tornadogenesis and Maintenance

Processes related to the production of vorticity in the forward and rear flank downdrafts and their interaction with the boundary layer are thought to play a role in tornadogenesis. We argue that an inverse energy cascade is a plausible mechanism for tornadogenesis and tornado maintenance and provides supporting evidence which is both numerical and observational. We apply a three-dimensional vortex gas model to supercritical vortices produced at the surface boundary layer possibly due to interactions of vortices brought to the surface by the rear flank downdraft and also to those related to the forward flank downdraft. Two-dimensional and three-dimensional vortex gas models are discussed, and the three-dimensional vortex gas model of Chorin, developed further by Flandoli and Gubinelli, is proposed as a model for intense small-scale subvortices found in tornadoes and in recent numerical studies by Orf et al. In this paper, the smaller scales are represented by intense, supercritical vortices, which transfer energy to the larger-scale tornadic flows (inverse energy cascade). We address the formation of these vortices as a result of the interaction of the flow with the surface and a boundary layer.