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.
We consider a modification of the fluid flow model for a tornado-like swirling vortex developed by Serrin [Phil. Trans. Roy. Soc. London, Series A, Math & Phys. Sci. 271(1214) (1972), 325-360], where velocity decreases as the reciprocal of the distance from the vortex axis. Recent studies, based on radar data of selected severe weather events [Mon. Wea. Rev. 133(9) (2005), 2535-2551; Mon. Wea. Rev. 128(7) (2000), 2135-2164; Mon. Wea. Rev. 133(1) (2005), 97-119], indicate that the angular momentum in a tornado may not be constant with the radius, and thus suggest a different scaling of the velocity/radial distance dependence.Motivated by this suggestion, we consider Serrin's approach with the assumption that the velocity decreases as the reciprocal of the distance from the vortex axis to the power b with a general b > 0. This leads to a boundary-value problem for a system of nonlinear differential equations. We analyze this problem for particular cases, both with nonzero and zero viscosity, discuss the question of existence of solutions, and use numerical techniques to describe those solutions that we cannot obtain analytically.
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.
Self-similarity in tornadic and some non-tornadic supercell flows is studied and power laws relating various quantities in such flows are demonstrated. Magnitudes of the exponents in these power laws are related to the intensity of the corresponding flow and thus the severity of the supercell storm. The features studied in this paper include the vertical vorticity and pseudovorticity, both obtained from radar observations and from numerical simulations, the tangential velocity, and the energy spectrum as a function of the wave number. Connections to fractals are highlighted and discussed. Figure 1. Hierarchy of known vortex scales in tornadic supercells; c AMS, [21].identify these vortices as supercritical in the sense of [27]. Analysis of the work in [5,9,15] suggests that the supercritical vortex below a vortex breakdown has its volume and its length decrease as the energy of the supercritical vortex increases. This suggests that the entropy (randomness of the vortex) is decreasing when the energy is increased [16]. Hence the inverse temperature, which is the rate of change of the entropy with respect to the energy of the vortex, is negative. This temperature has to be considered in the statistical mechanics sense and is not related to the molecular temperature of the atmosphere. Such vortices would be barotropic, however their origin could very well be baroclinic. Recent results suggest that vorticity is produced baroclinically in the rear-flank downdraft and then descends to the surface, where it is tilted into the vertical, contributing to tornadogenesis. Even more recently, simulations show vortices produced in the forward flank region contributing to tornadogenesis and maintenance [46]. Once these vortices come into contact with the surface, and the stretching and surface friction related swirl (boundary layer effects) are in the appropriate ratio, then by analogy with the work in [27] the vortex would have negative temperature and the vortex would now be barotropic [23].Geometric self-similarity is occasionally seen in high-resolution numerical simulations of tornadic supercells [1,14,36] and also in Doppler radar and reflectivity observations [11,47]. As an example, in the reflectivity image in Figure 3 we can see self-similarity on two different scales demonstrating itself as "hooks on a hook." This is likely due to the existence of subvortices within the larger vortex. High-quality video recordings of some recent tornadoes depict mini suction vortices (subvortices of suction vortices), confirming the smallest scale of the hierarchy in Figure 1 [12,54].Fractals are mathematical objects useful as idealizations of structures and phenomena in which features and patterns repeat on progressively smaller and smaller scales [43]. Such structures exhibit geometrical complexity that can be, in a simplified way, captured by a fractal dimension of the object, a number that describes how the fractal pattern changes with scale. For example, the fractal dimension of the well-known Koch snowflake shown in Figure 4 is ...
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