Possible implications of self-similarity for tornadogenesis and maintenance

Abstract

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 ...

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

“…Analysis and discussion of such smaller, violent vortices within a large tornadic flow is provided in [19]. Finally, a whole hierarchy of vortices within vortices is discussed in [20].…”

Equilibrium Energy and Entropy of Vortex Filaments on a Cubic Lattice: A Localized Transformations Algorithm

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