arXiv subjects

Mikhail M. Shvartsman

Publications and source records attributed to Mikhail M. Shvartsman.

5 recordsLinked to original sources

arXiv2025

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

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 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. 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 Schrödinger and Gross-Pitaevskii equations in describing vortex filaments.

physics.flu-dyn↗

arXiv2019

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 flows that one might see in the process of a vortex breakdown. The results in this paper are equally applicable in electromagnetics, where the equivalent concept is that of a force-free magnetic field.

physics.flu-dyn↗

arXiv2017

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 provide 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.

physics.flu-dyn↗

arXiv2015

Possible Implications of a Vortex Gas Model and Self-Similarity for Tornadogenesis and Maintenance

We describe tornadogenesis and maintenance using the 3-dimensional vortex gas model presented in Chorin (1994) and developed further in Flandoli and Gubinelli (2002). We suggest that high-energy, super-critical vortices in the sense of Benjamin (1962), that have been studied by Fiedler and Rotunno (1986), have negative temperature in the sense of Onsager (1949) play an important role in the model. We speculate that the formation of high-temperature vortices is related to the helicity inherited as they form or tilt into the vertical and their interaction with the surface and boundary layer. We also exploit the notion of self-similarity to justify power laws derived from observations of weak and strong tornadoes presented in Cai (2005); Wurman and Gill (2000); Wurman and Alexander (2005). Analysis of a Bryan Cloud Model (CM1) simulation of a tornadic supercell reveals scaling consistent with the observational studies.

math.DS↗

arXiv2014

Fractal powers in Serrin's swirling vortex solutions

We consider a modification of the fluid flow model for a tornado-like swirling vortex developed by J. Serrin, where velocity decreases as the reciprocal of the distance from the vortex axis. Recent studies, based on radar data of selected severe weather events, 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.

math-ph↗