Fractal powers in Serrin's swirling vortex solutions

Abstract

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.

Possible implications of self-similarity for tornadogenesis and maintenance

“…These results strongly indicate that established tornadoes exhibit a power-law decay in the tangential velocity and that the magnitude of the exponent correlates with the severity of the tornado. This is consistent with some of the authors' work [17], in which Serrin's "swirling vortex" model [53] is revisited and solutions to the Navier-Stokes and Euler equations are sought for which velocity is proportional to r −α with a general positive α. It is shown that only solutions with 0 < α ≤ 1 are physically reasonable, and that more violent storms would correspond to larger values of α and less violent ones to smaller values of α.…”

“…Self-similarity is also observed in [17], which revisits Serrin's "swirling vortex" model [53] and investigates solutions to the Navier-Stokes and Euler equations in spherical coordinates with the velocity, v, satisfying the power law |v| ∝ r −α , where r is the distance from the vertical coordinate axis and α is not necessarily equal to 1. The streamlines and other physical quantities of the modeled vortices, such as isobars, exhibit self-similarity.…”

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

“…In a related work, Bělík et al (2014) revisit the "swirling vortex" model of Serrin (1972) and investigate solutions to the Navier-Stokes and Euler equation v = r b , where b is not necessarily equal to −1. The streamlines of the modeled vortices exhibit self-similarity, i.e., both the power law and the geometric manifestations of self-similarity are addressed in this work.…”

Interaction of a Vortex Induced by a Rotating Cylinder with a Plane

“…Nevertheless, Serrin shows that these solutions have rich structures including two-celled vortex. Recently, there has been research on modifying the scaling of the velocity/radial distance dependence in Serrin's vortex solutions, based on radar data observation [4].…”