Fractal powers in Serrin's swirling vortex solutions

TL;DR

This paper explores a modified fluid flow model for tornado-like vortices, where velocity scales as a power of the distance from the vortex axis, analyzing solutions for different exponents and viscosities.

Contribution

It introduces a generalized velocity scaling in Serrin's vortex model, analyzing existence and solutions of the resulting nonlinear differential equations for various parameters.

Findings

Solutions exist for specific parameter ranges.

Numerical methods describe solutions not obtainable analytically.

Velocity scaling affects vortex structure and angular momentum distribution.

Abstract

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.