An analytical model of tornado generation

TL;DR

This paper presents a new analytical model for axisymmetric tornado vortices, describing their structure, velocity components, and development scenarios using ideal hydrodynamics and Bessel functions.

Contribution

It introduces an analytical solution for tornado generation, connecting internal, central, and external regions with velocity profiles based on Bessel functions.

Findings

Vortex is spatially localized by radius and height.

Radial and vertical velocities increase due to convective instability.

Maximum rotation occurs in the tornado wall at a specific height.

Abstract

A new analytical model for the generation of axisymmetric tornado-type vortices has been developed. A solution to the nonlinear equation for the stream function in an unstable stratified atmosphere is obtained and analyzed within the framework of ideal hydrodynamics. The solution is sought by smooth connecting continuous solutions for the internal region ("eye"), the central region ("wall" with maximum velocities) and the external region of the tornado. Expressions describing radial dependences for the radial and vertical velocity components include combinations of Bessel functions. The vortex is spatially localized by radius and height. Convective instability of a stratified atmosphere leads to an increase in the radial and vertical components of velocities according to the hyperbolic sine law. A downward flow is observed near the tornado axis. The maximum speed of the upward flow is achieved at a certain radial distance at a certain height. Below this height, radial flows converge towards the central part of the tornado, and above this height there is an outflow from the "wall" to the axis and to the periphery. The radial structure of the azimuthal velocity is determined by the structure of the initial disturbance and can change with height. Maximum rotation is achieved in the tornado "wall" at a certain height. The increase in azimuthal velocity can occur according to a superexponential law. Possible structures of movements, scenarios for the development of a tornado and its dynamics are discussed.