Generation Model of a Spatially Limited Vortex in a Stratified Unstable Atmosphere

TL;DR

This paper introduces a new analytical model for axisymmetric vortices in a stratified atmosphere, analyzing their structure, dynamics, and instability growth, with implications for understanding atmospheric vortex formation.

Contribution

The paper develops a nonlinear analytical model describing the structure and evolution of spatially limited vortices in an unstable stratified atmosphere, incorporating Bessel functions and analyzing instability growth.

Findings

Vortices are limited in height with maximum vertical velocity at a specific level.

Radial flows converge inward below the maximum height and diverge above it.

Azimuthal velocity grows super-exponentially with height and time.

Abstract

This paper presents a new model for the generation of axisymmetric concentrated vortices. The solution of a nonlinear equation for internal gravity waves in an unstable stratified atmosphere is obtained and analyzed within the framework of ideal hydrodynamics. The corresponding expressions describing the dependences on the radius for the radial and vertical velocity components in the inner and outer regions of the vortex include combinations of Bessel functions and modified Bessel functions. The proposed new nonlinear analytical model makes it possible to study the structure and nonlinear dynamics of vortices in the radial and vertical regions. The vortex is limited in height. The maximum vertical velocity component is reached at a certain height. Below this height, radial flows converge towards the axis, and above it, an outflow occurs. The resulting instability in the stratified atmosphere leads to an increase in the radial and vertical velocity components according to the hyperbolic sine law, which turns into exponential growth. The characteristic growth time is determined by the inverse growth rate of the instability. The formation of vortices with finite velocity components, which increase with time, is analyzed. The radial structure of the azimuthal velocity is determined by the structure of the initial perturbation and can change with height. The maximum rotation is reached at a certain height. The growth of the azimuth velocity occurs according to a super-exponential law.