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

Abstract

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