The evolution of localized vortex in stably stratified flows

TL;DR

This paper analyzes how localized vortices evolve in stably stratified flows using fluid impulse theory, revealing conditions for exponential growth and oscillatory behavior related to Richardson numbers.

Contribution

It introduces an integro-differential model for vortex evolution in stratified flows, highlighting the role of Richardson numbers in stability and oscillations.

Findings

Exponential growth of fluid impulse occurs within specific Richardson number ranges.

Critical Richardson number around 0.3 marks the onset of oscillatory behavior.

Upper limits of Richardson number for instability are approximately 1.23 (2D) and 0.89 (3D).

Abstract

The evolution of a localized vortex in stably stratified flow, within the Boussinesq approximation, is analyzed using the fluid impulse concept. The set of equations describing the temporal development of the fluid impulse has an integro-differential character where the terms representing the effect of stratification appear as convolution integral of the component of the fluid impulse and time-depended 'memory' functions. These functions are calculated for the case where the external parallel shear flow varies only in the direction gravitational force and is subjected to localized two- and three-dimensional disturbances. As follows from the solution of evolution equations, in both cases there is a range of Richardson numbers where the fluid impulse associated with the disturbance grows exponentially. The upper limit of this range for two- and three-dimensional cases are Ri ~ 1.23 and Ri ~ 0.89. Both cases are also characterized by a critical value of the Richardson number (around Ri ~ 0.3 for both cases), beyond which the solution exhibits oscillatory behavior. Indeed, this oscillatory behavior has been observed in turbulent flows and, as is shown in the present study, it is an inherent feature of a non-wavy localized vortex embedded in a stably stratified shear flow. The paper was written in 2001 and published now without changes and new additions.