Non-convergence of the rotating stratified flows toward the quasi-geostrophic dynamics

TL;DR

This paper demonstrates that rotating stratified flows do not always converge to quasi-geostrophic dynamics, especially when the rotation-stratification ratio approaches unity slowly, challenging the common approximation assumptions in geophysical fluid dynamics.

Contribution

The study reveals the non-convergence of rotating stratified Boussinesq flows to QG dynamics near a rotation-stratification ratio of unity, providing new insights into flow behavior in this regime.

Findings

Non-convergence when the ratio tends to unity slowly.

Convergence occurs when the ratio is fixed or converges quickly.

Lower bound of convergence rate diverges as ratio approaches unity.

Abstract

The quasi-geostrohpic (QG) equation has been used to capture the asymptotic dynamics of the rotating stratified Boussinesq flows in the regime of strong stratification and rapid rotation. In this paper, we establish the invalidity of such approximation when the rotation-stratification ratio is either fixed to be unity or tends to unity sufficiently slowly in the asymptotic regime: the difference between the rotating stratified Boussinesq flow and the corresponding QG flow remains strictly away from zero, independently of the intensities of rotation and stratification. In contrast, we also show that the convergence occurs when the rotation-stratification ratio is fixed to be a number other than unity or converges to unity sufficiently fast. As a corollary, we compute a lower bound of the convergence rate, which blows up as the rotation-stratification ratio goes to unity.