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Ellipsoidal vortices in rotating stratified fluids: beyond the quasi-geostrophic approximation

Published online by Cambridge University Press:  02 December 2014


Yue-Kin Tsang
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
David G. Dritschel
Affiliation:
School of Mathematics and Statistics, University of St Andrews, St Andrews KY16 9SS, UK
Corresponding
E-mail address:

Abstract

We examine the basic properties and stability of isolated vortices having uniform potential vorticity (PV) in a non-hydrostatic rotating stratified fluid, under the Boussinesq approximation. For simplicity, we consider a uniform background rotation and a linear basic-state stratification for which both the Coriolis and buoyancy frequencies, $f$ and $N$ , are constant. Moreover, we take $f/N\ll 1$ , as typically observed in the Earth’s atmosphere and oceans. In the small Rossby number ‘quasi-geostrophic’ (QG) limit, when the flow is weak compared to the background rotation, there exist exact solutions for steadily rotating ellipsoidal volumes of uniform PV in an unbounded flow (Zhmur & Shchepetkin, Izv. Akad. Nauk SSSR Atmos. Ocean. Phys., vol. 27, 1991, pp. 492–503; Meacham, Dyn. Atmos. Oceans, vol. 16, 1992, pp. 189–223). Furthermore, a wide range of these solutions are stable as long as the horizontal and vertical aspect ratios ${\it\lambda}$ and ${\it\mu}$ do not depart greatly from unity (Dritschel et al.,J. Fluid Mech., vol. 536, 2005, pp. 401–421). In the present study, we examine the behaviour of ellipsoidal vortices at Rossby numbers up to near unity in magnitude. We find that there is a monotonic increase in stability as one varies the Rossby number from nearly $-1$ (anticyclone) to nearly $+1$ (cyclone). That is, QG vortices are more stable than anticyclones at finite negative Rossby number, and generally less stable than cyclones at finite positive Rossby number. Ageostrophic effects strengthen both the rotation and the stratification within a cyclone, enhancing its stability. The converse is true for an anticyclone. For all Rossby numbers, stability is reinforced by increasing ${\it\lambda}$ towards unity or decreasing ${\it\mu}$ . An unstable vortex often restabilises by developing a near-circular cross-section, typically resulting in a roughly ellipsoidal vortex, but occasionally a binary system is formed. Throughout the nonlinear evolution of a vortex, the emission of inertia–gravity waves (IGWs) is negligible across the entire parameter space investigated. Thus, vortices at small to moderate Rossby numbers, and any associated instabilities, are (ageostrophically) balanced. A manifestation of this balance is that, at finite Rossby number, an anticyclone rotates faster than a cyclone.


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Tsang and Dritschel supplementary movie

A quasi-stable vortex remains close to its initial shape during the whole simulation of several hundreds inertial periods.

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Tsang and Dritschel supplementary movie

A strongly horizontally elongated vortex develops a dumbbell shape before breaking up into a binary system.

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Tsang and Dritschel supplementary movie

An unstable vortex becomes tilted as it evolves from its initial ellipsoidal shape into a columnar shape. The equilibrium shape is almost cylindrical with a tilt angle of about 35 degree.

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Tsang and Dritschel supplementary movie

An unstable vortex undergoes strong filament shedding. The filaments spun off from the top of the vortex form a long-lived ring which dissipates eventually.

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Tsang and Dritschel supplementary movie

This vortex starts to tumble as it becomes unstable. It then deforms and spins off filaments as it approaches an equilibrium state in which it develops a roughly ellipsoidal shape and continues to tumble.

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Tsang and Dritschel supplementary movie

After an initial period of steady rotation, this vortex starts to tumble while maintaining its original ellipsoidal shape.

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Tsang and Dritschel supplementary movie

A vortex oscillates between its initial ellipsoidal shape and a non-ellipsoidal shape. The top and bottom ends of the vortex are moving in phase.

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Tsang and Dritschel supplementary movie

After initial strong filament shedding, this vortex settles down in an oscillation between an upright and a slanted columnar shape, an example of shape oscillation in which the top and bottom ends of the vortex move out of phase.

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Ellipsoidal vortices in rotating stratified fluids: beyond the quasi-geostrophic approximation
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