Rotating Stratified Vortex Dynamics

Balance in geophysical flows

Fluid flows subject to the combined effects of rotation and stratification are generally radically different from flows where these effects are absent. Rotation tends to create vertical coherence or ‘two-dimensionality’ by strengthening the vertical component of vorticity, and thereby enabling the fluid to resist perturbations which bend vortex lines (those perturbations propagate away as ‘inertia waves’). Stratification tends to act just the opposite, breaking down vertical coherence and layering the fluid motion in approximately horizontal planes. Perturbations are resisted by the stable restoring force of gravity (and generally propagate away as (internal) gravity waves). Together, rotation and stratification compete and favour the formation of long-lived coherent vortices and jets (when the background planetary vorticity gradient is important). These structures are left behind as ‘inertia-gravity’ radiate away, and tend to evolve on a much slower time scale when the Rossby number (the ratio of the horizontal acceleration to the Coriolis acceleration) is small. This state of affairs is typical at least in geophysically-realistic situations in which the Coriolis frequency f is small compared to the buoyancy frequency N. The remnant, dominant, vortical-part of the flow is termed ‘balanced’, and is usefully thought of in terms of the dynamics of ‘potential vorticity’, a materially-conserved field in the absence of friction, forcing and diabatic heating.

Scale ratios

In turbulent geophysical flows, at scales small compared to the planetary scale (so f can be regarded as spatially uniform), coherent vortices tend to adopt a roughly spherical form in appropriately re-scaled coordinates in which the vertical z is stretched by the frequency ratio N/f (large).

The instability of an ellipsoidal vortex in a QG flow

This idea, due originally to Charney (1971), has been verified in numerous observations, experiments and numerical simulations. Charney considered a simplified set of equations, the ‘quasi-geostrophic’ (QG) equations, obtained by an asymptotic expansion of the full equations in Rossby number, assuming the Froude number (the ratio of horizontal vorticity magnitude to N) is comparable to the Rossby number (and small compared to 1). Our own research has verified Charney’s argument that typical vertical to horizontal scale ratios H/L obey an f/N scaling.

Beyond the QG approximation

The QG equations are, however, only a leading-order approximation to the full, inertia-gravity wave permitting equations. Formally, the QG equations apply only to the limit of zero Rossby number, though many researchers claim their validity is much greater, applying to O(1) Rossby numbers at least semi-quantitatively. Our own results support this claim, though there are significant new features arising at finite Rossby numbers, principally ageostrophic effects and inertia-gravity waves. It appears important to prepare initial conditions so that they are close to a state of ‘balance’ which radiates minimal inertia-gravity waves. Then, balance persists with minimal excitation of inertia-gravity waves even though the flow may exhibit pronounced ageostrophic features, such as enhanced stability for cyclonic vortices (spinning in the same direction as the background rotation) and reduced stability for anticyclonic vortices. This persistence of balance is thought to apply to all flows whose maximum PV anomaly magnitude is less than the background PV, at least when f/N is small (which can include values of up to 1/2).

Examples of ellipsoidal vortex instabilities at finite Rossby number (from Tsang & Dritschel 2013)

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