We revisit the problem of alignment of two co-rotating vortices of uniform potential vorticity in a three-dimensional, continuously stratified flow in the absence of external flow. The process of alignment is important as, together with vortex merger, it is responsible for the growth of coherent eddies in the oceans. Vortex alignment can occur as the result of the strong interaction between the two vortices. 

We first determine pairs of co-rotating vortices in mutual equilibrium and we address their stability. The vortices of the pair may strongly interact if the equilibrium is unstable. We first study equilibria using a simplified model where the vortices are modelled by ellipsoids and non-ellipsoidal deformations are filtered. We also determine equilibria using the full quasi-geostrophic equations. Results show that equilibrium prolate vortices (vortices with an aspect ratio height-to-radius, h/r exceeding unity) are tilted towards another. They nearly align obliquely for h/r ~ 1.5 N/f, where N and f are the buoyancy and Coriolis frequencies respectively. For small vertical separations and moderate horizontal separations, the vortex pairs can be unstable, and therefore strongly interact in the nonlinear regime.

We next examine the outcome of the nonlinear evolution of a pair of upright-standing spheroidal vortices of uniform vorticity. We show that prolate vortices naturally tilt towards each other, due to the existence of nearby tilted equilibria. This tilting cannot be captured in a two-layer simulation where each vortex are located in different layers, by the rigidity imposed in the vertical direction by the model. For a continuous stratification (or a many-layer model), the baroclinic modes available allow the vortices to tilt. In regimes where the nearby equilibria are unstable, the vortices may shed potential vorticity and partially break. The ejection of potential vorticity away from the main vortices, allows the vortices to align vertically while conserving the angular impulse, an invariant of the flow. Oblate vortices can also shed significant peripheral potential vorticity and partially align vertically.

We conclude that vortices separated in the vertical direction first tilt each other and may align at an angle. Such configuration may however be unstable and break. This, in turn, allows parts of the vortices to further align vertically.