Jean Noel Reinaud

We investigate the interaction between two equally signed neutral vortices, namely vortices with a vanishing area integral of vorticity in inviscid non–divergent two-dimensional flows or a vanishing volume integral of potential vorticity anomaly in three-dimensional quasi-geostrophic (QG) flows. The vortices have a continuous (potential) vorticity distribution, and are linear combinations of appropriately normalised cylindrical (or spherical) Bessel functions of order 0, truncated at a zero of the Bessel function of order 1. Some pairs of neutral vortices reach an oscillating near-equilibrium state, attracting and repelling each other as a result of the exchange of small amounts of vorticity in their peripheries. This vorticity exchange generates a dipolar moment within each vortex which separates the vortices slightly, whereas the subsequent radial redistribution of the vorticity causes the vortices to come back closer again. The interaction is slower and weaker in three-dimensional QG flows, as the potential vorticity exchange primarily takes place close to the horizontal mid-plane of the vortices. These results have been corroborated using two radically different numerical models, namely a pseudo-spectral model and a high-resolution contour-advection model, both in two and in three dimensions. The observed oscillation mechanism could explain the persistence of geophysical vortices under the influence of other vortices and their ability to form stable vortex structures without experiencing vortex merging.

Applying a variational analysis, a minimum-enstrophy vortex in three-dimensional (3-D) fluids with continuous stratification is found, under the quasi-geostrophic hypothesis. The buoyancy frequency is held constant. This vortex is an ideal limiting state in a flow with an enstrophy decay while energy and generalized angular momentum remain fixed. The variational method used to obtain two-dimensional (2-D) minimum-enstrophy vortices is applied here to 3-D integral quantities. The solution from the first-order variation is expanded on a basis of orthogonal spherical Bessel functions. By computing second-order variations, the solution is found to be a true minimum in enstrophy. This solution is weakly unstable when inserted in a numerical code of the quasi-geostrophic equations. After a stage of linear instability, nonlinear wave interaction leads to the reorganization of this vortex into a tripolar vortex. Further work will relate our solution with maximal entropy 3-D vortices.

1. Vortices are long-lived oceanic features, with a mostly horizontal circulation constrained by the Earth rotation and by the density stratification (Richardson 1983, McWilliams 1991). Vortices pl...

We investigate the nonlinear evolution of pairs of three-dimensional, equal-sized and opposite-signed vortices at finite Froude and Rossby numbers. The two vortices may be offset in the vertical direction. The initial conditions stem from relative equilibria obtained numerically in the quasi-geostrophic regime, for vanishing Froude and Rossby numbers. We first address the linear stability of the quasi-geostrophic opposite-signed pairs of vortices, and show that for all vertical offsets, the vortices are sensitive to an instability when close enough together. In the nonlinear regime, the instability may lead to the partial destruction of the vortices. We then address the nonlinear interaction of the vortices for various values of the Rossby number. We show that as the Rossby number increases, destructive interactions, where the vortices break into pieces, may occur for a larger separation between the vortices, compared to the quasi-geostrophic case. We also show that for well-separated vortices, the interaction is non-destructive, and ageostrophic effects lead to the deviation of the trajectory of the pair of vortices, as the anticyclonic vortex dominates the interaction. Finally, we show that the flow remains remarkably close to a balanced state, emitting only waves containing negligible energy, even when the interaction leads to the destruction of the vortices.