Dynamics of desingularized quasigeostrophic vortices

Abstract

A stratified ocean infinite in any direction with constant buoyancy frequency is considered. A quasigeostrophic approximation (the balance of pressure gradient with a Coriolis force) is used. The Coriolis factor is assumed constant ( f plane). Several problems are considered. (1) Evolution of an ellipsoidal vortex in nonuniform flow. The nonlinear problem of the behavior of an ellipsoidal region of constant potential vorticity is solved exactly. It is shown that, without an outflow at infinity, the ellipsoidal core rotates without deformation with constant angular velocity, dependent on the geometrical parameters of the ellipsoid and proportional to the vorticity. When the vertical axis tends to infinity, the rotation rate tends to that of the 2-D Kirchhoff vortex. When the outer plane flow is added (with velocities linear in the coordinates), the core becomes deformed, remaining ellipsoidal. Depending on the characteristic of the flow and parameters of the vortex, there are the following types of behavior: (a) rotation, and (b) nutation about a certain direction and infinite elongation in the direction of the flow strain. (2) Impact of a large vortex on a small eddy. The influence of a flow field, included by a large axisymmetric vortex on a small (probe) ellipsoidal eddy is considered. The probe eddy can be stretched into a filament around the large vortex or remain localized (survive) in its flow field. Which type of behavior is realized depends on the relative intensities of the large vortex and probe eddy and on the distance between them. The corresponding numerical criteria are given. To survive in the vicinity of a large vortex, the probe eddy of the same sign of potential vorticity must be much more intensive than the counter-signed eddy. The counter-signed eddy of the same value of potential vorticity can remain localized even up to the boundary of the large vortex, where the strain is strongest. On this basis, it is reasonable to suggest that, in the vicinity of large oceanic vortices, there can exist localized anomalies of potential vorticity of opposite sign, rather than the like-signed anomalies (which is consistent with observations in the ocean). These small eddies can influence the large vortex itself, for example, they can split from it filaments of its water (such structures are called ‘‘streamers’’ in oceanology) and mix them with the ambient liquid. (3) Distant interaction of ellipsoidal vortices, Hamiltonian formulation. The problem of distant interaction of an ensemble of desingularized quasigeostrophic vortices is considered. Each vortex has four degrees of freedom—two horizontal coordinates of the center (like point vortices) and parameters, describing the eccentricity and orientation of the ellipsoidal cores. The equations of motion are given using a Hamiltonian formalism. The integrals of motion, corresponding to momentum and torque conversation of the system, are found. It is shown that if all the vortices of the ensemble are like-signed, they cannot go far away from each other, and if they came close together, they must be more elongated and the weakest of them would be stretched into filaments.