Local Vortices in a Differentially Rotating Flow

A time-dependent local elliptic vortex in a differential two-dimensional incompressible fluid flow is considered. Nonlinear oscillations of vortices of the cyclonic and anticyclonic types are described. It is found that the evolutionary tracks can be both closed and unclosed. The former correspond to the azimuthal oscillations of the principal axis and the latter to the complete rotational state of the elliptic vortex. Steady-state solutions are also obtained; they are represented by ellipses elongated or compressed along the flow. Small oscillations of the vortex equilibrium figures are investigated and a general dispersion relation for arbitrary perturbations is derived. The stability criterion is found.