EXISTENCE AND STABILITY OF STATIONARY VORTICES IN A UNIFORM SHEAR-FLOW

Isolated vortices in a background flow of constant shear are studied. The flow is governed by the two-dimensional Euler equation. An infinit e family of integral invariants, the Casimirs, constrain the flow to a n isovortical surface. An isovortical surface consists of all flows th at can be obtained by some incompressible deformation of a given vorti city field. It is proved that on every isovortical surface satisfying appropriate conditions there exists a stationary solution, stable to a ll exponentially growing disturbances, which represents a localized vo rtex that is elongated in the direction of the external flow. The most important condition is that the vorticity anomaly q in the vortex has the same sign as the external shear. The validity of the proof also r equires that q is non-zero only in a finite region, and that 0 < q(min ) less-than-or-equal-to q less-than-or-equal-to q(max) < infinity in t his region (assuming the external shear to be positive).