Existence and stability of stationary vortices in a uniform shear flow

Abstract

Isolated vortices in a background flow of constant shear are studied. The flow is governed by the two-dimensional Euler equation. An infinite family of integral invariants, the Casimirs, constrain the flow to an isovortical surface. An isovortical surface consists of all flows that can be obtained by some incompressible deformation of a given vorticity field. It is proved that on every isovortical surface satisfying appropriate conditions there exists a stationary solution, stable to all exponentially growing disturbances, which represents a localized vortex 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 requires that q is non-zero only in a finite region, and that 0 < qmin ≤ q ≤ qmax ≤ ∞ in this region (assuming the external shear to be positive).