The stability of elliptical vortices in an external straining flow

Subject to uniform strain, an elliptical patch of vorticity in an inviscid, incompressible, two-dimensional fluid generally rotates or nutates and extends or compresses while retaining a precisely elliptical shape (the Kida solutions). This result is of interest because the uniform strain idealizes the leading-order distortional influence of distant vortices in a flow with many vortices. Because of the unsteady motion of the distant vortices, both the strain rate and the rotation rate of the strain axes typically vary with time. In the special case that the strain rate and rotation rate are steady, and when the strain rate is not too large, periodic motion of an elliptical vortex is possible. Larger strain rates lead to indefinite extension of the vortex. Uniform strain, however, only approximately mimics the effect of distant vortices. The local variations in the strain field around a vortex disturb the vortex, preventing it from retaining a simple, elliptical shape. These disturbances may amplify because of instabilities. In this paper, we examine the stability of periodic elliptical motion to small boundary disturbances, for the case of steady, uniform strain and rotation rate, first by linear Floquet theory and then by direct, high-resolution, nonlinear numerical integrations. It is discovered that a significant portion of the periodic solutions are linearly unstable. Instability can occur even when the strain rate is arbitrarily small and the basic motion arbitrarily close to circular. Extended nonlinear calculations exhibit recurrence, in some cases, and attrition of the vortex by repeated wave amplification, steepening, and breaking in others.