Vortex Patch Models
In this chapter we relax the assumption that the vorticity is distributed singularly and we describe solutions to the two-dimensional Euler equations in which it is uniformly distributed over compact planar regions. Since elliptical regions of constant vorticity play a particularly important role in the literature for reasons that are both physical and mathematical, much of the focus is on solutions and models based on this shape. We start by describing Kida’s (1981) exact solution of an unsteady elliptical region of constant vorticity, which persists in a background flowfield including strain and additional uniform vorticity. This solution generalizes the classical one described in Lamb (1932) of an isolated elliptical region undergoing self-induced rotation — a so-called Kirchhoff ellipse. It is also a time-dependent generalization of the steady state solutions discovered by Moore and Saffman (1971). The Kida vortex represents one of the few known exact, nonlinear, time-dependent solutions to the Euler equations, where vorticity is not distributed singularly. Our presentation combines the original approach of Kida (1981) with the more general approach of Neu (1984) that allows for out-of-plane strain and makes use of a Hamil-tonian formulation. We then use the Melnikov method to prove that in the presence of a three-dimensional time-periodic strain field, the elliptical vortex parameters oscillate chaotically, as was shown originally in the work of Bertozzi (1988) and more recently analyzed in Ide and Wiggins (1995).
KeywordsEuler Equation Geometric Phase Point Vortex Escape Time Melnikov Function
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