Elliptic vortex patches: coasts and chaos

We investigate the interaction of ocean vortex patches with boundaries via the elliptic moment model of Melander, Zabusky & Stysek [13], under which vortex patches are approximated as elliptic regions of uniform vorticity. The interaction with a straight boundary is shown to reduce to the problem of an ellipse in constant strain, as previously investigated by Kida [9]. Interactions with more complicated geometries, where we might have expected the motion to be chaotic, are found to be constrained by an adiabatic invariant arising from a separation of time scales between the rotation of the patch and its motion along the coast. A couple of examples where the presence of a background flow does produce clear chaotic motion are also given. Finally a conformal mapping technique is developed to find the motion of vortex patches in geometries where the flow is not so easily calculated.