  The deformation of two-dimensional vortex patches in the vicinity of fluid
boundaries is investigated. The presence of a boundary causes an initially
circular patch of uniform vorticity to deform. Sufficiently far away from the
boundary, the deformed shape is well approximated by an ellipse. This leading
order elliptical deformation is investigated via the elliptic moment model of
Melander, Zabusky & Styczek [M. V. Melander, N. J. Zabusky & A. S. Styczek, J.
Fluid. Mech., 167, 95 (1986)]. When the boundary is straight, the centre of the
elliptic patch remains at a constant distance from the boundary, and the motion
is integrable. Furthermore, since the straining flow acting on the patch is
constant in time, the problem is that of an elliptic vortex patch in constant
strain, which was analysed by Kida [S. Kida, J. Phys. Soc. Japan, 50, 3517
(1981)]. For more complicated boundary shapes, such as a square corner, the
motion is no longer integrable. Instead, there is an adiabatic invariant for
the motion. This adiabatic invariant arises due to the separation in times
scales between the relatively rapid time scale associated with the rotation of
the patch and the slower time scale associated with the self-advection of the
patch along the boundary. The interaction of a vortex patch with a circular
island is also considered. Without a background flow, conservation of angular
impulse implies that the motion is again integrable. The addition of an
irrotational flow past the island can drive the patch towards the boundary,
leading to the possibility of large deformations and breakup.
