The motion of a vortex near two circular cylinders of arbitrary radii-a problem of geophysical significance-is studied. The fluid motion is governed by the twodimensional Euler equations and the flow is irrotational exterior to the vortex. Two models are considered. First, the trajectories of a line vortex are obtained using conformal mapping techniques to construct the vortex Hamiltonian which respects the zero normal flow boundary condition on both cylinders. The vortex paths reveal a critical trajectory (i.e. separatrix) that divides trajectories into those that orbit both cylinders and those that orbit just one cylinder. Second, the motion of a patch of constant vorticity is computed using a combination of conformal mapping and the numerical method of contour surgery. Although the patch can deform, the results show that when the islands have comparable radii the patch remains remarkably coherent. Moreover, it is demonstrated that the trajectory of the centroid of the patch is well modelled by a line vortex. For the limiting case when one of the cylinders has infinite curvature (i.e. it becomes a straight line or wall) it is shown that the vortex patch, which propagates under the influence of its image in the wall, may undergo severe deformation as it collides with the smaller cylinder, with portions of the vortex passing around different sides of the cylinder.

Two models are presented for the motion of a vortex near a gap in an infinitely long straight barrier: a problem of geophysical significance. The first model considers a line vortex for which the trajectories are obtained analytically and are generalized to include simple ambient flows. The criterion determining whether in the absence of background flow a vortex originating far from the gap passes through or leaps across the gap is derived. The second model considers a finite area patch of constant vorticity and is based on conformal mapping and the numerical method of contour surgery. The two models enable a comparison of the trajectories of line vortices and vortex patches.In most examples the centroid of the patch is constrained to follow closely the trajectory of a line vortex of the same circulation. An exception occurs when flow through the gap forces the vortex patch close to one of the edges of the barrier where it splits into two with only one part of the vortex passing through the gap.

Two models are presented for the motion of vortices near gaps in infinitely long barriers. The first model considers a line vortex for which the exact nonlinear trajectories satisfying the governing two-dimensional Euler equations are obtained analytically. The second model considers a finite-area patch of constant vorticity and is based on conformal mapping and the numerical method of contour surgery. The two models enable a comparison of the trajectories of line vortices and vortex patches. The case of a double gap formed by an island lying between two headlands is considered in detail. It is noted that Kelvin's theorem constrains the circulation around the island to be a constant and thus forces a time-dependent volume flux between the islands and the headlands. When the gap between the island and a headland is small this flux requires arbitrarily large flow speeds through the gap. In most examples the centroid of the patch is constrained to follow closely the trajectory of a line vortex of the same circulation. Exceptions occur when the through-gap flow forces the vortex patch close to an edge of the island where it splits into two with only part of the vortex passing through the gap. In general the part squeezing through a narrow gap returns to near-circular to have a diameter significantly larger than the gap width.

It is argued that because shallow water cyclones on a β-plane drift westward at a speed equal to an available Rossby wave phase speed, they must radiate energy and cannot, therefore, be steady. The form of the Rossby wave wake accompanying a quasi-steady cyclone is calculated and the energy flux in the radiated waves determined. Further, an explicit expression for the radiation-induced northward drift of the cyclone is obtained. A general method for determining the effects of the radiation on the radius and amplitude of the vortex based on conservation of energy and potential vorticity is given. An example calculation for a cyclone with a 'top-hat' profile is presented, demonstrating that the primary effect of the radiation is to decrease the radius of the vortex. The dimensional timescale associated with the decay of oceanic vortices is of the order of several months to a year. IntroductionObservations of the Earth's ocean and also of the atmosphere's of Jupiter and Saturn show that there is a predominance of anticyclones over cyclones (see e.g. McWilliams 1985;Nezlin & Sutyrin 1994). This asymmetry is also evident in non-quasi-geostrophic β-plane numerical experiments such as those performed by . In particular, they demonstrated that anticyclones emerge in preference to cyclones from an initially turbulent field. Furthermore, experiments performed by Nezlin and co-workers (see Nezlin & Sutyrin 1994 for a review) in a rotating paraboloid also demonstrated the longevity of anticyclones relative to cyclones. The present work explores the possibility that the asymmetry may be due to the differing westward propagation speeds of cyclones and anticyclones. This is not a new idea. Indeed Nezlin & Sutyrin (1994) suggested that the asymmetry in the dispersive and nonlinear properties of cyclones and anticyclones was one of the essential reasons for the observed predominance of anticyclones. Nycander (1994) develops the idea further and advocates a necessary condition for the existence of steady vortex structures, namely that the centre-of-mass speed of the vortex lies outside the range of possible linear wave phase speeds. Otherwise the vortex will resonate with the linear wave field, radiate energy (sometimes referred to as Cerenkov radiation), and subsequently decay. Nycander illustrates the idea with examples from geophysical fluid dynamics and magnetohydrodynamics where he compares the drift velocity of the vortex obtained by global integration methods with the phase velocities of the waves calculated from the linear dispersion relation. The purpose of the present work is to calculate analytically the Rossby wave field accompanying a cyclone on a β-plane, the associated energy loss, the meridional velocity and the response of the cyclone to this energy loss.

Fluid of uniform vorticity is expelled from a line source against a wall. An exact analytical solution is obtained for the nonlinear problem determining the final steady state. Sufficiently close to the source, the flow is irrotational and isotropic, turning on the vortical scale Q∕ω (for area flux Q and vorticity ω) to travel along the wall to the right (for ω>0). The flow is linearly stable with perturbations propagating unattenuated along the interface between vortical and irrotational fluid. Fully nonlinear numerical integrations of the time-dependent equations of motion show that flow started from rest does indeed closely approach the steady state. Similar exact steady solutions are obtained for a vortical source-sink pair and vortical source doublet against a wall. Time-dependent integrations show that the steady state is unchanged at arbitrarily large times for sufficiently small disturbances but is disrupted by finite perturbations.