N. Robb McDonald

The motion of a vortex near two circular cylinders

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 two-dimensional 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.

Vortices near barriers with multiple gaps

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 Kelvins 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.

The motion of a vortex near a gap in a wall

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.

Maximal reduced-gravity flux in rotating hydraulics

Abstract A simple but general upper bound is given for the flux of a reduced-gravity fluid with non-negative, but otherwise arbitrary, potential vorticity through a sill of arbitrary shape. This bound is a modified form of the zero potential vorticity limit. A weaker bound is given for the flux in any layer of an (n + ½) layer fluid.

The point island approximation in vortex dynamics

The effect of approximating a circular island of non-zero radius by a point dipole – the point island approximation – on the dynamics of two-dimensional vortices is studied. The approximation enables the Hamiltonian for N point vortices moving among M point islands to be derived. Examples of exact (within the point island approximation) trajectories for a single point vortex in domains involving gaps and multiple islands are presented. In addition, vortex patch motion using the point island approximation is computed numerically, demonstrating that such patches, provided they stay close to a circular shape, are well-modelled by point vortices. The effect of making the point island approximation on the trajectories of a point vortex about, (i), two islands and, (ii), an island within a gap, is investigated by comparing them to the exact trajectories in each case. It is found that the point island approximation is an efficient and accurate way of determining the behaviour of the vortex provided that the vortex and islands all remain well-separated in comparison to the vortex and island radii.

Baroclinic geostrophic adjustment in a rotating circular basin

Baroclinic geostrophic adjustment in a rotating circular basin is investigated in a laboratory study. The adjustment process consists of a linear phase before advective and dissipative effects dominate the response for longer time. This work describes in detail the hydrodynamics and energetics of the linear phase of the adjustment process of a two-layer fluid from an initial step height discontinuity in the density interface

Coriolis effects and the thermal bar

\uDelta H

The motion of a point vortex near large-amplitude topography in a two-layer fluid

to a final response consisting of both geostrophic and fluctuating components. For a forcing lengthscale

The time-dependent behaviour of a spinning disc on a rotating planet: A model for geophysical vortex motion

r_f

Surf-zone vortices over stepped topography

equal to the basin radius