N. R. McDonald

The motion of an intense vortex near topography

The initial value problem for the motion of an intense, quasi-geostrophic, equivalent-barotropic, singular vortex near an infinitely long escarpment is studied in three parts. First, for times small compared to the topographic wave timescale the motion of the vortex is analysed by deriving an expression for the secondary circulation caused by the advection of fluid columns across the escarpment. The secondary circulation, in turn, advects the primary vortex and integral expressions are found for its velocity components. Analytical expressions in terms of integrals are found for the vortex drift velocity components. It is found that, initially, cyclones propagate away from the deep water region and anticyclones propagate away from the shallow water region. Asymptotic evaluation of the integrals shows that both cyclones and anticyclones eventually propagate parallel to the escarpment with shallow water on their right at a steady speed which decays exponentially with distance from the escarpment. Secondly, it is shown that for times comparable to, and larger than, the wave timescale, the vortex always resonates with the topographic wave field. The flux of energy in the topographic waves leads to a loss of energy in the vortex and global energy and momentum arguments are used to derive an equation for the distance (or, equivalently, the vortex velocity) of the vortex from the escarpment. It is shown that cyclones, provided they are initially within an O(1) distance (here a unit of distance is dimensionally equivalent to one Rossby radius of deformation) from the escarpment, drift further away from the deep water (i.e. toward higher ambient potential vorticity), possibly crossing the escarpment and accumulate at a distance of approximate to 1.2 on the shallow side of the escarpment. For distances larger than 1.2 there is essentially no drift of the vortex perpendicular to the escarpment. Anticyclones display similar behaviour except they drift in the opposite direction, i.e. away from the shallow water or toward lower ambient potential vorticity. Third, the method of contour dynamics is used to describe the evolution of the vortex and the interface representing the initial potential vorticity jump between the shallow and deep water regions. The contour dynamic results are in good quantitative agreement with the analytical results.

The motion of a singular vortex near an escarpment

McDonald (1998) has studied the motion of an intense, quasi-geostrophic, equivalent barotropic, singular vortex near an infinitely long escarpment. The present work considers the remaining cases of the motion of weak and moderate intensity singular vortices near an escarpment. First, the limit that the vortex is weak is studied using linear theory. For times which are short compared to the advective time scale associated with the vortex it is found that topographic waves propagate rapidly away from the vortex and have no leading-order influence on the vortex drift velocity. The vortex propagates parallel to the escarpment in the sense of its image in the escarpment. The mechanism for this motion is identified and is named the pseudoimage of the vortex. Large-time asymptotic results predict that vortices which move in the same direction as the topographic waves radiate non-decaying waves and drift slowly towards the escarpment in response to wave radiation. Vortices which move in the opposite direction to the topographic waves reach a steadily propagating state. Contour dynamics results reinforce the linear theory in the limit that the vortex is weak, and show that the linear theory is less robust for vortices which move counter to the topographic waves. Second, contour dynamics results for a moderate intensity vortex are given. It is shown that dipole formation is a generic feature of the motion of moderate intensity vortices and induces enhanced motion in the direction perpendicular to the escarpment.

A point vortex model for the formation of ocean eddies by flow separation

A simple model for the formation of ocean eddies by flow separation from sharply curved horizontal boundary topography is developed. This is based on the Brown–Michael model for two-dimensional vortex shedding, which is adapted to more realistically model mesoscale oceanic flow by including a deforming free surface. With a free surface, the streamfunction for the flow is not harmonic so the conformal mapping methods used in the standard Brown–Michael approach cannot be used and the problem must be solved numerically. A numerical scheme is developed based on a Chebyshev spectral method for the streamfunction partial differential equation and a second order implicit timestepping scheme for the vortex position ordinary differntial equations. This method is used to compute shed vortex trajectories for three background flows: (A) a steady flow around a semi-infinite plate, (B) a free vortex moving around a semi-infinite plate, and (C) a free vortex moving around a right-angled wedge. In (A), the inclusion of surface deformation dramatically slows the vortex and changes its trajectory from a straight path to a curved one. In (B) and (C), without the inclusion of flow separation, free vortices traverse fully around the tip along symmetrical trajectories. With the effects of flow separation included, very different trajectories are found: for all values of the model parameter—the Rossby radius—the free and shed vortices pair up and move off to infinity without passing around the tip. Their final propagation angle depends strongly and monotonically on the Rossby radius.

Vortex scattering by step topography

The scattering at a rectilinear step change in depth of a shallow-water vortex pair consisting of two patches of equal but opposite-signed vorticity is studied. Using the constants of motion, an explicit relationship is derived relating the angle of incidence to the refracted angle after crossing. A pair colliding with a step from deep water crosses the escarpment and subsequently propagates in shallow water refracted towards the normal to the escarpment. A pair colliding with a step from shallow water either crosses and propagates in deep water refracted away from the normal or, does not cross the step and is instead totally internally reflected by the escarpment. For large depth changes, numerical computations show that the coherence of the vortex pair is lost on encountering the escarpment.

Vortical source-sink flow against a wall: The initial value problem and exact steady states

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.

Steadily translating vortices near step topography

A family of finite area translating monopolar vortices which propagate steadily without change of shape (V states) is found for shallow flow near a finite step change in depth. Solutions are obtained numerically and are unique for a given volume and center of vorticity. Time-dependent integrations show that these vortices are robust: flows initialized with a V state remain close to the V state and flows initialized with a circular vortex shed vorticity to approach a V state. The translational velocity of the vortices is shown to be finite and, unlike that of a singular line vortex, not to increase without limit as the center of vorticity approaches the escarpment.

Finite Rossby radius effects on vortex motion near a gap

This work investigates the effect of the Rossby radius of deformation on the motion of a vortex near a gap in an infinitely long barrier. A key parameter determining the behaviour of the vortex is a, the ratio of the Rossby radius of deformation to the width of the gap. Assuming quasi-geostrophic dynamics for a single-layer, reduced-gravity fluid, an integral equation is derived whose solution gives the velocity at any point in the fluid. The integral equation is solved numerically and the velocity field is integrated to give the trajectories of point vortices. Combined with the method of contour dynamics, the method can be used to compute the evolution of finite area patches of constant vorticity. The trajectories of point vortices and vortex patches are compared. The patches are initially circular and the centroids of those vortex patches that remain close to circular follow the trajectory and speed of their equivalent point vortices when appropriately normalised. The critical point vortex trajectory (the separatrix) which divides vortices that leap across the gap and those that pass through, is computed for various a. Decreasing the Rossby radius of deformation increases the tendency of vortices to pass through the gap. The effect of various background flows on both point vortex and vortex patch motion is also described.

A Simple Model for Sheddies: Ocean Eddies Formed from Shed Vorticity

AbstractRecent studies show that vertical eddy diffusivity is sufficient on its own to introduce intense horizontal shear layers at sloping ocean margins (Molemaker et al.; Gula et al.; Dewar et al.). These layers influence mesoscale energy and potential vorticity budgets but cannot be fully represented in models without sloping boundaries, no-slip boundary conditions, and sufficiently high resolution. This paper investigates the detachment of these shear layers and their subsequent rolling up into concentrated eddies. These shed eddies, or “sheddies,” may have significant oceanographic impacts. Their growth is considered using a simple point vortex model that adapts the Brown–Michael model of vortex shedding to quasigeostrophic flow and allows detailed consideration of the vorticity fluxes. The model shows good qualitative agreement with observations and experimental and numerical results. It is applied to a number of examples of well-known cases of sheddy formation, including the Agulhas cyclones, Calif...

On the motion of unsteady translating bubbles in an unbounded Hele-Shaw cell

Unsteady propagating bubbles in an unbounded Hele-Shaw cell are considered numerically in the case of zero surface tension. The instability of elliptical bubbles and their evolution toward a stable circular boundary, with speed twice that of the fluid speed at infinity, is studied numerically and by stability analysis. Numerical simulations of bubbles demonstrate that the important role played by singularities of the Schwarz function of the bubble boundary in determining the evolution of the bubble. When the singularity lies close to the initial bubble, two types of topological change are observed: (i) bubble splitting into multiple bubbles and (ii) a finite fluid blob pinching off inside the bubble region.

Generalised Hele-Shaw flow: A Schwarz function approach

An equation governing the evolution of a Hele-Shaw free boundary flow in the presence of an arbitrary external potential - generalised Hele-Shaw flow - is derived in terms of the Schwarz function g(z, t) of the free boundary. This generalises the well-known equation ∂g/∂t = 2∂w/∂z, where w is the complex potential, which has been successfully employed in constructing many exact solutions in the absence of external potentials. The new equation is used to re-derive some known explicit solutions for equilibrium and time-dependent free boundary flows in the presence of external potentials, including those with singular potential fields, uniform gravity and centrifugal forces. Some new solutions are also constructed that variously describe equilibrium flows with higher order hydrodynamic singularities in the presence of electric point sources and an unsteady solution describing bubbles under the combined influence of strain and centrifugal potential.