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The time-dependent interaction of a point vortex with a vorticity jump separating regions of opposite signed and constant vorticities on the surface of a non-rotating sphere is examined. First, small amplitude interfacial waves are considered where linear theory is applicable. A point vortex in a region of same-signed vorticity will initially move away from the interface and a point vortex in a region of opposite-signed vorticity will move towards it. Weak vortices sufficiently far from the interface then undergo meridional oscillation while precessing about the sphere. The sense of azimuthal precession is determined by the sign of the vorticity jump at the interface. It is demonstrated by both linear and nonlinear theories that a vortex at a pole in a region of same-signed vorticity is a stable equilibrium whereas a vortex at a pole in a region of opposite-signed vorticity is an unstable equilibrium. Numerical computations using contour dynamics confirm these results and the dynamics of more nonlinear ca...

The time-dependent evolution of source-driven Hele-Shaw free boundary flows in the presence of an obstacle is computed numerically. The Baiocchi transformation is used to convert the Hele-Shaw Laplacian growth problem into a free boundary problem for a streamfunction-like variable u(x, y, t) governed by Poisson’s equation (with constant right-hand side) with the source becoming a point vortex of strength linearly dependent on time. On the free boundary, both u and its normal derivative vanish, and on the obstacle, the normal derivative of u vanishes. Interpreting u as a streamfunction, at a given time, the problem becomes that of finding a steady patch of uniform vorticity enclosing a point vortex of given strength such that the velocity vanishes on the free boundary and the tangential velocity vanishes on the obstacle. A combination of contour dynamics and Newton’s method is used to compute such equilibria. By varying the strength of the point vortex, these equilibria represent a sequence of source-driven growing blobs of fluid in a Hele-Shaw cell. The practicality and accuracy of the method is demonstrated by computing the evolution of Hele-Shaw flow driven by a source near a plane wall; a case for which there is a known exact solution. Other obstacles for which there are no known exact solutions are also considered, including a source both inside and outside a circular boundary, a source near a finite-length plate and the interaction of an infinite free boundary impinging on a circular disc.

A one-parameter family of exact solutions describing the bifurcation of a steady two-dimensional current with uniform vorticity near a gap in a thin barrier is found. The unsteady evolution of source-driven flows toward these steady states is studied using a version of contour dynamics, appropriately modified to take into account the presence of a barrier with a single gap. It is shown that some of the steady solutions are realizable as large-time limits of the source-driven flows, although some are not owing to persistent eddy-shedding events in the vicinity of the gap. For the special case when there is zero net flux through the gap, numerical experiments show that the through-gap flux of vortical fluid increases with the width of the gap and that this flux approaches a steady limit with time.

Techniques based on the conformal mapping and the numerical method of contour dynamics are presented for computing the motion of a finite area patch of constant vorticity on a sphere in the presence of a thin barrier with two gaps. Finite area patch motion is compared with exact point vortex trajectories and good agreement is found between the point vortex trajectories and the centroid motion of finite area patches when the patch remains close to circular. Patch centroids are, in general, closely constrained to follow point vortex trajectories. However, Kelvin’s theorem constrains the circulation about the barrier to be a constant of the motion, thus, forcing a time-dependent volume flux through the gaps. More exotic motion is observed when the through-gap flow forces the vortex patch close to an edge of a barrier, resulting in the vortex splitting with only part of the patch passing through the gap. As the gap width is decreased this effect becomes more dramatic.

The nonlinear interaction of vorticity driven coastal currents and eddies with topography is studied. The topography is either a semi-infinite escarpment perpendicular to the coast (such that topographic waves propagate toward the coast) or a semi-circular canyon or seamount attached to the coast. Assuming a piecewise constant potential vorticity distribution, the quasigeostrophic equations are solved using contour dynamics. Offshore propagating dipole eddies occur, whenever a coastal current or eddy interacts with escarpment and canyon topographies. The size and frequency at which dipoles form are found to depend on the vorticity of the current and amplitude of the topography. However, for a seamount, little eddy shedding is observed and the coastal current or eddy skirts around topography.

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.

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 vort...

This work examines the dynamics of point vortices in a two-layer fluid near large-amplitude, sharply varying topography like that which occurs in continental shelf regions. Topography takes the form of an infinitely long step change in depth, and the two-layer stratification is chosen such that the height of topography in the upper layer is a small fraction of the overall depth, enabling quasigeostrophic theory to be used in both layers. An analytic expression for the dispersion relation of free topographic waves in this system is found. Weak vortices are studied using linear theory and, if located in the lower layer, propagate mainly because of their image in the topography. Depending on their sign, they are able to produce significant topographic wave radiation in their wakes. Upper-layer vortices propagate much slower and produce relatively small amplitude topographic wave radiation. Contour dynamics results are used to investigate the nonlinear regions of parameter space. For lower-layer vortices, linear theory is a good approximation, but for upper-layer vortices complicated features evolve and linear theory is only valid for weak vortices.

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 $\uDelta H$ to a final response consisting of both geostrophic and fluctuating components. For a forcing lengthscale $r_f$ equal to the basin radius $R_0$ , the geostrophic component takes the form of a basin-scale double gyre while the fluctuating component is composed of baroclinic Kelvin and Poincare waves. The Burger number $S\,{=}\,R/r_f$ ( $R$ is the baroclinic Rossby radius of deformation) and the dimensionless forcing amplitude $\epsilon\,{=}\,\uDelta H/H_1$ ( $H_1$ is the upper-layer depth) characterize the response of the adjustment process. In particular, comparisons between analytical solutions and laboratory measurements indicate that for time $\tau$ : $1 \,{ ( $\tau$ is time scaled by the inertial period $2 \pi/f$ ), the basin-scale double gyre is established, followed by a period where the double gyre is sustained, given by $S^{-1} \,{ for a moderate forcing and $S^{-1}\,{ for a weak forcing ( $\tau_D$ is the dimensionless dissipation timescale due to Ekman damping). The analytical solution is used to calculate the energetics of the baroclinic geostrophic adjustment. The results are found to compare well with previous studies with partitioning of energy between the geostrophic and fluctuating components exhibiting a strong dependence on $S$ . Finally, the outcomes of this study are considered in terms of their application to lakes influenced by the rotation of the Earth.

The offshore spreading of coastal river outflows over shelf-like topography is inhibited by the demands of potential vorticity conservation. As a result, outflows tend to remain coastally trapped and flow parallel to the coast. Alongshore currents may also form when the outflow fluid has vorticity relative to the surrounding fluid, in which case the ‘image’ vorticity drives the outflow parallel to the coast. In the present work, the combined effects of topography and anomalous vorticity of the outflow in forming coastal currents are studied. Shelf-like topography is considered in the form of a sudden step change in depth, running parallel to a straight coast, some distance offshore. A simple model incorporating these effects is formulated using quasigeostrophic dynamics for a barotropic fluid with a rigid lid. The role of topographic wave radiation in establishing coastal currents is studied using the linearised barotropic potential vorticity equation for the case of a weak outflow of fluid with zero anomalous vorticity. The outflow is shown to be confined to the shelf and flows to the right of the source region. For stronger outflows, with relative vorticity, the method of contour dynamics is used to study the nonlinear evolution of the outflow plume and, in particular, its interaction with topography. The effects of outflow strength, magnitude and sign of the relative vorticity of the outflow fluid are considered in detail. In particular, the offshore penetration distance and rate of alongshore spreading of the outflow plume are studied. When the outflow comprises of fluid with positive (cyclonic) vorticity, the topographic and vorticity-driven image effects reinforce each other and increase the spreading rate of the outflow plume to the right of the source. For negative (anticyclonic) vorticity outflows the effects oppose each other leading to enhanced offshore spreading of the outflow. The numerical results demonstrate that the offshore transport is achieved by an efficient dipolar vortex mechanism.

The problem of vortical motions in the surf zone is simplified by taking the bottom topography to be piecewise flat while allowing finite-height jumps in depth between flat regions. The motion of an arbitrary number of singular vortices is cast into Hamiltonian form and the rule for relating Hamiltonians in conformally equivalent domains derived. Examples are given of a singular vortex pair colliding head-on with a step, of a vortex propagating along a curved coast to cross a step, and of a vortex being swept past a circular island straddling a step. Surf-zone vortices are then modelled as finite-area vortex patches and their motion followed by contour dynamics. It is shown that the paths of singular vortices can yield highly accurate explicit predictions of the paths of the centroids of vortex patches. Possible applications to surf-zone rip currents are noted.

A new nonlinear translating V-state of the barotropic quasigeostrophic equations with topography in the form of an infinitely long escarpment (or step) is computed numerically. Before the numerical computation, with the assumption of small excursions by fluid columns across the step, the linear equations of motion are solved analytically, demonstrating the possibility of constructing a translating V-state. The linear V-state has zero total circulation and self-propagates parallel to the step. It consists of two line vortices with positive circulation located on the deep side of the step and a finite-area patch of constant negative vorticity on the shallow side comprising of fluid which has crossed the step from the deep side. The linear solution motivates the numerical search for a nonlinear translating V-state comprising of two line vortices and a finite-area patch of constant vorticity. An algorithm based on contour dynamics and Newton's method is used to find such a V-state. Time-dependent contour dynamics is used to compute the evolution of the V-state and it is found to be unstable. However, the growth rate of disturbances is sufficiently small for the V-state to survive several turn-over times.

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.

Rotating, barotropic flow past a circular cylinder with circulation in the presence of a piecewise constant distribution of potential vorticity is considered. For a given cylinder radius, linear theory is used to show that there exists a unique value of the circulation for which the downstream wake vanishes. The cylinder experiences zero drag for such flows. A numerical method, based on contour dynamics, is then used to compute steady, nonlinear, waveless flows past the cylinder. For a given upstream velocity and different choices of the cylinder radius, waveless potential vorticity interface profiles are presented and corresponding values of the circulation calculated. Possible applications to propagating geophysical vortices are discussed, including the computation of nonlinear solutions which may be interpreted as freely propagating vortices.

Two motion of two interacting vortices in the presence of a background gradient of potential vorticity due to the β-effect is considered. The vortices, and their interaction, are modeled using a two-layer, quasigeostrophic model. The paper extends that of McDonald [Philos. Trans. R. Soc. London, Ser. A 456, 1029 (2000)] who, using a similar model, considered only steady configurations of zonally aligned vortices, i.e., lying along the same parallel of latitude. Here Green’s function techniques are used to calculate the velocity of one vortex due to the presence of the Rossby wave wake of the other. This enables consideration of the more general case of vortices having arbitrary relative location, the only requirement being that the vortices are well-separated (i.e., several Rossby radii apart). An additional quasisteady approximation enables the time evolution of vortices to be studied. Of particular interest is the existence of equilibrium configurations in which the vortices do not move relative to each...

The interaction of two anticyclonic baroclinic vortices on the beta-plane is studied, taking into account the effects of vortex-vortex advection and the radiation of barotropic Rossby waves. Quasi-geostrophic dynamics is assumed in each of the two layers and the vortices are represented by delta-function anomalies in the upper-layer potential vorticity. The vortices are strong (i.e, their swirl velocities are large compared with the long baroclinic Rossby wave speed), but are sufficiently widely separated so that the advection of one vortex due to another is of the same order as the meridional drift caused by Rossby wave radiation. Centre of mass arguments are used to demonstrate that the velocity of a vortex is due to the sum of (a) advection due to the other vortex and (b) drift due to wave radiation.The far-field structure of the Rossby wave wake of two zonally aligned vortices (i.e. they have zero meridional displacement relative to each other) is found as a function of their relative position enabling the total drag on the centre of mass to be found. The drag and subsequent meridional drift of each vortex due to the wave radiation alone is then deduced.Equilibria are found in which the vortices have the same southward and westward drift, thus preserving their zonal alignment. Depending on the vortex separation distance, the equilibrium configuration may have meridional drift significantly different from that of a single vortex by itself. For example, when the ratio of the upper-layer depth to bottom-layer depth is 0.4, the meridional drift may be as little as 35% or as much as 117% of the single-vortex drift speed. The stability of the equilibria as a function of the vortex separation distance is found.

The thin fluid layers constituting the Earth'oceans and atmosphere give rise to a wide variety of long–lived vortices, which are important in determining, amongst other things, our weather and climate. On a rotating planet, these vortices have the ability to self–propagate. This paper investigates this phenomenon by describing the fundamental physics involved and reviewing our current understanding of it. Also, areas of future research related to the motion of geophysical vortices are speculated upon.

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 ≈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 effect of topographic wave radiation on isolated eddies is modelled by an f-plane modon propagating parallel to an infinitely long escarpment. It is assumed that the lengthscale of the modon is much smaller than the lengthscale on which the topographic waves evolve. This enables the linearised equations of motion to be solved and the asymptotic (far-field, large-time) behaviour of the topographic wave field is subsequently described. A steady wave-like response is found when the modon moves within the range of possible topographic wave phase speeds. An anomalous case exists when the modon speed is the same as the long topographic wave group speed. This implies that energy cannot escape from the vicinity of the modon and consequently the response eventually becomes nonlinear. It is shown that the appropriate equation to describe the evolution of topographic waves in this case is a forced KdV equation. In the cases of the steady linear response and the nonlinear response (for positive forcing),...

The effect of mixing on a zonally stationary modon (Stern's modon) and rider is analysed using the method of multiple scales. Cases in which only the rider contributes to the mixing, or only the modon mixes, or a combination of both, are considered separately. In each case the mixing is shown to establish a dipole field aligned in such a manner to drive meridional motion in a direction dependent on the type of mixing. Moreover, the radius and amplitude of the modon and rider field are shown to decrease in response to mixing. The rule by which they decrease is dependent on the type of mixing. For the particular case when both the modon and rider fields contribute equally to the mixing, it is shown that cyclones propagate north and anticyclones propagate south.

A two-layer model of anticyclonic eddy propagation including the effects of diapycnal mixing is presented. The lower layer is assumed to be of finite volume, whereas the upper layer is infinite in horizontal extent, and its dynamics are quasigeostrophic and wavelike. Integral expressions for the zonal and meridional velocity am obtained using center of mass calculations. The meridional motion is a result of both southward and northward velocities in the upper layer generated by Sverdrup forcing due to diapycnal mixing and stretching of vortex lines by the collapsing eddy. Net meridional motion of the eddy arises by requiring the mixing eddy to collapse both vertically and horizontally as it adjusts cyclostrophically. Moreover, it is shown for anticyclones in solid body rotation such motion must be equatorward. The specific case of a parabolic eddy is used to illustrate the theory, and application is made to Mediterranean salt lenses and warm core rings.

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 time-dependent response of a 1½-layer, f-plane ocean with topography to a source of buoyancy is studied analytically and numerically. The topography consists of an infinitely long escarpment. Linear theory interprets the growth of a tube of fluid along the escarpment due to a point buoyancy source in terms of topographic waves The large-time, far-field asymptotic behavior of the topographic wave tube is discussed. In addition, there is a nonpropagating disturbance for which the interface elevation grows linearly with time. For an arbitrary source distribution only part of the source flux is carried away from the source in the topographic wave tube and nonlinear effects must eventually become important. Numerical results show that if the source area is located predominantly on the shallow side of the escarpment, eddies form at the source and self-propagate along the escarpment. The mass transport by these eddies compensates for the shortfall in the flux in the topographic wave tube. There is l...

Flows due to point sources and sinks of mass are considered on both f- and β-planes. The fluid is assumed to be continuously stratified with constant buoyancy frequency, N, and has no boundaries. Solutions to the flow field giving both horizontal and vertical structure are obtained on large timescales compared with an inertial period for the case when the source/sink flow is started from rest. Steady solutions are obtained through the inclusion of simple Rayleigh damping. On an f-plane the flow is radially symmetric, and a sink (source) generates cyclonic (anticyclonic) swirl due to angular momentum conservation. The nature of the flow of fluid toward the sink depends critically on the ratio f/N. On a β-plane a sink (source) also generates cyclonic (anticyclonic) swirl, but the velocity field is no longer radially symmetric and the circulation about the source/sink drifts westward. Also, the flow of fluid toward the sink intensifies to an eastward flowing jet. The behaviour is described both in terms of vorticity arguments and in terms of the westward propagation of energy by very low frequency Rossby waves. The solution is then used to calculate the surface circulation and vertical density structure in the vicinity of deep ocean convection events where the process is modelled by a simple mass transfer over depth.

The two-dimensional flow of an unbounded rotating stratified fluid towards a link sink is studied. The initial-value problem of suddenly initiating the sink flow is solved in Laplace space for a non-diffusive, inviscid fluid using the linearized Boussinesq equations. The solution shows that the sink flow is established by inertio–gravity waves radiated from the sink and that the initial development of the flow depends critically on the ratio of the inertial frequency, f, to the buoyancy frequency, N. For f < N the flow collapses to a horizontal withdrawal layer structure. The final steady state resembles potential flow in which the vertical axis is shrunk by a factor of f/N with a superimposed azimuthal velocity. Viscous, diffusive and nonlinear effects are studied using scaling analysis. A classification scheme based on two parameters delineating various force balance regimes and giving the corresponding withdrawal layer thicknesses is presented. The results show that under certain conditions rotation may cause a thicker withdrawal layer than would be observed if there were no rotation.

The streamfunction for a two-dimensional (line) mass sink in an unbounded rotating stratified fluid on an f-plane is derived using the linearized equations of motion. The solution is applied to the large scale circulation forced by the entrainment mechanism of a convective turbulent plane plume in a rotating stratified fluid. The linearity of the equations allow the effect of the plume on the ambient fluid to be modeled by an appropriate distribution of sources and sinks. A simple analytic expression for the streamfunction describing the far-field flow is obtained using this method. The streamfunction illustrates the modifying effect that entrainment by a turbulent plume has an the surrounding velocity and density fields. The theory is applied to predict the scale of the circulation in the vicinity of a hydrothermal plume and to successfully predict the surface cyclonic swirl observed in deep open-ocean convection events.

[1] A model for the thermal bar system in the rotating frame that includes unsteady inertia is formulated. Asymptotic solutions are found to the initial value problem in the frictionless, small bottom slope limit. These solutions include inertial oscillations that are significant enough to reverse the circulation ahead of the thermal bar. These asymptotic solutions are compared with numerical solutions of the full model that includes friction. The consequences of both sets of results on the thermal bar in lakes is discussed.

Techniques based on conformal mapping and the numerical method of contour dynamics are presented for computing the motion of a finite area patch of constant vorticity on a sphere with impenetrable boundaries. Several examples of impenetrable boundaries are considered including a spherical cap, longitudinal wedge, half-longitudinal wedge, and a thin barrier. Finite area patch motion is compared to exact point vortex trajectories and good agreement is found between the point vortex trajectories and the centroid motion of finite area patches when the patch remains close to circular. More exotic motion of the finite area patches, particularly in the thin barrier case, is then examined. In the case when background flow owing to a dipole located on the barrier is present, the vortex path is pushed close to one of the barrier edges, leading to vortex shedding and possible splitting and, in certain cases, to a quasisteady trapped vortex. A family of vortex equilibria bounded between the gap in the thin barrier is also computed.

Families of vortex equilibria, with constant vorticity, in steady flow past a flat plate are computed numerically. An equilibrium configuration, which can be thought of as a desingularized point vortex, involves a single symmetric vortex patch located wholly on one side of the plate. Given that the outermost edge of the vortex is unit distance from the plate, the equilibria depend on three parameters: the length of the plate, circulation about the plate, and the distance of the innermost edge of the vortex from the plate. Families in which there is zero circulation about the plate and for which the Kutta condition at the plate ends is satisfied are both considered. Properties such as vortex area, lift and free-stream speed are computed. Time-dependent numerical simulations are used to investigate the stability of the computed steady solutions.

A family of translating vortex equilibria for the two-dimensional Euler equations is investigated both analytically and numerically. The equilibria are comprised of a nonlinear wave propagating along an infinitely long vortical interface together with a comoving point vortex and dipole. The vortical interface separates the flow of uniform vorticity from a nontrivial irrotational flow. The analytical construction, though not an exact solution for the equilibria, is a relatively simple model having good quantitative agreement with the numerical solution well into the nonlinear regime. In the limit of small wave amplitude the analytical and numerical solutions approach those obtained using linear theory.

In this paper we consider the axisymmetric flow of a rotating stratified fluid into a point sink. Linear analysis of the initial value problem of flow of a linearly stratified fluid into a point sink that is suddenly switched on shows that a spatially variable selective withdrawal layer is established through the outward propagation of inertial shear waves. The amplitude of these waves decays with distance from the sink; the e-folding scale of a given mode is equal to the Rossby radius of that mode. As a consequence, the flow reaches an asymptotic state, dependent on viscosity and species diffusion, in which the withdrawal-layer structure only exists for distances less than the Rossby radius based on the wave speed of the lowest mode, R 1 . If the Prandtl number, Pr , is large, then the withdrawal layer slowly re-forms in a time that is O (δ 2 i κ -1 ), such that it extends out much farther to a distance that is O ( R 1 Pr δ 2 i δ -2 e ) rather than O ( R 1 ). Because there is no azimuthal pressure gradient to balance the Coriolis force associated with the radial, sinkward flow, a strong swirling flow develops. Using scaling arguments, we conclude that this swirl causes the withdrawal-layer thickness to grow like $(ft)^{\frac{1}{3}}$ , such that eventually there is no withdrawal layer anywhere in the flow domain. Scaling arguments also suggest that this thickening takes place in finite-size basins. These analyses of swirl-induced thickening and diffusive thinning can be combined to yield a classification scheme that shows how different types of flows are possible depending on the relative sizes of a parameter J , which we define as fQ ( Nhv ) -1 , E (the Ekman number fh 2 v -1 ), and Pr .

The flow of a stratified fluid toward a line sink in a rotating channel of finite width and depth is studied. The withdrawal flow is shown to be established by a set of Kelvin shear waves trapped within a distance of Nh/fn from the right-hand side wall (f > 0) looking in the direction of propagation, where n = 1, 2,… is the vertical mode number. In addition there are a set of waves (Poincare modes) which propagate away from the sink with a cross-channel modal structure. The withdrawal flow has a boundary-layer structure: far from the right-hand wall the flow resembles that of McDonald & Imberger (1991), whereas close to the right-hand wall the development of the vertical structure of the withdrawal flow resembles that of the non-rotating case due to the presence of Kelvin shear waves. In a narrow channel Kelvin shear waves dominate the establishment of the withdrawal flow. The withdrawal flow is investigated for large times compared to the inertial period, where it is shown that the width of the boundary layer is of the same order as the distance downstream from the sink. The flow within the boundary layer is unsteady as the withdrawal layer thickness δ continues to collapse indefinitely, while outside the boundary layer it is steady with δ ∼ fL/N, L being the horizontal lengthscale downstream from the sink. A scaling analysis is developed for the narrow channel case in which the cross-channel velocity can be ignored. The results are applied to actual field data, where it is shown that the effect of rotation may explain why previous non-rotating theories have been inaccurate in predicting withdrawal layer thickness.