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Oceanic internal waves can be decomposed into an infinite set of modes, and the dominant internal mode 1 waves have been extensively investigated. Although mode 2 waves have been observed, they have not received comparable attention, especially the generation mechanisms. In this work, we examine the generation of mode 2 internal waves by the interaction of mode 1 waves with topography. We use a coupled linear long-wave theory with mode coupling through topography, combined with evolution using a Korteweg–de Vries model, to predict the mode 2 wave amplitude, in an ideal three-layer fluid model, in a smooth density stratification and in two realistic oceanic settings. We find that the mode 2 wave amplitude is usually much smaller than the incident mode 1 wave amplitude and is quite sensitive to the pycnocline thickness, topographic slope and background stratification.

Hill's vortex is a classical solution of the incompressible Euler equations which consists of an axisymmetric spherical region of constant vorticity matched to an irrotational external flow. This solution has been shown to be a member of a one-parameter family of steady vortex rings and as such is commonly used as a simple analytic model for a vortex ring. Here, we model the decay of a Hill's vortex in a weakly rotating flow due to the radiation of inertial waves. We derive analytic results for the modification of the vortex structure by rotational effects and the generated wave field using an asymptotic approach where the rotation rate, or inverse Rossby number, is taken to be small. Using this model, we predict the decay of the vortex speed and radius by combining the flux of vortex energy to the wave field with the conservation of peak vorticity. We test our results against numerical simulations of the full axisymmetric Navier–Stokes equations.

A simple model is presented for the evolution of a dipolar vortex propagating horizontally in a vertical-slice model of a weakly stratified inviscid atmosphere, following the model of Flierl & Haines (Phys. Fluids, vol. 6, 1994, pp. 3487–3497) for a modon on the -plane. The dipole is assumed to evolve to remain within the family of Lamb–Chaplygin dipoles but with varying radius and speed. The dipole loses energy and impulse through internal wave radiation. It is argued, and verified against numerical solutions of the full equations, that an appropriately defined centre vorticity for the dipole is closely conserved throughout the flow evolution. Combining conservation of centre vorticity with the requirement that the dipole energy loss balances the work done on the fluid by internal wave radiation gives a model that captures much of the observed dipole decay. Similar results are noted for a cylindrical dipole propagating along the axis of a rotating fluid when the dipole axis is perpendicular to the axis of rotation and for a spherical vortex propagating horizontally in a weakly stratified fluid. The model extends to fluids of small viscosity and so provides an estimate for the relative importance of wave drag and dissipation in dipole decay.

This paper studies the hydraulic control of continental shelf waves using an inviscid barotropic quasi-geostrophic model with piecewise-constant potential vorticity, in which the shelf is represented by a flat step of variable width. A coastal-intensified geostrophic current generates topographic Rossby waves, which can become critical at a local decrease in shelf width when the background current opposes Rossby wave propagation. That is, the shelfbreak perturbation permanently modifies the flow field over arbitrarily large distances and the flow transitions from subcritical to supercritical as it crosses the perturbation. Critically controlled flows also lead to the exchange of significant volumes of water between the shelf and the deep ocean. We derive the boundaries for which critical control occurs in terms of a Froude number and the dimensionless magnitude of the perturbation, and analyse the possible transitions between controlled and far-field flow. When first-order dispersive terms are included in the model, transitions are resolved by dispersive shock waves, which remain attached to the forcing region when the Froude number is close to the boundary for critical flow. Contour dynamic simulations show that the dispersive long-wave model captures the quantitative behaviour of the full quasi-geostrophic system for slowly varying shelves, and replicates the qualitative behaviour even when the long-wave parameter is order one.

A number of recent results have shown that within the shallow water, rigid-lid approximation, alongshore variations in the bathymetry, i.e. a submerged ridge, can lead to continental shelf waves (CSWs) that are localised geographically at the ridge and then decay both along and away from the coast. Removing the rigid-lid assumption, introduces the superinertial Poincare waves and a Kelvin wave that is present at all frequencies. This implies that the spectrum of the associated wave operator, which is bounded for the rigid lid case, is continuous and unbounded for the free-surface case. In the rigid-lid case the localisation of modes are isolated eigenvalues lying above the continuous spectrum whereas any localised modes for the free-surface problem must necessarily be embedded in the continuous spectrum. The purpose of this work is to construct trapped CSWs analytically and numerically for a non-rectilinear shelf. A regular asymptotic method is employed by considering a slowly varying, non-rectilinear shelf with an approximate boundary condition at the shelf–ocean boundary. It is shown that even with the free-surface present, trapped CSWs do indeed exist for the submerged ridge topography. Comparison with highly accurate numerical results demonstrates the accuracy of the asymptotic method and also allows the consideration of shelves that abut an open ocean so avoiding the approximate boundary condition at the shelf–ocean boundary.

Submesoscale processes along coastal boundaries provide a potential mechanism for the dissipation of mesoscale kinetic energy in the ocean. Since these processes occur on scales not generally resolved by global ocean models, a physically motivated parametrisation is required to accurately describe their effects. Submesoscale dynamics is characterised by strong turbulent mixing, nonlinearity and topographic effects; all of which significantly modify the flow. A major component of the submesoscale boundary response to mesoscale forcing is the Kelvin – or coastally trapped – wave field, which has been shown to transport energy over large distances. This paper thus examines the influence of vertical mixing, nonlinearity and steep-slope topography on baroclinic Kelvin waves with the aim of assessing the importance of these effects. We consider the limit of a steep coastal boundary, weak mixing and weak nonlinearity and perform an asymptotic analysis to determine the modification of the classical Kelvin wave solution by these effects. Linear and nonlinear solutions are given and different mixing limits are discussed and compared with previous work. We find that vertical mixing acts to damp slowly propagating Kelvin waves while nonlinearity can cause wave breaking which may be important for fast waves. Steep-slope topography acts to modify the wave speed and structure consistent with previous work.

This paper studies the propagation of free, long waves on a potential vorticity front in the presence of a vertical coast, using a -layer, quasi-geostrophic model with piecewise-constant potential vorticity. The coastal boundary induces flow through image vorticity and a Kelvin wave, either of which can reinforce or oppose the Rossby wave dynamics at the front. The behaviour of the front depends strongly on the relative strengths of these three mechanisms, which are explicit in our model. The richest behaviour, which includes kink solitons (under-compressive shocks) and compound-wave structures, occurs in the regime where vortical effects are dominant. The evolution of the front is described by a fully nonlinear finite-amplitude equation including first-order dispersive effects, which is related to the modified Korteweg–de Vries equation. The different behaviours are classified using the canonical example of the Riemann problem, which we analyse using El’s technique of ‘dispersive shock-fitting’. Contour-dynamic simulations show that the dispersive long-wave theory captures the behaviour of the full quasi-geostrophic system to a high degree of accuracy.

The long-wave, reduced-gravity, shallow-water equations (the semi-geostrophic equations) are used to study the outflow of a river into the ocean. While previous models have studied dynamics driven by gradients in density, the focus here is on the effects of potential vorticity anomaly (PVa). The river water is taken to have the same density as a finite-depth upper layer of oceanic fluid, but the two fluids have different, uniform, potential vorticities. Under these assumptions, the governing equations reduce to two first-order, nonlinear partial differential equations which are integrated numerically for a prescribed efflux of river water and PVa. Results are found to depend strongly on the sign of the PVa, with all fluid turning downstream (in the direction of Kelvin-wave propagation) when the river water has positive PVa and anticyclonic flow upstream of the river mouth when the PVa is negative. In all cases, a nonlinear Kelvin wave propagates at finite speed ahead of the river water. Away from the river mouth, the uniformity of one set of Riemann invariants allows for similarity solutions that describe the shape of the outflow, as well as a theory that predicts properties of the Kelvin wave. A range of behaviours is observed, including flows that develop shocks and flows that continue to expand offshore. The qualitative behaviour of the outflow is strongly correlated with the value of a single dimensionless parameter that expresses the ratio of the speed of the flow driven by the Kelvin wave to that driven by image vorticity.

Based on a variable-coefficient Kadomtsev–Petviashvili (KP) equation, the topographic effect on the wave interactions between two oblique internal solitary waves is investigated. In the absence of rotation and background shear, the model set-up featuring idealised shoaling topography and continuous stratification is motivated by the large expanse of continental shelf in the South China Sea. When the bottom is flat, the evolution of an initial wave consisting of two branches of internal solitary waves can be categorised into six patterns depending on the respective amplitudes and the oblique angles measured counterclockwise from the transverse axis. Using theoretical multi-soliton solutions of the constant-coefficient KP equation, we select three observed patterns and examine each of them in detail both analytically and numerically. The effect of shoaling topography leads to a complicated structure of the leading waves and the emergence of two types of trailing wave trains. Further, the case when the along-crest width is short compared with the transverse domain of interest is examined and it is found that although the topographic effect can still modulate the wave field, the spreading effect in the transverse direction is dominant.

A study of the propagation of a mode-2 internal solitary wave over a slope-shelf topography is presented. The methodology is based on a variable-coefficient Korteweg–de Vries (vKdV) equation, using both analysis and numerical simulations, and simulations using the MIT general circulation model (MITgcm). Two configurations are considered. One is a mode-2 internal solitary wave propagating up the slope, from one three-layer system to another three-layer system. Depending on the height of the shelf, which determines the variation of the nonlinear coefficient of the vKdV equation, this can be classified into two cases. First, the case of a polarity change, in which the coefficient of the quadratic nonlinear term changes sign at a certain critical point on the slope, and second, the case with no such polarity change. In both these cases there is a small transfer of energy from the mode-2 wave to mode-1 waves. The other configuration is when the lower layer in the three-layer system goes to zero at a transition point on the slope, and beyond that point, there is a two-layer fluid system. A mode-2 internal solitary wave propagating up the slope cannot exist past this transition point. Instead it is extinguished and replaced by a mode-1 bore and trailing wave packet which moves onto the shelf.

Alongshore variations in coastline curvature or offshore depth profile can create localised regions of shelf-wave propagation with modes decaying outside these regions. These modes, termed localised continental shelf waves ($\ell$CSWs) here, exist only at certain discrete frequencies lying below the local maximum frequency, and above the far-field maximum frequency, for propagating shelf waves. The purpose of this paper is to obtain these frequencies and construct, both analytically and numerically, and discuss $\ell$CSWs for shelves with arbitrary alongshore variations in offshore depth profile and coastline curvature. If the shelf curvature changes by a small fraction of its value over the shelf section of interest or an alongshore perturbation in offshore depth profile varies slowly over the same length scale then $\ell$CSWs can be constructed using WKBJ theory. Two subcases are described: (i) if the propagating region is sufficiently long that the offshore structure of the $\ell$CSW varies appreciably alongshore then the frequency and alongshore structure are found from a sequence of local problems; (ii) if the propagating region is sufficiently short that the alongshore change in offshore structure of the $\ell$CSW is small then the alongshore modal structure is given in an explicit, uniformly valid form. A separate asymptotic theory is required for curvature perturbations to shelves that are otherwise straight rather than curved. Comparison with highly accurately numerically determined $\ell$CSWs shows that both theories are extremely accurate, with the WKBJ theory having a significantly wider range of applicability. An idealised model for the generation of $\ell$CSWs is also suggested. A localised time-periodic wind stress generates an evanescent continental shelf wave in the far field of a localised mode where the coast is almost straight and the response on the shelf is obtained numerically. If the forcing frequency is close to that of an $\ell$CSW then the wind stress excites energetic motions in the region of maximum curvature, creating a significant localised response possibly far from the forcing region.

We consider the geostrophic adjustment of a density-stratified fluid in a basin of constant depth on an $f$-plane in the context of linearized theory. For a single vertical mode, the equations are equivalent to those for a linearized shallow-water theory for a homogeneous fluid. Associated with any initial state there is a unique steady geostrophically adjusted component of the flow compatible with the initial conditions. This steady component gives the time average of the flow and is analogous to the adjusted flow in an unbounded domain without islands. The remainder of the response consists of superinertial Poincaré and subinertial Kelvin wave modes and expressions for the energy partition between the modes in arbitrary basins again follow directly from the initial conditions. The solution for an arbitrary initial density distribution released from rest in a circular domain is found in closed form. When the Rossby radius is much smaller than the basin radius, appropriate for the baroclinic modes, the interior adjusted solution is close to that of the initial state, except for small-amplitude trapped Poincaré waves, while Kelvin waves propagate around the boundaries, carrying, without change of form, the deviation of the initial height field from its average.

Since Trösch (Proceedings of the 4th International Conference on Applied Numerical Modeling, Tainan, Taiwan, 1984 (ed. H. M. Hsia, Y. L. Chou, S. Y. Wang & S. J. Hsieh) Science and Technology Series, vol. 63, 1986, pp. 307–311. American Astronautical Society) found trapped sub-inertial oscillations in computations of low-frequency variability in the Lake of Lugano, models of trapping have generally considered evenly spaced isobaths parallel to shorelines with approximate boundary conditions at any shelf–ocean boundary. Here an asymptotic analysis for slowly varying topography and accurate spectral computations demonstrate trapping on non-rectilinear shelves. It is shown that changes in any of three factors, isobath curvature, distance from the coast and the shelf-break, and the slope at the shelf-break, are sufficient on their own to give trapping. Continental shelves that abut smoothly onto the open ocean are considered thus avoiding the shelf–ocean boundary condition approximation and allowing the accuracy of previous approximations to be assessed.

An imposed normal temperature gradient on a sloping surface in a viscous stratified fluid can generate a slow steady flow along a thin ‘buoyancy layer’ against that surface, and in a contained fluid the associated mass flux leads to a broader-scale ‘outer flow’. Previous analysis for small values of the Wunsch–Phillips parameter R is extended to the nonlinear case in a contained fluid, when the imposed temperature gradient is comparable with the background temperature gradient. As for the linear case, a compatibility condition relates the buoyancy-layer mass flux along each sloping boundary to the outer-flow temperature gradient. This condition allows the leading-order flow to be determined throughout the container for a variety of configurations.

Wunsch (1970) and Phillips (1970) (Deep-Sea Res. vol. 17, pp. 293, 435) showed that a temperature flux condition on a sloping non-slip surface in a stratified fluid can generate a slow steady upward flow along a thin ‘buoyancy layer’. Their analysis is extended here to the more-general case of steady flow in a contained fluid where buoyancy layers may expel or entrain fluid from their outer edge. A compatibility condition that relates the mass flux and temperature gradient along that edge is derived, and this allows the fluid recirculation and temperature perturbation to be determined in the broader-scale ‘outer flow’ region. The analysis applies when the Wunsch–Phillips parameterRis small, in the linear case for which the density variations are dominated by a constant vertical gradient.

The flow of a one-and-a-half layer fluid over a three-dimensional obstacle of non-dimensional height M, relative to the lower layer depth, is investigated in the presence of rotation, the magnitude of which is measured by a non-dimensional parameter B (inverse Burger number). The transcritical regime in which the Froude number F, the ratio of the flow speed to the interfacial gravity wave speed, is close to unity is considered in the shallow-water (small-aspect-ratio) limit. For weakly rotating flow over a small isolated obstacle (M → 0) a similarity theory is developed in which the behaviour is shown to depend on the parameters Γ = (F−1)M−2/3 and ν = B1/2M−1/3. The flow pattern in this regime is determined by a nonlinear equation in which Γ and ν appear explicitly, termed here the ‘rotating transcritical small-disturbance equation’ (rTSD equation, following the analogy with compressible gasdynamics). The rTSD equation is forced by ‘equivalent aerofoil’ boundary conditions specific to each obstacle. Several qualitatively new flow behaviours are exhibited, and the parameter reduction afforded by the theory allows a (Γ, ν) regime diagram describing these behaviours to be constructed numerically. One important result is that, in a supercritical oncoming flow in the presence of sufficient rotation (ν ≳ 2), hydraulic jumps can appear downstream of the obstacle even in the absence of an upstream jump. Rotation is found to have the general effect of increasing the amplitude of any existing downstream hydraulic jumps and reducing the lateral extent and amplitude of upstream jumps. Numerical results are compared with results from a shock-capturing shallow-water model, and the (Γ, ν) regime diagram is found to give good qualitative and quantitative predictions of flow patterns at finite obstacle height (at least for M ≲ 0.4). Results are compared and contrasted with those for a two-dimensional obstacle or ridge, for which rotation also causes hydraulic jumps to form downstream of the obstacle and acts to attenuate upstream jumps.

Non-dispersive and weakly dispersive single-layer flows over axisymmetric obstacles, of non-dimensional height M measured relative to the layer depth, are investigated. The case of transcritical flow, for which the Froude number F of the oncoming flow is close to unity, and that of supercritical flow, for which F > 1, are considered. For transcritical flow, a similarity theory is developed for small obstacle height and, for non-dispersive flow, the problem is shown to be isomorphic to that of the transonic flow of a compressible gas over a thin aerofoil. The non-dimensional drag exerted by the obstacle on the flow takes the form D(Γ) M5/3, where Γ = (F-1)M−2/3 is a transcritical similarity parameter and D is a function which depends on the shape of the ‘equivalent aerofoil’ specific to the obstacle. The theory is verified numerically by comparing results from a shock-capturing shallow-water model with corresponding solutions of the transonic small-disturbance equation, and is found to be generally accurate for M≲0.4 and |Γ| ≲ 1. In weakly dispersive flow the equivalent aerofoil becomes the boundary condition for the Kadomtsev–Petviashvili equation and (multiple) solitary waves replace hydraulic jumps in the resulting flow patterns.For Γ ≳ 1.5 the transcritical similarity theory is found to be inaccurate and, for small M, flow patterns are well described by a supercritical theory, in which the flow is determined by the linear solution near the obstacle. In this regime the drag is shown to be $c_d M^2/(F\sqrt{F^2-1})$, where cd is a constant dependent on the obstacle shape. Away from the obstacle, in non-dispersive flow the far-field behaviour is known to be described by the N-wave theory of Whitham and in dispersive flow by the Korteweg–de Vries equation. In the latter case the number of emergent solitary waves in the wake is shown to be a function of ${\cal A}= 3M/(2\delta^2 \sqrt{F^2-1})$, where δ is the ratio of the undisturbed layer depth to the radial scale of the obstacle.

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.

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 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.

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.

This paper presents analytical and numerical solutions for steady flow past long obstacles on a β-plane. In the oceanographically-relevant limit of small Rossby and Ekman numbers nonlinear advection remains important but viscosity appears only through the influence of Ekman pumping. A reduced boundary-layer-type equation is derived giving the long-obstacle limit of an equation described in Page & Johnson (1990). Analytical solutions are presented or described in various asymptotic limits of this equation and compared with previous results for this or related flows. A novel technique for the numerical solution of the boundary-layer equation, based on a downstream–upstream iteration procedure, is described. Some modifications of the asymptotic layer structure described in Page & Johnson (1991) and Johnson & Page (1993) for the weakly nonlinear low-friction regime are outlined for the case of a lenticular obstacle.

This paper considers the role of long finite-amplitude Rossby waves in determining the evolution of flow along a rapidly rotating channel with an uneven floor. The Rossby waves travel on a potential vorticity interface in a channel with a cross-channel step change in depth, where step position varies slowly along the channel. A nonlinear wave equation is derived describing the evolution of the potential vorticity interface. To leading order this is the hydraulic equation derived by Haynes, Johnson & Hurst (1993). Dispersion appears at the next order. Various solution regimes are identified. As well as slowly varying hydraulic solutions, two further types of steady solutions appear: approach-controlled flows and twin supercritical leaps. Both these solutions are characterized by leaps between supercritical branches of the hydraulic function. It is shown how the position and size of these ‘supercritical leaps’ can be determined within the context of hydraulic theory. To fully resolve the internal structure of these leaps dispersive effects must be included and leaps are shown to correspond to kink soliton solutions of the steady unforced problem. It is also shown that increasing dispersion (decreasing topographic length scale) causes the loss of the subcritical solution branch in some subcritical flows. The only candidate for a steady solution in these regimes is then an approach-controlled flow. Integrations of initial value problems show that in general flows evolve towards the dispersive form of the solution predicted by hydraulic theory, at least near the topographic perturbation. However, in those subcritical flows where sufficiently large dispersion causes the subcritical branch to disappear, unsteady integrations evolve to approach-controlled flows even when the dispersion is sufficiently small that the subcritical branch still exists.

The flow of a constant-vorticity current past coastal topography is investigated in the long-wave weakly nonlinear limit. In contrast to other near-critical weakly nonlinear systems this problem does not exhibit hydraulically controlled solutions. It is shown that near criticality the evolution of the vorticity interface is governed by a forced BDA (Benjamin–Davis–Acrivos) equation. The solutions of this equation are discussed and two distinct near-critical flow regimes are identified. Owing to the non-local nature of the forcing, the first of these regimes is characterized by quasi-steady solutions controlled at the topography with some blocking of the upstream rotational fluid, while in the second regime steady nonlinear wavetrains form downstream of the obstacle with no upstream influence. In the hydraulic limit the velocity band for both of these critical regimes approaches zero.

This paper analyses the finite-amplitude flow of a constant-vorticity current past coastal topography in the long-wave limit. A forced finite-amplitude long-wave equation is derived to describe the evolution of the vorticity interface. An analysis of this equation shows that three distinct near-critical regimes occur. In the first the upstream flow is restricted, with overturning of the vorticity interface for sufficiently large topography. In the second quasi-steady nonlinear waves form downstream of the topography with weak upstream influence. In the third regime the upstream rotational fluid is partially blocked. Blocking and overturning are enhanced at headlands with steep rear faces and decreased at headlands with steep forward faces. For certain parameter values both overturning and partially blocked solutions are possible and the long-time evolution is critically dependent on the initial conditions. The reduction of the problem to a one-dimensional nonlinear wave equation allows solutions to be followed to much longer times and parameter space to be explored more finely than in the related pioneering contour-dynamical integrations of Stern (1991).

This paper gives analytical and numerical solutions for both westward and eastward flows past obstacles on a β-plane. The flows are considered in the quasi-geostrophic limit where nonlinearity and viscosity allow deviations from purely geostrophic flow. Asymptotic solutions for the layer structure in almost-inviscid flow are given for westward flow past both circular and more elongated cylindrical obstacles. Structures are given for all strengths of nonlinearity from purely linear flow through to strongly nonlinear flows where viscosity is negligible and potential vorticity conserved. These structures are supported by accurate numerical computations. Results on detraining nonlinear western boundary layers and corner regions in Page & Johnson (1991) are used to present the full structure for eastward flow past an obstacle with a Huff rear face, completing previous analysis in Page & Johnson (1990) of eastward flow past obstacles without rear stagnation points. Viscous separation is discussed and analytical structures proposed for separated flows. These lead to predictions for the size of separated regions that reproduce the behaviour observed in experiments and numerical computations on β-plane flows.

This paper continues the analysis of Johnson (1990, hereinafter referred to as I) of the scattering of Kelvin waves by collections of ridges and valleys. General results, flow patterns and explicit solutions follow by restricting attention to waves whose period is long compared to the inertial period but without the additional further simplification introduced in I of approximating general features by stepped topography. A simple direct method is presented giving explicit formulae for the amplitude of the transmitted Kelvin wave and the scattered topographic long waves. A simple but accurate approximation to the solution is also given. The accuracy and usefulness of the apparently crude method of I are confirmed and a superior method presented for choosing the positions of steps in the approximation of general topography. Inviscid flows, the effects of weak dissipation and weak stratification, the form and relevance of the short-wave field over downslopes, the partition of mass and energy flux between the long-wave and short-wave fields and the size and form of higher-order effects are also discussed.

The flow past a cylindrical obstacle in an enclosed channel is examined when the entire configuration is rotating rapidly about an axis which is aligned with that of the obstacle. When viewed from a frame of reference which is rotating with the channel, Coriolis forces dominate and act to constrain the motion to be two-dimensional. The channel is considered to have depth varying linearly across its width, producing effects equivalent to the so-called β-plane approximation and permitting waves to travel away from the obstacle, both upstream and downstream. For the eastward flow considered in this paper, this leads to the formation of a lee-wavetrain downstream of the obstacle and, under some conditions, a region of retarded, or ‘blocked’, flow upstream of the obstacle. The flow regime studied is essentially inviscid, although one form of frictional effect on the flow, introduced through the Ekman layers, is included. The properties of this system are examined numerically and compared with the theoretical predictions from other studies, which are applicable in asymptotic limits of the parameters. In particular, the relevance of ‘Long's model’ solutions is considered.

This paper presents analytical and numerical results for separated stratified inviscid flow over and around an isolated mountain in the limit of small Fronde number. The vertical density profile consists of a lower strongly stratified layer whose depth is just less than that of the mountain. It is separated from a semi-infinite upper stably stratified layer by a thin, highly stable, inversion layer. The paper aims to provide, for this particular profile, a thorough analysis of the three-dimensional separated flow over a mountain top with strong stratification. The Froude numbers F and F, of the lower layer and the interface are small with F-1 << F << 1, but the upper-layer Froude number is arbitrary. The flow at each height in the lower layer is governed by the two-dimensional Euler equations and moves horizontally around the mountain. It is given by a modification of a previous model using Kirchhoff free-streamline theory for the separated flow region downstream of the mountain. The pressure variations associated with the lower-layer flow are of the same order as the dynamic head and induce significant displacements of the inversion layer. When the inversion is near the top of the mountain these deflections are of the same order as the height of the projecting part of the mountain top and combine with the flow over the mountain top to excite vertically propagating internal waves in the upper layer. The resultant pressure field, vertical stream surface displacements, and surface streamlines in the upper layer are described consistently in the hydrostatic limit. Many of the features of the upper flow, including the perturbations of the critical dividing streamlines, are similar to those in flows with uniform stable stratification at low Froude number. Comparisons are made with experiments and approximate models for these summit flows based on the assumption that the dividing streamlines have small vertical displacement.

This paper considers hydraulic control and upstream influence in systems where the only wave propagation mechanism arises from the variation of vorticity or potential vorticity. These systems include two-dimensional shear flows as well as many simple paradigms for large-scale geophysical flows. The simplest is a flow in which the vorticity or potential vorticity is piecewise constant. We consider such a flow confined to a rotating channel and disturbed by a topographic perturbation. We analyse the behaviour of the system using steady nonlinear long-wave theory and demonstrate that it exhibits behaviour analogous to open-channel hydraulics, with the possibility of different upstream and downstream states. The manner by which the system achieves such states is examined using time-dependent long-wave theory via integration along characteristics and using full numerical solution via the contour-dynamics technique.The full integrations agree well with the hydraulic interpretation of the steady-state theory. One aspect of the behaviour of the system that is not seen in open-channel hydraulics is that for strong subcritical flows there is a critical topographic amplitude beyond which information from the control cannot propagate far upstream. Instead flow upstream of the topographic perturbation adjusts until the long-wave speed is zero, the control moves to the leading edge of the obstacle and flow downstream of the control is supercritical, with a transition from one supercritical branch to another on the downstream slope of the obstacle.

This paper extends recent theoretical work on sub-inertial trapped modes in bays to consider trapping of energy in the neighbourhood of estuary mouths on coastal shelves. The qualitative form of the theoretical predictions accords well with recent observations on the Scotian Shelf that show energy trapped near the Laurentian Channel at a frequency higher than that of the propagating waves on the shelf.The trapping and scattering of shelf waves is modelled for a shelf-estuary or shelf-headland system by considering barotropic waves in a straight, infinite channel with an attached rectangular estuary or interrupted by a rectangular headland. Taking the depth to increase exponentially with distance from the coast and expanding in cross-shelf modes reduces the problem to a system of real linear algebraic equations.Trapped modes with frequencies above the cutoff frequency of propagating waves are found near the mouth of the estuary. Waves propagating towards the estuary are strongly scattered and, for particular frequencies, incident energy can be either perfectly transmitted or totally reflected. An incident wave can be in resonance with the estuary causing energy to penetrate the estuary. Bounds on the frequencies of trapped and resonant solutions are given and allow an easy modal interpretation.If the frequency of an incident wave is sufficiently high, waves cannot propagate past a headland. Energy at these frequencies can however tunnel through the region and appear as an attenuated wave on the far side. For particular frequencies all energy passes the headland and none is reflected. For headlands long compared with the incident wave, transmission coefficients for single-mode scattering follow from spatially one-dimensional wave mechanics.

The adjustment of rotating free-surface flow over a step-like escarpment abutting a vertical wall is discussed in the context of the shallow-water equations. The problem is simplified by considering an escarpment of small fractional depth, so that on the slow topographic timescale the initial, fast Poincare and Kelvin wave adjustment of the free surface is effectively instantaneous, and further simplified by considering the surface displacement to be small compared with the escarpment height so that particle velocities are negligible during the topographic adjustment. Direct solution of the resulting linear system is not straightforward as arbitrarily small-scale motions are generated at sufficiently large times. The problem is reduced by a Green's function technique to one spatial dimension and the wall boundary layers resolved by introducing a scaling based on previously obtained limit solutions. Solutions verify the information-propagation arguments of Johnson (1985) and Gill et al. (1986) and also show interchange of fluid across the escarpment as eddies formed as the current crosses the step travel along the step with shallow water to their right. The pattern of evolution of the system is independent of the direction of the flow, depending solely on the sign of the topographic step. If the escarpment is such that topographic waves travel away from the wall, then a tongue of fluid moves outward along the step: the initial jet along the wall is diverted to flow parallel to, rather than across, the step. If waves travel towards the wall then the current is pinched into the wall and fluid crosses the escarpment in a thinning jet.

A straightforward method that yields explicit transmission amplitudes is presented for Kelvin wave scattering by topography whose isobaths are parallel sufficiently far from the vertical, but not necessarily planar, wall supporting the incident wave. These results are obtained by first restricting attention to the low-frequency limit in which the flow splits naturally into three regions: an outer- x region containing the incident and transmitted Kelvin waves, an outer- y region containing outwardly propagating long topographic waves and an inner quasi-steady geostrophic region whose structure follows from earlier time-dependent analyses. The present analysis is further simplified by approximating general smooth features by stepped profiles with no restriction on the size, number or order of steps. Various qualitative results on the transmission amplitudes and flow fields are deduced from the explicit solutions and results are given on orthogonality, completeness and direction of propagation of the scattered long waves.

An initial-value problem is considered for the oceanographically relevant case of slow flow over obstacles of small height and horizontal scale of order the fluid depth or larger. Previous work on starting flow over obstacles whose contours are closed (Johnson 1984) is extended to flow forced by a source–sink pair to cross a step change in depth bounded by a vertical sidewall. Bottom contours thus end abruptly and the near-periodic solutions of the earlier work are no longer possible. The relevant timescale for the motion is again the topographic vortex-stretching time h/2Ωh0, where h is the fluid depth, Ω the background rotation rate and h0 the step height. This time is taken to be long compared with the inertial period but short compared with the advection time. It is shown that if shallow water lies to the right (looking away from the wall) a wavefront moves outwards exponentially fast leaving behind a flow equivalent to that obtained by replacing the step by a rigid wall. If shallow water lies to the left the wavefront approaches the wall, forming at the wall–step junction an unsteady, exponentially thinning, singular region that transports the whole flux. The relevance of these solutions to experiments and steady solutions for free-surface and two-layer flows in Davey, Gill, Johnson & Linden (1984, 1985) is discussed.

The topographic wave equation is solved in a domain consisting of a channel with a terminating bay zone. For exponential depth profiles the problem reduces to an algebraic eigenvalue problem. In a flat channel adjacent to a shelf–like bay zone the solutions form a countably infinite set of orthogonal bay modes: the spectrum of eigenfrequencies is purely discrete. A channel with transverse topography allows wave propagation towards and away from the bay: the spectrum has a continuous part below the cutoff frequency of free channel waves. Above this cutoff frequency a finite number (possibly zero) of bay-trapped solutions occur. Bounds for this number are given. At particular frequencies below the cutoff the system is in resonance with the incident wave. These resonances are shown to be associated with bay modes.

The limiting process introduced by Stewartson & Cheng (1979) is used to obtain solutions in the limit of vanishing Rossby number for deep rotating flows at arbitrary Reynolds number over cross-stream ridges of finite slope. Examination of inviscid solutions for infinite-depth flow shows strong dependence on obstacle shape of not only the magnitudes but also the positions of disturbances in the far field. In finite-depth flow there is present the Stewartson & Cheng inertial wave wake, which may be expressed as a sum of vertical modes whose amplitudes depend on the obstacle shape but are independent of distance downstream; the smoother the topography and the shallower the flow, the fewer the number of modes required to describe the motion. For abrupt topography the strength of the wake does not, however, decrease monoton- ically with decreasing container depth (or Rossby number). In very deep flows viscosity causes the wake to decay on a length scale of order the Reynolds number times the ridge width. In shallower flows, where only a few modes are present, the decay of the wake is more rapid. For Reynolds numbers and depths of the order of those in the experiments of Hide, Ibbetson & Lighthill (1968)) viscosity causes the disturbance to take on the appearance of a leaning column.

The problem is considered of topographic waves propagating on depth gradients in rotating domains. A variational principle is derived for the eigenfunctions and eigenfrequencies of normal modes on a domain, and applied to subdomains of the whole domain. Considering suitable boundary conditions on the open boundary between the subdomain and the remainder of the whole domain gives upper and lower bounds and estimates for the frequencies of normal modes localized in the subdomain without the complication of solving over the whole domain. It is shown that applying a zero-mass-flux condition at the open boundary leads to a lower bound on the frequency whereas requiring a particular form of the energy flux to vanish identically at each point of the boundary provides an upper bound.

In a recent paper (Jansons & Johnson 1988) the authors discuss topographic Rossby waves over a random array of seamounts. It is noted that resonance is possible between a hill and an equal and opposite dale but such resonances are mentioned only briefly due to the small likelihood of correctly matched topography in the ocean. The present paper considers the resonances in detail showing how the normal modes formed by frequency splitting at resonance can be combined to give modes that slowly transfer energy from one region supporting topographic waves, across a region where such weaves are evanescent, to another region supporting waves. In addition to the simplest case of a hill—dale pair for which an exact energy-transferring mode is obtained, transferring modes are given for a three-hill system, for two hills near a coastal boundary, and for two-basin lakes. The analysis is simplified and the results generalized by extensive use of the invariance of the governing equation under conformal mappings. A transferring mode is given for a dale in a random array of seamounts showing energy leaking slowly from the dale to large distances and returning, with the rate of leakage depending on the area fraction of seamounts. It is concluded that although resonances and transferring modes are not likely to be important in random arrays on infinite planes, they are relevant to numerical models, which are necessarily restricted to finite domains, to coastal seamount chains, and to multi-basin lakes.

The initial-value problem for Taylor columns is considered for the oceanographically relevant case of slow flow over obstacles of small slope, and horizontal scale of order of the fluid depth or larger. In such flows depth variations cause, in the neighbourhood of the obstacle, a gradient of background potential vorticity. Topographic Rossby waves, forced in the starting flow, propagate across the gradient, cycling the obstacle in times of order of the topographic vortex-stretching time h /2Ω h 0 , where h is the fluid depth, Ω the background rotation rate and h 0 the obstacle height. The linear inertialwave equation and the quasigeostrophic equations are related by considering the case where the inertial period is small compared with the topographic time, which is in turn small compared with the advection time. When viscosity is present the waves decay to a steady motion in a time of order of the viscous spin-up time. In inviscid flow the waves do not decay. The flow oscillates about steady, irrotational, zero-circulation flow. The drag and lift on the obstacle oscillate with almost constant frequency and amplitude equal to the Coriolis force on a body of fluid whose volume matches that of the obstacle. Particles above a right cylinder move in closed orbits of diameter 1/ S , where S is the topographic parameter introduced by Hide (1961). Particles away from the cylinder move irrotationally, but asymmetrically, from upstream to downstream. A shear layer is present at the boundary of the cylinder throughout the motion. The initial vorticity distribution for flow over a paraboloid is everywhere finite. However, regions of positive and negative vorticity interlace ever more closely during the motion, causing at large time a similar shear layer at the column boundary. By this time the vorticity is increasing without bound above the obstacle and neglected nonlinear, horizontal viscous, or vertical effects are important.

A variational principle is presented which characterizes steady motions, at finite Rossby number, of rotating inviscid homogeneous fluids in which horizontal velocities are independent of depth. This is used to construct nonlinear solutions corresponding to stationary patches of distributed vorticity above topography of finite height in a uniform stream. Numerical results are presented for the specific case of a right circular cylinder and are interpreted using a series expansion, derived by analogy with a deformable self-gravitating body. The results show that below a critical free-stream velocity a trapped circular vortex is present above the cylinder and a smaller patch of more concentrated vorticity, of the opposite sign, maintains a position to the right (looking downstream) of the cylinder. An extension to finite Rossby number and finite obstacle height of Huppert's (1975) criterion for the formation of a Taylor column is presented in an appendix.

The barotropic potential vorticity equation or topographic wave equation is not linear in topography: the solution structure for topography formed from a sum of obstacles is not the sum of solutions for the obstacles in isolation, even when these individual solutions have identical frequencies. This paper considers the mechanism by which normal modes of oscillation above one mountain are modified by interactions with its neighbours. Exact explicit solutions for the normal modes above a pair of circular seamountains show that the interactions between the mountains rapidly approaches the large-separation approximation obtained by considering solely the first reflection of the disturbance of one mountain at the other. For mountains of one diameter separation at the closest point, the approximation is accurate to within 1%. Perhaps surprisingly, coupling between two identical mountains is weak and resonance occurs between mountains and dales of equal and opposite height. The accurate approximate solutions enable consideration of the effects on a mountain of an infinite set of randomly distributed neighbours. The ensembleaveraged frequency for a mountain of given height is determined in terms of the area fraction of the other mountains. The idea of an effective topography is introduced for the ensemble-averaged stream function: it is that (non-random) topography generating a stream function identical to the ensemble-averaged stream function. This differs markedly from the ensemble-averaged topography. The explicit form of the effective topography is derived for a set of right circular cylinders.