This work is an attempt to explain observations of vortices in experiments with shallow water in rotating paraboloidal vessels. The most long-lived vortices are invariably anticyclones, while cyclones quickly disperse, and they are larger than the Rossby radius. These experiments are designed to simulate geophysical flows, where large, long-lived, anticyclonic vortices are common.

The general condition for vortices to be steady is that they propagate faster than linear Rossby waves, so that the vortex energy is not dispersed by coupling to linear waves. The propagation velocity is determined by a general integral relation that gives the velocity of the centre of mass. In geophysical flows, to lowest order in the Rossby number, the difference between the centre-of-mass velocity and the maximum phase velocity of the Rossby waves is proportional to the relative perturbation of the fluid depth. Since for anticyclones the difference is positive they may be steady, whereas cyclones cannot be.

In the laboratory experiments this velocity difference is absent because of the latitudinal dependence of the effective gravity caused by the centrifugal force. However, to the next order in the Rossby number, there is another nonlinear contribution, so that anticyclones (but not cyclones) still propagate faster than the linear Rossby waves, and may thus be steady. The velocity difference is smaller than for geophysical flows, and vanishes in the limit of small Rossby number. The existence conditions also show that we can expect the experimental vortices to be smaller (as measured by the Rossby radius) than the planetary vortices. The theory does not apply to vortices that are much smaller than the Rossby radius.

We consider longshore motion generated within the surf zone by obliquely incident breaking waves, and seek to describe the effect on such motion of variations, caused by wave grouping, in the incident longshore momentum flux. The effects of associated variations in set-up are not considered.

We use the linear long-wave equations to describe the motion resulting from the longshore momentum contained in a wave group. This consists of a succession of edge waves which disperse along the beach, and, for the example considered, an eventual steady circulation cell at the position of the wave group. We suggest that such a cell is always likely to be formed if the wave group is sufficiently localized, and that higher-modenumber edge waves are more likely to be excited.

We find timescales for the dispersal of the edge waves, and for the decay, due to bottom friction, of the circulation cell: we suggest that the latter may more generally be used, as a timescale for the effect of friction on longshore motion.

We have studied oscillatory flow through a 180° curved tube with the ratio of tube radius to radius of curvature equal to 1/7. The flow rate is constrained to vary sinusoidally about a non-zero mean at a specified period T, and mean flow rate Q. At a certain parameter range of interest Hamakiotes & Berger (1990) predict that fully developed flow undergoes a period-tripling bifurcation. Our measurements using laser-Doppler velocimetry found no bifurcation. An additional experiment was done to ensure that the flow was fully developed.

Modulation of the growth rate of short capillary–gravity surface wind waves in the presence of a long wave with steepness much smaller than the maximum is studied theoretically. The Miles (1962) mechanism taking into account the viscous wave stresses in the air flow is considered to be the main process of short-wave generation. The short-wave growth rate is defined by the wind velocity gradient in the viscous sublayer of the logarithmic boundary layer. The long wave propagating on the wave surface induces an additional component of the wind velocity gradient oscillating with the length and time periods of the long wave, which results in modulation, with the same period, of the growth rate of the short wave riding on the long one. The growthrate modulation amplitude depends on the parameter M being of the order of the relation between the oscillating and the mean wind velocity gradients in the viscous sublayer \[M=\frac{2kac}{u^2_*}(ckv_{\alpha})^{1/2} \] (where c, k, a are the phase velocity, the wavenumber and the elevation amplitude of the long wave; va is the viscosity coefficient in the air; u* is the wind friction velocity). When M = O(1) (weak winds and long waves) the oscillating component of the shortwave growth rate is of the same order as the mean one. If M is much smaller than unity, then the relative amplitude of the growth rate is of the same order as the steepness of the long wave.

A numerical and analytical study is described for the divided flow field produced when the flows in two equal parallel channels, separated upstream by an aligned splitter plate, join together to form a single channel beyond the enclosed trailing edge of the plate. On the numerical side the second-order-accurate finite-difference scheme is based on a modified procedure to preserve accuracy and iterative convergence at higher Reynolds numbers R. Account is taken also of the influence of the boundedness of the computational domain and of the irregular behaviour of the flow solution at the trailing edge. The numerical solutions are presented for values of R up to 1000. On the analytical side the asymptotic structure of the solution for large R is governed mainly by a long $O(R^{\frac{1}{7}})$ relative lengthscale upstream and beyond the trailing edge. This is followed by a longer O(R) scale far downstream, but effects of practical significance also arise on the nominally tiny scale of O(R−½). Comparisons between the numerical and the asymptotic results for the wall shear stresses and the wake centreline velocity are made, and overall the agreement seems reasonable. The use of comparisons is believed to strengthen the value of both the numerical and the analytical approaches for these flows.

An earlier paper (Gaster 1981) discussed different asymptotic representations of the isolated wave packet that evolved from an impulsive point excitation of a laminar boundary layer. Comparisons of the various asymptotic representations of the integral describing these packets were made on the basis of numerical evbluations of the various approximations together with the direct numerical solutions of the integral. Here the problem is pursued by analytical means, and error estimates are obtained for the different methods used.

There exists a well-developed body of theory based on quasi-geostrophic (QG) dynamics that is central to our present understanding of large-scale atmospheric and oceanic dynamics. An important question is the extent to which this body of theory may generalize to more accurate dynamical models. As a first step in this process, we here generalize a set of theoretical results, concerning the evolution of disturbances to prescribed basic states, to semi-geostrophic (SG) dynamics. SG dynamics, like QG dynamics, is a Hamiltonian balanced model whose evolution is described by the material conservation of potential vorticity, together with an invertibility principle relating the potential vorticity to the advecting fields. SG dynamics has features that make it a good prototype for balanced models that are more accurate than QG dynamics.

In the first part of this two-part study, we derive a pseudomomentum invariant for the SG equations, and use it to obtain: (i) linear and nonlinear generalized Charney–Stern theorems for disturbances to parallel flows; (ii) a finite-amplitude local conservation law for the invariant, obeying the group-velocity property in the WKB limit; and (iii) a wave-mean-flow interaction theorem consisting of generalized Eliassen–Palm flux diagnostics, an elliptic equation for the stream-function tendency, and a non-acceleration theorem. All these results are analogous to their QG forms.

The pseudomomentum invariant – a conserved second-order disturbance quantity that is associated with zonal symmetry – is constructed using a variational principle in a similar manner to the QG calculations. Such an approach is possible when the equations of motion under the geostrophic momentum approximation are transformed to isentropic and geostrophic coordinates, in which the ageostrophic advection terms are no longer explicit. Symmetry-related wave-activity invariants such as the pseudomomentum then arise naturally from the Hamiltonian structure of the SG equations. We avoid use of the so-called ‘massless layer’ approach to the modelling of isentropic gradients at the lower boundary, preferring instead to incorporate explicitly those boundary contributions into the wave-activity and stability results. This makes the analogy with QG dynamics most transparent.

This paper treats the f-plane Boussinesq form of SG dynamics, and its recent extension to β-plane, compressible flow by Magnusdottir & Schubert. In the limit of small Rossby number, the results reduce to their respective QG forms. Novel features particular to SG dynamics include apparently unnoticed lateral boundary stability criteria in (i), and the necessity of including additional zonal-mean eddy correlation terms besides the zonal-mean potential vorticity fluxes in the wave-mean-flow balance in (iii).

In the companion paper, wave-activity conservation laws and stability theorems based on the SG form of the pseudoenergy are presented.

Two-dimensional (2-D) gravity–capillary solitary waves are generated using a moving pressure jet from a 2-D narrow slit as a forcing onto the surface of deep water. The forcing moves horizontally over the surface of the deep water at speeds close to the minimum phase speed $c_{min}=23~\text{cm}~\text{s}^{-1}$. Four different states are observed according to the forcing speed. At relatively low speeds below $c_{min}$, small-amplitude depressions are observed and they move steadily just below the moving forcing. As the forcing speed increases towards $c_{min}$, nonlinear 2-D gravity–capillary solitary waves are observed, and they move steadily behind the moving forcing. When the forcing speed is very close to $c_{min}$, periodic shedding of a 2-D local depression is observed behind the moving forcing. Finally, at relatively high speeds above $c_{min}$, a pair of short and long linear waves is observed, respectively ahead of and behind the moving forcing. In addition, we observe the transverse instability of free 2-D gravity–capillary solitary waves and, further, the resultant formation of three-dimensional gravity–capillary solitary waves. These experimental observations are compared with numerical results based on a model equation that admits gravity–capillary solitary wave solutions near $c_{min}$. They agree with each other very well. In particular, based on a linear stability analysis, we give a theoretical proof for the transverse instability of the 2-D gravity–capillary solitary waves on deep water.

When advection dominates diffusion, there are special directions (rays), distinct from but closely related to the contaminant flux vector, along which information is carried. The geometry of these ray paths is found to depend in a simple way upon the advection-diffusion vector ½u/D, where u is the flow velocity and D the cross-stream diffusivity. Simple models are used to show that for a steady point discharge the contaminant concentration is greatest in shallow water and towards the outside of bends.

The Stokes flow due to the motion of a small particle in arbitrary directions is investigated in the presence of a circular hole in an infinite thin plane wall separating a quiescent viscous fluid.

The solutions of the boundary-value problem are obtained in closed forms to the point-force approximation in toroidal coordinates, by the use of the Green and Neumann functions supplemented by the edge function to remove the singularity at the rim of the hole. The volume flux through the hole and the force and torque experienced by the small spherical particle are determined on the basis of this solution. The case of linear motion parallel to the plane of the wall is discussed in detail.

We consider the nonlinear problem of steady gravity-driven waves on the free surface of a two-dimensional flow of an inviscid, incompressible fluid (say, water). The water motion is supposed to be rotational with a Lipschitz continuous vorticity distribution, whereas the flow of finite depth is assumed to be unidirectional. We verify the Benjamin–Lighthill conjecture for flows with values of Bernoulli’s constant close to the critical one. For this purpose it is shown that a set of near-critical waves consists only of Stokes and solitary waves provided their slopes are bounded by a constant. Moreover, the subset of waves with crests located on a fixed vertical is uniquely parametrised by the flow force, which varies between its values for the supercritical and subcritical shear flows of constant depth. There exists another parametrisation for this set; it involves wave heights varying between the constant depth of the subcritical shear flow and the height of a solitary wave.

Drift of individual β-plane vortices confined to one layer of a two-layer fluid under the rigid-lid condition is considered. For this purpose, the theory of two-layer quasi-geostrophic singular vortices is employed. On a β-plane, any non-zonal displacement of a singular vortex results in the development of a regular flow. An individual singular β-plane vortex cannot be steady on its own: the vortex moves coexisting with a regular flow, be the drift steady or not. In this paper, both kinds of drift of a singular vortex are considered. A new steady exact solution is presented, a hybrid regular–singular modon. This hybrid modon consists of a dipole component and a circularly symmetric rider. The dipole is regular, and the rider is a superposition of the singular vortex and a regular circularly symmetric field. The unsteady drift of a singular vortex residing in one of the layers is considered under the condition that, at the initial instant, the regular field is absent. The development of barotropic and baroclinic regular β-gyres is examined. Whereas the barotropic and baroclinic modes of the singular vortex are comparable in magnitudes, the baroclinic β-gyres attenuate with time, making the trajectory of the vortex close to that of a barotropic monopole on a β-plane.

Flapping and revolving wings can produce attached leading-edge vortices when the angle of attack is large. In this work, a low-order model is proposed for the edge vortices that develop on a revolving plate at $90^{\circ }$ angle of attack, which is the simplest limiting case, yet shows remarkable similarity with the generally known leading-edge vortices. The problem is solved analytically, providing short closed-form expressions for the circulation and the position of the vortex. The good agreement with the numerical solution of the Navier–Stokes equations suggests that, for the conditions examined, the vorticity production at the sharp edge and its subsequent three-dimensional transport are the main effects that shape the edge vortex.

The stability of a compositionally buoyant plume, of circular cross-section, rising in a rotating infinite fluid is investigated. Both plume and fluid have the same non-zero kinematic viscosity, ν, and thermal diffusivity, κ. The growth rate of the instability depends on the Taylor number, Ta (which is a dimensionless number measuring the effect of the Coriolis force relative to the viscous force) and on the thickness, s0, of the plume in addition to the Prandtl number, σ(=ν/κ) and the Reynolds number, R (which measures the strength of the forcing). The analysis is restricted to the case of small R. It is found that the presence of rotation enhances instability. A simple model of a single interface separating the two parts of an infinite fluid is investigated first in order to isolate the mechanism responsible for the increase in the growth rate with rotation. It is shown that the Coriolis force interacts with the zonal velocity component to produce a velocity component normal to the interface. For the right choice of wave vector components, this normal velocity component is in phase with the displacement of the interface and this leads to instability. The maximum growth rate is identified in the whole space of the parameters σ, Ta, s0 when R[Lt ]1. While the maximum growth rate is of order R2 in the absence of rotation, it is increased to order R in the presence of rotation. It is also found that the Prandtl number, σ, which has a strong influence on the growth rate in the absence of rotation, plays a subservient role when rotation is present.

A large number of small spheres of the same size were injected successively into a horizontal pipe conveying water at constant mean velocity, and their times of transit were measured. The mean velocity of the spheres that were either somewhat heavier or lighter than water was less than that of those of neutral density; for those having a terminal velocity in water within ± 1% of the mean velocity of the water in the pipe, the discrepancy was only about 0.1%. The dispersion of the times of transit of the spheres was almost independent of their density. A theory is developed to show how the mean velocity of the spheres depends upon their relative density and size.

An isolated fluid mass travelling horizontally in a stratified layer is a phenomenon described alternatively as a detached gravity-current head or a strongly nonlinear solitary wave. A key feature of this flow is the transport of mass. Laboratory experiments examine the transition in time from a regime in which the flow is density driven, to one in which it is wave dominated. A simple means of creating this transitional regime, an isolated flow that exhibits both density and wave effects, is achieved by dropping a thermal into a linearly stratified layer. This transitional regime is called an ‘isolated propagating flow’. Parameters for which the transitional regime occurs are identified. Particle-tracking studies reveal the vertical flow structure. There is an upper zone that is wave dynamical, and a lower zone in which transport of mass occurs. The transported mass slowly leaks out, until the phenomenon resembles a weakly nonlinear solitary wave. The experiments mimic a thunderstorm microburst impacting a temperature inversion, which has aviation safety implications. In the ocean, cracks in the ice cap (polar leads) cause similar flows impacting the thermocline.

The characteristics of the initial formation of streamwise vortices and their relation to the spanwise structures are experimentally investigated at several phases of the initial roll-up and pairing in a two-stream mixing layer at high Reynolds numbers (up to Reθ = 1.25 × 103). The spanwise structures were stabilized by acoustically exciting the mixing layer at the natural instability frequency and its first subharmonic. No artificial spanwise forcing was applied. Quantitative information was obtained through conditional sampling with the forcing signal as the phase reference. Time traces of the three velocity components were recorded by hot-wire anemometry at a grid of locations on planes normal to the flow at four successive streamwise locations which include the roll-up and first pairing of the spanwise structures. From these measurements, three-dimensional distributions of ensemble-average vorticity components were computed and analysed.

The results show that the spanwise structures remain mostly two-dimensional from roll-up to the end of the first pairing, except for small oscillations in curvature of their axes. Concentrated streamwise vortices evolve at locations corresponding to small ‘kinks’ in the spanwise rollers and first appear at the beginning of the pairing process. Their subsequent development bears many similarities with the results of small-perturbation simulations. They expand laterally, forming counter-rotating rib vortices that grow almost exponentially in the braid region. In the core region, a complex but organized streamwise vorticity pattern emerges. The pattern consists of layered streamwise vortices whose orientation suggests that the oscillation of the spanwise roller core is π radians out of phase with that of the streamwise vortex lines that join the rib vortices. The peak streamwise vorticity and circulation in the core region increase until the completion of the spanwise structure pairing when the layered pattern collapses. Measurements at large spanwise locations conducted downstream show that the ‘kink’ is not the only location of streamwise vorticity generation: streamwise vortices develop at nearly periodic spanwise locations other than that of the initial ‘kink’. The initial spacing between streamwise vortices agrees with previous stability analysis. At the end of the spanwise structure pairing, isolated pairings between streamwise vortices take place, but a global doubling in spanwise wavelength does not occur.

Scorer (1950) has described an extension of the method of stationary phase. This method is used to derive an approximation to the wave pattern at large distances from the ship in a simple manner. The approximation is valid in the vicinity of the critical angle and the nature of the disturbance in this region is readily seen. The divergent and transverse wave systems in a critical region are shown, together with their variation of amplitude and phase.

In stably stratified and rotating fluids, an axisymmetric columnar vortex can be unstable to a special instability with an azimuthal wavenumber $m=1$ which bends and slices the vortex into pancake vortices (Gent & McWilliams Geophys. Astrophys. Fluid Dyn., vol. 35 (1–4), 1986, pp. 209–233). This bending instability, called the ‘Gent–McWilliams instability’ herein, is distinct from the shear, centrifugal or radiative instabilities. The goals of the paper are to better understand the origin and properties of this instability and to explain why it operates only in stratified rotating fluids. Both numerical and asymptotic stability analyses of several velocity profiles have been performed for wide ranges of Froude number $\mathit{Fr}_{h}={\it\Omega}_{0}/N$ and Rossby number $\mathit{Ro}=2{\it\Omega}_{0}/f$, where ${\it\Omega}_{0}$ is the angular velocity on the vortex axis, $N$ the Brunt–Väisälä frequency and $f$ the Coriolis parameter. Numerical analyses restricted to the centrifugally stable range show that the maximum growth rate of the Gent–McWilliams instability increases with $\mathit{Ro}$ and is independent of $\mathit{Fr}_{h}$ for $\mathit{Fr}_{h}\leqslant 1$. In contrast, when $\mathit{Fr}_{h}>1$, the maximum growth rate decreases dramatically with $\mathit{Fr}_{h}$. Long axial wavelength asymptotic analyses for isolated vortices prove that the Gent–McWilliams instability is due to the destabilization of the long-wavelength bending mode by a critical layer at the radius $r_{c}$ where the angular velocity ${\it\Omega}$ is equal to the frequency ${\it\omega}$: ${\it\Omega}(r_{c})={\it\omega}$. A necessary and sufficient instability condition valid for long wavelengths, finite Rossby number and $\mathit{Fr}_{h}\leqslant 1$ is that the derivative of the vertical vorticity of the basic vortex is positive at $r_{c}$: ${\it\zeta}^{\prime }(r_{c})>0$. Such a critical layer $r_{c}$ exists for finite Rossby and Froude numbers because the real part of the frequency of the long-wavelength bending mode is positive instead of being negative as in a homogeneous non-rotating fluid ($\mathit{Ro}=\mathit{Fr}_{h}=\infty$). When $\mathit{Fr}_{h}>1$, the instability condition ${\it\zeta}^{\prime }(r_{c})>0$ is necessary but not sufficient because the destabilizing effect of the critical layer $r_{c}$ is strongly reduced by a second stabilizing critical layer $r_{c2}$ existing at the radius where the angular velocity is equal to the Brunt–Väisälä frequency. For non-isolated vortices, numerical results show that only finite axial wavenumbers are unstable to the Gent–McWilliams instability.