Viscous fluid flows with curved streamlines can support both centrifugal and viscous travelling wave instabilities. Here the interaction of these instabilities in the context of the fully developed flow in a curved channel is discussed. The viscous (Tollmien–Schlichting) instability is described asymptotically at high Reynolds numbers and it is found that it can induce a Taylor–Görtler flow even at extremely small amplitudes. In this interaction, the Tollmien–Schlichting wave can drive a vortex state with wavelength either comparable with the channel width or the wavelength of lower-branch viscous modes. The nonlinear equations which describe these interactions are solved for nonlinear equilibrium states.

We consider the region of agitated grains that is at rest in the neighbourhood of its interface with a dense granular flow. We suppose that this region can be modelled as an amorphous solid of nearly elastic spheres in which both momentum and energy are transferred and energy is dissipated in collisions. Making rather rough assumptions about the collision probability, we calculate the stress and energy flux in the solid and use their continuity at the interface to obtain boundary conditions on the flow. We employ them with existing kinetic theory for nearly elastic spheres to solve boundary-value problems for shearing between two such interfaces and between such an interface and a flat plate to which spheres have been rigidly attached. For the latter, we compare the predictions of the theory with the results of experiments.

Explicit relations are derived for the dependence of the longshore wavenumber on the wave frequency of symmetric trapped waves or edge waves travelling near the cutoff frequency over submerged horizontally symmetric thin bodies or near-vertical cliffs. Results for particular geometries are presented and shown to agree with certain explicit solutions for edge waves over sloping beaches or trapped waves over a submerged narrow shelf, or a semicircular mound on the sea bed. Similar results are obtained for thin bodies extending vertically throughout the depth in open channels or for thin cross-sections in an acoustic wave guide.

Evaluation of spectral closure theory and direct numerical simulation are used to examine the eddy transport of a passive scalar in barotropic β-plane flow. When a large-scale gradient of scalar concentration is imposed, the implied scale separation between fixed background gradient and eddies supports the concept of ‘eddy diffusion’. The results can be cast in terms of an eddy diffusion tensor K, whose behaviour as a function of mean vorticity gradient β is examined. Earlier theoretical work by Holloway & Kristmannsson (1984) is extended to include cases where strong vorticity–scalar correlations are observed, and corrected in order to restore random Galilean invariance.

The anisotropy of eddy energy and the direct influence of Rossby wave propagation contribute to the overall anisotropy of K, The resulting suppression of meridional diffusivity Kyy, and enhancement of zonal diffusivity Kxx, with increased β is examined. The variation in simulation Kyy is closely reproduced in the closure equations. However, the increased Kxx is the result of zonal jets whose persistence is not accounted for in the statistical theory.

The effect of curvature on longitudinal dispersion in an axially uniform toroidal tube during oscillatory flow is investigated. The regimes of dispersion and the rate of longitudinal transport are estimated by order-of-magnitude arguments. Experiments are reported for the range, 0.66 < Dn2/α4 < 2.4, 5.4 < α < 26, Sc = 0.68, where Dn is the Dean number, α is the Womersley number and Sc is the Schmidt number. For β2 = α2Sc > 30, curvature causes a sharp increase in the effective diffusivity, relative to that for a straight tube, by a factor of about 6 at Dn2/α4 ≈ 2. The results from two numerical simulation methods are also presented. One, a Monte Carlo simulation (0.01 < Dn < 10, 0.01 < α < 0.32, Sc = 104), predicts the spread of a bolus in quasi-steady flow. The other, a spectral-element method (1 < Dn < 1000, 1 < α < 100, Sc = 0.68), is used to find the dispersion in unsteady oscillatory flow subjected to a constant longitudinal concentration gradient. Two mechanisms are identified by which axial transport is modified by curvature. First, the enhanced lateral transport due to secondary flow decreases axial transport by a factor of up to 5 for low β2 and increases axial transport by an even greater amount for high β2. Second, axial transport is enhanced owing to a form of resonance when the secondary flow circulation time is equal to the cycle period.

A theory is presented to describe the propagation and structure of coherent cold-core (mesoscale) eddies on a sloping bottom including dynamical and thermodynamical interaction with the surrounding fluid. Based on parameter values suggested by oceanographic and rotating-tank experimental data, the evolution of the baroclinic eddy is modelled with nonlinear ‘intermediate lengthscale’ geostrophic dynamics which is coupled to the surrounding fluid. The process of ventilation is modelled with a simple cross-interfacial mass flux parameterization. The surrounding fluid is governed by nonlinear quasi-geostrophic dynamics including eddy-induced vortex-tube compression. Assuming a relatively weak ventilation rate, a multiple-scale asymptotic theory is constructed to describe the propagation of an initially isolated or coherent baroclinic eddy. Throughout the evolution the eddy is assumed to be interacting strongly with the surrounding fluid. To leading order, the eddy and surrounding fluid satisfy the Stern isolation constraint. The magnitude of the Eulerian velocity field in the surrounding fluid above the eddy is shown to be larger than the swirl velocities in the eddy interior as suggested by experimental data. Also, to leading order, the along-shelf translation speed is given by the Nof formula. The process of ventilation is shown to induce a slowly decaying upslope translation in the propagating eddy, and acts to stimulate a weak slowly decaying topographic Rossby wave field in the surrounding fluid. The important features of the theory are illustrated with a simple example calculation.

The dynamics of a nonlinear modulated cross-wave of resonant frequency ω1 and carrier frequency ω ≈ ω1 is considered. The wave is excited in a long channel of width 6 that contains water of depth d, which is subjected to a vertical oscillation of frequency 2ω. As has been shown by Miles (1984b), the complex amplitude satisfies a cubic Schrödinger equation with weak damping and parametric driving. The stability of its solitary wave solution is considered here in various parameter regions. We find that in a certain regime the solitary wave is stable. Completely new is the result of instability outside this parameter regime. The instability has also been verified numerically. It is shown that the final stage of solitary wave instability is a cnoidal-wave-type solution.

Nonlinear transfer due to wave-wave interactions was first described by the Boltzmann integrals of Hasselmann (1961) and has been the subject of modelling ever since. We present an economical method to evaluate the complete integral, which uses selected scaling properties and symmetries of the nonlinear energy transfer integrals to construct the integration grid. An important aspect of this integration is the inherent smoothness and stability of the computed nonlinear energy transfer. Energy fluxes associated with the nonlinear energy transfers and their behaviour within the equilibrium range are investigated with respect to high-frequency power law, peak frequency, peakedness, spectral sharpness and angular spreading. We also compute the time evolution of the spectral energy and the nonlinear energy transfers in the absence of energy input by wind or dissipated by wave breaking. The response of nonlinear iterations to perturbations is given and a formulation of relaxation time in the equilibrium range is suggested in terms of total equilibrium range energy and the nonlinear energy fluxes within the equilibrium range.

The unsteady boundary layer of a rotating, stratified, viscous, and diffusive flow along an insulating slope is investigated using theory, numerical simulation, and laboratory experiment. Previous work in this field has focused either on steady flow, or flow over a conducting boundary, both of which yield Ekman-type solutions. After the onset of a circulation directed along constant-depth contours, Ekman-type flux up or down the slope is opposed by buoyancy forces. In the unsteady, insulating case, it is found that the cross-slope transport decreases in time as (t/τ)−½ where \[ \tau = \frac{1}{S^2f\cos\alpha}\left(\frac{1/\sigma + S}{1+S}\right), \] may be called the ‘shut-down’ time. Here S = (N sin α/f cos α)2, f is the Coriolis frequency, α is the slope angle, N is the buoyancy frequency, and σ is the Prandtl number. Subsequently the along-slope flow, $\hat{v}$, approximately obeys a simple diffusion equation \[ \frac{\partial\hat{v}}{\partial t} = \nu\left(\frac{1/\sigma + S}{1+S}\right)\frac{\partial^2\hat{v}}{\partial\hat{z}^2}, \] where t is time, ν is the kinematic viscosity, and $\hat{z}$ is the coordinate normal to the slope. By this process the boundary layer diffuses into the interior, unlike an Ekman layer, but at a rate that may be much slower than would occur with simple non-rotating momentum diffusion. The along-slope flow, $\hat{v}$, is nevertheless close to thermal wind balance, and the much-reduced cross-slope transport is balanced by stress on the boundary. For a slope of infinite extent the steady asymptotic state is the diffusively driven ‘boundary-mixing’ circulation of Thorpe (1987). By inhibiting the cross-slope transport, buoyancy virtually eliminated the boundary stress and hence the ‘ fast’ spin-up of classical theory in laboratory experiments with a bowl-shaped container of stratified, rotating fluid.