We present a study of nonlinear gravity–capillary waves with surface forcing and viscous dissipation. Based on a viscous boundary layer approximation near the water surface, the theory permits the efficient calculation of steady gravity–capillary waves with parasitic capillary ripples. To balance the viscous dissipation and thus achieve steady solutions, wind forcing is applied by adding a surface pressure distribution. For a given wavelength the properties of the solutions depend upon two independent parameters: the amplitude of the dominant wave and the amplitude of the pressure forcing. We find two main classes of waves for relatively weak forcing: Class 1 and Class 2. (A third class of solution requires strong forcing and is qualitatively different.) For Class 1 waves the maximum surface pressure occurs near the wave trough, while for Class 2 it is near the crest. The Class 1 waves are associated with Miles' (1957, 1959) mechanism of wind-wave generation, while the Class 2 waves may be related to instabilities of the subsurface shear current. For both classes of waves, steady solutions are possible only for forcing amplitudes greater than a certain threshold. We demonstrate how parasitic capillary ripples affect the dissipative and dispersive properties of the solutions. In particular, there may be a significant deviation from the linear phase speed for gravity–capillary waves. Also, wave damping is strongly enhanced by the parasitic capillaries (by as much as two orders of magnitude when compared to the case with no capillary waves). Preliminary experimental results from a wind-wave channel give good agreement with the theory. We find a sharp cut-off in the wavenumber spectra of the solutions which is similar to that observed in laboratory measurements of short gravity–capillary waves. Finally, for large wave amplitudes we find a sharp corner in the wave profile which may separate an overhanging wave crest from a train of parasitic capillaries.

Previous Space-based experiments (Wang et al. 1994a) showed that a rotating liquid drop bifurcates into a two-lobed shape at a lower critical angular velocity, if it is flattened acoustically by the leviating sound field. In this work, we undertake a systematic experimental study of the effect of acoustic flattening on the rotational bifurcation of a liquid drop. We also look into the complementary effect of rotation on the equilibrium of an acoustically drastically flattened drop. Theoretical models are developed for each of the two effects and then woven into a unified picture. The first effect concerns neutral equilibrium, while the second concerns loss of equilibrium, neither of them involving instability. The theories agree well with the experiments.

The dynamics of a vortex ring moving orthogonally to the rotation vector of a uniformly rotating fluid is analysed by laboratory experiments and numerical simulations. In the rotating system the vortex ring describes a curved trajectory, turning in the opposite sense to the system's anti-clockwise rotation. This behaviour has been explained by using the analogy with the motion of a sphere in a rotating fluid for which Proudman (1916) computed the forces acting on the body surface. Measurements have revealed that the angular velocity of the vortex ring in its curved trajectory is opposite to the background rotation rate, so that the vortex has a fixed orientation in an inertial frame of reference and that the curvature increases proportionally to the rotation rate.

The dynamics of the vorticity of the vortex ring is affected by the background rotation in such a way that the part of the vortex core in clockwise rotation shrinks while the anti-clockwise-rotating core part widens. By this opposite forcing on either side of the vortex core Kelvin waves are excited, travelling along the toroidal axis of the vortex ring, with a net mass flow which is responsible for the accumulation of passive scalars on the anti-clockwise-rotating core part. In addition, the curved motion of the vortex ring modifies its self-induced strain field, resulting in stripping of vorticity filaments at the front of the vortex ring from the anti-clockwise-rotating core part and at the rear from the core part in clockwise rotation. Vortex lines, being deflected by the main vortex ring due to induction of relative vorticity, are stretched by the local straining field and form a horizontally extending vortex pair behind the vortex ring. This vortex pair propagates by its self-induced motion towards the clockwise-rotating side of the vortex ring and thus contributes to the deformation of the ring core. The deflection of vortex lines from the main vortex ring persists during the whole motion and is responsible for the gradual erosion of the coherent toroidal structure of the initial vortex ring.

Penetrative turbulent convection from a localized circular top source into a rotating, linearly stratified ambient fluid of strength N has been investigated in a laboratory tank. Initially, the induced three-dimensional convective flow penetrated rapidly into the stratified water column until it reached an equilibrium depth at which the convective flow began to propagate radially outward. At this stage, the usual cyclonic vortices were generated around the convection source at the edge of the radially propagating flow. Soon after, a thin ‘subsurface anticyclone’ was formed at the level of equilibrium depth beneath the convection source. Later, this anticyclone dominated the central part of the convective regime and did not allow new cyclones to be injected into the system. After reaching its maximum mean diameter Da/R ≈10(R0;R)2/3 and swirl velocity va ≈(B0R)1/3, an anticyclone became unstable and split into two new vortices that left the area beneath the source, allowing a new anticyclone to form at its original place (here, R0,R =(B0/f3R2) 1/2 is the Rossby number based on R the radius of the source, B0 is the surface negative buoyancy flux, and f is the Coriolis parameter). These observations provide crucial evidence that many of the ‘subsurface anticyclonic’ vortices detected in the stratified pycnocline of the central Arctic Ocean are indeed generated as a result of convective processes occurring in this region.

Twin jet plumes on aircraft can couple, producing dynamic pressures significant enough to cause structural fatigue. For closely spaced jets with a moderate aspect ratio (e.g. 5), previous work has established that two coupling modes (antisymmetric and symmetric) are kinematically permissible. However, the dynamics of twin-jet coupling have remained unexplored. In this paper a more fundamental assessment of the steady and unsteady aspects of twin-jet coupling is attempted. While we document and discuss the nozzle spacings and Mach numbers over which phase-locked coupling occurs, our concentration is much more on answering the following questions: (a) What mechanism causes the jets to couple in one mode or the other? (b) Why do the jets switch from one mode to another? (c) Are the two modes mutually exclusive or do they overlap at the transition point? Our results reveal, among many things, the following. (i) For very closely spaced twin jets in the side-by-side configuration phased feedback based on source to nozzle exit distance of adjacent jets does not fully explain the coupling modes. However, the ‘null’ phase regions surrounding the jets where the phase of an acoustic wavefront (arriving from downstream) does not vary appears to correlate well with the existence of the symmetric mode. When the ‘null’ regions of adjacent jets do not overlap antisymmetric coupling occurs and when they do overlap the jets couple symmetrically. We provide a simple correlation using a parameter (α) that can be used as a simple test to determine the mode of coupling. (ii) The switch from the antisymmetric to the symmetric mode of coupling appears to occur because of an abrupt shift in the effective screech source from the third to the fourth shock, which in turn causes the ‘null’ phase region surrounding the jets to grow abruptly and overlap. (iii) The two modes are mutually exclusive. Our results provide considerable insight into the twin-jet coupling problem and offer hope for designing twin-jet configurations that minimize damage to aircraft components.

The steady streaming flow due to vibration in capillary bridges is considered in the limiting case when both the capillary Reynolds number and the non-dimensional vibration frequency (based on the capillary time) are large. An asymptotic model is obtained that provides the streaming flow in the bulk, outside the thin oscillatory boundary layers near the disks and the interface. Numerical integration of this model shows that several symmetric and non-symmetric streaming flow patterns are obtained for varying values of the vibration parameters. As a by-product, the quantitative response of the liquid bridge to high-frequency axial vibrations of the disks is also obtained.

Direct numerical simulation is used to examine turbulent mixing in a shear-free stably stratified fluid. Energy is continuously supplied to a small region to maintain a well-developed kinetic energy profile, as in an oscillating grid flow (Briggs et al. 1996; Hopfinger & Toly 1976; Nokes 1988). A microscale Reynolds number of 60 is maintained in the source region. The turbulence forms a well-mixed layer which diffuses from the source into the quiescent fluid below. Turbulence transport at the interface causes the mixed layer to grow under weakly stratified conditions. When the stratification is strong, large-scale turbulent transport is inactive and pressure transport becomes the principal mechanism for the growth of the turbulence layer. Down-gradient buoyancy flux is present in the large scales; however, far from the source, weak counter-gradient fluxes appear in the medium to small scales. The production of internal waves and counter-gradient fluxes rapidly reduces the mixing when the turbulent Froude number is lower than unity. When the stratification is weak, the turbulence is strong enough to break up the density interface and transport fluid parcels of different density over large vertical distances. As the stratification intensifies, turbulent eddies flatten against the interface creating anisotropy and internal waves. The dominant entrainment mechanism is then scouring. Mixing efficiency, defined as the ratio of buoyancy flux to available kinetic energy, exhibits a similar dependence on Froude number to other stratified flows (Holt et al. 1992; Lienhard & Van Atta 1990). However, using the anisotropy of the turbulence to define an alternative mixing efficiency and Froude number improves the correlation and allows local scaling.

The work is concerned with long nonlinear internal waves interacting with a shear flow localized near the sea surface. The study is focused on the most intense resonant interaction occurring when the phase velocity of internal waves matches the flow velocity at the surface. The perturbations of the shear flow are considered as ‘vorticity waves’, which enables us to treat the wave–flow resonance as the resonant wave–wave interaction between an internal gravity mode and the vorticity mode. Within the weakly nonlinear long-wave approximation a system of evolution equations governing the nonlinear dynamics of the waves in resonance is derived and an asymptotic solution to the basic equations is constructed. At resonance the nonlinearity of the internal wave dynamics is due to the interaction with the vorticity mode, while the wave's own nonlinearity proves to be negligible. The equations derived are found to possess solitary wave solutions of different polarities propagating slightly faster or slower than the surface velocity of the shear flow. The amplitudes of the ‘fast’ solitary waves are limited from above; the crest of the limiting wave forms a sharp corner. The solitary waves of amplitude smaller than a certain threshold are shown to be stable; ‘subcritical’ localized pulses tend to such solutions. The localized pulses of amplitude exceeding this threshold form infinite slopes in finite time, which indicates wave breaking.

Results from direct numerical simulations of an unsteady turbulent shear layer with a free surface are presented. The emphasis is on the interaction dynamics of the free surface with the coherent vortices in the underlying turbulent shear flow as well as the resulting free-surface signatures. Instantaneous vortex lines and isosurfaces of enstrophy indicate that coherent horseshoe vortical structures emerge from the random initial vorticity field. These horseshoe vortices impinge, break and reconnect onto the free surface, and then appear as two vortex connections with opposite signs on the surface. The two identified vortical structures correspond to ‘splatting’ and ‘swirling’ events, which have been observed in other experiments and simulations of free-surface/turbulence flows. Though free-surface depressions form near the vertical-vorticity centres in the connection processes, only a low correlation (≈50% to 60%) is found between the free-surface roughness (vertical deformation) and the connected normal vorticity. On the other hand, the free-surface curvatures and the tangential free-surface vorticities are better correlated (≈80% to 90%). The balance of enstrophy and the vorticity transport show that stretching and viscous dissipation along the direction of the vorticity vector dominate the vortex dynamics near the free surface. These two transport mechanisms are found to be responsible for the cancellation of the spanwise vorticity of the horseshoe-vortex heads and the annihilation of the surface-connected normal vorticities.

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.

Idealized model examples of non-dissipative wave–mean interactions, using small-amplitude and slow-modulation approximations, are studied in order to re-examine the usual assumption that the only important interactions are dissipative. The results clarify and extend the body of wave–mean interaction theory on which our present understanding of, for instance, the global-scale atmospheric circulation depends (e.g. Holton et al. 1995). The waves considered are either gravity or inertia–gravity waves. The mean flows need not be zonally symmetric, but are approximately ‘balanced’ in a sense that non-trivially generalizes the standard concepts of geostrophic or higher-order balance at low Froude and/or Rossby number. Among the examples studied are cases in which irreversible mean-flow changes, capable of persisting after the gravity waves have propagated out of the domain of interest, take place without any need for wave dissipation. The irreversible mean-flow changes can be substantial in certain circumstances, such as Rossby-wave resonance, in which potential-vorticity contours are advected cumulatively. The examples studied in detail use shallow-water systems, but also provide a basis for generalizations to more realistic, stratified flow models. Independent checks on the analytical shallow-water results are obtained by using a different method based on particle-following averages in the sense of ‘generalized Lagrangian-mean theory’, and by verifying the theoretical predictions with nonlinear numerical simulations. The Lagrangian-mean method is seen to generalize easily to the three-dimensional stratified Boussinesq model, and to allow a partial generalization of the results to finite amplitude. This includes a finite-amplitude mean potential-vorticity theorem with a larger range of validity than had been hitherto recognized.

Inviscid equilibrium mean flows over topography are considered for continuously stratified quasi-geostrophic models, in contrast to previous work which has dealt with two-layer models. From the constraint of maximum entropy, an equation for the equilibrium mean flow is derived. Analytical solutions are obtained for uniform and piecewise-constant stratifications. With increasing stratification, the mean streamfunction becomes increasingly bottom intensified. Bottom trapping becomes ever more pronounced on smaller scales, but can remain significant even on the largest scales. When boundary temperature is uniform, transport is shown to be independent of stratification, other factors being equal. Although two-layer models share this property, they represent poorly the energetics of the continuous system when bottom trapping is significant.

The similarity solution describing the motion of converging spherical and cylindrical shocks is governed by a set of three ordinary differential equations. Previous descriptions of the shock motion have been based on numerical solutions of these differential equations. In the present paper a study of the singular points of the differential equations leads to an analytic description of the flow and a determination of the similarity exponent which is in excellent agreement with the earlier numerical values. Limiting values of the ratio of specific heats are considered. It is shown that as the ratio tends to unity the shock becomes ‘freely propagating’ and the first terms in a power series for the similarity exponent are obtained. Large values of the ratio of specific heats are briefly considered and provide a further check on the analytic description of this paper. Finally in the Appendix the condition for the pressure to have a maximum is clarified and the location of the maximum provides further strong evidence of the high accuracy of the analytic approach of this paper.