A theoretical wave-action model, based on the work of Grimshaw (1974), is presented which describes broadbanded internal wave motions in a stratified shear flow with the viscous terms included. The model shows that in the neighbourhood of the critical level viscosity can have important effects and can stabilize the flow with respect to convective overturning. The predictions of the model are compared with the results of an experimental study where detailed measurements of the velocity and density fields are made within an internal wave packet propagating through a continuously stratified shear flow. The results show that the model accurately predicts the occurrence or non-occurrence of wave overturning due to interactions with the shear, and provides an accurate description of the structure of the wave-induced density and velocity fluctuations in those regions of the flow where the assumptions of the model are satisfied.

The formation and destruction of a gravity current in a turbulent fluid is examined in laboratory experiments. The gravity current is produced by lock exchange and the fluid is kept turbulent by bubbling air from the base of the tank. When the lock is released the buoyancy forces associated with the reduced gravity g′ between the fluid on the two sides of the lock drives a counterflow, with the dense fluid slumping underneath the less-dense fluid, and a gravity current is formed. The current has a sharp density front at its leading edge, and a stable density stratification is established behind the front. The turbulence, characterized by a longitudinal turbulent diffusion coefficient K, tends to mix this stable stratification. Once the fluid is vertically mixed the gravity current front is destroyed, and the density varies smoothly with horizontal distance over a zone whose length increases with time owing to the continuing longitudinal turbulent diffusion and buoyancy driving. It is found that the gravity current propagates over a distance L1 before it is destroyed, where L1/H ≈ 0.08(g′H)½H/K, and H is the fluid depth. At this point turbulent dissipation balances the buoyancy driving and frontogenesis is inhibited. The turbulent dispersion coefficient is found to increase with the buoyancy driving with K∞ Ri½, where Ri = g′H/q2 and q is the r.m.s. turbulence velocity fluctuations. It is also shown that when the turbulence level is reduced nonlinearities in the horizontal density gradient can sharpen up to form a front. The implications of these frontogenetical processes to the sea-breeze front and fronts in shallow seas is discussed.

The Lagrangian L for gravity waves of small but finite amplitude in an N-layer stratified fluid is constructed as a function of the generalized coordinates qv(t) ≡ {qvn(t)} of the N + 1 interfaces, where the qvn are the Fourier coefficients of the expansion of the interfacial displacement ηv(x, t) in a complete, orthogonal set {ψn(x)}. The density is constant in each layer, by virtue of which a velocity potential exists for that layer (even though the full flow is rotational). The explicit expansion of L is constructed through fourth-order in qν and $\dot{\boldmath q}_{\nu}$ through an extension of the surface-wave formulation (Miles 1976), in which the pressure appears as the Lagrangian density (Luke 1967). Three-dimensional progressive and standing interfacial waves in a two-layer fluid are treated as general examples, and the two-dimensional results of Hunt (1961) and Thorpe (1968) are recovered as explicit examples. It is shown that the spatial resonance between surface and internal waves conjectured by Mahony & Smith (1972) is impossible for the two-layer Boussinesq model.

The joint limit N ↑ ∞ and layer thickness ↓ 0 yields the Lagrangian density L for a continuously stratified, Boussinesq fluid as a functional of qn([yscr ]) and $\dot{q}_n$([yscr ]), where [yscr ], the counterpart of the layer index, is a Lagrangian (rather than Eulerian) coordinate. The coefficient C in the nonlinear dispersion relation (ω/ω1)2 = 1 + Ck2A2 for progressive waves of frequency ω, wavenumber k and amplitude A, where ω1 = ω1(k) for infinitesimal waves, is determined for any density profile for which the (linear) vertical structure problem can be solved. Explicit results are given for a fluid of finite vertical extent in which the buoyancy frequency is constant and for a vertically unbounded fluid in which the buoyancy frequency varies like sech ([yscr ]/h) and C = C(kh).

The Lagrangian L for gravity waves of finite amplitude in an N-layer, stratified shear flow is constructed as a functional of the generalized coordinates qv(t)≡ {qnv(t)} of the N + 1 interfaces, where the qnv are the Fourier coefficients in the expansion of the interfacial displacement ηv(x, t) in a complete, orthogonal set {ψn(x)}. The explicit expansion of L is constructed through fourth-order in the qnν and $\dot{q}_{n}^{\nu}$. Progressive interfacial waves and Kelvin–Helmholtz instability in a two-layer fluid are examined, and the earlier results of Drazin (1970), Nayfeh & Saric (1972) and Weissman (1979) are extended to finite depth. It is found that the pitchfork bifurcation associated with the critical point for Kelvin–Helmholtz instability, which is supercritical for infinitely deep layers, may be subcritical (inverted) for finite depths. The evolution equations that govern Kelvin–Helmholtz waves in the parametric neighbourhood of this critical point are shown to be equivalent to those for a particle in a two-parameter, central force field. The effect of surface tension is examined in an Appendix. Finally, the wave motion forced by flow over a sinusoidal bottom (as in Thorpe's tilting tank) is examined and the corresponding resonance curves and Hopf bifurcations determined. Numerical integrations reveal that stable limit cycles exist in some parametric neighbourhoods of these bifurcations. Period doubling was observed but did not lead to chaotic motion.

In this paper an attempt is made to explain the period-doubling of wind-generated gravity-capillary waves as observed in the experiment of Choi (1977). It is conjectured that period-doubling is closely related to the phenomenon of second-harmonic resonance. In order to obtain a simple dynamical model, results of McGoldrick (1970) and Simmons (1969) are extended to include the effect of wind input and shear in the current. For pure gravity–capillary waves (no wind, no current) the condition for energy transfer from the second harmonic to the fundamental wave of Chen & Saffman (1979) is recovered. We also discuss the effect of wind and we find that wind input gives rise to a very sudden period-doubling. Qualitative agreement with experiment is obtained.

We derive exact criteria for the highest capillary waves on sheets of fluid and present the profiles for both symmetric and antisymmetric waves. In addition we provide correct definitions of the phase speed of these waves. Previous ‘essentially nonlinear’ solutions to the problem are shown to have been misinterpreted.

Trajectory properties of particles in both types of wave are presented. We also give results for the drift velocity ratios as a function of the mean displacement of a streamline from the free surface. Many of the results are exact, being given in terms of elliptic functions and elliptic integrals. An application to wave motion in thin films on pipe walls is considered.