This paper continues an investigation of the effects of surface tension on steep water waves in deep water begun in Hogan (1979a). A Stokes-type expansion method is given which can be applied to most wavelengths. For capillary waves (2 cm or less) it is found that the surface of the highest wave encloses a bubble of air, as was found for pure capillary waves by Crapper (1957). For intermediate waves (20 cm) the wave profiles are similar to those of pure gravity waves and the wave properties increase monotonically. For gravity waves (200 cm) the wave properties all exhibit a maximum just short of the maximum wave height obtained by the method. The integral properties for all the waves are drawn and given in numerical form in the appendix.

Two models are discussed to account for the motion of the concentration interface in turbulent mixing layers. In the first one the interface is treated as a vortex sheet and its roll-up is studied. It is argued that this situation represents only the first stages of layer growth and another model is studied in detail in which a row of vortex cores entrains an essentially passive concentration interface with no vorticity. Both models give values of the spreading rate in approximate agreement with observations, and their relation is discussed.

The heat transfer and free convective motion, in inclined air layers heated from below, for angles of incidence 0 [les ] ϕ [les ] 30°, and Rayleigh numbers 100 < Ra cos ϕ < 10000, are studied experimentally. Results of both heat-transfer measurements and flow-visualization studies are reported. The purpose of the study was to investigate the fact, first noted by Hollands et al. (1976), that the experimental heat-transfer data, for ϕ > 20°, is not a function of the product Ra cos ϕ only, as expected from theoretical consideration. This discrepancy between theory and experiment is here attributed to a hypothesized secondary transition in the convective motion, due primarily to perturbation velocities in the upslope direction. This secondary transition appears to be the same as that predicted theoretically by Clever & Busse (1977); qualitative agreement with their theory is observed.

The heat transfer across an inclined air layer in the longitudinal roll regime often falls below expected values. A companion paper (Ruth et al. 1980) presents the heat transfer measurement and flow visualization observations; the present paper discusses the mechanisms which are thought to be responsible for this anomalous behaviour and formulates a simple model which correlates the results of our previous study and that of Hart (1971). The observed suppression of heat transfer below values expected for pure longitudinal roll motion is explained in terms of a shear instability. A stabilizing mechansim, which would restore the longitudinal rolls at higher Rayleigh numbers, is described; the return of the heat transfer to expected levels at higher Rayleigh numbers is ascribed to this mechanism. Comparisons are made between the results of the present model and the analyses of Clever & Busse (1977).

Compressibility is added to Busse's (1976) study of convection in a rotating, electrically conducting layer of fluid, of varying depth, which is permeated by an azimuthal magnetic field that is orthogonal to both the rotation vector Ω and the gravitational acceleration g. On the basis of a linear theory, we investigate the kinds of infinitesimal wave motion which the fluid can support when the dominant balance of forces is between pressure gradient and Coriolis force, so that the Proudman–Taylor theorem holds. Unlike Busse, however, we assume that the fluid is statically stable in the sense that the temperature gradient is sub-adiabatic.

In the absence of diffusion, these waves are dynamically neutral and take the form of Rossby waves modified by compressibility and magnetic field. The waves are examined in two limits, the adiabatic and the isothermal, and we define two distinct frequencies at which a pure Rossby wave can oscillate. When diffusion is restored, the disparity between these frequencies makes the fluid susceptible to overstability. We prove that all such amplifying waves must propagate eastward, i.e. in the direction of g ∧ Ω, irrespective of the sign of the depth gradient or the magnitudes of the diffusivities. The theorem does not apply if the fluid is incompressible or convectively unstable at the outset, and therefore does not contradict Busse's result that the direction of azimuthal propagation can be altered by the diffusivities. Nevertheless, we suggest reasons why Busse's method of regarding the imaginary part of the marginal stability equation as a dispersion relation is not in general a reliable one.

We examine the instabilities by subjecting the neutral waves to a weakly diffusive perturbation. We discover, in particular, a new kind of magnetic instability which is crucially dependent upon both compressibility and the depth gradient. In agreement with the general result described above, the instability takes the form of a slow, eastward propagating, amplifying wave.

The principal source for all instabilities is elastic energy, which cannot be tapped in a Boussinesq fluid, since the work done by compression is neglected under that approximation.

An approximate set of equations is derived for a compressible liquid of infinite Prandtl number. These are referred to as the anelastic-liquid equations. The approximation requires the product of absolute temperature and volume coefficient of thermal expansion to be small compared to one. A single parameter defined as the ratio of the depth of the convecting layer, d, to the temperature scale height of the liquid, HT, governs the importance of the non-Boussinesq effects of compressibility, viscous dissipation, variable adiabatic temperature gradients and non-hydrostatic pressure gradients. When d/HT [Lt ] 1 the Boussinesq equations result, but when d/HT is O(1) the non-Boussinesq terms become important. Using a time-dependent numerical model, the anelastic-liquid equations are solved in two dimensions and a systematic investigation of compressible convection is presented in which d/HT is varied from 0·1 to 1·5. Both marginal stability and finite-amplitude convection are studied. For d/HT [les ] 1·0 the effect of density variations is primarily geometric; descending parcels of liquid contract and ascending parcels expand, resulting in an increase in vorticity with depth. When d/HT > 1·0 the density stratification significantly stabilizes the lower regions of the marginal state solutions. At all values of d/HT [ges ] 0·25, an adiabatic temperature gradient proportional to temperature has a noticeable stabilizing effect on the lower regions. For d/HT [ges ] 0·5, marginal solutions are completely stabilized at the bottom of the layer and penetrative convection occurs for a finite range of supercritical Rayleigh numbers. In the finite-amplitude solutions adiabatic heating and cooling produces an isentropic central region. Viscous dissipation acts to redistribute buoyancy sources and intense frictional heating influences flow solutions locally in a time-dependent manner. The ratio of the total viscous heating in the convecting system, ϕ, to the heat flux across the upper surface, Fu, has an upper limit equal to d/HT. This limit is achieved at high Rayleigh numbers, when heating is entirely from below, and, for sufficiently large values of d/HT, Φ/Fu is greater than 1·00.

A global asymptotic solution for the linear stability of this flow with respect to axisymmetric disturbances is developed without invoking the usual quasi-parallel flow assumption. This solution predicts the occurrence of quasi-periodic spatially amplified disturbances relatively near the apex, which ultimately are stabilized further down the cone by the relative increase in viscous forces associated with the progressive thinning of the film. The wave speed and wavelength of these disturbances are found to decrease with increasing distance from the apex.

Steady two-dimensional potential flow past a semi-infinite flat-bottomed body is considered. This stern flow is assumed to separate tangentially from the body. Gravity waves of finite amplitude occur on the free surface. An exact relation between the amplitude of these waves and the Fronde number F is derived. It shows that these waves can exist only for F greater than the value F* = 2·23. This is slightly less than the value Fc = 2·26 at which breaking occurs. For F slightly larger than F*, the steepness is a multi-valued function of F, indicating the existence of more than one solution for these values of F. In addition, a numerical scheme based on an integro-differential equation formulation is derived to solve the problem in the fully nonlinear case. The shape of the free surface profile is computed for different values of F. As a check on the numerical results, they are shown to satisfy the exact relation between steepness and the Froude number.

A small region (called the surface tension region) where pressure differences across the oil–water and oil–air interfaces are important is shown to exist between the gravity viscous and monolayer regions in a spreading oil slick (Di Pietro, Huh & Cox 1978). The importance of this new region is that (i) it is necessary in order to connect the gravity–viscous and monolayer regions and (ii) it is a region where slopes of interfaces are large. This idea is used to find the thickness profile of an oil layer contained upstream of a barrier (an oil boom) placed across a channel in which water is flowing at a constant velocity. The assumption is also made that the velocity difference across the oil layer is small compared with the water velocity. The general conditions for the validity of the results are then discussed together with the modifications to the theory which are necessary if the boundary layer in the water below the oil should be turbulent rather than laminar. Good agreement is found to exist between experimental results for unsteady spreading on quiescent water in a channel and the results of the theory applied to this situation assuming quasi-steady spreading.