The generation and growth of small water waves by a turbulent wind has been investigated in a laboratory channel. The evolution of these oscillations with fetch was traced from their inception with amplitudes in the micron range under conditions of steady air flow. The experiments revealed that the waves are generated at all air velocities in small bursts consisting of groups of waves of nearly constant frequency. After travelling for some distance downstream, these wavelets attain sufficient amplitude to become visible. For this condition, a wind speed critical to raise waves is well defined. After the first wavelets appear, two new stages of growth are identified at longer fetches if the air speed remains unchanged. In the first of these, the wave component associated with the spectral peak grows faster with fetch than any other part of the wave spectrum of the initial waves until its amplitude attains an upper limit consistent with Phillips's equilibrium range, which appears to be universal for wind waves on any body of water. If the air flow is not changed, then the frequency of this dominant wave remains constant with fetch up to equilibrium. This frequency tends to decrease, however, with increasing wind shear on the water. In the second stage of growth, only the energy of wave components with spectral densities lower than the equilibrium limit tend to increase with fetch so that the wave spectrum is maintained near equilibrium in the high-frequency range of the spectrum.

The origin of the first waves and the rate of their subsequent growth was examined in the light of possible generating mechanisms. There was no indication that they were produced by direct interaction of the water surface with the air turbulence. Neither could any significant feedback of the waves into the turbulence structure be detected. The growth of the waves was found to be in better agreement with theoretical predictions. Under the shearing action of the wind, the first waves were found to grow exponentially. The growth rates agreed with the estimates from the viscous shearing mechanism of Miles (1962a) to a fractional error of 61% or less. A slight improvement was obtained with the viscous theory of Drake (1967) in which Miles’ model is extended to include the effect of the drift current induced by the wind in the water. Since the magnitude of the water currents observed in the tunnel is very small, this improvement is not significant.

We first consider a steady laminar model of salt fingers and show that the latter become unstable with respect to internal gravity waves when the finger Reynolds number exceeds a critical value. The criterion is then used in speculations about the statistically steady state in a fully developed similarity model where horizontally averaged temperature and salinity gradients are constant at all depths. Dimensional reasoning is used to obtain the asymptotic dependence of the turbulent flux on the molecular salt diffusivity. From this and other relationships order-of-magnitude estimates are obtained and compared with laboratory experiments and ocean observations.

Experimental measurements were made of the variation of water-surface height across the crests of solitary waves travelling along trapezoidal channels. The channels had one vertical side and the other side sloped at either 30°, 45° or 60° to the vertical. Reliable results were not obtained with the last channel, but for the other channels there was reasonable agreement with recent theoretical results.

The evolution of a model of the flow in a layer of fluid suddenly heated from below at a Rayleigh number sufficient for a laboratory flow to ultimately become turbulent is investigated with numerical experiments. The numerical study simulates the flow by means of the mean field equations along the lines of Herring's (1963, 1964) pioneering study but uses a different finite difference technique and concentrates attention on the flow development rather than on the final statistically steady state. These solutions are compared with previous and some new simulations in two dimensions. The solutions confirm Herring's work and in addition show that the mean field equations, and in particular the weak-coupling approximations, describe the gross features of the model sufficiently well for the mean field equations to be used with reasonable confidence in evolutionary studies.

The boundary-layer flow over a semi-infinite vertical flat plate, heated to a constant temperature in a uniform free stream, is discussed in the two cases when the buoyancy forces aid and oppose the development of the boundary layer. In the former case, two series solutions are obtained, one of which is valid near the leading edge and the other is valid asymptotically. An accurate numerical method is used to describe the flow in the region where the series are not valid. In the latter case, a series, valid near the leading edge is obtained and it is extended by a numerical method to the point where the boundary layer is shown to separate.

The primary objective of this work was to study the response of gravity water waves to wind excitation and, in particular, the applicability of the Miles inviscid shear-flow theory of gravity wave growth, by conducting experiments in a laboratory wind-wave channel under conditions approximating the assumptions of the mathematical model. Mechanically generated wave profiles subjected to wind action were measured with capacitance wire sensors and wave energy was calculated at seven stations spaced at 10ft. intervals along the channel test section. Waves varied in length from about 2·5 to 6·5 ft. and maximum wind speeds ranged from 12 to 44 ft./sec. Vertical mean air velocity profiles were taken at six stations in the channel, fitted near the air-water interface with semi-logarithmic profiles, and used in a stepwise computation of theoretical wave growth. The results show that the measured wave energy growth is exponential but considerably larger than the growth predictions of Miles's theory. Derived experimental values of the phase-shifted pressure component β are greater than theoretical values by a factor varying from 1 to 10, with a mean of about 3. Wind mean velocity profiles appear to be closely logarithmic near the air-water interface. Wind-generated ripples superposed on mechanically generated waves created a rough water surface with standard deviation larger, in all cases, than the respective critical-layer thickness.

Laboratory experiments of non-steady penetrative convection in water are performed that closely simulate the lifting of an atmospheric inversion above heated ground. Vertical profiles of horizontally averaged temperature and heat flux are measured and interpreted. The rate at which kinetic energy is destroyed by the downward heat flux in the vicinity of the inversion base is found to be a very small fraction of the rate at which it is generated in the lower convective region. The interface separating the convective region from the stable region is examined and its rate of rise explained.

Experiments are presented in which axisymmetric internal waves are generated by an oscillating sphere moving vertically in a stably stratified salt solution. The Reynolds numbers for the sphere based on the diameter and the mean velocity are between 10 and 200. Lighthill's theory for dispersive waves is used to calculate the phase configuration of the internal waves. The agreement between experiment and theory is reasonably good.

Experiments are described in which the stability of a liquid stabilized by a gradient of concentration of a solute has been investigated. In all the cases observed, convection began at marginal stability with an oscillatory motion. The boundary of stability in the (R, Rs)-plane departs somewhat from that predicted by the simplest theories; the boundary is a curve, concave to the origin, and the critical Rayleigh number is high. The total rate of transport of solute increases much more rapidly than that of heat when the oscillations begin. The motions begin in cells which are separate from one another.

Experiments on reverse transition were conducted in two-dimensional accelerated incompressible turbulent boundary layers. Mean velocity profiles, longitudinal velocity fluctuations $\tilde{u}^{\prime}(=(\overline{u^{\prime 2}})^{\frac{1}{2}})$ and the wall-shearing stress (TW) were measured. The mean velocity profiles show that the wall region adjusts itself to laminar conditions earlier than the outer region. During the reverse transition process, increases in the shape parameter (H) are accompanied by a decrease in the skin friction coefficient (Cf). Profiles of turbulent intensity (u’2) exhibit near similarity in the turbulence decay region. The breakdown of the law of the wall is characterized by the parameter \[ \Delta_p (=\nu[dP/dx]/\rho U^{*3}) = - 0.02, \] where U* is the friction velocity. Downstream of this region the decay of $\tilde{u}^{\prime}$ fluctuations occurred when the momentum thickness Reynolds number (R) decreased roughly below 400.

Expressions are derived for a three-dimensional polar in the pressure ratio, deflexion angle space to represent conditions downstream of a detonation wave with given upstream conditions. An analysis of reflected waves is undertaken and the representation of three wave confluences in a three-dimensional hodograph space studied. These techniques are applied to experimental results available for spin detonation in gaseous mixtures of acetylene-oxygen-argon and carbon monoxide-oxygen. Assumptions made in earlier papers concerning local extinction of combustion at the confluence point are not necessary in this picture.

Thermally induced convection in a rotating cylinder of fluid heated from above and strongly influenced by centrifugal accelerations is treated using boundary-layer methods. As in the theory of homogeneous rotating fluids, the horizontal Ekman layers control the inviscid axial flow. The solution also largely depends upon the thermal conditions assumed at the side wall, and if these be insulated, consideration of the side-wall boundary layers is necessary for complete specification of the problem. For perfectly conducting side walls, the side layers do not influence the zeroth-order flow, but contribute a second-order correction, which would be absent if the lateral boundaries were ignored. The critical parameters governing the solutions in both cases are found to be γ and the group σβε−½, where γ is the aspect ratio, σ the Prandtl number, ε the Ekman number, and β the thermal Rossby number for the flow. Boundary-layer solutions are given for a wide range of parameters, and gravity is seen to have at most only a local effect on the flow near the side walls.

The stability of a horizontal layer of fluid heated from below is examined when, in addition to a steady temperature difference between the walls of the layer, a time-dependent sinusoidal perturbation is applied to the wall temperatures. Only infinitesimal disturbances are considered. The effects of the oscillating temperature field are treated by a perturbation expansion in powers of the amplitude of the applied field. The shift in the critical Rayleigh number is calculated as a function of frequency, and it is found that it is possible to advance or delay the onset of convection by time modulation of the wall temperatures.

A uniform electric arc column is held at rest against an imposed low-speed flow perpendicular to its length by an applied magnetic field transverse to both the arc and the flow. The situation is represented mathematically by two regions separated by an isothermal boundary, the arc periphery, across which certain gas properties change discontinuously. It is assumed that the arc has low power gradient so that the Nusselt number is small compared with unity. The Reynolds number is then small also and the methods of the theory of flow at low Reynolds number are used to obtain solutions for the temperature, magnetic field, velocity and pressure inside and outside the arc. It is found that the periphery of the arc is a circle and its radius is determined by heat transfer. The flow near the periphery, and the drag of the arc, are found to depend on a final boundary condition at the periphery, the form of which is not yet clear. Several examples of possible flow patterns are given, and it is shown that the arc may be likened to a slippery porous body for which the slipperiness and porousness are governed by the final boundary condition. The electric and magnetic characteristics of the arc are derived and shown to be amenable to examination by experiment and to empirical extension for arcs of higher power gradient.

The stratified shear flow over a two-dimensional obstacle of semi-elliptical crosssection is considered. The shear flow is assumed to be inviscid with constant upstream values of the density gradient and dynamic pressure (Long's model). Two complete sets of lee-wave functions, each of which satisfies the condition of no upstream reflexion, are determined in elliptic co-ordinates for ε ≤ 1 and ε ≥ 1, where ε is the ratio of height to half-width of the obstacle. These functions are used to determine the lee-wave field produced by, and the consequent drag on, a semi-elliptical obstacle as functions of ε and the reduced frequency (reciprocal Froude number) within the range of stable flow. The reduced frequency at which static instability first occurs is calculated as a function of ε.

The problem considered is that of a strong shock propagating from a point energy source into a cold atmosphere whose density varies exponentially with altitude. An explicit analytic solution is obtained by taking the flow field as ‘locally radial’ and using an integral method with an energy constraint. A scaling law is given which eliminates the parametric dependence of the solution on the explosion energy, scale height, and atmospheric density at the point of the explosion. The scaling law also transforms the infinity of solutions for various polar angles into two distinct solutions which show that all motions of the ascending portion of the shock may be scaled from the vertically upward behaviour and all motions of the descending portion of the shock may be scaled from the vertically downward behaviour. The limit in the lateral direction of both of the fundamental solutions corresponds to the case of the uniform density atmosphere. The results for the uniform density atmosphere show remarkable agreement with the exact Taylor—Sedov results. Comparison with finite difference calculations of Troutman & Davis for the vertically upward and downward directions shows excellent agreement with respect to the prediction of shock propagation velocity, position, and the flow variables behind the shock. A scaling law for the time, shock velocity, and pressure for different values of the adiabatic exponent γ is proposed which correlates the results of the present analysis for different values of γ over the entire range of shock positions where the analysis applies. The solution shows that, contrary to the result obtained by Kompaneets, there is no theoretical limit as to how far downward a strong shock may propagate. The far field behaviour of the shock wave in the upward and downward directions is found to be of the same form as the self-similar asymptotic solutions obtained by Raizer for a plane shock. It is shown by relaxing the energy constraint in the vertically downward direction that the asymptotic result obtained agrees closely with that obtained by Raizer. The energy constraint, however, is the appropriate one for all but the far field behaviour. The far field limit of the present solution in the upward direction is found to compare favourably with the approximate asymptotic calculations of Hayes for an ascending curved shock. The empirical concept of ‘modified Sachs scaling’ for calculating the overpressure is considered and shown within this model to have a justification in the downward direction but a limited range of applicability in the upward direction.

General qualitative rules are derived for the behaviour of the volume of an ascending spherical bubble and of the gas pressure within it. Three modes of behaviour are discerned, corresponding to as many possible orderings of the relative influences of ascent velocity, gas leakage and surface tension on the volume and the pressure balance. These general results are nearly independent of the particular forms of the ascent velocity and gas exchange functions. Quantitative results are presented for the Stokes law régime.

A two-dimensional stratified flow over an obstacle in a half space is considered on the assumptions that the upstream dynamic pressure and density gradient are constant (Long's model). A general solution of the resulting boundary-value problem is established in terms of an assumed distribution of dipole sources. Asymptotic solutions for prescribed bodies are established for limiting values of the slenderness ratio ε (height/breadth) of the obstacle and the reduced frequency k (inverse Froude number based on the obstacle breadth) as follows: (i) ε → 0 with k fixed; (ii) k → 0 with ε fixed; (iii) k → ∞ with kε fixed. The approximation (i) is deveoped to both first (linearized theory) and second order in ε in terms of Fourier integrals. The approximation (ii), which constitutes a modification of Rayleigh-scattering theory, is obtained by the method of matched asymptotic expansions and depends essentially on the dipole form (which is proportional to the sum of the displaced and virtual masses) of the obstacle with respect to a uniform flow. A simple approximation to this dipole form is proposed and validated by a series of examples in an appendix. The approximation (iii) is obtained through the reduction of the original integral equation to a singular integral equation of Hilbert's type that is solved by the techniques of function theory. A composite approximation to the lee-wave field that is valid in each of the limits (i)-(iii) also is obtained. The approximation (iii) yields an estimate of the maximum value of kε for which completely stable lee-wave formation behind a slender obstacle is possible. The differential and total scattering cross-sections and the wave drag on the obstacle are related to the power spectrum of the dipole density. It is shown that the drag is invariant under a reversal of the flow in the limits (i) and (ii), but only for a symmetric obstacle in the limit (iii). The results are applied to a semi-ellipse, an asymmetric generalization thereof, the Witch of Agnesi (Queney's mountain), and a rectangle. The approximate results for the semi-ellipse are compared with the more accurate results obtain by Huppert & Miles (1969). It appears from this comparison that the approximate solutions should be adequate for any slender obstacle within the range of kε for which completely stable lee-wave formation is possible. The extension to obstacles in a channel of finite height is considered in an appendix.

This paper considers the large-amplitude symmetric and asymmetric irrota-tional motion of an inviscid incompressible fluid with a liquid—vapour interface in an accelerating container of revolution. A combined analytical—numerical method which involves no linearizations in the hydrodynamical equations and applies to all but surface-tension dominated motions is used to compute a variety of such motions. One important aspect of this non-linear method is that it accurately determines the initial development of surface instabilities such as breakers near the wall of the container.

A mechanism is proposed whereby planetary zonal flows can be generated by the resonant interaction of Rossby wave packets whose amplitudes are slowly varying functions of both space and time. Equations are derived describing the long-time behaviour of a resonantly interacting triad. At the first closure certain properties analogous to those already known for discrete waves are deduced. At the second closure, in the particular case when one of the members of the triad is a zonal flow, it is shown that the sideband resonance mechanism can cause energy to be gained or lost by this zonal flow. It is also shown that a single Rossby wave packet can exchange energy with a zonal flow with weak shear. In the final section a resonant quarter mechanism for producing zonal flows is discussed. A numerical estimate of the acceleration of a zonal current from a zero initial state gives values of a few km/day per day.