An integral equation for unsteady inviscid surface waves has been obtained. Existing known approximations are all derived from the one equation. The Boussinesq equation is obtained and criticized.

Some experiments are described in which steady-state shearing flows are developed in stratified brine solutions contained in a cyclically continuous tank of rectangular cross-section. Over the range of overall Richardson numbers studied, the results suggest that whenever turbulent layers are present on either side of a region of fluid with a gravitationally stable density gradient, they cause erosion of this region to occur. The erosion leads to the formation of two homogeneous layers separated by a thin layer of strong density and velocity gradients. The gradient Richardson number, computed by using the velocity and density gradients in this transition layer, tends to have a value of order one.

If we define an overall Richardson number Ri* by averaging the velocity and density gradients over the entire depth of fluid in the tank, we find that the non- dimensional buoyancy flux, Q, is functionally related to Ri* by an equation of the form Q = C1(Ri*)−1 where C1 is a constant, approximately, and Ri* ranges in value between one and thirty.

To check the effect of a large variation of the molecular diffusivity coefficient on flow conditions, we ran a limited number of experiments with thermally stratified fluid. Over a restricted range, 1·0 < Ri* < 5·0, velocity profiles very similar to those measured in the brine-stratified experiments at like values of Ri* were obtained. This suggests that the coefficient of molecular diffusion is not an important parameter in either type of experiment.

Other experiments, made in the same apparatus, describe the entrainment by a turbulent, homogeneous layer of an initially quiescent layer of fluid with a linear density gradient. The depth of the turbulent layer, D, increases with time, t, according to the relation. \[ D^3\propto t. \] This result is consistent with that found by Kato & Phillips (1969), although the turbulent layer in the present experiment is generated in a different manner.

The velocity measuring technique described in this paper consists of measuring the time of flight of a tracer of heated air from an electrically pulsed wire to one of two sensor wires which are operated as resistance thermometers. These sensor wires are at right angles to the pulsed wire and are placed one on either side of the pulsed wire. The instrument may be used in highly turbulent flows including regions in which flow reversals occur. The paper discusses the theoretical behaviour of the probe and the results of some calibration experiments.

A thermal boundary layer is established by heating a vertical plate in a dielectric liquid. An alternating voltage is applied between the heated plate and another plate which is not parallel to the heated plate. This voltage produces a non-uni- form electric field which in turn produces electrical forces acting on the gradients in dielectric permittivity which result from the temperature gradients. These electrical forces alter the boundary layer. In this paper approximate equations are developed which allow one to calculate the boundary-layer,thickness, velocity, and Nusselt numbers for the boundary layer in the presence of the non-uniform electric field. Numerical calculations show that the heat-transfer coefficient can be either increased or decreased by the non-uniform field, depending on whether the field is strongest at the top or bottom of the plates and also on the field strength. Experiments were performed which demonstrate the change in heat transfer caused by the non-uniform field.

This paper deals with a system of equations which includes as special cases the equations governing such hydrodynamic stability problems as the Taylor problem, the Bénard problem, and the stability of plane parallel flow. A non-linear analysis is made of disturbances to a basic flow. The basic flow depends on a single co-ordinate η. The disturbances that are considered are represented as a superposition of many functions each of which is periodic in a co-ordinate ξ normal to η and is independent of the third co-ordinate direction. The paper considers problems in which the disturbance energy is initially concentrated in a denumerable set of ‘most dangerous’ modes whose wave-numbers are close to the critical wave-number selected by linear stability theory. It is a major result of the analysis that this concentration persists as time passes. Because of this the problem can be reduced to the study of a single non-linear partial differential equation for a special Fourier transform of the modal amplitudes. It is a striking feature of the present work that the study of a wide class of problems reduces to the study of this single fundamental equation which does not essentially depend on the specific forms ofthe operators in the original system of governing equations. Certain general conclusions are drawn from this equation, for example for some problems there exist multi-modal steady solutions which are a combination of a number of modes with different spatial periods. (Whether any such solutions are stable remains an open question.) It is also shown in other circumstances that there are solutions (at least for some interval of time) which are non-linear travelling waves whose kinematic behaviour can be clarified by the concept of group speed.

A general theoretical relationship between the (three-dimensional) vcLacity field and density field is established for geostrophic flow of a non-homogeneous fluid. The practical value of the relationship – which reduces to the celebrated two-dimensional theorem due to Proudman and Taylor when the fluid is homogeneous – is illustrated by means of three examples of flows in rapidly rotating fluids, namely (i) baroclinic waves in laboratory systems and in the atmosphere, (ii) the gyroscopic ‘steering’ processes in Jupiter's atmosphere that are implied by the ‘Taylor column’ theory of the Great Red Spot, and (iii) certain striking properties of ocean currents revealed by recent observations in the Atlantic and Mediterranean.

The impulsive linear spin-down of a stably stratified Boussinesq fluid in a circular cylinder is analyzed under the assumption that the Prandtl number P = ν/κ is small (ν is the kinematic viscosity, κ the thermal diffusivity). The nature of the spin-down process depends on the ordering of P with respect to the Ekman number E. For P < E½, the spin-down is similar to that of an unstratified fluid. For P > E½ the process is similar to that for a stratified fluid with P = O(1). The distinctive case P = O(E½) is analyzed in detail. For that case it is shown that for N [Gt ] Ω, the asymptotic state of rigid rotation is reached in a time of the order of (N/Ω)2τ, where N is the Brunt-Väisälä frequency, Ω the angular velocity and τ the thermal diffusion time for the cylinder. We calculate spin-down times for parameter values corresponding to the solar interior. For an angular velocity as large as that suggested by Dicke (5·74 × 10−5 sec −1) the spin-down time is less than the age of the sun. For an angular velocity comparable to the surface value (2middot;87 × 10−6 sec −1), the spin-down time is greater than the age of the sun. These results suggest that a uniformly and rapidly rotating solar interior is not possible, but we cannot rule out a state of non-uniform rotation producing an oblateness as large as that measured by Dicke & Goldenberg (1967).

A theory is presented which describes the propagation of large amplitude tsunamis across a basin of variable depth in the limit when this depth is varying slowly on a scale deiined by the wavelength. In part 1 only the off-shore behaviour is considered; in part 2 some features of the final run up are described.

The technique used is to regard the wave as a slowly modulated simple wave with a slowly changing Riemann invariant. One of the most significant results is that over distances where the effect of depth variation modulates the amplitude of the wave, but does not disperse it, the variations of the amplitudes of the flow variables, such as maximum surface elevation, can be calculated as functions of the undisturbed depth without knowing how this depth varies in distance and without knowing the wave profile. These variations are fully calculated.

The work continues the investigation on large amplitude acoustic pulses in stratified media described in an earlier paper by Varley & Cumberbatch (1970). It is a generalization of Whitham's work (1953) on the sonic boom.

An investigation of the mechanism responsible for the self-excited oscillations of a harmonium reed is presented. Experiments show that the amplitude of vibration of the reed grows exponentially, and measurements of the growth rate as a function of the flow past the reed are reported. Flow visualization studies lead to the conclusion that jet or wake instability is not important in exciting the reed vibration. An analysis of the flow around the reed as an unsteady potential flow results in the evaluation of the aerodynamic forces exciting the reed. The analysis shows that the pressure which excites the reed motion is of the order of $\rho U_0\dot{a}$, where U0 is the flow velocity and $\dot{a}$ is the reed velocity.

Three problems are considered. In the first, a uniform magnetic field is suddenly switched on outside an infinitely long circular cylinder of incompressible con- ducting fluid. As the field diffuses into the fluid, electric currents are generated and hence a Lorentz force field. Under the assumptions of small magnetic Reynolds number and small magnetic Prandtl number, the initial flow (before it has been modified by convection or viscous diffusion of vorticity) is oalculated. In the second problem an initially uniform field is suddenly switched off outside a similar cylinder of fluid. It is shown that the switching off of the field produces a vorticity distribution identical with that produced by the switching on. The third problem considered is that of a circular cylinder of conducting fluid placed in a rapidly alternating magnetic field. It is shown that the alternating field produces vorticity at a constant rate, and a qualitative description of the resulting flow is given.