A nonlinear solution is constructed representing the steady flow field generated in viscous incompressible fluid in a spherical envelope by a constant point force F0 acting at the centre O of the envelope. Our analysis shows that when F0 = O(3ν2ρ), where v is the coefficient of kinematic viscosity and ρ the density of the fluid, the linear solution, which is symmetric about a plane through O perpendicular to the force, represents a reasonable approximation to the velocity field. As F0 increases the velocity field develops an asymmetry and the centre of the eddy, that exists in a meridian section, is displaced towards the direction of the force and is closer to the boundary. Also as F0 increases, on the axis of symmetry, the fluid speed per unit force decreases behind the force and increases ahead of it and percentage-wise the increase is larger further from O.

Definitions of energy density, energy flux and momentum flux for capillary—gravity waves are derived by integration of the equations of motion and also by Whitham's averaged Lagrangian method. We then confirm recent results due to Hogan (1979) both in the general case and in the case of pure capillary waves. Comparison with the Lagrangian results also allows us to give general definitions of ‘wave action density’ and ‘wave action flux’.

In an effort to determine the characteristics of the various types of convection that can occur in a fluid-saturated porous medium heated from below, a Galerkin approach is used to investigate three-dimensional convection in a cube and two-dimensional convection in a square cross-section. Strictly two-dimensional, single-cell flow in a square cross-section is steady for Rayleigh numbers R between 4π2 and a critical value which lies between 300 and 320; it is unsteady at higher values of R. Double-cell, two-dimensional flow in a square cross-section becomes unsteady when R exceeds a value between 650 and 700, and triple-cell motion is unsteady for R larger than a value between 800 and 1000. Considerable caution must be exercised in attributing physical reality to these flows. Strictly two-dimensional, steady, multicellular convection may not be realizable in a three-dimensional geometry because of instability to perturbations in the orthogonal dimension. For example, even though single-cell, two-dimensional convection in a square cross-section is steady at R = 200, it cannot exist in either an infinitely long square cylinder or in a cube. It could exist, however, in a cylinder whose length is smaller than 0.38 times the dimension of its square cross-section. Three-dimensional convection in a cube becomes unsteady when R exceeds a value between 300 and 320, similar to the unicellular two-dimensional flow in a square cross-section. Nusselt numbers Nu, generally accurate to 1%, are given for the strictly two-dimensional flows up to R = 1000 and for three-dimensional convection in cubes up to R = 500. Single-cell, two-dimensional, steady convection in a square cross-section transports the most heat for R < 97; this mode of convection is also stable in square cylinders of arbitrary length including the cube for R < 97. Steady three-dimensional convection in cubes transports more heat for 97 [lsim ] R [lsim ] 300 than do any of the realizable two-dimensional modes. At R [gsim ] 300 the unsteady modes of convection in both square cylinders and cubes involve wide variations in Nu.

This paper makes use of the ease of modelling topographic Rossby waves in a laboratory context to investigate the ability of these waves to generate strong zonal mean flows when the geostrophic (f/H) contours are closed. A zonally travelling wave is forced in a narrow latitude band of a ‘polar beta plane’. Stronger signals occur when the motion of the driving is retrograde and at the phase speed of the gravest free modes. An important zonal westward mean flow occurs in the free interior while a compensating eastward jet is found at forced latitudes. The dependence of the mean flow strength upon the wave steepness indicates that genuine rectification processes are indeed taking place when the fluid is stirred by purely oscillating devices.

This general tendency for topographic Rossby waves to transfer energy to zonal components is first analysed theoretically by investigating a side-band instability mechanism within an unforced fluid. Among the products of the interactions between a primary wave of wavenumber k and its side bands of wavenumber k ± δk, the zonal flow is prominent. Wave steepnesses of order (|δk|/|k|)½ only are required for zonal energy to grow whereas non-zonal components of scale longer or shorter than the primary wave need huge steepnesses [of order (|δk|/|k|−3/2] for amplification. This supplements the earlier notion that ‘nearly zonal’ waves may be generated by weak resonant interaction.

For gentle driving certain classical aspects of Rossby wave propagation can be checked against the experiments. The linear theory provides also a convenient framework to discuss the meridional structure of the wave-induced Reynolds stress. For more energetic driving, a test of the potential vorticity mixing theory can be carried out and sheds further light upon the rectification mechanisms.

The flow of a homogeneous viscous liquid towards a sink in the interior of a rotating basin with a free surface, a horizontal bottom and a vertical side wall is considered. The conditions assumed are such that an Ekman layer occurs at the bottom beyond a small distance from the sink. A first-order correction to the Ekman model accounting for the influence of the inertial terms in the equations of motion is given for a special case. It is shown theoretically and experimentally that eccentric withdrawal from a circular basin causes a vortex at the sink and a counter-rotating gyre attached to the far wall.

In 1965 Sullivan made many measurements of concentration in dye plumes in the surface layer of Lake Huron with the primary purpose of estimating the distance–neighbour function (Sullivan 1965, 1971). This paper presents the results of a recent analysis of the concentration fluctuations in these experiments for, despite their great practical and theoretical importance, there are very few published reports of such measurements from natural environments. One reason for this apparent neglect has undoubtedly been the anticipated high noise level, and the present results confirm this expectation. The experimental analysis uses the framework of relative diffusion since this has great advantages compared with that of absolute diffusion. Despite the noise, the results are consistent, to the degree of spatial resolution attained, with the self-similar structure anticipated for relative, but not absolute, diffusion. Further interesting features of the results are that changes in the form of the statistical properties across the plume indicate an unexpectedly strong influence of the central regions, and that certain statistical properties have much less noisy profiles than that of the mean square fluctuations. The influence of molecular diffusion is shown to be strong. Interpretation of the results is based partly on the extension of the theory recently developed by Chatwin & Sullivan (1979a) for a cloud, although the limited spatial resolution attained did not allow direct critical examination of this work.

We have used the technique of laser-Doppler velocimetry to study the transition to turbulence in a fluid contained between concentric cylinders with the inner cylinder rotating. The experiment was designed to test recent proposals for the number and types of dynamical regimes exhibited by a flow before it becomes turbulent. For different Reynolds numbers the radial component of the local velocity was recorded as a function of time in a computer, and the records were then Fourier-transformed to obtain velocity power spectra. The first two instabilities in the flow, to time-independent Taylor vortex flow and then to time-dependent wavy vortex flow, are well known, but the present experiment provides the first quantitative information on the subsequent regimes that precede turbulent flow. Beyond the onset of wavy vortex flow the velocity spectra contain a single sharp frequency component and its harmonics; the flow is strictly periodic. As the Reynolds number is increased, a previously unobserved second sharp frequency component appears at R/Rc = 10·1, where Rc is the critical Reynolds number for the Taylor instability. The two frequencies appear to be irrationally related; hence this is a quasi-periodic flow. A chaotic element appears in the flow at R/Rc ≃ 12, where a weak broadband component is observed in addition to the sharp components; this flow can be described as weakly turbulent. As R is increased further, the component that appeared at R/Rc= 10·1 disappears at R/Rc = 19·3, and the remaining sharp component disappears at R/Rc = 21·9, leaving a spectrum with only the broad component and a background continuum. The observance of only two discrete frequencies and then chaotic flow is contrary to Landau's picture of an infinite sequence of instabilities, each adding a new frequency to the motion. However, recent studies of nonlinear models with a few degrees of freedom show a behaviour similar in most respects to that observed.

A method is outlined by which high-order solutions are obtained for steadily progressing shallow water waves. It is shown that a suitable expansion parameter for these cnoidal wave solutions is the dimensionless wave height divided by the parameter m of the cn functions: this explicitly shows the limitation of the theory to waves in relatively shallow water. The corresponding deep water limitation for Stokes waves is analysed and a modified expansion parameter suggested.

Cnoidal wave solutions to fifth order are given so that a steady wave problem with known water depth, wave height and wave period or length may be solved to give expressions for the wave profile and fluid velocities, as well as integral quantities such as wave power and radiation stress. These series solutions seem to exhibit asymptotic behaviour such that there is no gain in including terms beyond fifth order. Results from the present theory are compared with exact numerical results and with experiment. It is concluded that the fifth-order cnoidal theory should be used in preference to fifth-order Stokes wave theory for wavelengths greater than eight times the water depth, when it gives quite accurate results.

This paper applies the test field model developed by Kraichnan to the study of an isotropic, passive scalar contaminant convected by decaying isotropic turbulence. Test field model predictions of scalar and velocity dissipation spectra at large Reynolds and Péclet numbers are shown to be in excellent agreement with atmospheric data, after intrinsic scale constants in the model are adjusted to give valid inertial range coefficients. Theoretical values for the inertial range coefficients are obtained for large and small Prandtl numbers. Simulation results for velocity and scalar energy, dissipation and transfer spectra and second- and third-order velocity, scalar and velocity–scalar correlations at moderate Reynolds and Péclet numbers are shown to agree moderately well with heated grid turbulence data. Simulation results are presented for the normalized decay rates of the scalar and velocity dissipation rates and for the ratio of the velocity to scalar decay time scales; these quantities are employed in second-order modelling. In the self-similar decay mode the simulations yield unity levels of the normalized decay rates and of the ratio of decay time scales over the moderate range of Reynolds and Prandtl numbers investigated. These results are compared with data from heated grid turbulence experiments and are discussed in the light of asymptotic decay of concomitant scalar and velocity fields.

The development in space and time of a plane initial disturbance to a spatially uniform exploding atmosphere is analysed on the assumption that the disturbance amplitude is comparable in magnitude with the inverse (dimensionless) activation energy of the explosion reaction. Particular attention is focused on the shock-fitting problem, which has features that distinguish it from its inert-atmosphere counterpart.

Using the positive half of a sine wave to typify an isolated compression perturbation, it is found that the amplifying effect of the ambient reaction leads to very rapid shock wave development, which depends significantly on the spatial extent of the disturbance. The latter also influences the question of whether local explosion (local explosion is recognized here as a logarithmically unbounded growth of the disturbance amplitude; in other words as a local breakdown of the present approximations) occurs at the shock wave or some distance behind it. The subsequent evolution of these two states will no doubt be significantly different, but the answer to this speculation must await extension of the present theory to encompass the rapid events that ensue near the local explosion regions.