The influence of slow time variations of the Brun-Väisälä frequency N upon the energy of internal gravity waves is investigated. It is found that, when time variations in N are produced by a mean deformation field (reversible mean state), the wave energy can vary in either direct or inverse proportion, depending on the wavenumber orientation. When N changes owing to a certain type of irreversible process, the wave energy varies with only inverse proportionality.

The nocturnal planetary boundary layer (NPBL) provides an example where N = N(z, t). The full initial/boundary-value problem for an N(z, t) similar to the climatological mean for the NPBL is solved.

A mathematical model of rectilinear surface waves incident on a submerged circular sill is developed. The class of waves considered consists of those for which the ratio of wavelength to water depth is large but which do not necessarily belong in the longwave category. A friction damping term is introduced into the equations of motion and the solutions obtained for the regions over the sill and in the ocean are matched by assuming a continuous surface and energy flux at the sill edge. The results show large reductions in the Q-factor of the resonance peaks brought about by friction damping. It is also found that, except at low frequency, a large number of overlapping resonance peaks which are out of phase with one another occupy a relatively narrow frequency band such that these resonance peaks effectively cancel one another. Experiments were performed to determine the friction constant used in the equations of motion and, using this friction constant, the theoretical results of wave resonance are verified.

It is known that head seas cannot travel without deformation along a cylinder of full constant cross-section, and recent calculations have indicated that the wave amplitude near the cylinder ultimately decreases as the waves travel along the cylinder, i.e. that the waves are refracted away from the axis of the cylinder. It was assumed in these calculations that the cross-section was a half-immersed circle of radius a of the same order as the wavelength 2π/K, but the method can probably be adapted to arbitrary full constant cross-sections. (There is however another calculation which indicates that for a thin ship the wave amplitude ultimately increases.) In the present paper these calculations are extended. The circular section is again studied but it is now supposed that the wavenumber Ka may be small. Uniformly valid expressions for the wave potential are obtained which show that for small Ka the refraction becomes significant only when Kx (the dimensionless distance along the cylinder) is so large that the product (Kx)½ v0(Ka) is also large; here the function v0(Ka) ∼ 2Ka arises in the solution of a certain eigenvalue problem. (The uniformly valid expressions also suggest an interpretation of the thin-ship calculation which resolves the apparent inconsistency.) The same method is applied to the waves generated by a pulsating source on an infinite cylinder, and similar results are obtained.

A new technique for generating a pair of line vortices in the laboratory has been developed. The mean flow of these vortices is highly two-dimensional, although most of the flow field is turbulent. This two-dimensionality permits the study of vortex motions in the absence of the Crow mutual induction instability and other three-dimensional effects. The vortices are generated in a water tank of dimensions 15 × 122 × 244 cm. They propagate vertically and their axes span the 15 cm width of the tank. One wall of the tank is transparent, and the flow is visualized using fluorescein dye. High speed photography is used to study both the transition to turbulence during the vortex formation process and the interaction of the turbulent vortices with a simulated ground plane.

Transition occurs first in an annular region surrounding the core of each vortex, starting with a shear-layer instability on the rolled-up vortex sheet. The turbulent region then grows both radially inwards and radially outwards until the entire recirculation cell is turbulent. A ‘relaminarization’ of the vortex core appears to take place somewhat later.

The interaction of the vortex pair with the ground plane does not follow the predictions of potential-flow theory for line vortices. Although the total circulation is apparently conserved, the vortices remain at a larger distance from the ground than is expected and eventually ‘rebound’ or move away from the ground. Differences between a free-surface boundary condition and a smooth or rough ground plane are discussed. The ground-plane interaction is qualitatively very similar to that of aircraft trailing vortices observed in recent flight tests.

The scattering of short surface waves by a partially immersed cylinder is considered. The cylinder is taken to be circular and to pass through two fixed points on the free surface with its centre on or above the free surface. Of particular interest is the behaviour of the solution as the cross-section of the immersed part of the cylinder approaches a semicircle. The method of matched asymptotic expansions is used.

The paper develops and discusses some new additions to the available stock of analytical solutions of the nonlinear equations of fluid motion. The motions are steady, two-dimensional and devoid of viscous or other rotational forces (although such forces must have been significant during any starting process). The fluid density is constant.

The solutions are in two groups, referred respectively to Cartesian and polar co-ordinates. In both the stream function is of separable form, i.e. expressible as a product of two functions, each dependent on one co-ordinate. A remarkable variety of motions is revealed. Those that are most significant physically are described as bends (rapid transitions from one rectilinear flow to another) or as loops (closed, non-circular, vortex-type flows). The effects of boundary layers at walls or instability are not explored.

The paper closes with a mention of some preliminary experiments on loop flows in which all streamlines are ellipses and some discussion of the applicability of bend flows. Generalizations to axisymmetric flows and compressible flows are also mentioned briefly.

This paper is the second of a pair describing two-point velocity measurements in fully developed pipe flow. A method of processing hot-film anemometer signals to identify intervals of high energy production (‘bursts’) in wall turbulence is presented. The method uses filtered cross-stream spatial derivatives of the axial velocity fluctuations. It is demonstrated to be more sensitive to ‘bursts’ than several other methods of indentification. The bursts identified in this manner are shown to have similar characteristics to those observed in visual studies.

The technique has been applied to the wall region of turbulent pipe flow. Mean burst rates have been obtained at various distances from the wall for three Reynolds numbers. It is shown that the mean burst rate cannot be reliably obtained from a previously used technique based on the autocorrelation of the axial velocity fluctuations.

On the basis of our experiments, the mean burst rate and the turbulent shear stress have been found to vary similarly with distance from the wall. In the region near the wall where the shear stress is constant the mean burst rate is independent of the kinematic viscosity.

Some characteristics of the velocity fluctuations during burst intervals have been studied. All the bursts began with a relative minimum in the axial velocity fluctuations followed by a peak in the cross-stream spatial derivative. A second peak always occurred midway through the burst. The sequence of events is somewhat similar to that in the last stage of laminar-to-turbulent transition.

The evolution of long, finite amplitude Rossby waves in a horizontally sheared zonal current is studied. The wave evolution is described by the Korteweg–de Vries equation or the modified Korteweg-de Vries equation depending on the atmospheric stratification. In either case, the cross-stream modal structure of these waves is given by the long-wave limit of the neutral eigensolutions of the barotropic stability equation. Both non-singular and singular eigensolutions are considered and the appropriate analysis is developed to yield a uniformly valid description of the motion in the critical-layer region where the wave speed matches the flow velocity. The analysis demonstrates that coherent, propagating, eddy structures can exist in stable shear flows and that these eddies have peculiar interaction properties quite distinct from the traditional views of turbulent motion.

An extension of the turbulence ‘test-field model’ (Kraichnan 1971 a) is given for two-dimensional flow with Rossby-wave propagation. Such a unified treatment of waves and turbulence is necessary for flows in which the relative strength of nonlinear terms depends upon the length scale considered. We treat the geophysically interesting case in which long, fast Rossby waves propagate substantially without interaction while short Rossby waves are thoroughly dominated by advection. We recover the observations of Rhines (1975) that the tendency of two-dimensional flow to organize energy into larger scales of motion is inhibited by Rossby waves and that an initially isotropic flow develops anisotropy preferring zonal motion. The anisotropy evolves to an equilibrium functional dependence on the isotropic part of the flow spectrum. Theoretical results are found to be in quantitative agreement with numerical flow simulations.

Temporal and spatial development of short gravity waves in a linear wind-wave tank has been measured for wind speeds up to 15 m/s using microwave Doppler spectrometry. Surface waves of wavelength 4·1 cm, 9·8 cm, 16·5 cm and 36 cm were observed as a function of fetch, wind speed and wind duration. The waves grew exponentially from inception until they were about 10 dB smaller than their maximum height, and the temporal growth and spectral transport (spatial growth) rates were about equal when the wave amplitude was sufficiently small. The amplitude of a short gravity wave of fixed wavelength was found to decrease substantially at winds, fetches or durations greater than those at which the short gravity wave was approximately the dominant wave; such phenomena are sometimes referred to as overshoot. The dominant short gravity wave was observed to reach a maximum amplitude which depended only on wavelength, showing that wave breaking induced by an augmented wind drift cannot be the primary limitation to the wave height. Waves travelling against the wind were observed for wavelengths of 9·8 cm, 16·5 cm and 36 cm and were shown to be generated by the air flow at low wind speeds.

Measured initial growth rates for 16·5 cm and 36 cm waves were greater than expected, suggesting the existence of a growth mechanism in addition to direct transfer from the wind via linear instability of the boundary-layer flow. Initial temporal growth rates and spectral transport rates were compared to yield an experimental determination of the magnitude of the sum of nonlinear interactions and dissipation in short gravity waves. If the steady-state energy input in the neighbourhood of the dominant wave occurs at the measured initial temporal growth rates, then most of the energy input is locally dissipated; relatively little is advected away. Calculated gravitycapillary nonlinear energy transfer rates match those determined from initial growth rates for 9·8 cm waves and the gravity–capillary wave interaction continues to be significant for waves as long as 16·5 cm. For longer waves the gravity–capillary interaction is too small to bring the short gravity wave to a steady state when it is the dominant wave of the wind-wave system.