A variational principle is presented which characterizes steady motions, at finite Rossby number, of rotating inviscid homogeneous fluids in which horizontal velocities are independent of depth. This is used to construct nonlinear solutions corresponding to stationary patches of distributed vorticity above topography of finite height in a uniform stream. Numerical results are presented for the specific case of a right circular cylinder and are interpreted using a series expansion, derived by analogy with a deformable self-gravitating body. The results show that below a critical free-stream velocity a trapped circular vortex is present above the cylinder and a smaller patch of more concentrated vorticity, of the opposite sign, maintains a position to the right (looking downstream) of the cylinder. An extension to finite Rossby number and finite obstacle height of Huppert's (1975) criterion for the formation of a Taylor column is presented in an appendix.

The stability of a plane Rossby wave in a homogeneous fluid is considered. When the two-dimensional equation which governs fluid flow on a beta-plane is linearized in the disturbance stream function, a partial differential equation with a periodic coefficient results. Substitution of a solution dictated by the Floquet theory leads to a determinant equation, and it may be shown from its symmetry properties that disturbances to a Rossby wave may be of only two types: (i) neutrally stable modes not necessarily contiguous to a stability boundary and (ii) a pair of temporally unstable waves, one growing and the other decaying.

The determinant is solved numerically for the neutral-stability boundaries and curves of constant disturbance growth rate; two distinct types of instability emerge. The first is the parametric instability, which renders all waves unstable, and is shown to be asymptotic to the classical nonlinear resonant interaction in the limit of vanishing basic-state amplitude. The details of the disturbance frequency bifurcation for zero-amplitude basic-state waves are presented, and calculations for waves with eastward and westward group velocities are made and discussed in the context of Rhines’ (1975) results for waves and turbulence on a beta-plane. In addition, a second type of instability is computed which is separate and distinct from the parametric instability. The very limited evidence presented suggests that this second kind of instability may possess characteristics which are identifiable in part with the shearing of the fluid by the large-amplitude basic state and in part with the overturning of the ambient vorticity gradient.

Laboratory experiments were conducted to study the motion of miscible buoyant fluid parcels in both homogeneous and stratified media, and were complemented with numerical experiments using the model of Orlanski & Ross (1973). Parcel motion in the homogeneous case was found to be shape-preserving, in agreement with the large body of data for such motion. The stratified medium experiments were such that the parcel attained equilibrium positions. It was found that the time for the parcel to reach maximum depth was approximately 0·7 times the buoyancy period, independent of that depth. The subsequent collapse of the parcel and generation of internal gravity waves was observed. Ray-like patterns of waves propagated out from the collapse region, and wave frequencies near 0·7 times the Brunt-Väisälä frequency were predominant. The internal wave energy radiated away from the collapsed parcel was estimated and found to be 20 to 25% of the change in the system potential energy.

Natural modes of oscillation of a vanishingly thin spherical rotating fluid shell, with frequencies σ less than twice the angular velocity Ω, were found by Haurwitz (1940). Their validity is, however, put in question by the presence of a singularity at critical co-latitudes θc: 2Ω cosθc = σ in the O(ε) term of an expansion in the relative shell thickness ε (Stewartson & Rickard 1969). The problem is investigated here by considering the evolution of flow from a specified initial distribution. The principal features are as follows:

(i) The O(1) natural modes of Haurwitz, decaying on a time scale Ω−1ε−2.

(ii) Corrections to (i), regular and of magnitude O(ε) except near critical latitudes.

(iii) Essentially transient inertial waves of magnitude O(ε).

(iv) Inertial waves of magnitude O(ε½) with the natural-mode frequencies σ and generated by (i) at critical latitudes.

On a time scale Ω−1ε−1, (iii) and (iv) develop vertical and horizontal length scales ε and propagate throughout the ocean. The continuing energy transfer from (i) to (iv), at a rate O(ε2Ω), appears to be the principal respect in which (i) and (ii) fail to constitute a conventional normal mode.

In a previous paper (Booker 1976) we found experimentally that the convective heat transport in a fluid with temperature-dependent viscosity decreased significantly as the ratio of the viscosities at the top and bottom boundaries increased. In this note, we show that this decrease in heat transport can be entirely accounted for by an increase in the critical Rayleigh number with variable viscosity.

A study is made of the entrainment interface in a two-layer flow with one layer turbulent owing to an imposed surface stress. Turbulent and wave parameters were measured from films of an experiment carried out by Kantha, Phillips & Azad (1977) and are analysed to determine relationships between wave and turbulent length and velocity scales and bulk scales.

A measure of interface distortion is calculated and it is found to be related to the mixed-layer depth only at small overall Richardson number. The local velocity of advance of the interface is determined and is found to vary with overall Richardson number tending to the Kolmogorov scale at small Ri and to Phillips’ (1966) quasilaminar scale at large Ri.

Kelvin-Helmholtz type instabilities are observed and their wavelength, amplitude and lifetime are measured. The wavelength is not found to be related to the mixed-layer depth but some dependence on overall Richardson number is observed. An interfacial mean flow Richardson number is estimated and the variation of wave slope at maximum amplitude with this Richardson number is found to be comparable with observations of Thorpe (1973a) in a laminar two-layer flow. A relationship between the wavelength of an instability and the distortion length scale is found and is discussed.

The flow field of an ‘axisymmetric’ vortex breakdown has been mapped using a laser-Doppler anemometer. The interior of the recirculation zone is dominated by energetic, non-axisymmetric, low frequency periodic fluctuations. Spectra for a number of points inside this zone, as well as time-averaged swirl and axial velocity profiles both inside and outside the recirculation zone, have been obtained. The time-averaged streamlines in the interior show an unexpected two-celled structure attributed to the action of the fluctuations. Although the present experiment deals with one particular breakdown, flow-visualization studies indicate that the case examined is typical of the ‘axisymmetric’ form of breakdown over a range of flow conditions.

Kelvin and Poincaré waves are generated when an ocean wave arrives at a nominally rectilinear coastline and interacts with coastal irregularities. The discussion of this problem given by Howe & Mysak (1973) is extended in this paper in order to examine the role of multiple scattering of the Kelvin and Poincaré waves. An integro-differential kinetic equation is derived to describe these processes in the limit in which the irregularities are small compared with the characteristic wavelength. In the absence of dissipative mechanisms it is verified that this description of interactions with the coast conserves total wave energy. The theory is applied to a variety of idealized problems which model tidal and storm surge events, including the generation and decay of Kelvin waves by extensive and localized Poincaré-wave forcing, and the influence of multiple scattering on the radiation of Poincaré-wave noise into the open ocean.

Analyses of fluid flow over a wavy wall are of interest because of their applications to the physical problems mentioned in § 1. The authors have therefore devoted their attention to the effect of waviness of one of the walls on the flow and heat-transfer characteristics of an incompressible viscous fluid confined between two long vertical walls and set in motion by a difference in the wall temperatures. The equations governing the fluid flow and heat transfer have been solved subject to the relevant boundary conditions by assuming that the solution consists of two parts: a mean part and a disturbance or perturbed part. To obtain the perturbed part of the solution use has been made of the long-wave approximation. The mean (zeroth-order) part of the solution has been found to be in good agreement with that of Ostrach (1952) after certain modifications resulting from the different non-dimensionalizations employed by Ostrach and the present authors respectively. The perturbed part of the solution is the contribution from the waviness of the wall. The zeroth-order, the first-order and the total solution of the problem have been evaluated numerically for several sets of values of the various parameters entering the problem. Certain qualitatively interesting properties of the flow and heat transfer, along with the changes in the fluid pressure on the wavy and flat wall, are recorded in §§ 5 and 6.

Hot-wire measurements are presented of the onset of instability in developed axial and tangential flow due to inner-cylinder rotation in annuli of radius ratios 0·9, 0·81 and 0·58 for axial-flow Reynolds numbers between 86 and 2000. Measurements of the minimum critical Taylor number are reported along three radii at azimuthal locations 90° apart. At the largest radius ratio these azimuthal measurements show considerable variation but the sensitivity of marginal stability to angular orientation becomes negligible at a radius ratio of 0·58, when measurements and the Galerkin predictions of Hasoon & Martin (1977) are in close accord. The greater sensitivity to orientation as the radius ratio increases appears to correlate with measured percentage circumferential variations in annulus width arising from manufacturing tolerances and non-uniform curvature of the surfaces.

The applicability of Howard's formula is considered in cases where there exist critical layers at the boundaries in addition to one inside the flow field. It is shown that in a few particular cases Howard's formula can be applied provided that the finite part of an otherwise diverging integral is taken. In general, however, this revised formula is not valid and an expression analogous to the one given by Banks, Drazin & Zaturska for a particular example is found.