When vortices are generated in one layer of a rotating, two-layer density stratification, the velocity field of each vortex is strongly baroclinic within a distance of order one Rossby radius from its centre. In this system there are two classes of vortex pairs: those pairs (consisting of vortices of opposite signs) for which the vortices are in the same layer, and those for which the vortices are in opposite layers. We pay particular attention to a laboratory demonstration of the properties of the latter class. These vortex pairs have the ability to transport density (or heat) in the horizontal, and provide a means for describing the release of potential energy by baroclinic instability. We also observe that interactions of real vortices and real vortex pairs differ from those computed for point vortices.

The Sun and the giant planets rotate and possess deep shells of convection. Some basic aspects of prototypical global convection have been studied with a laboratory model operated in the microgravity environment of Spacelab 3 that flew on the space shuttle Challenger in May 1985. This experiment studied thermally driven circulations within a rotating hemispherical shell of fluid across which are imposed radial and latitudinal temperature gradients. The radial force of gravity is modelled by imposing a strong electric field across the shell, with dielectric polarization forces producing radial accelerations proportional to temperature. When the influence of rotation is large, the experiments yield north-south oriented columnar convection in equatorial and subequatorial regions. As the differential heating is increased, these roll-like cells interact with mid-latitude waves, ultimately being destroyed by turbulent, horizontally isotropic convection that moves down from the pole. When a significant equator-to-pole temperature difference is imposed on the boundaries, spiral waves develop on top of a strong meridional circulation. Intricate, non-axisymmetric, convective patterns that propagate in longitude and evolve in time are described. Schlieren visualizations of these laboratory flows are compared with three-dimensional nonlinear simulations.

The problem of convection in a rotating cylindrical annulus heated from the outside and cooled from the inside is considered in the limit of high rotation rates. The constraint of rotation enforces the two-dimensional character of the motion when the angle of inclination of the axisymmetric end surfaces with respect to the equatorial plane is small. Even when the angle of inclination is large only the dependences on the radial and the azimuthal coordinates need to be considered. The dependence on time at the onset of convection is similar to that of Rossby waves. But at higher Rayleigh numbers a transition to vacillating solutions occurs. In the limit of high rotation rates simple equations can be derived which permit the reproduction and extension of previous numerical results.

During the last few years, a number of fluid dynamicists, often in collaboration with geologists, have been investigating problems suggested by geology and creating the new field of geological fluid mechanics. The research has uncovered many new fluid-mechanical phenomena and the studies’ have ranged widely over theoretical contributions, laboratory experiments and confirmatory field observations. This paper presents an overview of part of the field, with a strong emphasis on the novel fluid-mechanical concepts. The presentation includes discussion of some of the effects of cooling and crystallization and the resulting convective motions, the flow of gravity currents at both high and low Reynolds number, and the erosion of solid retaining walls by hot, turbulent flows.

Bending waves, perturbation modes leading to deflections of the vortex centreline, are considered for an infinitely long straight vortex embedded in an irrotational flow of unlimited extent. We first establish the general form of the dispersion relation for long waves on columnar vortices with arbitrary distributions of axial and azimuthal vorticity by a singular perturbation analysis of the Howard-Gupta equation. The asymptotic results are shown to compare favourably with numerical solutions of the Howard-Gupta equation for wavelengths comparable to the vortex core radius and longer. Dispersion relations are then found numerically for specific models of vortex core structures observed experimentally; here no restrictions are placed on wavelength. The linear dispersion relation has an infinite number of branches, falling into two families; one with infinite phase speed at zero wavenumber (which we call ‘fast’ waves), the other with zero phase speed (‘slow’ waves). In the long-wave limit, slow waves have zero group velocity, while the fast waves may have finite non-zero group speeds that depend on the form of the velocity profiles on the axis of rotation. Weakly nonlinear waves are described under most circumstances by the nonlinear Schrödinger equation. Solitons are possible in certain ‘windows’ of wavenumbers of the carrier waves. An example has already been presented by Leibovich & Ma (1983), who compute solitons and soliton windows on a fast-wave branch for a vortex with a particular core structure. Experimental data of Maxworthy, Hopfinger & Redekopp (1985) reveal solitons which appear to be associated with the slow branch, and these are computed for velocity profiles fitting their data. The nonlinear Schrödinger equation is shown to fail for long waves, and to be replaced by a nonlinear integro-differential equation.

G. I. Taylor's solution in 1910 for the interior structure of a weak shock wave is, with appropriate generalization, an essential component of weak-shock theory. The Taylor balance between nonlinear convection and thermoviscous diffusion is, however, endangered when other linear mechanisms - such as density stratification, geometrical spreading effects, tube wall attenuation and dispersion, etc. - are included. The ways in which some of these linear mechanisms cause the Taylor shock structure to break down when a weak shock has propagated over a large (and in some cases quite moderate) distance will be studied. Different forms of breakdown of the Taylor shock structure will be identified, both for quadratic (gasdynamic) nonlinearity and also for cubic nonlinearity appropriate to transverse waves in solid media or electromagnetic waves in nonlinear dielectrics. From this a description will be given of the fate of a nonlinear wave containing a pattern of weak shock waves, as it propagates over large ranges under the influence of linear and nonlinear mechanisms.

Fundamental aspects of the acoustic emission by vortex motions are considered by summarizing our recent work. Three typical cases are presented as illustrative examples: (i) head-on collision of two vortex rings, (ii) a vortex ring moving near a circular cylinder, and (iii) a vortex ring moving near a sharp edge of a semi-infinite plate. The theory of aerodynamic sound for low-Mach-number motion of an inviscid fluid predicts that the amplitude of the acoustic pressure in the far field is proportional to U4, U3 and U2.5 for (i)-(iii) respectively, where U is the translation velocity of a single vortex ring. Therefore the vortex-edge interaction generates the most powerful sound among the three cases at low Mach numbers. Our observations have confirmed these scaling laws. In addition to the scaling properties, we show the wave profiles of the emission as well as the directionality pattern. The head-on collision radiates waves of quadrupole directionality, whereas waves of dipole property are originated by the vortex-cylinder interaction. The third, vortex-edge, interaction generates waves of a cardioid directionality pattern. The wave profiles of all three cases are related to the time derivatives of the volume flux (through the vortex ring) of an imaginary potential flow which is characteristic of each configuration, although the orders of the time derivatives are different for each case. The observed profiles are surprisingly well fitted to the curves predicted by the theory, except the final period of the first case, in which viscosity is assumed to play an important role. The observed wave profiles are shown in a perspective diagram.

This article is aimed at relating a certain substantial body of established material concerning wave loading on offshore structures to fundamental principles of mechanics of solids and of fluids and to important results by G. I. Taylor (1928a, b). The object is to make some key parts within a rather specialised field accessible to the general fluid-mechanics reader.

The article is concerned primarily to develop the ideas which validate a separation of hydrodynamic loadings into vortex-flow forces and potential-flow forces; and to clarify, as Taylor (1928b) first did, the major role played by components of the potential-flow forces which are of the second order in the amplitude of ambient velocity fluctuations. Recent methods for calculating these forces have proved increasingly important for modes of motion of structures (such as tension-leg platforms) of very low natural frequency.

Surface waves can be recorded in two kinds of ways, either with a fixed (Eulerian) probe or with a free-floating (Lagrangian) buoy. In steep waves, the differences between corresponding properties can be very marked.

By a simple physical model and by accurate calculation it is shown that the Lagrangian wave period may differ from the Eulerian wave period by as much as 38 %. The Lagrangian mean level is also higher than the Eulerian mean, leading to possible discrepancies in remote sensing of the ocean from satellites.

Surface accelerations are of interest in relation to the incidence of breaking waves, and for interactions between short (gravity or capillary) waves and longer gravity waves. Eulerian accelerations tend to be very non-sinusoidal, with large downwards peaks, sometimes exceeding - g in magnitude, near to sharp wave crests. Lagrangian accelerations are much smoother; for uniform gravity waves they lie between −0.388g and +0.315g. These values are verified by laboratory experiments. In wind-generated waves the limits are probably wider.

In progressive gravity waves in deep water the horizontal accelerations generally exceed the vertical accelerations. In steep waves, the subsurface accelerations can slightly exceed those at the free surface.

A novel application is made to the rolling motion of ships. In very steep, irrotational waves it is shown theoretically that the flow near the wave crest can lead to the rotation of the hull through angles up to 120° by a single wave, even if the wave is not breaking. This is confirmed by simple experiments. The efficiency of the keel appears to promote capsizing.