This note provides numerical values for the long-wave limit of the virtual-mass coefficient relating to the heaving motion of a half-immersed circular cylinder on water of finite depth, found analytically by Ursell in the preceding paper; some preliminary analysis is needed, however.

The study of nonlinear magnetosonic waves in a turbulent plasma is extended to include the effects of the Hall term. The turbulence and Hall effect are characterized by an effective electrical conductivity and an ion gyrofrequency respectively. It is shown that the magnetosonic waves are governed by a nonlinear equation which can be considered as the generalization of a Korteweg & de Vries (1895) equation with dispersion. For a stationary solution two cases are considered in detail: (a) an unperturbed magnetic field is almost parallel to a wave vector, and (b) they are almost perpendicular. In the case (a) it is shown that the presence of the Hall term can lead to an oscillatory solution which decays due to the finite conductivity. In the second case the Hall effect does not affect the monotonous character of a decaying Taylor-shock profile.

This paper discusses the long-wave limit of the asymptotic theory of hypersonic boundary-layer stability for a gas with the Prandtl number ½ < σ < 1 and with the viscosity–temperature law being a power function. The investigation is confined to the local-parallel approximation.

In the long-wave limit the vorticity mode starts to interact with the acoustic disturbances in the boundary-layer region. The general solution of the linear problem in the boundary-layer inner region is analysed numerically and analytically. This solution is matched with the long-wave vorticity-mode solution near the transition layer. As a result, the inviscid instability problem for a hypersonic boundary layer is formulated. The analytical solution of this problem is found and analysed. The different limits of the solution are considered and the universal forms of the dependence are obtained. A similarity parameter is found which is a function of the Prandtl number and the power in the viscosity–temperature law. A significant change of the solution behaviour is noticed when this parameter passes a critical value. The asymptotic structure of the amplification rate, as a function of the wavenumber, is described and discussed.

It is shown that certain similarity solutions relating to axisymmetric vortices in a viscous fluid over a plane wall can be associated with a point source or sink of vorticity at the origin and a line vortex along the symmetry axis of the system. The rotationality of the nonlinear terms in the momentum equation, due to the radial vorticity and the azimuthal flow field, induces a poloidal flow which relates to one-cell or two-cell configurations. It is shown that a small imput of radial vorticity into a strong line vortex can induce an intense up-draught. There are ranges of values of the parameters yielding two solutions. The extremeties of these ranges are associated with values that yield velocity breakdown.

The flow structures generated and drag experienced by a rigid cylinder moving in an arbitrary direction through a rotating electrically conducting fluid in the presence of an applied magnetic field are investigated, with he aim of understanding better the nature of the small-scale flow in the core of the Earth which may be responsible for maintaining the geomagnetic field through dynamo action. Three cases are considered in the limit of small Rossby and magnetic Reynolds numbers. In the case of very weak rotation, the possible flow structures consist of a thin Hartmann layer and a long wake extending in the direction of the magnetic field, in which Lorentz and viscous forces balance, but only the long wake plays a dynamical role. The dominant drag force is experienced for motion that cuts magnetic lines of force. Motion of the cylinder parallel to its axis induces a much weaker drag, while that in the direction of the magnetic field induces none to dominant order. The cylinder also experiences weak lateral forces due to the Coriolis effect. In the case of weak rotation, the balance in the long wake is now magnetostrophic: between Lorentz and Coriolis forces. The drag is qualitatively identical to that in the first case, but the drag induced by motion parallel to the axis of the cylinder is increased, though still smaller than that for motions cutting magnetic lines of force. In the case of strong rotation, the flow structures consist of a thin Ekman layer and a foreshortened Taylor column extending in the direction of the rotation axis. In this column, the force balance is again magnetostrophic. Again only the large-scale structure plays a dynamical role. Motion of the cylinder perpendicular to its axis induces a larger drag than does motion parallel to its axis. The cylinder also experiences large lateral Coriolis forces.

Here we study the spin-up and spin-down of a homogeneous fluid with a free surface on an experimental ‘β-plane’ and describe the important features for both cases over a range of parameters. Quantitative values are found for the velocity fields using a new image processing technique that analyses a video record of particle motion and stores the results digitally. Streamlines, pressure fields and vorticity values are found by interpolation techniques and result in a complete description of the flow characteristics. We discuss the relationship between the results of these experiments and those observed in large-scale homogeneous models of ocean circulation, e.g. Moore (1963). This study extends the work of van Heijst et al. (1990) to the case of spin-up in a rectangular container but of non-uniform depth and we note the differences to and similarities with their observations. It is related, also, to more recent results of Maas et al. (1992), who considered spin-up on a β-plane but in a tank of very different proportions to the one considered here.

Observations, numerical simulations and theoretical scaling arguments suggest that in mid-latitudes, away from the source regions and the equator, the meridional transport of abyssal water masses along a continental slope corresponds to planetary geostrophic flows that are gravity- or density-driven and topographically steered. We investigate these dynamics using a nonlinear reduced-gravity model that can describe grounded abyssal meridional flow over sloping topography that crosses the planetary vorticity gradient. Exact nonlinear steady and time-dependent solutions are obtained. The general steady theory is illustrated for a non-parallel equatorward flow that possesses a single along-slope grounding along the upslope flank of the current (complementing previous work). Four specific nonlinear time-dependent solutions are described. Two initial-value problems are solved exactly. The first initial configuration corresponds to an equatorward abyssal flow that has no cross-slope shear in the along-slope velocity and possesses a single grounding along the upslope flank of the current. The nonlinear time-dependent evolution of this initial current into a non-parallel shear flow is described. The second initial condition corresponds to an isolated radially symmetric grounded abyssal pool or dome. The nonlinear time-dependent evolution of this abyssal dome, which propagates equatorward with unsteady along- and cross-slope velocities while deforming into an elliptically shaped abyssal dome with $\unicode[STIX]{x1D6FD}$-induced diminishing height, is described. Finally, the nonlinear time-dependent boundary-value problem can be solved exactly in which the in-flow boundary condition on the poleward boundary of the mid-latitude domain corresponds to a time-dependent abyssal current with both an upslope and downslope grounding. Two specific time-dependent boundary conditions are examined. The first corresponds to a time-limited surge in the equatorward volume transport in the abyssal current along the poleward boundary. The second configuration corresponds to the nonlinear evolution of a finite-amplitude downslope plume or loop that forms in the abyssal current that is reminiscent of those seen in baroclinic instability simulations.

The linear spin-up problem for a rapidly rotating viscous diffusive ideal gas is considered in the limit of vanishing Ekman number E. Particular attention is given to gases having a large molecular weight. The gas is enclosed in a cylindrical annulus, with flat top and bottom walls, which is rotating around its axis of symmetry with rotation rate Ω. The walls of the container are adiabatic. In a rotating gas (of any molecular weight), the Ekman layers on adiabatic walls are weak, which implies that there is no distinct non-diffusive response of the gas outside the Ekman and Stewartson boundary layers on the timescale E−1/2Ω−1 for spin-up of a homogeneous fluid. For the case of adiabatic walls, it is shown that the spin-up mechanisms due to viscous diffusion and Ekman suction are, from a formal point of view, equally strong. Therefore, the gas will adjust to the increased rotation rate of the container on the diffusive timescale E−1Ω−1. However, if E1/3 [Lt ] γ – 1 [Lt ] 1 and M [siml ] 1, which characterizes rapidly rotating heavy gases (where γ is the ratio of specific heats of the gas and M the Mach number), it is shown that the gas spins up mainly by Ekman suction on the shorter timescale (γ–1)2E−1Ω−1. In such cases, the interior motion splits up into a non-diffusive part of geostrophic character and diffusive boundary layers of thickness (γ – 1) outside the Ekman and Stewartson layers. The motion approaches the steady state of rigid rotation algebraically instead of exponentially as is usually the case for spin-up.