Past study of the large-scale consequences of forced small-scale motions in electrically conducting fluids has led to the ‘α-effect’ dynamos. Various linear kinematic aspects of these dynamos have been explored, suggesting their value in the interpretation of observed planetary and stellar magnetic fields. However, large-scale magnetic fields with global boundary conditions can not be force free and in general will cause large-scale motions as they grow. I n this paper the finite amplitude behaviour of global magnetic fields and the large-scale flows induced by them in rotating systems is investigated. In general, viscous and ohmic dissipative mechanisms both play a role in determining the amplitude and structure of the flows and magnetic fields which evolve. In circumstances where ohmic loss is the principal dissipation, it is found that determination of a geo- strophic flow is an essential part of the solution of the basic stability problem. Nonlinear aspects of the theory include flow amplitudes which are independent of the rotation and a total magnetic energy which is directly proportional to the rotation. Constant a is the simplest example exhibiting the various dynamic balances of this stabilizing mechanism for planetary dynamos. A detailed analysis is made for this case to determine the initial equilibrium of fields and flows in a rotating sphere.

The resonant response of a harbour H of depth scale d and area A to excitation of frequency ω through a mouth M of width a is calculated in the joint limit a2/Aω2A/gd↓0. The results are relevant to the tsunami response of narrow-mouthed harbours. 16 is assumed that an adequate approximation to the radiation impedance of the external domain is available (Miles 1972). The boundary-value problem for H is reduced to the solution of ∇. (h∇ϕ) = −1/A, where h is the relative depth, the normal derivative of ϕ is prescribed in M and vanishes elsewhere on the boundary of H, and the spatial mean of ϕ must vanish. The kinetic energy in H is proportional to an inertial parameter M that is a quadratic functional of ϕ. It is demonstrated that decreasing/increasing h increases/decreases M. Explicit lower bounds to M are developed for both uniform and variable depth. The results are extended to coupled basins (inner and outer harbours). Several examples are considered, including a model of Long Beach Harbor, for which the calculated resonant frequency of the dominant mode is within 1% of the measured value. The effects of entry-separation and bottom-friction losses are considered; the latter are typically negligible, whereas the former may be comparable with, or dominate, radiation losses.

It is shown how a two-dimensional surface wave can be either totally reflected or totally transmitted in the presence of two parallel vertical barriers each containing a small gap. Total transmission of a surface wave past obstacles has been known to occur in many situations in water-wave theory, but total reflexion is a comparatively new phenomenon which could be of practical use in the design of breakwaters.

It was shown theoretically in part 1 (Shercliff 1969) that, when a horizontal magnetic field and electric current field are imposed upon a conducting liquid with a free surface, waves excited at the surface have an anisotropic dispersion relation. Here the possibility of demonstrating this phenomenon in the laboratory is considered. The analysis is completed for waves a t the interface between a conducting and non-conducting fluid of similar density, taking full account of surface tension, and then viscosity. Surface tension is found to have a considerable influence in reducing the degree of anisotropy to be expected. The problem of choosing suitable experimental parameters is discussed, and an apparatus is described in which it was possible to demonstrate the existence of anisotropic surface waves, and to compare the phase velocity with the theoretical results. Also, an experiment is described which verifies the relationship between the orientation of the anisotropy and the relative orientation of the imposed magnetic and current fields.

The surface elevations in a strip 0 < y < b and in a rectangle -d < y < 0, |x| < a are expressed as a Fourier integral and Fourier series respectively. Using a Galerkin technique to match boundary conditions, the solution to the general problem of the response of a rectangular resonator to inviscid shallow-water waves in infinite and semi-infinite channels is obtained.

Theoretical results are compared with laboratory experiments of James (1970, 1971a, b). Agreement is generally very good, except that the inviscid and linear theory does not predict the observed large energy losses in the resonator at resonance.

The theory is also applied to a geometry corresponding to the Gulf of Carpentaria, and the calculated response of the Gulf to semi-diurnal tides gives a zero response at Kuramba, in agreement with observations. The full response of the Gulf is calculated in subsequent work (Williams 1974) which takes the effects of the earth's rotation into account.

It is well known that the small perturbation equation governing steady or mildly unsteady potential flow in a sonic gas jet is nonlinear. However, for a sonic gas jet submerged in a liquid with a disturbance on the gas-liquid interface, it is shown that the transient motion of the gas dominates, and the nonlinear term due to accumulation of disturbances in the basic flow becomes negligible; the condition necessary for the applicability of the linearized governing equation is obtained. It is demonstrated that most gas-jet/liquid systems of physical interest satisfy this condition and that the margin with which this condition is satisfied improves as the wave velocity of the disturbance or, more particularly, as the stagnation pressure or density of the gas for a given gas-liquid system increases. The Kelvin-Helmholtz instability of the gas-liquid interface of a sonic gas jet submerged in a liquid is predominantly governed by the transfer of energy from the gas phase to the liquid layer, both through wave-drag and ‘lift’ components of the pressure perturbation; at and above the cut-off wavenumber, which only exists for very low viscosity liquids owing to the stabilizing effect of surface tension, the pressure perturbation becomes in phase with the wave amplitude. It is shown that for low viscosity liquids the phase angle between the pressure perturbation exerted by the gas phase on the liquid a t the gas-liquid interface and the wave amplitude, which is the measure of the relative effectiveness of the ‘lift’ and wave-drag components of the pressure perturbation, is a function of the density ratio (ratio of gas density a t throat conditions to liquid density). At low density ratios both of these components are operative; however, at high density ratios the wave-drag component becomes dominant. The analysis further shows that the cut-off wave- number and the wavenumber a t maximum instability decrease with increasing density ratio. For highly viscous liquids and liquids having finite viscosity the pressure perturbation is always out of phase with the wave amplitude, and no cut-off wavenumber exists, i.e. the gas-liquid interface is always unstable in spite of the stabilizing effect of viscosity and surface tension.

Measurements of the fine-structure of the energy spectrum of the velocity fluctuations were made in the transition region of a two-dimensional wake. Line and continuous spectra were determined separately with a high-selectivity band-pass filter. The transition was initiated by an external sinusoidal sound. The sound-induced periodic fluctuation and the natural random fluctuation in the wake were added and a system of line and continuous spectra was formed. Higher harmonics of the periodic component were produced by the nonlinear interaction. As a result of the interaction between discrete and continuous components, a continuous spectrum was generated at low wavenumbers. A simple model for the interaction is proposed. The evolution of the spectrum is explained by three fundamental rules concerning the nonlinear interaction between spectral components: (i) the growth of a spectral component is suppressed by the presence of another strong component, (ii) mutual interaction is more effective when the amplitudes of interacting components are closer, and (iii) a stronger interaction takes place between components of closer wavenumbers. The randomization of the regular fluctuation is properly expressed as the growth of the ‘randomness factor’, the ratio of the energy of the random components to the total fluctuation energy.

In isotropic ‘box’ turbulence without a mean flow, the Lagrangian frequency spectrum extends to frequencies of order $(\epsilon/\nu)^{\frac{1}{2}}$ (ε is the rate of dissipation of kinetic energy per unit mass and ν is the kinematic viscosity of the fluid). This leads to an estimate that makes the r.m.s. value of du/dt of order $(\epsilon^3/\nu)^{\frac{1}{4}}$. The Eulerian frequency spectrum, however, extends to higher frequencies than its Lagrangian counterpart; this is caused by spectral broadening associated with large-scale advection of dissipative eddies. As a consequence, the r.m.s. value of ∂u/∂t at a fixed observation point is (apart from a numerical factor) $R_{\lambda}^{\frac{1}{2}}$ times as large as the r.m.s. value of du/dt (RΛ is the turbulence Reynolds number based on the Taylor microscale). The results of a theoretical analysis based on these premises agree with data obtained by Comte-Bellot, Shlien and Corrsin. The analysis also suggests that the Eulerian frequency spectrum has a $\omega^{-\frac{5}{3}} $ behaviour in the inertial subrange, and that it is not governed by Kolmogorov similarity.

This paper suggests a simple way of including gravitational effects in the pres-sure-containing correlations that appear in the equations for the transport of Reynolds stress and heat flux. The predicted changes in structure due to the gravitational field are shown to agree closely with Webster's (1964) measurements in a stably stratified shear flow.

A finite-difference procedure is employed to predict the development of turbulent flow in curved pipes. The turbulence model used involves the solution of two differential equations, one for the kinetic energy of the turbulence and the other for its dissipation rate. The predicted total-velocity contours for the developing flow in a 180° bend are compared with the experimental data. Predictions of fully developed velocity profiles for long helically wound pipes are also presented and compared with experimental measurements.

This paper examines the effect of steady flow on the generation of sound by aerodynamic sources located in the neighbourhood of a scattering body. The analysis is facilitated by the use of a reverse-flow reciprocal theorem which is valid for a certain class of inhomogeneous flow problems. In the case of acoustic ‘bremsstrahlung’, in which sound is generated during the passage of a silent fluid inhomogeneity, such as an entropy spot, past an obstacle in the flow, an elegant discussion of the interaction can be given in terms of a renormalized Green's function. Here the effect of the obstacle is equivalent to a local distortion of the geometry of the three-dimensional space occupied by the medium, the extent of which governs the properties of the acoustic radiation. This is illustrated by means of a detailed analysis of the sound generated during the unsteady convection of a source of constant strength through a neck in a duct.

The effect of gravity acting perpendicularly to the stratification induced by rotation in a compressible fluid is investigated. The γ = 1 approximation used by Gans (1974) in an investigation of the free modes of a rotating gas is used to find an inviscid solution. The boundary layers are calculated, and their suctions are shown to lead to a resonance. The amplitude of the resonance mode is calculated and seen to be of the same order as the simple inviscid solution, so that a corrected solution can be obtained.

The stability of the system is discussed in a simple fashion. It seems likely that a turbulent core will form in a cylinder. The interaction of such a core with the remainder of the fluid is beyond the scope of this paper. Finally a procedure by which a zonal flow field can be found is given.