Experimental measurements were made of the variation of water-surface height across the crests of solitary waves travelling along trapezoidal channels. The channels had one vertical side and the other side sloped at either 30°, 45° or 60° to the vertical. Reliable results were not obtained with the last channel, but for the other channels there was reasonable agreement with recent theoretical results.

Laboratory experiments of non-steady penetrative convection in water are performed that closely simulate the lifting of an atmospheric inversion above heated ground. Vertical profiles of horizontally averaged temperature and heat flux are measured and interpreted. The rate at which kinetic energy is destroyed by the downward heat flux in the vicinity of the inversion base is found to be a very small fraction of the rate at which it is generated in the lower convective region. The interface separating the convective region from the stable region is examined and its rate of rise explained.

Thermally induced convection in a rotating cylinder of fluid heated from above and strongly influenced by centrifugal accelerations is treated using boundary-layer methods. As in the theory of homogeneous rotating fluids, the horizontal Ekman layers control the inviscid axial flow. The solution also largely depends upon the thermal conditions assumed at the side wall, and if these be insulated, consideration of the side-wall boundary layers is necessary for complete specification of the problem. For perfectly conducting side walls, the side layers do not influence the zeroth-order flow, but contribute a second-order correction, which would be absent if the lateral boundaries were ignored. The critical parameters governing the solutions in both cases are found to be γ and the group σβε−½, where γ is the aspect ratio, σ the Prandtl number, ε the Ekman number, and β the thermal Rossby number for the flow. Boundary-layer solutions are given for a wide range of parameters, and gravity is seen to have at most only a local effect on the flow near the side walls.

The problem considered is that of a strong shock propagating from a point energy source into a cold atmosphere whose density varies exponentially with altitude. An explicit analytic solution is obtained by taking the flow field as ‘locally radial’ and using an integral method with an energy constraint. A scaling law is given which eliminates the parametric dependence of the solution on the explosion energy, scale height, and atmospheric density at the point of the explosion. The scaling law also transforms the infinity of solutions for various polar angles into two distinct solutions which show that all motions of the ascending portion of the shock may be scaled from the vertically upward behaviour and all motions of the descending portion of the shock may be scaled from the vertically downward behaviour. The limit in the lateral direction of both of the fundamental solutions corresponds to the case of the uniform density atmosphere. The results for the uniform density atmosphere show remarkable agreement with the exact Taylor—Sedov results. Comparison with finite difference calculations of Troutman & Davis for the vertically upward and downward directions shows excellent agreement with respect to the prediction of shock propagation velocity, position, and the flow variables behind the shock. A scaling law for the time, shock velocity, and pressure for different values of the adiabatic exponent γ is proposed which correlates the results of the present analysis for different values of γ over the entire range of shock positions where the analysis applies. The solution shows that, contrary to the result obtained by Kompaneets, there is no theoretical limit as to how far downward a strong shock may propagate. The far field behaviour of the shock wave in the upward and downward directions is found to be of the same form as the self-similar asymptotic solutions obtained by Raizer for a plane shock. It is shown by relaxing the energy constraint in the vertically downward direction that the asymptotic result obtained agrees closely with that obtained by Raizer. The energy constraint, however, is the appropriate one for all but the far field behaviour. The far field limit of the present solution in the upward direction is found to compare favourably with the approximate asymptotic calculations of Hayes for an ascending curved shock. The empirical concept of ‘modified Sachs scaling’ for calculating the overpressure is considered and shown within this model to have a justification in the downward direction but a limited range of applicability in the upward direction.

This paper considers the large-amplitude symmetric and asymmetric irrota-tional motion of an inviscid incompressible fluid with a liquid—vapour interface in an accelerating container of revolution. A combined analytical—numerical method which involves no linearizations in the hydrodynamical equations and applies to all but surface-tension dominated motions is used to compute a variety of such motions. One important aspect of this non-linear method is that it accurately determines the initial development of surface instabilities such as breakers near the wall of the container.

The stability of almost fully developed viscous flow in a rotating pipe is considered. In cylindrical polar co-ordinates (r, ø, z) this flow has the velocity components \[ \{W_0o(1),\quad\Omega r[1+o(\epsilon)],\quad W_0[1-r^2/r^2_0+o(1)]\},_{+}^{+} \] where ε = Wo/2Ωr0 and is bounded externally by the rigid cylinder r = r0, which rotates about its axis with angular velocity Ω. In the limit of small ε, the disturbance equations can be solved in terms of Bessel functions and it is shown that, in that limit, the flow is unstable for Reynolds numbers R = Wor0/v greater than Rc [asymp ] 82[sdot ]9. The unstable disturbances take the form of growing spiral waves, which are stationary relative to the rotating cylinder and the critical disturbance at R = Rc has azimuthal wave-number 1 and axial wavelength 2πr0/ε. Furthermore, it is shown that the most rapidly growing disturbance for R > Rc has an azimuthal wave-number which increases with R. Some of the problems involved in testing the results by experiment are discussed and a possible application to the theory of vortex breakdown is mentioned. In an appendix this instability is shown to be an example of inertial instability.

Let u(x) be the velocity field in a fluid of infinite extent due to a vorticity distribution w(x) which is zero except in two closed vortex filaments of strengths K1, K2. It is first shown that the integral \[ I=\int{\bf u}.{\boldmath \omega}\,dV \] is equal to αK1K2 where α is an integer representing the degree of linkage of the two filaments; α = 0 if they are unlinked, ± 1 if they are singly linked. The invariance of I for a continuous localized vorticity distribution is then established for barotropic inviscid flow under conservative body forces. The result is interpreted in terms of the conservation of linkages of vortex lines which move with the fluid.

Some examples of steady flows for which I ± 0 are briefly described; in particular, attention is drawn to a family of spherical vortices with swirl (which is closely analogous to a known family of solutions of the equations of magnetostatics); the vortex lines of these flows are both knotted and linked.

Two related magnetohydrodynamic invariants discovered by Woltjer (1958a, b) are discussed in ±5.

The refraction of progressive internal waves on sloping bottoms is treated for the case of constant Brunt—Väisälä frequency. In two dimensions simple, explicit expressions for the changing wavelengths and amplitudes are found. For small slopes, the solutions reduce to simple propagating waves at infinity.

The singularity along a characteristic is shown to be removable, though the solutions are now inhomogeneous waves. The viscous boundary layers of the wedge geometry are briefly considered with the inviscid solutions remaining as interior solutions.

A theory valid for small slopes is obtained for three-dimensional waves. The waves are refracted in the usual manner, turning parallel to the beach in shallow water.

Treated in the present paper are the second-order effects, i.e. the effect of displacement thickness, longitudinal curvature, external shear and slip, on the two-dimensional laminar boundary-layer flow of incompressible fluid. The analysis is developed in terms of the stream-function co-ordinates proposed for the analysis of the rotational flow by the senior author previously. An inverse problem is set in the co-ordinates plane, and, within the framework of the second-order approximation, the problem is reduced to solving a parabolic partial differential equation for the total head. An implicit finite-difference scheme is then devised to solve the equation, and the practical computations are carried out for the flow along a flat plate or the one around a parabolic body, by making use of an electronic computer.

The vibrational relaxation frequency of carbon dioxide has been determined by measuring the rate of change of thermal emission in shock waves near 4±3μ. This method of measuring the relaxation frequency depends mainly on the degree of excitation of the asymmetric stretching mode of the molecule, and the results are compared with those of earlier density measurements made in the same shock tube. The gas samples used are not optically thin, and it is shown that self-absorption can be taken into account. The results imply that the relaxation frequency of the asymmetric stretching mode is about 70% of that of the bending mode.

A linear model of the circulation of a stratified ocean, in a closed basin, driven by both wind stress and heating is presented. Particular attention is given to the interdependence of the primary features of the oceanic circulation. The upwelling process is studied in detail and it is shown that the complete determination of the mid-ocean thermocline solution depends on the upwelling in the boundary layers on the ocean basin's side walls. The morphology of the side wall boundary layers as a function of the stratification is also discussed.