In this paper a theoretical model of the motion of isolated buoyant elements in turbulent surroundings is introduced, which takes into account both the growth due to turbulent entrainment and a loss of buoyant fluid to the environment. On dimensional grounds the outflow velocity is taken to be constant and proportional to some characteristic turbulent velocity in the environment, while the entrainment velocity is proportional to the upward velocity of the element. Numerical solutions of the resulting non-dimensional equations of motion are presented, corresponding to a wide range of stabilities. Typically, an element in stable, neutral or moderately unstable surroundings at first grows and then is eroded away, but at a certain value of a stability parameter γ elements become absolutely unstable and continue to grow and rise indefinitely. The value of γ is extremely sensitive to the level of turbulence in the environment, which could therefore exert a controlling influence on the growth of buoyant elements in unstable conditions; large elements are more likely to grow when the level of turbulence is low.

Laboratory experiments have been carried out in order to test one of the predictions of this theory, the form of the dependence of the height attained on total buoyancy and level of turbulence in uniform surroundings. The agreement is good, and numerical comparison of theory and experiment suggests that the assumed outflow velocity is of the same order as, but somewhat less than, the r.m.s. turbulent velocity.

A study is made of some two-dimensional flows of an inviscid, uniformly conducting fluid in the presence of applied magnetic fields, for the case in which the magnetic Reynolds number (the ratio of induced fields to applied fields) is small but the magnetic force coefficient (the ratio of electromagnetio forces to inertia forces) is of order unity.

Perturbations of uniform incompressible and compressible flows through aligned and crossed fields are considered. All these perturbations are strongly dependent upon the magnitude of the magnetic force coefficient. An experimental analogy of the flow through an aligned field is described.

This paper is a continuation of previous work (Ogura 1962a, b) on the dynamical consequence of the hypothesis that fourth-order mean values of the fluctuating velocity components are related to second-order mean values as they would be for a normal joint-probability distribution. The equations derived by Tatsumi (1957) for isotropic turbulence on the basis of this hypothesis are integrated numerically for specific intitial conditions. The initial values of the Reynolds number, $R_ \lambda = (u^{\overline{2}})^{\frac {1}{2}} \lambda|v$, assigned in this investigation are 28·8, 14·4, 7·2 and 1·8, where $(u^{\overline{2}})^{\frac {1}{2}}$ is the root-mean-square turbulent velocity, λ the dissipation length and v the kinematic viscosity coefficient.

The result of such computations is that the energy spectrum does develop negative values for Rλ = 28·8 and 14·4. This first occurs at a time approximately 2·8 for Rλ = 28·8 and 4·2 for Rλ = 14·4 The time-scale here is $(E_0 k^3_0)^-{\frac{1}{2}}$, where k0 is a wave-number scale typical of the energy-containing velocity component and E10, a typical value of the energy spectrum, is given by $4 \pi ^-{\frac{1}{2}}k_0^{-1}\overline{u^2}$

There is no evidence of the energy distribution tending to become negative for Rλ = 7·2 and 1·8. It is observed that inertial effects are relatively weak at Rλ = 7·2 and the decay process is largely controlled by viscous effects. For Rλ = 1·8 a purely viscous calculation is found to be adequate to account for the numerically integrated results.

The pressure fluctuations caused by a turbulent boundary layer on a solid surface have been obtained over the Mach number range 1·33-5·00. The root-mean-square values of the pressures are found to be proportional to the local skin friction. The proportionality constant is not a strong function of Mach number in the range investigated but is larger than the previously obtained subsonic values by about a factor of 2. The space-time correlations with the space separation in the direction of the mean flow are characterized by a convection speed Uc, and this speed falls from 0·8 of the free-steam velocity at M = 1·33 to 0·6 at M = 5. The peak value of the correlation coefficient falls to one-half for a spatial separation of the measuring points of about two-tenths of the boundary layer thickness.

The problem of surface waves due to the interaction of a blast-generated shock wave with an ideal incompressible heavy (or light) fluid of infinite depth has been investigated in both two and three dimensions. The wave integrals have been evaluated exactly for arbitrary as well as special pressure distributions on the fluid surface. Asymptotic values of the surface wave elevation have been obtained for large values of time at a large distance from the seat of the applied pressure. Certain peculiarities of the motion are discussed.

Experiments have been carried out in a cylindrical magnetic shock tube on the propagation of a shock wave into a fluid containing a uniform transverse magnetic field. Cylindrical shock waves in the Mach number range from 20 to 100 have been produced which are very stable and reproducible. Theory predicts basic differences between the case where the gas is initially highly conducting (pre-ionized) and the case where a gas-ionizing shock propagates into a cold gas. Good agreement between theory and experiment has been obtained for the gas ionizing shocks.

The effect of radiative transfer on thermal stability when a Coriolis force is also acting has been examined. Two asymptotic approximations of the radiative transfer equation have been used to study the stability problem in detail. The necessary and sufficient conditions for the validity of the principle of exchange of stabilities are also obtained.

The investigations to the shoreward travel of a bore into water at rest is extended to beaches of non-uniform slope. It is shown that the shore singularity established in Part 1 (Ho & Meyer 1962) still furnishes an approximate solution. The shape of the beach, like the shape of the wave forming the bore, is found to influence the bore development close to shore almost only in so far as it determines the basic velocity scale of the bore.

When a bore travels shoreward into water at rest on a beach, then according to the first-order non-linear long-wave theory, the bore accelerates and decreases in height, until it collapses at the shore. The investigation here reported concerns the question, what happens next? It is formulated as a singular characteristic boundary-value problem with somewhat unusual mathematical properties. Its asymptotic solution predicts a rather thin sheet of run-up and back-wash with some unexpected features.

A conductivity tensor is calculated for a partially ionized gas subject to electric fields which are harmonic in space and time, and to a uniform magnetic field. The collisions between charged particles and neutral molecules are included in the theory in an approximate way by means of a simplified form of Boltzmann's equation, as proposed by Bhatnagar, Gross and Krook. A simplified expression for the longitudinal component of the tensor is derived. Some applications of the results are mentioned, and will be described in detail in further papers.

A theory is given for the non-linear transfer of energy from gravity waves on water to capillary waves. When a progressive gravity wave approaches its maximum steepness it develops a sharp crest, at which the surface tension must be locally important. This gives rise to a travelling disturbance which produces a train of capillary waves ahead of the crest, i.e. on the foward face of the gravity wave. The capillary waves, once formed, then take further energy from the gravity wave through the radiation stresses, at the same time losing energy by viscosity.

The steepness of the capillary waves is calculated and found to be in substantial agreement with some observations by C. S. Cox. An approximate expression for the ripple steepness near the crest of the gravity wave is $(2 \pi|3)e^{-g|6T^\prime k^2},$ where T’ is the surface tension constant and K Is the curvature at the wave crest. The ripple steepness also varies with distance from the wave crest.

Under favourable circumstances the dissipation of energy by the capillary waves can be many times the dissipation in the gravity waves. The capillary waves may therefore play a significant role in the generation of waves by wind, in that they tend to delay the onset of breaking.