Internal gravity waves of small amplitude propagate in a Boussinesq inviscid, adiabatic liquid in which the mean horizontal velocity U ( z ) depends on height z only. If the Richardson number R is everywhere larger than t, the waves are attenuated by a factor exp(-2n(R -if&) as they pass through a critical level at which U is equal to the horizontal phase speed, and momentum is transferred to the mean flow there. This effect is considered in relation to lee waves in the airflow over a mountain, and in relation to transient localized disturbances. It is significant in considering the propagation of gravity waves from the troposphere to the ionosphere, and possibly in transferring horizontal momentum into the deep ocean without substantial mixing. 33Fluid Mech. 37 33-2

This paper presents a broad investigation into the properties of steady gravity currents, in so far as they can be represented by perfect-fluid theory and simple extensions of it (like the classical theory of hydraulic jumps) that give a rudimentary account of dissipation. As usually understood, a gravity current consists of a wedge of heavy fluid (e.g. salt water, cold air) intruding into an expanse of lighter fluid (fresh water, warm air); but it is pointed out in § 1 that, if the effects of viscosity and mixing of the fluids at the interface are ignored, the hydrodynamical problem is formally the same as that for an empty cavity advancing along the upper boundary of a liquid. Being simplest in detail, the latter problem is treated as a prototype for the class of physical problems under study: most of the analysis is related to it specifically, but the results thus obtained are immediately applicable to gravity currents by scaling the gravitational constant according to a simple rule.In § 2 the possible states of steady flow in the present category between fixed horizontal boundaries are examined on the assumption that the interface becomes horizontal far downstream. A certain range of flows appears to be possible when energy is dissipated; but in the absence of dissipation only one flow is possible, in which the asymptotic level of the interface is midway between the plane boundaries. The corresponding flow in a tube of circular cross-section is found in § 3, and the theory is shown to be in excellent agreement with the results of recent experiments by Zukoski. A discussion of the effects of surface tension is included in § 3. The two-dimensional energy-conserving flow is investigated further in § 4, and finally a close approximation to the shape of the interface is obtained. In § 5 the discussion turns to the question whether flows characterized by periodic wavetrains are realizable, and it appears that none is possible without a large loss of energy occurring. In § 6 the case of infinite total depth is considered, relating to deeply submerged gravity currents. It is shown that the flow must always feature a breaking ‘head wave’, and various properties of the resulting wake are demonstrated. Reasonable agreement is established with experimental results obtained by Keulegan and others.

This paper presents a general theoretical treatment of a new class of long stationary waves with finite amplitude. As the property in common amongst physical systems capable of manifesting these waves, the density of the (incompressible) fluid varies only within a layer whose thickness h is much smaller than the total depth, and it is h rather than the total depth that must be considered as the fundamental scale against which wave amplitude and length are to be measured. Internal-wave motions supported by the oceanic thermocline appear to be the most promising field of practical application for the theory, although applications to atmospheric studies are also a possibility.The waves in question differ in important respects from long waves of more familiar kinds, and in § 1 their character is discussed on the basis of a comparison with solitary-wave and cnoidal-wave theories on customary lines, such as apply to internal waves in fluids of limited depth. A summary of some simple experiments is included at the end of § 1. Then, in § 2, the comparatively easy example of two-fluid systems is examined, again to illustrate principles and to prepare the way for the main analysis in § 3. This proceeds to a second stage of approximation in powers of wave amplitude, and its leading result is an equation (3·51) determining, for arbitrary specifications of the density distribution, the form of the streamlines in the layer of heterogeneous fluid. Periodic solutions of this equation are obtained, and their properties are discussed with regard to the interpretation of internal bores and wave-resistance phenomena. Solutions representing solitary waves are then obtained in the form \[ f(x) = a\lambda^2/(x^2+\lambda^2), \] where xis the horizontal co-ordinate and where aΛ = O(h2). (The latter relation between wave amplitude and length scale contrasts with the customary one, aΛ2 = O(h3)). The main analysis is developed with particular reference to systems in which the heterogeneous layer lies on a rigid horizontal bottom below an infinite expanse of homogeneous fluid; but in § 4 ways are given to apply the results to various other systems, including ones in which the heterogeneous layer is uppermost and is bounded by a free surface. Finally, in §5, three specific examples are treated: the density variation with depth is taken, respectively, to have a discontinuous, an exponential and a ‘tanh’ profile.

Several topics are studied concerning mathematical models for the unidirectional propagation of long waves in systems that manifest nonlinear and dispersive effects of a particular but common kind. Most of the new material presented relates to the initial-value problem for the equation u t + u x + u u x − u x x t = 0 , ( a ) , whose solution u ( x,t ) is considered in a class of real nonperiodic functions defined for ࢤ∞ < x < ∞, t ≥0. As an approximation derived for moderately long waves of small but finite amplitude in particular physical systems, this equation has the same formal justification as the Korteweg-de Vries equation u t + u x + u u x − u x x x = 0 , ( b ) with which ( a ) is to be compared in various ways. It is contended that ( a ) is in important respects the preferable model, obviating certain problematical aspects of ( b ) and generally having more expedient mathematical properties. The paper divides into two parts where respectively the emphasis is on descriptive and on rigorous mathematics In §2 the origins and immediate properties of equations ( a ) and ( b ) are discussed in general terms, and the comparative shortcomings of ( b ) are reviewed. In the remainder of the paper (§§ 3,4) - which can be read independently Preceding discussion _ an exact theory of ( a ) is developed. In § 3 the existence of classical solutions is proved: and following our main result, theorem 1, several extensions and sidelights are presented. In § 4 solutions are shown to be unique, to depend continuously on their initial values, and also to depend continuously on forcing functions added to the right-hand side of ( a ). Thus the initial-value problem is confirmed to be classically well set in the Hadamard sense. In appendix 1 a generalization of ( a ) is considered, in which dispersive effects within a wide class are represented by an abstract pseudo-differential operator. The physical origins of such an equation are explained in the style of § 2, two examples are given deriving from definite physical problems, and an existence theory is outlined. In appendix 2 a technical fact used in § 3 is established.