Stratified Liquid Research Articles

On waves generated in rotating stratified liquids by travelling forcing effects

The pattern and propagation of waves excited by forcing effects that may oscillate with a frequency σo and are travelling with a uniform velocity U along the axis of rotation in a rotating stratified liquid are studied by employing the technique of Lighthill. The fluid is assumed to be an unbounded, in viscid, incompressible, non-diffusive, rotating stratified liquid, rotating with a constant angular velocity Ω about a vertical axis and with a density decreasing vertically upwards. It is found that the effect of rotation on stratification or vice versa is to reduce the region of disturbance and to split the wave crests for Ω0 ≠ 0. The periodic nature of the forcing effect excites various systems of waves which are otherwise coincident.

Similarities between Ion Waves in Plasmas and Gravity Waves in Incompressible Fluids

Some similarities between ion waves in plasmas and gravity waves in incompressible fluids are investigated. It is shown that for zero ion temperature the ion-wave dispersion relation is similar to that of gravity waves in a stratified liquid between rigid, horizontal walls; for large wavelength the ion waves behave as the surface gravity waves of shallow-water theory. The general character of the pattern of ion waves arising in steady plasma flows is analyzed for arbitrary ion temperature, wavelength, and acoustic Mach number (which is based on the ion-acoustic speed), and is compared to the pattern of surface gravity waves in steady water flows when surface tension is taken into account.

The flow of a stratified viscous liquid down an inclined plane

The problem of the flow of a viscous liquid layer down an inclined plane is analyzed in the case in which the local viscosity depends linearly upon the height above the plane. By an application ofYih's technique, the velocity of long waves is calculated and the criterion for their stability is established. These are then compared with the corresponding results for a homogeneous layer.

Two automatic systems to generate gravitationally stable density stratifications in bodies of liquid at rest

Two automatic control systems for filling large tanks with gravitationally stable density stratified liquids are described. They are: (i) an analogue method using gate and mixture control valves operated through servomechanisms by rotating cams; (ii) a digital method with a numerically controlled metering-type pump. Measurements taken of a typical stratification showed that the actual density distribution was within 0?02% of that expected using the analogue method and within 0?01% using the digital method.

On the mean motion induced by internal gravity waves

A train of internal gravity waves in a stratified liquid exerts a stress on the liquid and induces changes in the mean motion of second order in the wave amplitude. In those circumstances in which the concept of a slowly varying quasi-sinusoidal wave train is consistent, the mean velocity is almost horizontal and is determined to a first approximation irrespective of the vertical forces exerted by the waves. The sum of the mean flow kinetic energy and the wave energy is then conserved. The circulation around a horizontal circuit moving with the mean velocity is increased in the presence of waves according to a simple formula. The flow pattern is obtained around two- and three-dimensional wave packets propagating into a liquid at rest and the results are generalized for any basic state of motion in which the internal Froude number is small. Momentum can be associated with a wave packet equal to the horizontal wave-number times the wave energy divided by the intrinsic frequency.

Surface and internal waves in a liquid of variable depth

Consider a stably stratified liquid, whose density varies exponentially with the vertical co-ordinate, that is bounded above by a free surface and below by a bed whose height depends on only one of the horizontal co-ordinates. Suppose that a gravity wave, that may be either a surface or an internal one, is travelling in a direction normal to the lines of constant depth. It is shown that if the frequency is below a certain value an infinite number of waves, all of the same frequency but having differing wave lengths, are generated and expressions for their amplitude are given in terms of the changes in depth which are assumed to be small.

Acoustic Wave Scattering and Transmission by Corrugated Surfaces

An exact theoretical model for the interaction between an acoustic plane wave and a stationary internal wave of a two-layer fluid is presented. Weber's two-dimensional analog of Helmholtz's integral formula is employed to represent the scattered and the transmitted acoustic fields. The boundary velocity potentials and their normal derivatives are expanded into complex Fourier series. The complex boundary Fourier coefficients are related by a pair of simultaneous, doubly infinite algebraic equations. Generalized Rayleigh scattering and transmission coefficients are found and exact expressions for the acoustic velocity potentials for the scattered and transmitted fields are given. The principal features of this development are its wide applicability to internal waves of arbitrary amplitude and wavelength and to strongly as well as weakly stratified liquids.

On the interaction between internal gravity waves and a shear flow

The theory of wave action conservation is summarized, and its interpretation in terms of the working, against the rate of strain of the basic flow, of an interaction stress associated with the waves is discussed. Usually this interaction stress is identical with the radiation stress of a uniform plane wave. The problem of internal gravity wave propagation in an incompressible, stratified Boussinesq liquid is considered in detail for a more general basic flow than has hitherto been treated, and the interaction stress is derived. One component of the interaction stress tensor is only equal to the corresponding component of the radiation stress tensor if we include in the latter, in addition to the Reynolds stress, a term associated with the redistribution of matter, on the average, by the wave. Two other components of the radiation stress tensor are modified in a similar manner, but the corresponding components of the interaction stress tensor are undefined, and so no comparison is possible.

The internal wave pattern produced by a sphere moving vertically in a density stratified liquid

Experiments were conducted to test the linear theory of internal gravity waves produced in a stably stratified liquid by the steady vertical motion of a sphere. The measurements of the phase configuration in a medium whose density increased linearly with depth were made by means of a Toepler-schlieren system. The agreement between observation and prediction was found to be good.

The two-dimensional wave pattern produced by a disturbance moving in an arbitrary direction in a density stratified liquid

The two-dimensional internal wave pattern produced by a small disturbance moving with constant velocity along a line of arbitrary inclination in an inviscid, density stratified liquid is studied. It is shown how the far field wave pattern evolves as the inclination changes from the extreme of horizontal motion to that of vertical motion. It is found that, in general, waves propagate ahead of the disturbance.

A theoretical and experimental investigation of the phase configuration of internal waves of small amplitude in a density stratified liquid

Experiments were conducted to test the linear theory of internal gravity waves produced in a stably stratified liquid by the forced oscillations and the initial impulsive motion of a two-dimensional stationary disturbance. The measurements of the wave configuration in a medium whose density increased linearly with depth were made by means of a Toepler-schlieren system. The agreement between observation and prediction was found to be good.

The use of schlieren and shadowgraph techniques in the study of flow patterns in density stratified liquids

Various methods of establishing fluid media stably stratified in either temperature or the concentration of a foreign constituent are compared. A method used to construct a salt solution with precisely predetermined stratification is described. The Euler-Lagrange equations are solved exactly for a medium whose refractive index varies linearly in one dimension; the solution shows that a schlieren system may be used. For media with non-linear one-dimensional variation of refractive index, a modified shadowgraph technique may be employed. A device for the quantitative evaluation of a one-dimensional stratification is outlined and an example of its use is given.

A method for generating linear density profiles in laboratory tanks

A by-product of an investigation of turbulent wakes in density stratified liquids [Stockhausen et al., 1966] conducted at the MIT Hydrodynamics Laboratory has been the development of a fast and efficient method for generating a linear density profile in a liquid contained in a large laboratory tank. The method that has been generally used heretofore [e.g., Schooley and Stewart, 1963; Wu,, 1965] consists of carefully placing in the tank a series of fluid layers, with an incremental density difference between successive layers, and then allowing molecular diffusion to smooth the profile. This procedure is very time consuming and tedious and is not practical for use in a large tank, of the dimensions (15.24 meters long, 1.37 meters wide, and 1.07 meters deep) used in the present investigation. Oster [1965] has described a profile generating technique that involves mixing together two fluids of different density while introducing them at different and varying rates into the experimental vessel.

Flow and Stress Near an Interface between Stratified Liquids

Observations are made of a density current system in which salt water flows turbulently under a pool of fresh water. The density and rate of flow of the salt water are varied, resulting in varying degrees of agitation of the interface. Measurements include the interface slope, the velocity and density profiles, and the rate of mixing. Profiles of stress and effective viscosity are developed from the observations and the equation of motion. In the zone of stable stratification the effective viscosity has a minimum equal to or greater than the molecular viscosity depending upon whether the interface is laminar or agitated. Dimensionless relationships between the observed or computed quantities and the given fluid properties and flow characteristics are investigated. The principal independent variables are a Reynolds number and a Froude number. An interfacial stress coefficient is found to depend upon both. With fair accuracy, the velocity profile can be related to the interfacial stress in a manner analogous to that for turbulent flow near a rigid boundary.

Long waves, lee waves and gravity waves

AbstractIt is shown that the equation determining the streamlines of large‐scale atmospheric motions (jet streams and long waves) is similar to the equation determining the streamlines of small‐scale gravitational motions of a stratified liquid. Both motions are characterized by the occurrence of jets, the number of jets being determined by the value of a dimensionless parameter. Atmospheric flow at 500 mb is compared with theoretical solutions (derived by Long 1955) of the non‐linear equations for the flow of a stratified liquid. Linearization of the equation for large‐scale atmospheric motions leads to the Rossby wave equation, whereas linearization of the equation for the motion of a stratified liquid leads to Scorer's lee‐wave equation.