Theory Of Internal Waves Research Articles

Linear and weakly nonlinear internal wave theories applied to "morning glory" waves

Linear and weakly nonlinear internal wave theories applied to "morning glory" waves

On some possible interactions between internal waves and sea ice in the marginal ice zone

The ice edges of the world ocean are generally the site, at times when winds are blowing off the ice, of regularly spaced surface bands of ice floes. These bands have size scales of the order of 1–10 km, and their long axes are oriented approximately normal to the wind direction. Available oceanic temperature and salinity data from the Bering and Greenland Sea ice edge regions suggest that these ice bands are commonly underlain by a two‐layered density structure which is maintained by net melting along the ice edges. Linear internal wave theory is applied to these data to compute first‐mode interfacial wave phase speeds. A simple analytical model is developed that demonstrates the feasibility of generation of such interfacial internal waves by the stress discontinuity due to off‐ice winds blowing over either a stationary or a moving ice edge. It is qualitatively shown that the computed internal wave phase speeds and wavelengths are, under many conditions, compatible with the speeds and spacings of the surface ice bands. This compatibility suggests, in turn, that coupling between internal waves and ice bands may commonly occur. Some possible implications of the generation and presence of these internal waves upon other ice edge processes, such as air‐sea heat and momentum transfer, are qualitatively discussed.

A Non-Reflective Upper Boundary Condition for Limited-Height Hydrostatic Models

Abstract A simple upper boundary condition for hydrostatic, Boussinesq models is derived from a linear internal wave theory, assuming a uniform stratification and no Coriolis effects. This condition is applied in a two-dimentional nonlinear model of the planetary boundary layer. The numerical implementation and some stability problems are discussed. A comparison of the results of numerical experiments using different vertical extensions with analytical solutions is used to show that the condition provides a satisfactory solution to the problems of radiation of upward propagating energy.

Nonlinear Theory of Gravity Waves: Momentum Deposition, Generalized Rayleigh Friction, and Diffusion

Abstract A nonlinear theory of internal gravity waves is extended to slowly varying mean winds u0. This theory determines the wave amplitudes, enhanced diffusion and momentum deposition, for a broad spectrum of waves as well as for a single wave. It is shown that the momentum deposition is not constant with altitude, even above the altitude where a wave saturates (breaks). The deposition is seen to decrease as a critical level is approached. The rate of decrease varies with spectral width. In general, the behavior of saturated waves near a critical level is shown to differ greatly from that of linearly growing waves. It is then proven that the momentum deposition is always related to the diffusion coefficient in a very simple and general way. Hence, knowledge of one implies knowledge of the other. Both quantities vary with altitude in nearly the same way. A feature of the theoretical momentum deposition is that it sometimes behaves like Rayleigh friction and sometimes not. It is referred to as “generalize...

Low-Frequency Currents and Continental Shelf Waves in the Southern Weddell Sea

Abstract The salient features of low-frequency current fluctuations, obtained from an analysis of eight current meter records from the continental shelf and slope of the southern Weddell Sea, are compared to baroclinic and barotropic theories. A simple baroclinic theory of internal waves is used successfully to predict high-frequency spectral cutoff values from low-frequency velocity ellipse calculations made from the continental slope mooring data. The success of this theory indicates that the higher spectral energy levels observed over the slope compared to the shelf are probably due to baroclinic motions. A barotropic model of free continental shelf waves probably by Saint-Guily (1976) is adapted for the local topography and the predictions of the model compared to observations. Coherences and phases between moorings separated by 10 and 160 km in the along-shelf direction provide substantial evidence of the existence of shelf waves as predicted by the theory for periods of 3–60 days and for the lowest ...

Self-consistent effective-medium theory of random internal waves

In the first part of this paper we introduce a path-integral formalism for the internal-wave field of the ocean. The intent is to show that this type of formalism may be useful in suggesting improvements to current calculations, as it provides a framework for applying a wide variety of approximations that have been and are currently being developed in other areas of physics. We demonstrate the method by deriving equations for a self-consistent field approach (also known as the direct-interaction approximation). The experience in other areas of physics is that the self-consistent field approximation is more reliable than lowest-order perturbation theory. The end result of the DIA is the determination of an effective linear model for the description of internal waves in the deep-ocean environment. In the second part of the paper we obtain Hasselmann's source function by a prescribed limiting process and are able to indicate possible improvements in related calculations by comparing the limiting assumptions with numerically computed values.

Large Amplitude Internal Waves of Permanent Form

In this paper the weakly nonlinear theory of long internal gravity waves propagating in stratified media is extended to the fully nonlinear case by treating Long's nonlinear partial differential equation for steady inviscid flows without restriction to small amplitudes and long wavelengths. The existence of finite amplitude solutions of “permanent form” is established analytically for a large class of stratification profiles, and properties are calculated numerically for the case of a hyperbolic tangent density profile in a large range of fluid depths. The numerical results agree well with the experimental data of Davis and Acrivos over the full range of wave amplitudes measured; such agreement is not obtainable with existing weakly nonlinear theories.

An investigation of internal solitary waves in a two-fluid system

The results of an experimental investigation dealing with finite-amplitude internal solitary waves in a two-fluid system are presented. Particular attention is paid to characterizing solitons in terms of their shape and amplitude-wavelength scale relationship. Two cases are considered, viz., a shallow- and a deep-water configuration, in order to study the depth effect upon the propagational characteristics of the waves. Comparisons are made between the experimental results and existing internal-wave theories. In addition, discussion is presented describing how these existing theories may be extended to include higher-order nonlinear and viscous effects.

Preliminary results of underwater acoustic measurements at a mid-ocean site, 1977

The Mid-ocean Acoustics Transmission Experiment 1977 (MATE), was conducted to measure the acoustic variability at four frequencies from 2 to 13 kHz over an 18 km path at Cobb Seamount. Preliminary results indicate time of arrival fluctuations for a single refracted path are well predicted by internal wave theories. The variation in acoustic intensity of the single path, however, is not well predicted. Simultaneous environmental measurements make possible a careful comparison between the observed acoustics and the physical fields. The environmental data sets suggest alternative mechanisms which must be considered in predicting the observed acoustic variability. [Work supported by ONR.]

Viscous effects in lee waves

Abstract A horizontal cylinder is traversed horizontally normal to its longitudinal axis in density stratified brine, and the lee waves which are produced over a range of model velocities are viewed with a six mirror Mach Zehnder interferometer which allows the density distribution to be evaluated. At the lowest model velocities, viscous effects limit the number of visible waves to no more than three. The viscous internal wave theory of Chang and Stevenson is extended and used to calculate lee waves over a ridge. Although the linear theory does not include the no-slip condition at the lower boundary and does not include the upstream wake, which is clearly visible in the experiments, the way in which the waves attenuate away from the model is reproduced quite well by the theory. The viscous results will be compared with the standard inviscid calculations.

Radiation damping of inertial oscillations in the upper ocean

Turbulent motions within the wind-mixed layer, which is advected by near-surface inertial oscillations, excite internal gravity waves in the underlying ocean layers. Momentum transport in the radiated wave field results in a drag force on the inertial currents. Because the magnitude of the inertial currents is large compared with the turbulence intensity, the resultant rate of dissipation of inertial oscillation energy is approximately equal to the energy flux in the radiated wave field. Using linear internal wave theory, asymptotic results are derived for the energy flux in terms of the Brunt-Väisälä frequency N below the mixed layer, the magnitude U0 of the inertial current, the integral length scale l of the mixed-layer turbulence and the mean-square displacement 〈ζ20〉 of the base of the mixed layer. For representative conditions, we estimate an energy flux of 1-10 erg/cm2 s into relatively short (wavelength of order 2πU0/N) high frequency (of order, but less than, N) internal waves. The resultant decay times for inertial oscillation energy range from a day to a week or so, in agreement with reported observations on the decay of inertial oscillations in the upper ocean. The estimated energy flux is comparable in magnitude to estimates for other internal wave generation mechanisms, indicating that, in addition to being a significant sink of inertial energy, this process may locally represent a significant source of internal wave energy in the open ocean.

A kinetic theory for internal waves in a randomly stratified fluid : MYSAK L. A. and M. S. HOWE, 1976. Dynam. Atmos. Oceans, 1 (1): 3–31

A kinetic theory for internal waves in a randomly stratified fluid : MYSAK L. A. and M. S. HOWE, 1976. Dynam. Atmos. Oceans, 1 (1): 3–31

A kinetic theory for internal waves in a randomly stratified fluid

Abstract We discuss the transport of energy of internal waves propagating in a stratified unbounded fluid with randomly varying buoyancy frequency N of the form N 2 = N 0 2 [1 + ϵΞ( )] . Here N0 = constant, 0 = (x,z) where x and z are respectively horizontal and vertical coordinates. In the limit of small ϵ, a linear kinetic equation (transport equation) is derived which describes the space—time evolution of the mean wave energy density in such a medium. When it is integrated over all wave number space, the kinetic equation implies the total conservation of wave energy. The approach used is reminiscent of the nonlinear wave interaction theories of Phillips (1960), Benny (1962) and Hasselmann (1966) and others, in which the random microstructure Ξ would be regarded as a nonpropagating (zero-frequency) random wave field. Our analysis is based on the Eulerian equations of motion in which no a priori assumptions are made regarding the scattered wave field — on the contrary, it is a deduction of the theory presented here that only the propagating internal wave modes participate in the energy exchange processes. In particular we do not assume that the internal wave field constitutes a homogenous assembly of random wave packets evolving in time alone — unlike much of the earlier work — and this enables us to treat the scattering of individual internal waves by the microstructure. The kinetic equation is used to determine the energy transmitted through and reflected by a horizontally oriented random slab which models a layer of microstructure of finite thickness in the ocean. Specifically, we show that significant reflection can occur when Ψ(2k0z) is sufficiently large, where Ψ is the vertical wavenumber spectrum of the microstructure fluctuations Ξ(z) and k0z is the vertical wavenumber of the incident wave. We also show that the reflection coefficient increases monotonically with increasing frequency, which is in qualitative agreement with recent measurements at site D(39°N, 70°W) which indicate that in regions where density inhomogeneities are present, the vertical coherence decreases with increasing frequency. Actual numerical estimates for the reflection coefficient are obtained for vertical microstructure data from station P(50°N, 145°W). It is found that for intermediate wavelengths — 0(102m) — and a broad band of frequencies (0.6 ⩽ ω/N0

Progressive waves of permanent form in continuously stratified fluids

A theory for internal progressive waves of permanent form in any continuously stratified fluid is presented, and a calculation for the flow and the wave velocity is carried out for an exponential stratified fluid. The most important conclusion from this calculation, also valid for other weak stratifications, is that the wave velocity always decreases with the amplitude, provided the density gradient is weak and the wavelength is not too short. This conclusion is significant because it entails the existence or nonexistence of solitary waves in weakly stratified fluids. The validity of the Boussinesq approximation and the significance of the well-known exactly linear cases are also discussed.

Solitary and cnoidal planetary waves

Abstract A class of long planetary waves in a zonal channel analogous to the solitary and cnoidal waves of surface and internal gravity wave theory is discussed. On a mid-latitude β-plane, such waves exist as the result of divergence, non-uniform zonal velocity fields or bottom topography. In all cases studied the wave profile along the channel was found to satisfy the Korteweg-de Vries equation.

Internal waves in a wedge-shaped region

The Green's functions are found for a line source of internal waves in a wedge of stratified fluid of constant Brunt–Väisälä frequency, and are used to discuss the diffraction of internal waves by a wedge in all cases when the vertex angle of the wedge of fluid exceeds the acute angle between a characteristic and the horizontal. Robinson's (1970) results are confirmed and extended.It is found that the diffracted waves are as important as the incident and reflected ones at all points that lie within a quarter-wavelength or so of either characteristic that passes through the apex. Also, in cases when all the reflected waves are inclined forwards, the diffracted waves lead to a positive backscatter of energy. When the vertex angle of the fluid wedge is less than the characteristic angle, the diffraction problem appears to be ill-posed, and, instead, the motion due to a vibrating body in the wedge of fluid is considered.A general conclusion is that the so-called ray theory for internal waves, in which the incident and reflected waves alone are considered, has similar limitations to the geometrical theory of optics. Both theories involve the assumption that the typical dimensions in the problem are large compared to the wavelength.

Internal Equatorial Planetary-Scale Waves in Shear Flow

Abstract The theory of internal equatorial Planetary-scale waves as developed by Matsuno and by Lindzen is extended to include the effects of shear in the basic state. By means of a numerical study we find that both the equatorial Kelvin wave (the gravest symmetric westerly mode) and the Yanai wave (the gravest anti- symmetric easterly wave), the two most commonly observed internal equatorial waves, are absorbed shortly before reaching critical levels where their frequencies are Doppler shifted to zero. Away from such levels, these waves retain their identity within shear zones. However, they become more closely confined to the equator as their Doppler-shifted frequencies are reduced. It is found, moreover, for a given frequency and vertical wavelength, that the vertical group velocity of the Yanai wave is smaller than that of the Kelvin wave. In addition, as the Doppler-shifted frequency ωˆ is reduced, the vertical group velocity of the Yanai wave diminishes as ωˆ3 while the vertical group velocity of Ke...

A theory of the propagation of internal gravity waves of finite amplitude

A non-linear theory of internal gravity waves of finite amplitude is developed in terms of conservation equations averaged with respect to the phase. The theory overcomes the failure of linear ray theory in regions in which waves are trapped and establishes the conditions under which finite amplitude waves may propagate. It gives a geometrical representation of the degeneration of waves into quasi-turbulence and predicts the dependence of the energy density on its parameters.

Evidence for reversible heating in theEregion from radar Thomson scatter observations of ion temperature

Values of ion temperature (Ti) and ion-neutral collision frequency (ν) in the daytime E region have been obtained from radar Thomson scatter measurements at the Arecibo Ionospheric Observatory. These quantities were deduced from the shape of the autocorrelation function of the backscattered signal that was measured with a height resolution of 15 km using a double-pulse technique. Usually the daily variation of Ti at 115 and 130 km exhibited wave-like structure that must also have been present in the neutral atmosphere because the ion and neutral constituents have the same temperature at these altitudes. The observed temperature oscillations are interpreted as a manifestation of the reversible adiabatic heating accompanying long-period internal gravity waves in the neutral atmosphere. A simple harmonic analysis of the temperatures for a number of days gives dominant wave periods of from 7 to 13 hours and fractional amplitudes as large as 22%. On the reversible heating explanation, the observed amplitudes of the temperature waves imply average horizontal wind amplitudes of 60 to 70 m/sec. A downward phase progression (corresponding to upward energy propagation) is evident with an average vertical wavelength near 60 km. A significant wave-like variation is also detected in some of the ion-neutral collision frequency results at 115 km. The period is similar to that of the corresponding temperature wave, but the fractional amplitude is larger, and there is an appreciable phase difference. The phase and amplitude of the variation in the neutral concentration, as predicted from the collision frequency variation using the polarization ion-neutral interaction, does not agree with that deduced from the temperature variation using simple internal gravity wave theory with upward energy propagation. The resolution of this discrepancy may depend on the removal of Some of the idealizations in the gravity wave theory or the inclusion of additional ion-neutral interaction terms.

Thomson scatter observations of a gravity wave in the ionospheric F-region

Abstract Thomson scatter observations of the ionospheric F have revealed important oscillations of the ionization density, velocity and temperatures. These oscillations are interpreted using the theory of internal gravity waves originating in the auroral zone and their presence appears to be correlated with strong magnetic activity.