Dispersion Relation For Internal Waves Research Articles

Estimating the In Situ Stratification via Remotely Sensed Internal Wave Speeds

Abstract This work tests a methodology for estimating the ocean stratification gradient using remotely sensed, high temporal and spatial resolution field measurements of internal wave propagation speeds. The internal wave (IW) speeds were calculated from IW tracks observed using a shore-based, X-band marine radar deployed at a field site on the south-central coast of California. An inverse model, based on the work of Kar and Guha, utilizes the linear internal wave dispersion relation, assuming a constant vertical density gradient is the basis for the inverse model. This allows the vertical gradient of density to be expressed as a function of the internal wave phase speed, local water depth, and a background average density. The inputs to the algorithm are the known cross-shore bathymetry, the background ocean density, and the remotely sensed cross-shore profiles of IW speed. The estimated density gradients are then compared to the synchronously measured vertical density profiles collected from an in situ instrument array. The results show a very good agreement offshore in deeper water (∼50–30 m) but more significant discrepancies in shallow water (20–10 m) closer to shore. In addition, a sensitivity analysis is conducted that relates errors in measured speeds to errors in the estimated density gradients. Significance Statement The propagation speed of ocean internal waves inherently depends on the vertical structure of the water density, which is termed stratification. In this work, we evaluate and test with real field observations a technique to infer the ocean density stratification from internal wave propagation speeds collected from remote sensing images. Such methods offer a way to monitor ocean stratification without the need for extensive in situ measurements.

Experimental generation of axisymmetric internal wave super-harmonics

In this paper, we present an experimental study of weakly non-linear interaction of axisymmetric internal gravity waves in a resonant cavity, supported by theoretical considerations. Contrary to plane waves in Cartesian coordinates, for which self-interacting terms are null in a linear stratifiation, the non-linear self-interaction of an internal wave mode in axisymmetric geometry is found to be efficient at producing super-harmonics, i.e. waves whose frequencies are integer multiples of the excitation frequency. Due to the range of frequencies tested in our experiments, the first harmonic frequency is below the cut-off imposed by the stratification so the lowest harmonic created can always propagate. The study shows that the super-harmonic wave field is a sum of standing waves satisfying both the dispersion relation for internal waves and the boundary conditions imposed by the cavity walls, while conserving the axisymmetry.

Моделирование аккумуляции кинетической энергии внутренних волн в областях с большим отношением горизонтального и вертикального масштабов

Tidal forcing excites internal waves in the bulk of the ocean. Deep ocean is an example of a system with continuous stratification subject to large-scale periodic forcing. Owing to specific dispersion relation of internal waves, the domains bounded by sloping boundaries may support wave patterns with wave rays converging to closed trajectories (geometric attractors) as result of iterative focusing reflections. Previously the behavior of kinetic energy in wave attractors has been investigated in two-dimensional domain with comparable depth and length. As the geometric aspect ratio of the domain increases, the dynamic pattern of energy focusing may significantly evolve both in laminar and turbulent regimes. The present paper shows that the energy density in domains with large aspect ratio can significantly increase. In numerical simulations the input forcing has been introduced at global scale by prescribing small-amplitude deformations of the upper bound of the liquid domain. The evolution of internal wave motion in such system has been computed numerically for different values of the forcing amplitude. The behavior of the large-aspect-ratio system has been compared to the well-studied case of the system with depth-to-length ratio of order unity. A number of most typical situations has been analysed in terms of behavior of integral mechanical quantities such as total dissipation, mean kinetic energy and energy fluctuations in laminar and turbulent cases.

Selection of internal wave beam directions by a geometric constraint provided by topography.

Direct numerical simulations are performed to investigate the generation of internal waves in a linearly stratified fluid by oscillating barotropic flows over a model continental shelf-slope topography. The presence of a third wave-beam emitted from an abrupt shelf break and transverse to the topography, which has not been adequately interpreted, is now explained in terms of a geometric constraint provided by the topography. This explanation applies to wave beam selection at any abrupt topographic junction point, no matter whether it is convex or concave, or its nearby slope is subcritical or supercritical. One exception is that, at an abrupt concave point with a nearby supercritical slope, the blocking effect leads to the presence of "dead water" (i.e., no flow) and thus no wave beam is emitted. On a critical slope, two beams with opposite directions are emitted from an amphidromic point that has a distinct distance from the shelf break. In addition to the internal wave dispersion relation that restricts possible wave beam directions to form an X-pattern, the geometric constraint proposed in the present work serves as a second selection mechanism that further restricts wave beam directions. The reflective direction of a wave beam incident onto a slope can also be explained by this geometric constraint. The present work provides an updated explanation of internal wave beams emitted at abrupt topographic junction points and unifies the explanation of the wave beam direction for both wave generation and reflection processes.

Geophysical flows with anisotropic turbulence and dispersive waves: flows with stable stratification

The quasi-normal scale elimination (QNSE) is an analytical spectral theory of turbulence based upon a successive ensemble averaging of the velocity and temperature modes over the smallest scales of motion and calculating corresponding eddy viscosity and eddy diffusivity. By extending the process of successive ensemble averaging to the turbulence macroscale one eliminates all fluctuating scales and arrives at models analogous to the conventional Reynolds stress closures. The scale dependency embedded in the QNSE method reflects contributions from different processes on different scales. Two of the most important processes in stably stratified turbulence, internal wave propagation and flow anisotropization, are explicitly accounted for in the QNSE formalism. For relatively weak stratification, the theory becomes amenable to analytical processing revealing just how increasing stratification modifies the flow field via growing anisotropy and gravity wave radiation. The QNSE theory yields the dispersion relation for internal waves in the presence of turbulence and provides a theoretical reasoning for the Gargett et al. (J Phys Oceanogr 11:1258–1271, 1981) scaling of the vertical shear spectrum. In addition, it shows that the internal wave breaking and flow anisotropization void the notion of the critical Richardson number at which turbulence is fully suppressed. The isopycnal and diapycnal viscosities and diffusivities can be expressed in the form of the Richardson diffusion laws thus providing a theoretical framework for the Okubo dispersion diagrams. Transitions in the spectral slopes can be associated with the turbulence- and wave-dominated ranges and have direct implications for the transport processes. We show that only quasi-isotropic, turbulence-dominated scales contribute to the diapycnal diffusivity. On larger, buoyancy dominated scales, the diapycnal diffusivity becomes scale independent. This result underscores the well-known fact that waves can only transfer momentum but not a scalar and sheds a new light upon the Ellison–Britter–Osborn mixing model. It also provides a general framework for separation of the effects of turbulence and waves even if they act on the same spatial and temporal scales. The QNSE theory-based turbulence models have been tested in various applications and demonstrated reliable performance. It is suggested that these models present a viable alternative to conventional Reynolds stress closures.

The Surface Expression of Semidiurnal Internal Tides near a Strong Source at Hawaii. Part II: Interactions with Mesoscale Currents*

Abstract Observations of semidiurnal surface currents in the Kauai Channel, Hawaii, are interpreted in the light of the interaction of internal tides with energetic surface-intensified mesoscale currents. The impacts on internal tide propagation of a cyclone of 55-km diameter and ∼100-m vertical decay scale, as well as of vorticity waves of ∼100-km wavelength and 100–200-m vertical decay scales, are investigated using 3D ray tracing. The Doppler-shifted intrinsic frequency is assumed to satisfy the classic hydrostatic internal wave dispersion relation, using the local buoyancy frequency associated with the background currents through thermal-wind or gradient-wind balance. The M2 internal tide rays with initial horizontal wavelength of 50 km and vertical wavelength of O(1000 m) are propagated from possible generation locations at critical topographic slopes through idealized mesoscale currents approximating the observed currents. Despite the lack of scale separation between the internal waves and background state, which is required by the ray-tracing approximation, the results are qualitatively consistent with observations: the cyclone causes the energy of internal tide rays propagating through its core to increase near the surface (up to a factor of 15), with surfacing time delayed by up to 5 h (∼150° phase lag), and the vorticity waves enhance or reduce the energy near the surface, depending on their phase. These examples illustrate the fact that, even close to their generation location, semidiurnal internal tides can become incoherent with astronomical forcing because of the presence of mesoscale variability. Internal tide energy is mainly affected by refraction through the inhomogeneous buoyancy frequency field, with Doppler shifting playing a secondary but not negligible role, inducing energy transfers between the internal tides and background currents. Furthermore, the vertical wavelength can be reduced by a factor of 6 near the surface in the presence of the cyclone, which, combined with the energy amplification, leads to increased vertical shear within the internal tide rays, with implications for internal wave-induced mixing in the ocean.

Effects of rotation on self-resonant internal gravity waves in the ocean

The Resonant Triad Model (RTM) developed in ( Ibragimov, 2007), is used to study the Thorpe’s problem ( Thorpe, 1997) on the existence of self-resonant internal waves, i.e., the waves for which a resonant interaction occurs at second order between the incident and reflected internal waves off slopes. The RTM represents the extension of the McComas and Bretherton’s three wave hydrostatic model ( McComas and Bretherton, 1977) which ignores the effects of the earth’s rotation to the case of the non-hydrostatic analytical model involving arbitrarily large number of rotating internal waves with frequencies spanning the range of possible frequencies, i.e., between the maximum of the buoyancy frequency (vertical motion) and a minimum of the inertial frequency (horizontal motion). The present analysis is based on classification of resonant interactions into the sum, middle and difference interaction classes. It is shown in this paper that there exists a certain value of latitude, which is classified as the singular latitude, at which the coalescence of the middle and difference interaction classes occurs. Such coalescence, which apparently had passed unnoticed before, can be used to study the Thorpe’s problem on the existence of self-resonant waves. In particular, it is shown that the value of the bottom slope at which the second-order frequency and wavenumber components of the incident and reflected waves satisfy the internal wave dispersion relation can be approximated by two latitude-dependent parameters in the limiting case when latitude approaches its singular value. Since the existence of a such singular latitude is generic for resonant triad interactions, a question on application of the RTM to the modeling of enhanced mixing in the vicinity of ridges in the ocean arises.

Minimum and maximum propagation frequencies for internal gravity waves

Estimates of the effects of vorticity, baroclinicity, and rate of strain on the minimum and maximum propagation frequencies of gravity waves are given using a general dispersion relation for internal waves in the atmosphere and ocean. Rotating the coordinate axes to eliminate from the dispersion relation cross terms in the wavenumber makes it much easier to determine the minimum and maximum frequencies for which gravity waves can propagate. Corrections to the minimum frequency from breakdown of the WKB approximation are included.

A quasinormal scale elimination model of turbulent flows with stable stratification

A new spectral model for turbulent flows with stable stratification is presented. The model is based on a quasi-Gaussian mapping of the velocity and temperature fields using the Langevin equations and employs a recursive procedure of small-scale mode elimination that results in a coupled system of differential equations for effective, horizontal and vertical, viscosities and diffusivities. With increasing stratification, the vertical viscosity and diffusivity are suppressed while their horizontal counterparts are enhanced thus explicitly accounting for the anisotropy introduced by stable stratification. The new model is used to derive various spectral characteristics of stably stratified turbulent flows. It accounts for energy accumulation in the horizontal components at the expense of the energy reduction in the vertical component. The scale elimination algorithm explicitly accounts for the combined effect of turbulence and internal waves. A modified dispersion relation for internal waves, a relationship for internal wave frequency shift, and a threshold criterion for internal wave generation in the presence of turbulent scrambling are derived. It is shown how the model can be utilized to derive parameters needed for Reynolds-averaged Navier-Stokes modeling. Implemented in the K-ϵ format, the new model produces results that agree very well with the observational data on stably stratified atmospheric boundary layers and that are difficult to obtain using traditional two-equation turbulence modeling.

Blind prediction of broadband coherence time at basin-scales

A blind comparison with data is made with a model for the coherence time of broadband sound (133 Hz, 17 Hz bandwidth) at 3709 km. Coherence time is limited by changes in the ocean because the acoustic instruments are fixed to the Earth on the bottom of the sea with time bases maintained by atomic clocks. Although the modeled coherence time depends a bit on the difficult problem of correctly modeling relative signal-to-noise ratios, normalized correlation coefficients of the broadband signals for the data (model) are 0.90 (0.83), 0.72 (0.59), and 0.51 (0.36) at lags of 2, 4.1, and 6.2 mins respectively. In all these cases, observed coherence times are a bit longer than modeled. The temporal evolution of the model is based on the linear dispersion relation for internal waves. Acoustic propagation is modeled with the parabolic approximation and the sound speed insensitive operator.

Blind prediction of broadband coherence time at basin scales.

A blind comparison with data is made with a model for the coherence time of broadband sound (133 Hz, 17-Hz bandwidth) at 3709 km. Coherence time is limited by changes in the ocean because the acoustic instruments are fixed to the Earth on the bottom of the sea with time bases maintained by atomic clocks. Although the modeled coherence time depends a bit on the difficult problem of correctly modeling relative signal-to-noise ratios, normalized correlation coefficients of the broadband signals for the data (model) are 0.90 (0.83), 0.72 (0.59), and 0.51 (0.36) at lags of 2, 4.1, and 6.2 min, respectively. In all these cases, observed coherence times are a bit longer than modeled. The temporal evolution of the model is based on the linear dispersion relation for internal waves. Acoustic propagation is modeled with the parabolic approximation and the sound-speed insensitive operator.

The dispersion relation for internal acoustic-gravity waves in a baroclinic fluid

The dispersion relation for internal waves in a fluid is generalized from the barotropic approximation to the baroclinic case to allow for the inclination of surfaces of constant density to surfaces of constant pressure. This generalization allows the barotropic approximation to be tested in a variety of situations. The dispersion relation applies to both acoustic waves and internal gravity waves propagating in either the ocean or the atmosphere. Imaginary terms in the dispersion relation proportional to the baroclinic vector indicate energy exchange between the wave and the mean flow, a result of buoyancy being a nonconservative force in a baroclinic fluid. A buoyancy calculation shows that the baroclinic generalization of the Brunt–Väisälä frequency N is given by N2=∇ρθ⋅∇p/ρ2, where ρ is density, ρθ is potential density, and p is pressure. The baroclinicity in a weather front or cyclonic ring can sometimes have as large an effect as the Earth’s rotation on the propagation of internal gravity waves.

Internal seiches in the strongly stratified Dead Sea

and 15° due West of South, respectively. The waves propagate southward along the lake's western shore, and this cyclonic propagation sense, as well as their dispersion relation, are reminis- cent of internal Kelvin waves. The extremely large density jump at the pycnocline and the accurate determination of its depth yield a very precise fit of the measured period of the internal seiches to theoretical predictions. These data show, for the first time, that the widely used linear dispersion relation of internal waves is valid even at large reduced gravity of the order of 10 -1 m·s -2 .

On the Barber dispersion relation for internal waves

We investigate the dispersive behaviour of internal waves supported by a density gradient in a fluid which has an upper boundary surface. By identifying a realistic model for the density gradient which leads to an analytically tractable internal wave eigenvalue equation we are able to verify the general applicability of, and to characterize the deviations from, the simple dispersion relation proposed recently by Barber. We also consider the effect of a linear background density variation superimposed on the localized pycnocline; this establishes a minimum frequency and wavenumber consistent with the propagation of the internal wave and leads to significant deviations from the Barber result.

The effects of laterally sloping upper and lower boundaries on waves and instability in stratified shear flows

The dispersion relation for internal waves travelling in the horizontal direction of a two-dimensional stratified shear flow is affected by the presence of topography, or upper and lower boundaries, which vary only in the horizontal direction normal to that of the flow. This is in accordance with Grimshaw's (1978) findings for long internal waves. Such topography, however, imposes boundary conditions which affect both the growth rates and orientation of the crests of unstable disturbances. Predictions are derived for the particular case of instability of a fluid with a thin interface separating two fluids of different densities in relative flow, which are confined in a tube with horizontal generators but which has upper and lower parallel plane boundaries inclined to the horizontal at an angle α. These predictions are compared with laboratory experiments. There is good qualitative agreement and some quantitative agreement, at least when the predicted angle, γ, between the crest lines of the disturbances and the cross-flow direction, is small. Relatively small disturbances, or billow, amplitudes are found along the centreline of the tube. At large γ, however, the disturbance crests becomes less well-ordered, with billows on one side of the tube centreline connecting with neighbouring pairs on the other, so evolving a zigzag pattern. The results have application to naturally occurring flows in estuaries and channels, and to the development of instability in the stratified atmosphere.

On the dispersion relation for trapped internal waves

An analysis is constructed in order to estimate the dispersion relation for internal waves trapped in a layer and propagating linearly in a fluid of infinite depth with a rigid surface. The main interest is in predicting the structure of internal wave wakes, but the results are applicable to any internal waves. It is demonstrated that, in general 1/cp= 1/CpO+k/ωmax+ ∈(k) wherecpis the wave phase speed for a particular mode,CpOis the phase speed atk= 0, ωmaxis the maximum possible wave angular frequency and ωmax≤NmaxwhereNmaxis the maximum buoyancy frequency. Also, ∈(0) = 0, ∈(k) =o(k) forklarge, and is bounded for finitek.In particular, when ∈(k) can be neglected, the dispersion relation for a lowest mode wave is approximately 1/cp≈ (∫∞0N2(y)ydy)-½+k/ωmax. The eigenvalue problem is analysed for a class of buoyancy frequency squared functionsN2(x) which is taken to be a class of realvalued functions of a real variablexwhere O ≤x∞ such thatN2(x) =O(e-βx) asx→ ∞ and 1/β is an arbitrary length scale. It is demonstrated thatN2(x) can be represented by a power series in e-βx. The eigenfunction equation is constructed for such a function and it is shown that there are two cases of the equation which have solutions in terms of known functions (Bessel functions and confluent hypergeometric functions). For these two cases it is shown that ∈(k) can be neglected and that, in addition, ωmax=Nmax. More generally, it is demonstrated that whenk→ ∞ it is possible to approximate the equation uniformly in such a way that it can be compared with the confluent hypergeometric equation. The eigenvalues are then, approximately, zeros of the Whittaker functions. The main result which follows from this approach is that ifN2(x) isO(e-βx) asx→ ∞ and has a maximum valueN2maxthen a sufficient condition for 1/cp∼k/Nmaxto hold for largekfor the lowest mode is thatN2(t)/tis convex for O ≤t≤ 1 wheret= e-βx.

Internal wave eigenmodes and their weak nonlinear resonant interaction in a nonlinearly stratified fluid—A preliminary theoretical and experimental study

A preliminary theoretical and experimental study was conducted on internal wave modes and their weak nonlinear resonant interaction in a nonlinearly stratified fluid. An asymptotical solution of the modes and a dispersion relation of internal waves in a stratified fluid with density profile\(\rho = \rho _m - \Delta \rho \pi ^{ - 1} tg^{ - 1} \left( {b^{ - 1} y} \right)\) similar to that in our experiment were obtained theoretically. The resonant-interaction mechanism to 2nd order approximation is also discussed. The resonant interaction of the 3rd and 4th mode internal waves excited by the unstable 1st mode wave is analyzed on the basis of data obtained by conductivity probes. The resonant-interaction condition, i.e.,\(\omega _1 - \omega _3 - \omega _4 = 0\),k1+k3+k4=0, is examined. It is shown that the resonant instability increases with pycnocline thickness and wave maker driving frequency.

Parameterization of the dispersion relation of internal waves using the data of hydrological soundings in the Tropical Atlantic

Dependences have been determined which connect the parameters of the dispersion relation of the lowest mode of internal waves with the integral characteristics of the seasonal thermocline when 10 min≤τ≤30 min, 20 m≤h≤150 m, and 0·4 m2/s2 ≤Q≤5·2 m2/s2.

Directional spectra and eigen-solutions of vertically standing wavemodes of internal waves in shallow seas

Some approximate formulas, based on the internal- wave directional spectral model established by Schott and Willebrand (1973), of vertically standing wavemode eigenfunctions and a dispersion relation of internal waves in shallow seas are presented. An optimization method to estimate internal wave directional spectra is described and the confidence interval expression of the estimates is established.

Observations on the Vertical Polarization and Energy Flux of Near-Inertial Waves

Abstract Vertical profiles of horizontal ocean currents are used to study the vertical structure and temporal behavior of internal waves in the ocean, particularly those near the local inertial frequency. The polarization, or direction in which the horizontal velocity vector of an internal wave rotates with depth, is an important feature of the vertical structure, since it provides information on the direction and magnitude of the vertical wave energy flux. Analysis of a time series of profiles at one location over smooth topography shows that the observed wave polarization and phase propagation in the vertical are consistent, at least within the limits of the observational technique that was employed, with the linear dispersion relation for internal waves. The fact that the waves are polarized in the clockwise sense with increasing pressure shows that they have a net downward energy flux. A spectral decomposition of the profiles into clockwise and anti-clockwise components provides an estimate of 0.2–0.4...