Internal Wave Spectrum Research Articles

Nonlinear wave–wave interactions in stratified flows: Direct numerical simulations

To investigate the formation mechanism of energy spectra of internal waves in the oceans, direct numerical simulations are performed. The simulations are based on the reduced dynamical equations of rotating stratified turbulence. In the reduced dynamical equations only wave modes are retained, and vortices and horizontally uniform vertical shears are excluded. Despite the simplifications, our simulations reproduce some key features of oceanic internal-wave spectra: accumulation of energy at near-inertial waves and realistic frequency and horizontal wavenumber dependencies. Furthermore, we provide evidence that formation of the energy spectra in the inertial subrange is dominated by scale-separated interactions with the near-inertial waves. These findings support observationally based intuition that spectral energy density of internal waves is the result of predominantly wave–wave interactions.

Reconstruction of spatial spectra of the isotropic field of background internal waves

Model reconstruction of the two-dimensional spatial spectra of isotropic background internal waves is considered in the framework of computer modeling. The solution of the inverse problem is based on information about the frequency shifts of interference maxima of an acoustic field. The results of reconstruction of spectra with and without focusing of the inverse wave field are presented. The possibilities of monitoring are illustrated in relation to the interference pattern formed by different modal composition.

Anisotropy of the Wavefront Distortion for Acoustic Pulse Propagation Through Ocean Sound-Speed Fluctuations: A Ray Perspective

Observations of broadband sound propagation through the deep ocean, rich in sound-speed inhomogeneities, show that the double accordion acoustic wavefront pattern expected from model predictions without inhomogeneities is remarkably stable. This stability is found for propagation ranges up to 5000 km for acoustic frequencies of 28-84 Hz, and up to 1200-km range for 250 Hz. While the observed wavefront pattern is stable, the acoustic intensity along the wavefront is not. Furthermore, significant vertical extension of turning point caustics has been observed. This line of evidence suggests that the scattering is anisotropic in the sense that it is primarily <i xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">along</i> the wavefront, rather than across it. In addition, ray and parabolic equation simulations of acoustic propagation through ocean internal waves obeying the Garrett-Munk (GM) internal wave spectrum reinforce this notion of the anisotropy of the wavefront distortion. This paper presents a ray-based physical model for this phenomenon based on small angle forward scattering and provides analytic formulas to predict the wavefront distortions caused by ocean internal waves and other ocean processes. Further applications include out-of vertical-plane scattering and wavefront healing near seamounts or islands.

Interharmonics in internal gravity waves generated by tide-topography interaction

The dynamics and spectrum of internal gravity waves generated in a linearly stratified fluid by tidal flow over a flat-topped ridge are investigated at five different latitudes using an inviscid two-dimensional numerical model. The resulting wave field includes progressive freely propagating waves which satisfy the dispersion relation, and forced waves which are trapped non-propagating oscillations with frequencies outside the internal wave band. The flow is largely stable with respect to shear instabilities, and, throughout the runs, there is a negligibly small amount of overturning which is confined to the highly nonlinear regions along the sloping topography and where tidal beams reflect from the boundaries. The wave spectrum exhibits a self-similar structure with prominent peaks at tidal harmonics and interharmonics, whose magnitudes decay exponentially with frequency. Two strong subharmonics are generated by an instability of tidal beams which is particularly strong for near-critical latitudes where the Coriolis frequency is half the tidal frequency. When both subharmonics are within the free internal wave range (as in cases 0°–20° N), they form a resonant triad with the tidal harmonic. When at least one of the two subharmonics is outside of the range (as in cases 30°–40° N) the observed instability is no longer a resonant triad interaction. We argue that the two subharmonics are generated by parametric subharmonic instability that can produce both progressive and forced waves. Other interharmonics are produced through wave–wave interactions and are not an artefact of Doppler shifting as assumed by previous authors. As the two subharmonics are, in general, not proper fractions of the tidal frequency, the wave–wave interactions are capable of transferring energy to a continuum of frequencies.

Method of acoustic sweep monitoring of oceanic inhomogeneities (Review)

Theoretical grounds of the new method of monitoring of the temporal variability of oceanic inhomogeneities, which uses the data on frequency shifts of interference maxima of the sound field, were described. The method is free of limitations on both the resolution of signals coming in individual modes (rays) and the adiabatic approximation underlying the conventional methods of inhomogeneity reconstruction. The monitoring sensitivity was estimated, which allows us to estimate minimum detectable changes in the speed of sound by measurement data on frequency shifts of local maxima. Experimental data on shifts of the frequency spectrum of a broadband low-frequency signal on a stationary path in a shallow sea were presented. On their basis, the possibility of applying this method to diagnose tidal variations was shown. Within a numerical simulation, model reconstruction of the frequency spectrum of background internal waves was considered on the basis of the data on measurements of the spectrum of frequency shifts of the interference maximum. The results of the spectrum reconstruction with and without focusing of the conjugate wave field are presented. The problems of monitoring stability and efficiency with respect to the interference pattern formed by various groups of modes were discussed.

Absolute intensities of acoustic shadow zone arrivals

Qualitative observations from bottom-mounted US Navy SOSUS receiving stations in the North Pacific reveal anomalously deep acoustic arrivals at travel times directly corresponding with timefronts expected to have turned much higher in the water column. The vertical structure of these shadow zone arrivals was studied during SPICEX, a long-range propagation experiment conducted from June to November 2004 in the North Pacific, utilizing moored sources 500 and 1000 km distant from two vertical line array receivers, which together virtually spanned the full ocean depth. Comparison of the measured absolute intensities of shadow zone arrivals with Monte Carlo parabolic equation simulations suggest that the amount of internal wave scattering associated with the standard Garrett-Munk (GM) internal wave spectrum is not adequate to account for the extent of scattering into the acoustic shadow evident in the experimental data, suggesting either that the GM spectrum is not an appropriate representation of the internal wave field or that some other mechanism, such as oceanic spice, may be also be contributing to the scattering.

Coherence of tracked arrivals in SPICEX

In the fall of 2004, 250 Hz broadband signals were received at 500 km and 1000 km ranges on a near full water‐column vertical array in the North Pacific ocean. Individual ray arrivals of very high SNR could easily be identified and tracked using a turning‐point filter (time‐delay beamforming accounting for channel structure), thus providing accurate vertical coherence estimates. The observations can be compared to standard Monte Carlo estimates of coherence made using accurate parabolic‐equation acoustic propagation in an ensemble of ocean states consistent with the standard Garrett‐Munk ocean internal‐wave spectrum. Acoustic coherence can also be expressed as a depth‐ dependent structure function. This is naturally estimated by the full‐wave travel‐time sensitivity kernel (TSK) which provides a linearized transformation from the internal wave spectrum to the structure function. Environmental measurements were conducted almost concurrently with the acoustic trials, constraining the acceptable ocean variability. Allowances must be made for scattering by sound‐speed variability along isopycnals (spiciness) in the upper mixed layer. The most important conclusion to date is that the vertical coherence is depth‐dependent but this statement must be qualified by the ability of the beamformer to separate arrivals.

Doppler spreading of internal gravity waves by an inertia‐wave packet

Present Doppler‐spreading models for the high wave number end of atmospheric and oceanic internal wave spectra either neglect the time dependence of the background long‐wave shear entirely, or ignore time‐dependent effects in their parameterization of dissipation. Through ray tracing and numerical simulations the Doppler spreading of an idealized interaction between a short internal‐wave packet by an inertia‐wave packet is examined. The results are sufficient to show that time dependence in the long‐wave shear can make a significant difference to short‐wave behavior, and will need to be taken into account in future modeling efforts.

Forced Resonant Oscillations as a Response to Periodic Winds in a Stratified Reservoir

The response of the Sau reservoir to a wind field characterized by having marked periodicities of 12 h and 24 h has been studied. Measurements of temperature, with a thermistor string, and currents, with an acoustic Doppler current meter, show that the reservoir also responds with the same water periodicities. During certain times of the stratified period, some of the natural oscillation modes of the reservoir are close to these forcing wind periods. In particular, in mid-July the vertical Mode V2 is close to 12 h and in mid- to end of September the vertical Mode V3 is close to 24 h. In these situations, these modes are selected out of the spectrum of possible internal waves and the reservoir behaves as a forced oscillator in resonance with the wind. The structure and the period of these vertical modes have been elucidated by using the 3D model ELCOM. Both modes are affected by the Earth's rotation at the widest part of the reservoir.

Spectral modification and geographic redistribution of the semi-diurnal internal tide

A realistic-geometry global baroclinic tidal model forced with a single tidal constituent ( M 2 ) is used to investigate the generation of the internal tide and the associated radiated baroclinic energy flux. The model internal wave spectrum is populated at discrete frequency multiples ( 1 / 2 , 1 , 3 / 2 , 2 , 5 / 2 , … ) of M 2 . The 1 / 2 M 2 subharmonic is particularly energetic at its turning latitude of ± 28.8 ° . Poleward only integer superharmonics of M 2 are significantly excited. The subharmonic turning latitude (SHTL) disturbance has high vertical wavenumber and shear, provided that internal tide energy level exceeds a threshold value. Under these circumstances, Richardson numbers smaller than 1/4 occur in the upper few hundred meters in both the realistic-geometry model and in a complimentary idealized geometry two-dimensional (2D) model. In the 2D model, the disturbance enables Richardson number dependent diapycnal entrainment to effect a modification of the stratification of the upper 400 m of the ocean, and poleward cross-SHTL energy flux falls to 10% of its pre-instability value due to energy transfer to the non-propagating (i.e., inertial) subharmonic. Realistic-geometry simulations suggest a more modest 40% decrease in net flux, although the strongest beams are almost entirely shut down. The predicted energy flux-convergence implies a thermocline dissipation rate in the 28.5–30.0°N latitude band of 5 × 10 - 9 W kg - 1 , with an associated diapycnal diffusivity of 10 - 4 m 2 s - 1 . North of Hawaii the implied regional dissipation rate reaches 4 × 10 - 8 W kg - 1 with an associated thermocline diffusivity of 8 × 10 - 4 m 2 s - 1 . Investigations of subgridscale parameterization and resolution sensitivity suggest that the basic character and magnitude of the predictions are robust to details of the numerical solutions. The present results are taken as further evidence that an increase in shear-driven turbulent mixing in the upper ocean is predicted at special latitudes. It is suggested that the search should be directed to regions where intense low-mode internal tide beams cross their subharmonic turning latitude.

On a spectrum of nonlinear internal waves in the oceanic coastal zone

Abstract. This paper studies the internal wave band of temperature fluctuation spectra in the coastal zone of Pacific ocean. It is observed that on the central Mexican Pacific Shelf in the high-frequency band of temperature spectra the spectral exponent tends to ~ω-1 at the time of spring tide and ω-2 at the time of neap tide. On the western shelf of the Japan/East Sea, in the Ω

Application of time-reversed wave field focusing to reconstructing the frequency spectrum of background internal waves

A numerical experiment is used to analyze the possibility of focusing the time-reversed wave field for reconstructing the frequency spectrum of the vertical displacements of water layers by measuring the frequency shifts of the sound field maximum at the focal spot. The focusing of the field is controlled by varying the transmitted frequency at a fixed distribution of the reversed field, which is formed in the unperturbed waveguide, over the array aperture. The data of computations are compared with those obtained without focusing.

Numerical prediction of coherent integration time at 75 Hz, 0.03 s temporal resolution at 3250 km

Coherence time of sound is modeled for a 3250 km section in the Pacific at 75 Hz, 0.03 s resolution. The model is based on the temporal evolution of a standard spectrum of internal gravity waves and an accurate approximation for the acoustic wave equation. Modeled coherence time extends up to about 40 min. The probability is 0.6 that the signal-to-noise ratio first decreases at times of 13 min or less. Afterward, the s/n usually increases, reaching a peak near 40 min. The 40 min time is longer than experimentally reported coherence times of 12.7 min [J. Acoust. Soc. Am., 105, 3185–3201 (1999)]. The experimental value is similar to the modeled time of 13 min or less at which the probability is 0.6 for finding the s/n ratio to decrease for the first time. The experimental value of 12.7 min was found using a single Doppler correction speed yielding maximum output of the impulse response. Analysis of accelerations of the source and receiver indicates a variable Doppler correction scheme should be used to estimate coherence time. The experimental procedure for assuming a constant Doppler speed correction appears to be invalid. [Work supported by the ONR.]

Oceanic Isopycnal Slope Spectra. Part I: Internal Waves

Abstract Horizontal tow measurements of internal waves are rare and have been largely supplanted in recent decades by vertical profile measurements. Here, estimates of isotherm displacements and turbulence dissipation rate from a towed vehicle deployed near Hawaii are presented. The displacement data are interpreted in terms of horizontal wavenumber spectra of isopycnal slope. The spectra span scales from 5 km to 0.1 m, encompassing both internal waves and turbulence. The turbulence subrange is identified using a standard turbulence fit, and the rest of the motions are deemed to be internal waves. The remaining subrange has a slightly red slope (ϕ ∼ k−1/2x) and vertical coherences compatible with internal waves, in agreement with previous towed measurements. However, spectral amplitudes in the internal wave subrange exhibit surprisingly little variation despite a four-order-of-magnitude change in turbulence dissipation rate observed at the site. The shape and amplitude of the horizontal spectra are shown to be consistent with observations and models of vertical internal wave spectra that consist of two subranges: a “linear” subrange (ϕ ∼ k0z) and a red “saturated” subrange (ϕ ∼ k−1z). These two subranges are blurred in the transformation to horizontal spectra, yielding slopes close to those observed. The saturated subrange does not admit amplitude variations in the spectra yet is an important component of the measured horizontal spectra, explaining the poor correspondence with the dissipation rate.

Stochastic differential equation analysis for sound scattering by random internal waves in the ocean

The scattering of a weakly divergent narrow sound beam by random inhomogeneities of a fluctuating ocean is considered in the coupled-mode approximation. The random index of sound refraction is described using the Garrett-Munk internal wave spectrum. The problem is solved using the stochastic differential equations for the first-and second-order statistical moments of the acoustic field. The equations are formulated according to the cumulant expansion method. The existence of weakly divergent narrow sound beams in long-range sound propagation was one of the last discoveries of L.M. Brekhovskikh, to which he attached much importance. The concentration of sound into narrow beams away from the axis of the underwater sound channel was first observed experimentally and then explained by Brekhovskikh and his former students Goncharov, Kurtepov, and Petukhov. In the present paper, the scattered field intensity of a sound beam is calculated for different frequencies and source depths. Analytical expressions are obtained for the coefficients of the differential equation. The intermode energy transfer that accompanies the long-range propagation of a weakly divergent sound beam is analyzed. A comparison with the conventionally used Monte Carlo simulation in the parabolic equation approximation is performed.

Towed spectra of internal waves in the pycnocline of the Baltic sea

Towed spectra of internal waves in the pycnocline of the Baltic sea

Influence of stratification and topography upon internal wave spectra in the region of sills

A non‐hydrostatic model in vertical cross section form is used to examine the role of sill width and buoyancy upon the energy cascade into the internal tidal band and higher frequencies in sill regions. Power spectra analysis of computed time series show that as sill width is reduced the energy transfer from the baroclinic tide to its higher harmonics is reduced. However, there is an increase in energy at higher frequencies corresponding to lee wave periods. In addition internal mixing on the lee side of the sill increases. The addition of small scale topography on the sill slope produces a similar spectral change in the internal wave distribution and mixing. For a narrow sill the addition of a surface buoyant layer with a higher buoyancy frequency, significantly reduces the energy cascade to higher frequencies with a corresponding reduction in mixing. The associated spectra have characteristics resembling wide sill topography.

Vertical coherence of low‐frequency sound waves propagating through a fluctuating ocean

Low‐frequency sound waves propagating over large distances in the ocean exhibit significant scattering due to internal waves. Studies of low‐frequency sound propagation through a fluctuating ocean are important for many practical concerns such as communication, acoustic tomography, and source detection, ranging, and recognition. In a recent work [A. G. Voronovich and V. E. Ostashev, J. Acoust. Soc. Am. 63, 1406–1419 (2006)] a 3‐D modal theory of sound propagation through a fluctuating ocean was developed. Closed equations for the first two statistical moments of a sound field were derived using the Chernov method and assuming that the spectrum of internal waves is given by the Garret‐Munk spectrum. In the present paper, we numerically solve the derived equation for the second moment. The results obtained are used to study the dependence of the vertical coherence of a sound field on several parameters of the problem: range, frequency, source and receiver depths, and parameters that characterize the Garret‐...

Resonant triad model for studying evolution of the energy spectrum among a large number of internal waves

Internal waves are generally accepted to be responsible for a large fraction of mixing in the deep ocean. Internal waves interact nonlinearly with one another, exchanging energy among themselves to create the background internal wave spectrum. The most important mechanism resulting in the transfer of energy from one wave to another is believed to be resonant triad interactions. In this paper we consider a large number of resonantly interacting triads in order to investigate the evolution of the energy spectrum due to solely resonant triad interactions. To this end we solve the evolution equations for a large number of resonant triads to determine the temporal evolution of the energy distribution among the various possible wave numbers and frequencies. Our model involves internal waves with frequencies spanning the range of possible frequencies, i.e., between a maximum of the buoyancy frequency N for horizontal wave vectors (vertical motion) to a minimum of the inertial frequency f for vertical wave vectors (horizontal motion) [two limiting cases]. Because of the inclusion of high-frequency waves we cannot make the hydrostatic approximation. We investigate the evolution of the wave’s amplitudes to predict the evolution of the internal wave energy spectrum.

Global Abyssal Mixing Inferred from Lowered ADCP Shear and CTD Strain Profiles

Abstract Internal wave–wave interaction theories and observations support a parameterization for the turbulent dissipation rate ɛ and eddy diffusivity K that depends on internal wave shear 〈Vz2〉 and strain 〈ξz2〉 variances. Its latest incarnation is applied to about 3500 lowered ADCP/CTD profiles from the Indian, Pacific, North Atlantic, and Southern Oceans. Inferred diffusivities K are functions of latitude and depth, ranging from 0.03 × 10−4 m2 s−1 within 2° of the equator to (0.4–0.5) × 10−4 m2 s−1 at 50°–70°. Diffusivities K also increase with depth in tropical and subtropical waters. Diffusivities below 4500-m depth exhibit a peak of 0.7 × 10−4 m2 s−1 between 20° and 30°, latitudes where semidiurnal parametric subharmonic instability is expected to be active. Turbulence is highly heterogeneous. Though the bulk of the vertically integrated dissipation ∫ɛ is contributed from the main pycnocline, hotspots in ∫ɛ show some correlation with small-scale bottom roughness and near-bottom flow at sites where strong surface tidal dissipation resulting from tide–topography interactions has been implicated. Average vertically integrated dissipation rates are 1.0 mW m−2, lying closer to the 0.8 mW m−2 expected for a canonical (Garrett and Munk) internal wave spectrum than the global-averaged deep-ocean surface tide loss of 3.3 mW m−2.