Internal Wave Energy Research Articles

The Importance of Remote Forcing for Regional Modeling of Internal Waves

AbstractRegional ocean general circulation models are generally forced at the boundaries by mesoscale ocean dynamics and barotropic tides. In this work we provide evidence that remotely forced internal waves can be a significant source of energy for the dynamics. We compare global and regional model solutions within the California Current System. Both models have similar inputs, forcings, and identical grids and numerics. The global model has a steric height power spectrum consistent with mooring observations at superinertial frequencies, while the regional model spectrum is weaker. The regional model also has less sea surface height variance at high wavenumber than the global model. The vertical velocity variance is significantly larger in the global model, except in the sheltered Southern California Bight. While the regional model has roughly equal high‐pass baroclinic and barotropic kinetic energy levels, the global model high‐pass baroclinic kinetic energy is 28% (0.39 PJ) greater than the barotropic energy. An internal wave energy flux analysis reveals that the regional model domain boundaries act as a sink of 183 MW, while in the global model the analysis domain boundaries act as a source of 539 MW. This 722 MW difference can account for the relative increase of 0.39 PJ high‐pass baroclinic energy in the global model, assuming a baroclinic kinetic energy dissipation time in the domain of approximately 6.3 days. The results here imply that most regional ocean models will need to account for internal wave boundary fluxes in order to reproduce the observed internal wave continuum spectrum.

On the wave turbulence theory for stratified flows in the ocean

After the pioneering work of Garrett and Munk, the statistics of oceanic internal gravity waves has become a central subject of research in oceanography. The time evolution of the spectral energy of internal waves in the ocean can be described by a near-resonance wave turbulence equation, of quantum Boltzmann type. In this work, we provide the first rigorous mathematical study for the equation by showing the global existence and uniqueness of strong solutions.

Investigation of the energy of wave motions in a three-layer hydrodynamic system

In this paper, the propagation of the energy of internal waves in a three-layer hydrodynamic liquid system ‘a layer with a solid bottom–a layer–a layer with a lid’ is investigated. The problem statement is executed in a dimensionless form. Using the method of multidimensional development, the first three linear problems were obtained. For the first approximation problem, a dispersion equation of the first order is derived, and two pairs of independent solutions are obtained. The dependence of total energy on the wave number and thickness at different values of physical parameters were graphically illustrated and analyzed. The limiting case in which the obtained results are compared with the calculation of the energy that transferring the wave in the hydrodynamic system ‘liquid layer–a layer with a lid’, which was investigated earlier is considered. Graphically illustrates the transition of this system to a two-layer one. Data obtained as a result of the study of the problem of propagation of internal wave energy in a liquid three-layer hydrodynamic system can be used in the study of similar areas in the ocean and in the development of appropriate technologies and devices using the energy of water waves or converted into electricity.

Strong Internal Waves Generated by the Interaction of the Kuroshio and Tides over a Shallow Ridge

AbstractThe Kuroshio and tides significantly influence the oceanic environment off the Japanese mainland and promote mass/heat transport. However, the interaction between the Kuroshio and tides/internal waves has not been examined in previous works. To investigate this phenomenon, the two-dimensional high-resolution nonhydrostatic oceanic Stanford Unstructured Nonhydrostatic Terrain-Following Adaptive Navier–Stokes Simulator (SUNTANS) model was employed. The results show that strong internal tides propagating upstream in the Kuroshio are generated at a near-critical internal Froude number (Fri = 0.91). The upstream internal wave energy flux reaches a magnitude of 12 kW m−1, which is approximately 3 times higher than that of internal waves without the Kuroshio. On the other hand, under supercritical conditions, the Kuroshio suppresses the internal wave energy flux. The interaction of internal tides and the Kuroshio also generates upstream propagating high-frequency internal waves and solitary wave packets. The high-frequency internal waves contribute to the increase in the total internal wave energy flux up to 40% at the near-critical Fri value. The results of this study suggest that the interaction of internal tides and the Kuroshio enhances the upstream propagating internal tides under the specified conditions (Fri ~ 1), which may lead to deep ocean mixing and transport at significant distances from the internal wave generation sites.

Internal Wave Radiation through Surface Mixed Layer Turbulence

AbstractIn a series of large-eddy simulations with different forcing, we study the generation of internal gravity waves at the base of the surface mixed layer. If turbulent eddies act as obstacles and undulate the base of the mixed layer, horizontal velocities associated with inertial oscillations and Ekman dynamics can move the obstacles relative to the stratified interior, exciting internal gravity waves similar to lee waves. We find strong evidence that the “obstacle mechanism” is able to excite large parts of the internal wave spectrum, including near inertial waves. The high-frequency part of the excited wave spectrum is filtered by the increased stratification in the transition layer between the mixed layer and lower stratified interior, but a substantial part of the wave spectrum is able to overcome this barrier, hence contributing to interior mixing. The magnitude of the downward-radiated energy below the transition layer depends on the source of turbulence, but we show that the obstacle mechanism, especially under destabilizing heat fluxes, has the potential to contribute considerably to the internal wave energy in the interior ocean.

Toward global maps of internal tide energy sinks

Internal tides power much of the observed small-scale turbulence in the ocean interior. To represent mixing induced by this turbulence in ocean climate models, the cascade of internal tide energy to dissipation scales must be understood and mapped. Here, we present a framework for estimating the geography of internal tide energy sinks. The mapping relies on the following ingredients: (i) a global observational climatology of stratification; (ii) maps of the generation of M2, S2 and K1 internal tides decomposed into vertical normal modes; (iii) simplified representations of the dissipation of low-mode internal tides due to wave-wave interactions, scattering by small-scale topography, interaction with critical slopes and shoaling; (iv) Lagrangian tracking of low-mode energy beams through observed stratification, including refraction and reflection. We thus obtain a global map of the column-integrated energy dissipation for each of the four considered dissipative processes, each of the three tidal constituents and each of the first five modes. Modes ≥6 are inferred to dissipate within the local water column at the employed half-degree horizontal resolution. Combining all processes, modes and constituents, we construct a map of the total internal tide energy dissipation, which compares well with observational inferences of internal wave energy dissipation. This result suggests that tides largely shape observed spatial contrasts of dissipation, and that the framework has potential in improving understanding and modelling of ocean mixing. However, sensitivity to poorly constrained parameters and simplifying assumptions entering the parameterized energy sinks calls for additional investigation. The attenuation of low-mode internal tides by wave-wave interactions needs particular attention.

Generation and evolution of internal solitary waves in the southern Taiwan Strait

ABSTRACTThe generation processes and potential energy sources of internal solitary waves (ISWs) in the southern Taiwan Strait are investigated by driving a high resolution non-hydrostatic numerical model with realistic background conditions. Two main types of ISWs are clarified according to their different energy sources. One is generated by the nonlinear disintegration of remote internal tides emanating from Luzon Strait, and the other type is generated by local tide-topography interaction at the continental slope. The basic properties and evolution processes differ between these two kinds of ISWs. The waves originated from the remote internal tides at Luzon Strait have amplitudes comparable to previous field observations. In contrast, the ISWs generated locally are much weaker than observed waves, even in the presence of a steady offshore background current, which intensifies the generation of onshore ISWs. The ISWs induced by remotely generated M2 internal tides are stronger than those induced by K1 internal tides, and the fraction of internal wave energy transmitted onto the shelf is not significantly influenced by the intensity of remotely generated internal tides.

Temporal Coherence for Sound Transmission Through Fluctuating Deep Water with Adiabatic Perturbed Modes

Adiabatic approximation (AP) combined with perturbation theory gives a fast normal-mode solution of temporal coherence for sound field in a two-dimensional deep water with time-varying random internal waves. Internal waves induced mode changes are deduced using the first-order perturbation theory [C. T. Tindle, L. M. O’Driscoll and C. J. Higham, Coupled mode perturbation theory of range dependence, J. Acoust. Soc. Am. 108(1) (2000) 76–83]. And mode perturbations in amplitude are neglected by the adiabatic method with wavenumber perturbations in phase merely considered. The AP expression of temporal coherence function is theoretically identical to the adiabatic transport equation theory [J. A. Colosi, T. K. Chandrayadula, A. G. Voronovich and V. E. Ostashev, Coupled mode transport theory for sound transmission through an ocean with random sound speed perturbations: Coherence in deep water environments, J. Acoust. Soc. Am. 134(4) (2013) 3119–3133]. Numerical results of the adiabatic temporal coherence function for several low frequencies and ranges up to 1000[Formula: see text]km are calculated. Then the coherence time scales obtained from the calculations are examined by a one-way coupled theory considering forward scattering [A. G. Voronovich, V. E. Ostashev and J. A. Colosi, Temporal coherence of acoustic signals in a fluctuating ocean, J. Acoust. Soc. Am. 129(6) (2011) 3590–3597]. Comparisons demonstrate that the range and frequency dependence of coherence time for both methods are quite close. And this shows good agreement with the well-known inverse frequency and inverse square root range laws. In addition, the internal wave energy dependence of coherence time is also studied.

Internal wave energy flux from density perturbations in nonlinear stratifications

Internal gravity wave energy contributes significantly to the energy budget of the oceans, affecting mixing and the thermohaline circulation. Hence it is important to determine the internal wave energy flux $\boldsymbol{J}=p\,\boldsymbol{v}$, where $p$ is the pressure perturbation field and $\boldsymbol{v}$ is the velocity perturbation field. However, the pressure perturbation field is not directly accessible in laboratory or field observations. Previously, a Green’s function based method was developed to calculate the instantaneous energy flux field from a measured density perturbation field $\unicode[STIX]{x1D70C}(x,z,t)$, given a constant buoyancy frequency $N$. Here we present methods for computing the instantaneous energy flux $\boldsymbol{J}(x,z,t)$ for an internal wave field with vertically varying background $N(z)$, as in the oceans where $N(z)$ typically decreases by two orders of magnitude from the pycnocline to the deep ocean. Analytic methods are presented for computing $\boldsymbol{J}(x,z,t)$ from a density perturbation field for $N(z)$ varying linearly with $z$ and for $N^{2}(z)$ varying as $\tanh (z)$. To generalize this approach to arbitrary $N(z)$, we present a computational method for obtaining $\boldsymbol{J}(x,z,t)$. The results for $\boldsymbol{J}(x,z,t)$ for the different cases agree well with results from direct numerical simulations of the Navier–Stokes equations. Our computational method can be applied to any density perturbation data using the MATLAB graphical user interface ‘EnergyFlux’.

Flexural-gravity wave dynamics in two-layer fluid: blocking and dead water analogue

Flexural-gravity wave characteristics are analysed, in the presence of a compressive force and a two-layer fluid, under the assumption of linearized water wave theory and small amplitude structural response. The occurrence of blocking for flexural-gravity waves is demonstrated in both the surface and internal modes. Within the threshold of the blocking and the buckling limit, the dispersion relation possesses four positive roots (for fixed wavenumber). It is shown that, under certain conditions, the phase and group velocities coalesce. Moreover, a wavenumber range for certain critical values of compression and depth is provided within which the internal wave energy moves faster than that of the surface wave. It is also demonstrated that, for shallow water, the wave frequencies in the surface and internal modes will never coalesce. It is established that the phase speed in the surface and internal modes attains a minimum and maximum, respectively, when the interface is located approximately in the middle of the water depth. An analogue of the dead water phenomenon, the occurrence of a high amplitude internal wave with a low amplitude at the surface, is established, irrespective of water depth, when the densities of the two fluids are close to each other. When the interface becomes close to the seabed, the dead water effect ceases to exist. The theory developed in the frequency domain is extended to the time domain and examples of negative energy waves and blocking are presented.

Observations of Shelf Exchange and High‐Frequency Variability in an Alaskan Fjord

AbstractHigh‐frequency observations collected over 3 years are used to describe upper‐ocean variability in Behm Canal, a nonglacial fjord in Southeast Alaska. The fjord is sheltered by surrounding topography and connected to the outer continental shelf by a 400‐m‐deep strait. Summer conditions are characterized by strong near‐surface stratification, a sea breeze wind regime, and tidally dominated flows. In nonsummer months, baroclinic subinertial flows exceeding 0.5 m/s dominate the velocity record. The flow events represent a wind‐driven response to low‐pressure systems that impact the coast as they propagate across the Gulf of Alaska. The observations suggest that the storm systems generate downwelling events that propagate into the fjord with a mode‐one‐like vertical structure in the upper 60 m. Following the initial up‐fjord current pulse lasting approximately a day, a down‐fjord flow occurs lasting several days, the duration of the downwelling anomaly. These subinertial downwelling events likely are the dominant mechanism of shelf‐fjord exchange, with estimated fjord‐flushing times of 50 days. The downwelling events are accompanied by enhanced near‐inertial shear, which exhibits downward energy propagation. Evidence for enhanced energy near the maximum buoyancy frequency in the thermocline suggests high‐frequency internal wave trapping, with the primary energy source in the semidiurnal band associated with tidal and wind forcing. The observations highlight the importance of shelf‐fjord coupling through meteorological forcing and the existence of internal wave energy at a range of frequencies, which have implications for mixing and transport within the fjord.

ВНУТРЕННИЙ ПРИЛИВ НА ПЛАТО САНТОС

We analyze internal tides over the Santos Plateau in the South Atlantic. The study is based on the measurements of temperature and currents on the line of moorings normal to the South America coast in the latitudinal band between 28°S and 31°S. The moorings were deployed at distances between 25 and 888 km (from the 500 m isobath) southeast of the continental slope. The instruments were set at 900 m. Numerical modeling revealed the properties of internal tides (amplitudes, wavelength, and decay of amplitudes) along the line of their propagation from the continental slope. The amplitudes of internal tide decrease from the continental slope in the direction to the Vema Channel from 36 m to 15 m. The densities of the energy of tidal internal waves were calculated from the semidiurnal tidal components of current and temperature time series taking into account the vertical gradients of temperature and the Brunt-Vaisala frequency. Decay of the energy of internal tide occurs according to an inverse power law. Numerical modeling of the generation and propagation of internal tides reveals that beams of internal tides are formed near the continental slope. The energy decay based on modeling is similar to the measured data.

Internal wave attractors in three-dimensional geometries: trapping by oblique reflection

We study experimentally the propagation of internal waves in two different three-dimensional (3D) geometries, with a special emphasis on the refractive focusing due to the 3D reflection of obliquely incident internal waves on a slope. Both studies are initiated by ray tracing calculations to determine the appropriate experimental parameters. First, we consider a 3D geometry, the classical set-up to get simple, two-dimensional (2D) parallelogram-shaped attractors in which waves are forced in a direction perpendicular to a sloping bottom. Here, however, the forcing is of reduced extent in the along-slope, transverse direction. We show how the refractive focusing mechanism explains the formation of attractors over the whole width of the tank, even away from the forcing region. Direct numerical simulations confirm the dynamics, emphasize the role of boundary conditions and reveal the phase shifting in the transverse direction. Second, we consider a long and narrow tank having an inclined bottom, to simply reproduce a canal. In this case, the energy is injected in a direction parallel to the slope. Interestingly, the wave energy ends up forming 2D internal wave attractors in planes that are transverse to the initial propagation direction. This focusing mechanism prevents indefinite transmission of most of the internal wave energy along the canal.

Analysis of energy of internal waves in a three­layer semi­infinite hydrodynamic system

Energy characteristics of waves propagation along the contact surfaces in a hydrodynamics system "liquid half-space – layer –layer with a rigid lid" are explored. Based on the solutions of first approximation to a weakly non-linear model, the integral relations for wave motion energy in each layer and for the total energy of the system were obtained. An analysis of energy of wave processes revealed that an increase in wave number causes a decrease in energy of wave motion of the upper layers, and energy of wave motion of the lower half-space at some values of wave number reaches extreme values. In this case, total energy of the system is descending in nature and rather quickly approaches its limit value.The numerical values of energy for three different cases of propagation of progressive waves were obtained: only along the upper contact surface, only along the lower contact surface, along both surfaces simultaneously. Comparison of the obtained values of energy revealed that in the case of waves propagation along both contact surfaces simultaneously, the total energy of the system is close to the sum of energies of the system at waves propagation along one of the surfaces

Dynamics of Internal Tides Over a Shallow Ridge Investigated With a High‐Resolution Downscaling Regional Ocean Model

AbstractThis study investigates the dynamics of tidally induced internal waves over a shallow ridge, the Izu‐Ogasawara Ridge off the Japanese mainland, using a downscaled high‐resolution regional ocean numerical model. Both the Kuroshio and tides contribute to the field of currents in the study area. The model results show strong internal tidal energy fluxes over the ridge, exceeding 3.5 kW m−1, which are higher than the fluxes along the Japanese mainland. The flux in the upstream side of the Kuroshio is enhanced by an interaction of internal waves and currents. The tidal forcing induces 92% of the total internal wave energy flux, exhibiting the considerable dominance of tides in internal waves. The tidal forcing enhances the kinetic energy, particularly in the northern area of the ridge where the Kuroshio Current is not a direct influence. The tidal forcing contributes to roughly 30% of the total kinetic energy in the study area.

Response of the Internal Wave Field to Remote Wind Forcing by Tropical Cyclones

AbstractIn the tropical North Atlantic, mean winds introduce relatively little energy into the internal wave field, but hurricanes act as very energetic sources for near-inertial waves. In addition to the eventlike passage of such tropical cyclones, changes in the wind speed north of the trade wind system induce a seasonal cycle in surface swell, with potential implications for the high-frequency part of the internal wave field. Using a 5-yr mooring time series in the interior of the tropical North Atlantic at 16°N, the temporal variability of internal wave energy south of the main hurricane track in different frequency bands is studied, and the magnitude of its variability, along with possible energy transfer mechanisms, is analyzed. The results show that changes in near-inertial energy are dominated by the passage of internal waves generated by hurricanes centered several hundred kilometers north of the mooring. The major role of hurricanes in the generation of near-inertial waves is also seen in an extended slab model that takes the horizontal divergence of the near-inertial current field at the mixed layer base into account. A seasonal cycle is observed in the energy at the high-frequency end (frequencies above 6 cpd) of the internal wave spectrum. It is not in phase with the near-inertial energy variability but covaries with changes in the local surface waves. These high-frequency internal waves are most energetic at times when large-amplitude surface swell with long periods and correspondingly long wavelengths is observed.

Oblique internal-wave chain resonance over seabed corrugations

Here we show that monochromatic long-crested corrugations on an otherwise flat seafloor can coherently scatter the energy of an oblique incident internal wave to multiple multi-directional higher-mode internal waves via a series of resonant interactions. We demonstrate that a resonance between seabed corrugations and a normally or slightly oblique incident internal wave results in a series of follow-up resonant interactions, which take place between the same corrugations and successively resonated shorter waves. A chain resonance of internal waves that carries energy to small scales is thus obtained, and we find that the Richardson number decreases by several orders of magnitude over the corrugated patch. If the incidence angle is large, and the incident wave perfectly satisfies a resonance condition with the topography, it turns out that not many higher-mode resonance or near-resonance conditions can be satisfied, such that energy stays confined within the first few modes. Nevertheless, if the incident waves are sufficiently detuned from satisfying a perfect resonance condition with the seabed corrugations, then we show that this frequency detuning may balance off the large detuning due to oblique incidence, leading to a chain resonance that again carries energy to small scales. The evolution of the incident and resonated wave amplitudes is predicted from the envelope equation for internal waves over resonant seabed topography in a three-dimensional rotating fluid, which we derive considering the Boussinesq and $f$-plane approximations with $f$ the Coriolis frequency, linear density stratification and small-amplitude corrugations. Our results suggest that topographic features on the ocean floor with a well-defined dominant wavenumber vector, through the chain resonance mechanism elucidated here, may play a more important role than previously thought in the enhancement of diapycnal mixing and energy dissipation.

Three-dimensional evolution of internal waves reflected from a submarine seamount

The interaction between internal waves and submarine seamounts is three-dimensional (3D). In this work, we conduct laboratory experiments to investigate the role of the three-dimensional seamount on the reflection of internal waves. Experimental results show that the energy of internal waves is focused after the reflection and the phase of the internal wave front become curved. The energy of the focusing region was nearly 2 times as that of the scattering region. There is a strong internal wave induced mean flow whose strength is nearly as high as 50% of that of the reflected internal wave. The mean flow was generated by the variation of the velocity amplitude in both the x axis and y axis. These results indicate that the reflection from the 3D seamount could lead to a complex spatial distribution of the internal wave and a different energy conversion process.

Dynamics and Energetics of Trapped Diurnal Internal Kelvin Waves around a Midlatitude Island

AbstractThe generation of trapped and radiating internal tides around Izu‐Oshima Island located off Sagami Bay, Japan, is investigated using the three-dimensional Stanford Unstructured Nonhydrostatic Terrain-following Adaptive Navier–Stokes Simulator (SUNTANS) that is validated with observations of isotherm displacements in shallow water. The model is forced by barotropic tides, which generate strong baroclinic internal tides in the study region. Model results showed that when diurnal K1 barotropic tides dominate, resonance of a trapped internal Kelvin wave leads to large-amplitude internal tides in shallow waters on the coast. This resonance produces diurnal motions that are much stronger than the semidiurnal motions. The weaker, freely propagating, semidiurnal internal tides are generated on the western side of the island, where the M2 internal tide beam angle matches the topographic slope. The internal wave energy flux due to the diurnal internal tides is much higher than that of the semidiurnal tides in the study region. Although the diurnal internal tide energy is trapped, this study shows that steepening of the Kelvin waves produces high-frequency internal tides that radiate from the island, thus acting as a mechanism to extract energy from the diurnal motions.

Recent progress and problems in the studies of the global distribution of internal wave energy available for turbulent mixing in the deep ocean

Recent progress and problems in the studies of the global distribution of internal wave energy available for turbulent mixing in the deep ocean