Internal Waves in a Nonuniformly Stratified Ocean

The model ‘Internal Wave Dissipation, Energy and Mixing’ (IDEMIX) provides an energetically consistent representation for the diapycnal diffusivity induced by breaking of internal gravity waves in ocean and atmosphere circulation models. IDEMIX predicts the internal wave energy, dissipation rates, and diapycnal diffusivities. Such small-scale processes cannot be resolved but have to be parameterized due to their relevance for the large-scale circulation. The basic version of the model has been shown to be generally successful in ocean and atmosphere applications. However, in regions of strong forcing deviations from observational estimates were found. To evaluate the local performance of the model we analyzed the agreement with observational estimates of full-depth profiles of both stratification and horizontal velocity collected by several cruises around 47°N and 16°N in the Atlantic. The hydrographic profiles come from two dynamically different regions: the subpolar North Atlantic with energetic wind-induced near-inertial waves and the western subtropical Atlantic where the strong Deep Western Boundary Current interacts with the continental slope producing lee waves. Internal wave energy, dissipation rates and diapycnal diffusivities estimates are obtained using the finestructure method. These estimates can be calculated using shear and strain or strain only in lack of velocity data. In this study, both formulations have been calculated and contrasted between each other to evaluate the importance of shear information for a realistic energy budget. The results when comparing IDEMIX with observations show that the higher differences are close to the surface and over the rough topography where internal gravity waves are more predominant. The analysis of the observational data will increase our understanding of the spatiotemporal variability of the ocean’s internal gravity wavefield and will complement the model-based investigation of the responsible processes.

Tidal currents flowing over benthic relief (e.g., banks, shelf break) can produce large internal waves. These waves propagate away from their origin and are capable of crossing the continental shelf and seas. Studies of shoreward transport of larval invertebrates and fish by these internal waves unintentionally tested whether they can capture, concentrate and transport floating plastic. Plastic surface drifters deployed in front of sets of internal wave convergences were often captured (>90% captured) and transported kilometers by the waves. There are, however, few investigations into how internal tidal waves may affect the fate and distribution of floating plastic waste. A number of areas of future research are suggested: (1) How much floating plastic is found in internal wave convergences? (2) How buoyant must floating plastic be to be captured by internal waves? (3) Why did only some sets of internal waves cause concentration and transport of surface material? (4) Do concentration and transport of floating plastic vary over the spring/neap tidal cycle? (5) Do seasonal changes in the depth of the pycnocline alter the transport of floating plastic by internal waves? (6) Plastic debris deposited on shore may not be evenly distributed, but may be more abundant landward of sites on the shelf break that more readily generate large internal waves. (7) Internal waves that travel long distances (10–100 s of km) have the potential to accumulate large amounts of plastic debris. (8) At locations where internal waves cross the continental shelf, how far offshore does transport commence?

The generation of internal gravity waves in the ocean is largely driven by tides, winds, and interaction of currents with the seafloor. Models and observations indicate a global energy supply for the internal wave field of about 1 TW by the conversion of barotropic tides at mid-ocean ridges and abrupt topographic features. Winds acting on the oceanic mixed layer contribute 0.3–1.5 TW, and mesoscale flow over rough topography adds about 0.2 TW. Globally, 1–2 TW are needed to maintain the observed stratification of the deep ocean by diapycnal mixing that results from the breaking of internal waves. Ocean circulation models show significant impact of the spatial distribution of internal wave dissipation and mixing on the ocean state, e.g., thermal structure, stratification, and meridional overturning circulation. Observations indicate that the local ratio of generation and dissipation of internal waves is often below unity, and thus, the energy available for mixing must be redistributed by internal tides and near-inertial waves at low vertical wavenumber that can propagate thousands of kilometers from their source regions. Eddy-permitting global ocean circulation models are able to quantify the different sources of energy input and can also simulate the propagation of the lowest internal wave modes. However, the variation of the internal wave energy flux along its paths by wave–wave interaction, topographic scattering, and refraction by mesoscale features as well as its ultimate fate by dissipation remains to be parameterized.

Oceanic turbulent mixing plays a fundamental role in global circulation and climate regulation, yet its spatial variability remains poorly observed, especially in the deep ocean. This study presents a detailed observational analysis of finescale turbulence in two North Atlantic regions, using repeated full‐depth profiles of stratification and horizontal velocity. We characterize spatial patterns in the dissipation of turbulent kinetic energy by examining their relationship with topographic features, and the sources of internal gravity wave energy. Elevated dissipation values are found over rough topography such as the Mid‐Atlantic Ridge and at the western boundaries, consistent with regions of enhanced internal wave generation. Observed shear‐to‐strain ratios further illuminate the contributions of different wave sources and highlight contrasts between the northern and tropical Atlantic. To complement these observations, we compare dissipation rates with predictions from the model IDEMIX (Internal Wave Dissipation, Energy and Mixing), with a particular focus on the impact of lee wave forcing (LWF). Results demonstrate that incorporating LWF significantly improves agreement with observations in the northern Atlantic, particularly in deeper layers. IDEMIX effectively reproduces large‐scale mixing patterns, though its performance is more sensitive to parameter choices in regions of enhanced near‐bottom velocity gradients and rough topography. Though IDEMIX is a model framework, it offers a physically grounded reference to interpret mixing patterns where direct measurements are limited.

Interfacial internal wave excitation in the wake of towed ships is studied experimentally in a quasi-two-layer fluid. At a critical ‘resonant’ towing velocity, whose value depends on the structure of the vertical density profile, the amplitude of the internal wave train following the ship reaches a maximum, in unison with the development of a drag force acting on the vessel, known in the maritime literature as ‘dead water’. The amplitudes and wavelengths of the emerging internal waves are evaluated for various ship speeds, ship lengths and stratification profiles. The results are compared to linear two- and three-layer theories of freely propagating waves and lee waves. We find that despite the fact that the observed internal waves can have considerable amplitudes, linear theories can still provide a surprisingly adequate description of subcritical-to-supercritical transition and the associated amplification of internal waves. We argue that the latter can be interpreted as a coalescence of frequencies of two fundamental stable wave motions, namely lee waves and propagating interfacial wave modes.Graphic abstract

As part of a National Oceanographic Partnership Program (NOPP) project, seven teams—comprising investigators from universities, federal laboratories, and industry—are collaboratively investigating the generation, propagation, and dissipation of internal waves in the global ocean using complementary, state-of-the-art observations and model simulations. Internal waves, generated by the interaction of tides, winds, and mean flows, permeate the ocean and influence its physical state. Internal waves transport scalar and vector properties—both geographically and across scales—and contribute to irreversible mixing, modulate acoustic propagation, and complicate the identification of subinertial (e.g., geostrophic) flows in observations. For these reasons, accurately representing internal waves in global ocean forecast models is a high priority. The collaborations reported here are improving the understanding of the internal wave life cycle and enhancing model skill in simulating it. Three observational teams are collecting in situ data using 1) redeployable moored arrays that resolve internal waves from multiple directions, 2) global deployments of profiling floats that measure internal wave energy fluxes, shear, and mixing, and 3) high-resolution arrays that focus on bottom boundary layer processes. Four modeling teams are guiding the design and placement of these observation platforms and are using the collected observations to 1) improve internal wave representation and dissipation in ocean models, 2) conduct high-resolution process studies, and 3) implement data assimilation in idealized, regional, and global simulations. These efforts are further supported by high-resolution sea surface height measurements from the new Surface Water and Ocean Topography (SWOT) satellite, which provide context for in situ observations and improve ocean forecasting systems. Significance Statement A collaboration among scientists from U.S. universities, national laboratories, and industry is advancing our understanding and prediction of internal waves in the global ocean. These waves—characterized by vertical scales of tens to hundreds of meters and horizontal scales of tens to hundreds of kilometers—play a critical role in maritime commerce, naval operations, and ocean circulation. The team integrates novel observational approaches, including internal wave–resolving moored arrays, ship-of-opportunity float deployments, bottom boundary layer–distributed sensor networks, and satellite wide-swath altimetry, with cutting-edge global, regional, and process-model simulations. Together, these efforts are improving the representation of internal wave processes in ocean models and enhancing their predictive capabilities for operational forecasts.

We employ the parallel, unstructured-grid, nonhydrostatic coastal ocean model SUNTANS to simulate the formation and evolution of internal gravity waves in the South China Sea, a site of some of the largest observed internal waves on Earth. The use of unstructured grids enables the simulations to capture the multiscale behavior of the waves, while the nonhydrostatic capability of the model enables simulation of solitary-like internal gravity waves. The results compare well to satellite imagery and reveal that the dominant internal wave signal in the South China Sea arises from internal waves that are generated daily at the eastern ridge of a two-ridge system in the Luzon Strait, on the eastern boundary of the sea.

Breaking internal gravity waves cause small-scale turbulent mixing, which changes water mass properties, affects biogeochemical cycles, and contributes to driving the large-scale overturning circulation. Ocean general circulation models do not resolve this process and thus rely on a parameterization. The state-of-the-art IDEMIX (Internal wave Dissipation, Energy and MIXing) model predicts the propagation and dissipation of internal wave energy based on external forcing functions that represent the main generation mechanisms, notably the internal tide generation at the sea floor and the near-inertial wave generation at the sea surface. By linking small-scale mixing to internal wave energetics, IDEMIX allows the consistent parameterization of wave-induced mixing in ocean models. Its basic incarnation treats all internal waves as part of a horizontally homogeneous continuum and was shown to successfully reproduce observed turbulent kinetic energy dissipation rates and internal wave energy levels. In a newer configuration (IDEMIX2), the internal wave field is compartmentalized, distinguishing between a high-mode continuum on the one hand and low-mode near-inertial wave and internal tide compartments, whose horizontal propagation is explicitly resolved in wavenumber angle space, on the other hand. We present the evaluation of the IDEMIX2 model with a particular focus on the impact of applying an anisotropic internal tide forcing. So far, parameterizations of internal tide-driven mixing have not taken the strong anisotropy of the internal tide generation process into account. We demonstrate the need for doing so, showing a notable impact on the modeled internal wave energetics and predicted mixing when changing from the previous isotropic to the new anisotropic tidal forcing in IDEMIX2. 

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.

Small-scale turbulent mixing affects large-scale ocean processes such as the global overturning circulation but remains unresolved in ocean models. Since the breaking of internal gravity waves is a major source of this mixing, consistent parameterizations take internal wave energetics into account. The model Internal Wave Dissipation, Energy and Mixing (IDEMIX) predicts the internal wave energy, dissipation rates, and diapycnal diffusivities based on a simplification of the spectral radiation balance of the wave field and can be used as a mixing module in global numerical simulations. In this study, it is evaluated against finestructure estimates of turbulent dissipation rates derived from Argo float observations. In addition, a novel method to compute internal gravity wave energy from finescale strain information alone is presented and applied. IDEMIX well reproduces the magnitude and the large-scale variations of the Argo-derived dissipation rate and energy level estimates. Deficiencies arise with respect to the detailed vertical structure or the spatial extent of mixing hot spots. This points toward the need to improve the forcing functions in IDEMIX, both by implementing additional physical detail and by better constraining the processes already included in the model. A prominent example is the energy transfer from the mesoscale eddies to the internal gravity waves, which is identified as an essential contributor to turbulent mixing in idealized simulations but needs to be better understood through the help of numerical, analytical, and observational studies in order to be represented realistically in ocean models.

In this chapter we study trivial nonlinear effects appearing while long internal waves propagate. There may be two situations and each of them needs its own analysis. The first is the propagation of internal waves in shallow water (e.g., in a shallow sea like the Baltic Sea) when the wavelength λ considerably exceeds the sea’s depth H. In this case equations of internal waves dynamics are reduced to the Korteweg-de Vries equation (KdV) (see a review of its evolution in the Appendix). Derivation and analysis of this equation for long internal waves is given in Section 8.1. In Section 8.2 stationary solutions of the KdV equation describing long nonlinear internal waves of the stable type are analyzed. Here we also give the numerical results for particular stratification profiles. The second possible situation is propagation of long waves within a thin pycnocline (the wavelength λ is much greater than the pycnocline’s thickness h). This is quite typical for natural conditions and is studied in Section 8.3. In this case internal waves are described by an equation of the KdV type but with integral dispersion. An analysis of the stationary solutions for this equation is presented as well.

The internal wave–vortex interaction was investigated for a broad parameter range except near inertial waves, by 1) scaling, 2) numerical experiments, and 3) the estimation of possible occurrences. By scaling, we identified a nondimensional parameter, δ = (V/c)[1/(kR)], where V is the vortex flow speed, R is the radius, c is the incident wave phase speed, and k is the horizontal wavenumber. As δ appears in all terms related to the interaction, it is important in the classification of the wave–vortex interaction. Numerical experiments were conducted on internal waves incident on a stable barotropic vortex with a parameter range of δ = [0.001, 1.7], which is much broader than that used in previous studies (δ ≪ 1). We found new phenomena for δ > 0.15, in addition to previously known scattering for δ ≤ 0.15 (scattering regime). For 0.15 < δ ≤ 0.4, part of the incident internal wave is trapped in a vortex, forming a wheel-like shape maintaining a superinertial frequency (wheel-trapping regime). When δ > 0.4, incident waves are trapped, but with a spiral shape (spiral-trapping regime). Spiral-shaped trapped waves release momentum by wave breaking, which deforms the vortex into a zigzag shape in the vertical direction. Vortex deformation produces vertical shear, which rapidly increases the vertical wavenumber of the incident wave. The distribution of δ in the Pacific Ocean was estimated using a high-resolution (1/30°) ocean general circulation model output. We found the occurrences of all three regimes. The scattering and wheel-trapping regimes are distributed broadly and varied seasonally, thus affecting mixing variability. Significance Statement Oceanic internal waves constitute the fundamental forcing of overturning and material circulation, because internal waves eventually break and cause vertical mixing. Interactions between internal waves and vortices affect wave properties and, therefore, mixing. However, as far as we are aware, all previous studies have focused on large weak vortices relative to waves. Here, we investigated such interactions for a much larger parameter space and identified two new regimes, in which vertical mixing is caused by newly found internal wave trapping and vortex deformation processes. We identified a nondimensional parameter that classifies the regimes and estimated their spatiotemporal distribution. These results suggest new energy routes from internal waves to turbulence and are applicable to other types of waves and vortices.

Application of different internal solitary wave theories for SAR remote sensing inversion in the northern South China Sea

The nonhydrostatic pressure effects on the generation and propagation of wind-forced internal waves are studied with a two-dimensional numerical ocean model. A one-way directed wind pulse over a stratified ocean initiates surface and internal waves in a closed basin. The studies are performed with horizontal grid sizes in the range from 1 km to 62.5 m. The experiments are performed with both a hydrostatic and a nonhydrostatic model, facilitating systematic studies of the sensitivity of the numerical model results to the grid size and to the nonhydrostatic pressure adjustments. The results show that the nonhydrostatic pressure effects are highly dependent on the grid size and grow with increased resolution. In the internal depression wave, the horizontal nonhydrostatic pressure gradients reach the same order of magnitude as the hydrostatic gradients in the high-resolution nonhydrostatic studies. In these studies, the nonhydrostatic pressure gradients approximately balance the corresponding hydrostatic pressure gradients in the internal depression wave, and the wave degenerates into a train of soliton waves. The time for the soliton form to develop agrees with the steepening timescale calculated from Korteweg-de Vries theory. In the high-resolution hydrostatic model, the internal depression wave takes the form of a single wave front. When the internal waves are generated in the boundary layers, the nonhydrostatic pressure gradients are much smaller than the hydrostatic gradients and the generation processes are not effected by the nonhydrostatic pressure with the present range of grid sizes.

Oceanic flows are turbulent and multi-scale in nature, and are composed of fast internal waves and slowly evolving balanced eddies. Contrary to conventional wisdom in physical oceanography, the past two decades of in situ , satellite altimeter and realistically forced global scale ocean model outputs have revealed that internal gravity waves can have comparable or higher energy levels than geostrophically balanced flows at 10–100 km scales in different parts of the world’s oceans. These relatively recent findings have fuelled a wide range of research activities aimed at understanding how fast internal gravity waves interact with slowly evolving balanced flows, particularly with the goal of deducing whether internal waves can form an energy sink for oceanic balanced flows. In this paper, we comprehensively review theoretical, numerical and observational investigations undertaken to study internal wave-balance flow exchanges. Theoretical calculations, inspired by different wave-balance regimes seen in observational and global ocean model outputs, are used to point out that internal waves can affect balanced flow dynamics. The theoretical results are followed up by a detailed discussion of numerical results on wave-balance interactions in a broad set of parameter regimes. The numerical results reveal how different kinds of waves exchange energy with balance flow, affect energy flux across scales of balanced flow and facilitate the generation of small-scale dissipative balanced flow structures. The numerical simulation results and global internal wave energy and balanced energy maps are used to conjecture that out of the 0.8 TW of power going to balanced flow kinetic energy in the ocean, at least 0.1 TW could be dissipated by internal gravity waves. We therefore hypothesize that internal waves can form a non-negligible energy sink for balanced flow in the world’s oceans.