Oceanic Internal Waves Research Articles

Higher‐order integrable models for oceanic internal wave–current interactions

AbstractIn this paper, we derive a higher‐order Korteweg–de Vries (HKdV) equation as a model to describe the unidirectional propagation of waves on an internal interface separating two fluid layers of varying densities. Our model incorporates underlying currents by permitting a sheared current in both fluid layers, and also accommodates the effect of the Earth's rotation by including Coriolis forces (restricted to the Equatorial region). The resulting governing equations describing the water wave problem in two fluid layers under a “flat‐surface” assumption are expressed in a general form as a system of two coupled equations through Dirichlet–Neumann (DN) operators. The DN operators also facilitate a convenient Hamiltonian formulation of the problem. We then derive the HKdV equation from this Hamiltonian formulation, in the long‐wave, and small‐amplitude, asymptotic regimes. Finally, it is demonstrated that there is an explicit transformation connecting the HKdV we derive with the following integrable equations of a similar type: KdV5, Kaup–Kuperschmidt equation, Sawada–Kotera equation, and Camassa–Holm and Degasperis–Procesi equations.

A One-Dimensional Model for Investigating Scale-separated Approaches to the Interaction of Oceanic Internal Waves

Abstract High-frequency wave propagation in near-inertial wave shear has, for four decades, been considered fundamental in setting the spectral character of the oceanic internal wave continuum and for transporting energy to wave-breaking. We compare idealized ray tracing numerical results with metrics derived using a wave turbulence derivation for the kinetic equation and a path integral to study this specific process. Statistical metrics include the changing ensemble mean vertical wavenumber, referred to as a mean drift, dispersion about the mean drift, time lagged correlation estimates of wavenumber and phase locking of the wave packets with the background. The path integral permits us to identify the mean drift as a resonant process and dispersion about that mean drift as non-resonant. At small inertial wave amplitudes; ray tracing, wave turbulence and the path integral provide consistent descriptions for the mean drift of wavepackets in the spectral domain and dispersion about the mean drift. Extrapolating these results to the background internal wavefield over-predicts downscale energy transports by an order of magnitude. At oceanic amplitudes, however, the numerics support diminished transport and dispersion that coincide with the mean drift time scale becoming similar to the lagged correlation time scale. We parse this as the transition to a non-Markovian process. Despite this decrease, numerical estimates of downscale energy transfer are still too large. We argue that residual differences result from an unwarranted discard of Bragg scattering resonances. Our results support replacing the long standing interpretive paradigm of extreme scale-separated interactions with a more nuanced slate of ‘local’ interactions in the kinetic equation.

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.

The modified cubic Benjamin–Ono equation describing internal solitary waves in the deep ocean and its related properties

In this article, we investigate the propagation of internal solitary waves in deep ocean. Based on the principles of nonlinear theory, perturbation expansion, and multi-scale analysis, a time-dependent modified cubic Benjamin–Ono (mCBO) equation is derived to describe internal solitary waves in the deep ocean with stronger nonlinearity. When the dispersive term ∂3f∂X3 vanishes, the mCBO equation transforms into the cubic BO equation. Similarly, when the dispersive term ∂3f∂X3 becomes zero and the nonlinear term ∂f3∂X degenerates into ∂f2∂X, the mCBO equation reduces to the BO equation. Furthermore, if the integral term ∂2∂X2ℵ(f) disappears, it simplifies to the mKdV equation. To gain deeper insight into the characteristics of solitary waves, conservation of mass and momentum associated with them are discussed. By employing Hirota's bilinear method, we obtain soliton solutions for the mCBO equation and subsequently investigate interactions between two solitary waves with different directions, leading to the occurrence of important events such as rogue waves and Mach reflections. Additionally, we explore how certain parameters influence Mach stem while drawing meaningful conclusions. Our discoveries reveal the complex dynamics of internal solitary waves within the deep ocean and contribute to a broader understanding of nonlinear wave phenomena.

Eddy–Internal Wave Interactions: Stimulated Cascades in Cross-Scale Kinetic Energy and Enstrophy Fluxes

Abstract The interactions between oceanic mesoscale eddies, submesoscale currents, and internal gravity waves (IWs) are investigated in submesoscale-resolving realistic simulations in the North Atlantic Ocean. Using a novel analysis framework that couples the coarse-graining method in space with temporal filtering and a Helmholtz decomposition, we quantify the effects of the interactions on the cross-scale kinetic energy (KE) and enstrophy fluxes. By systematically comparing solutions with and without IW forcing, we show that externally forced IWs stimulate a reduction in the KE inverse cascade associated with mesoscale rotational motions and an enhancement in the KE forward cascade associated with divergent submesoscale currents, i.e., a “stimulated cascade” process. The corresponding IW effects on the enstrophy fluxes are seasonally dependent, with a stimulated reduction (enhancement) in the forward enstrophy cascade during summer (winter). Direct KE and enstrophy transfers from currents to IWs are also found, albeit with weaker magnitudes compared with the stimulated cascades. We further find that the forward KE and enstrophy fluxes associated with IW motions are almost entirely driven by the scattering of the waves by the rotational eddy field, rather than by wave–wave interactions. This process is investigated in detail in a companion manuscript. Finally, we demonstrate that the stimulated cascades are spatially localized in coherent structures. Specifically, the magnitude and direction of the bidirectional KE fluxes at submesoscales are highly correlated with, and inversely proportional to, divergence-dominated circulations, and the inverse KE fluxes at mesoscales are highly correlated with strain-dominated circulations. The predominantly forward enstrophy fluxes in both seasons are also correlated with strain-dominated flow structures.

Generalized transport characterizations for short oceanic internal waves in a sea of long waves

Wave turbulence provides a conceptual framework for weakly nonlinear interactions in dispersive media. Dating from five decades ago, applications of wave turbulence theory to oceanic internal waves assigned a leading-order role to interactions characterized by a near equivalence between the group velocity of high-frequency internal waves with the phase velocity of near-inertial waves. This scale-separated interaction leads to a Fokker–Planck (generalized diffusion) equation. More recently, starting four decades ago, this scale-separated paradigm has been investigated using ray tracing methods. These ray methods characterize spectral transport of energy by counting the amplitude and net velocity of wave packets in phase space past a high-wavenumber gate prior to ‘breaking’. This explicitly advective characterization is based on an intuitive assignment and lacks theoretical underpinning. When one takes an estimate of the net spectral drift from the wave turbulence derivation and makes the corresponding assessment, one obtains a prediction of spectral transport that is an order of magnitude larger than either observations or reported ray tracing estimates. Motivated by this contradiction, we report two parallel derivations for transport equations describing the refraction of high-frequency internal waves in a sea of random inertial waves. The first uses standard wave turbulence techniques and the second is an ensemble-averaged packet transport equation characterized by the dispersion of wave packets about a mean drift in the spectral domain. The ensemble-averaged transport equation for ray tracing differs in that it contains the intuitively motivated advective term. We conclude that the aforementioned contradiction between theory, numerics and observations needs to be taken at face value and present a pathway for resolving this contradiction.

A Texture Enhancement Method for Oceanic Internal Wave Synthetic Aperture Radar Images Based on Non-Local Mean Filtering and Texture Layer Enhancement

Synthetic aperture radar (SAR) is an important tool for observing the oceanic internal wave phenomenon. However, owing to the unstable imaging quality of SAR on oceanic internal waves, the texture details of internal wave images are usually unclear, which is not conducive to the subsequent applications of the images. To cope with this problem, a texture enhancement method for oceanic internal wave SAR images is proposed in this paper, which is based on non-local mean (NLM) filtering and texture layer enhancement (TLE). Since the strong speckle noise commonly present in internal wave images is simultaneously enhanced during texture enhancement, resulting in degraded image quality, NLM filtering is first performed to suppress speckle noise. Then, the denoised image is decomposed into the structure layer and the texture layer, and a texture layer enhancement method oriented to the texture characteristics of oceanic internal waves is proposed and applied. Finally, the enhanced texture layer and the structure layer are combined to reconstruct the final enhanced image. Experiments are conducted based on the Gaofen-3 real SAR data, and the results demonstrate that the proposed method performs well in suppressing speckle noise, maintaining overall image brightness, and enhancing internal wave texture details.

Bright traveling breathers in media with long-range nonconvex dispersion.

The existence and properties of envelope solitary waves on a periodic traveling-wave background, called traveling breathers, are investigated numerically in representative nonlocal dispersive media. Using a fixed-point computational scheme, a space-time boundary-value problem for bright traveling breather solutions is solved for the weakly nonlinear Benjamin-Bona-Mahony equation, a nonlocal, regularized shallow water wave model, and the strongly nonlinear conduit equation, a nonlocal model of viscous core-annular flows. Curves of unit-mean traveling breather solutions within a three-dimensional parameter space are obtained. Resonance due to nonconvex, rational linear dispersion leads to a nonzero oscillatory background upon which traveling breathers propagate. These solutions exhibit a topological phase jump and so act as defects within the periodic background. For small amplitudes, traveling breathers are well approximated by bright soliton solutions of the nonlinear Schrödinger equationwith a negligibly small periodic background. These solutions are numerically continued into the large-amplitude regime as elevation defects on cnoidal or cnoidal-like periodic traveling-wave backgrounds. This study of bright traveling breathers provides insight into systems with nonconvex, nonlocal dispersion that occur in a variety of media such as internal oceanic waves subject to rotation and short, intense optical pulses.

Stripe Extraction of Oceanic Internal Waves Using PCGAN with Small-Data Training

Playing a crucial role in ocean activities, internal solitary waves (ISWs) are of significant importance. Currently, the use of deep learning for detecting ISWs in synthetic aperture radar (SAR) imagery is gaining growing attention. However, these approaches often demand a considerable number of labeled images, which can be challenging to acquire in practice. In this study, we propose an innovative method employing a pyramidal conditional generative adversarial network (PCGAN). At each scale, it employs the framework of a conditional generative adversarial network (CGAN), comprising a generator and a discriminator. The generator works to produce internal wave patterns as authentically as possible, while the discriminator is designed to differentiate between images generated by the generator and reference images. The architecture based on pyramids adeptly captures the encompassing as well as localized characteristics of internal waves. The incorporation of upsampling further bolsters the model’s ability to recognize fine-scale internal wave stripes. These attributes endow the PCGAN with the capacity to learn from a limited amount of internal wave observation data. Experimental results affirm that the PCGAN, trained with just four internal wave images, can accurately detect internal wave stripes in the test set. Through comparative experiments with other segmentation models, we demonstrate the effectiveness and robustness of PCGAN.

Oceanic internal solitary wave interactions via the KP equation in a three-layer fluid with shear flow

Oceanic internal solitary wave interactions via the KP equation in a three-layer fluid with shear flow

On the steady-state interfacial waves with two-dimensional type-A double exact resonance

Steady-state interfacial waves under two-dimensional (2D) type-A exact triad resonance and other related resonances are researched in a two-layer liquid model with a free surface in contact with air. Five groups (groups 1–5) of convergent series solutions are achieved via the homotopy analysis method. It is found that the phenomenon of double exact resonance could exist in periodic interfacial waves if physical parameters correspond to the intersection of two exact resonance curves. The double exact resonance considered here contains a 2D type-A triad resonance and an other resonance. Under the 2D type-A exact triad resonance, the other resonant triad could obviously enlarge or reduce the wave amplitudes and energy proportions of primary and resonant components. Nevertheless, other resonant quartet, quintet, sextet, and septet all produce no influence on interfacial waves when the 2D type-A exact triad resonance occurs. The above-mentioned results indicate that in the neighborhood of the double exact triad resonance, small perturbations of wave vector of a primary component can cause huge changes on wave profiles of free surface and interface, wave amplitude spectrum, and energy distribution of internal waves in real ocean. In addition, the closer the interfacial waves are to the double exact triad resonance, the more possible energy combinations exist in the wave system, and the greater the number of steady-state interfacial wave solutions. All of this should deepen our understanding of nonlinear resonance interactions in short-crested internal waves.

Ocean Internal Wave Detection in SAR Images Based on Improved YOLOv7

Ocean Internal Wave Detection in SAR Images Based on Improved YOLOv7

Oceanic eddies induce a rapid formation of an internal wave continuum

Oceanic internal waves are a major driver for turbulent mixing in the ocean, which controls the global overturning circulation and the oceanic heat and carbon transport. Internal waves are observed to have a continuous energy distribution across all wave frequencies and scales, commonly known as the internal wave continuum, despite being forced at near-inertial and tidal frequencies at large scales. This internal wave continuum is widely thought to be developed primarily through wave-wave interactions. Here we show, using realistic numerical simulations in the subpolar North Atlantic, that oceanic eddies rapidly distribute large-scale wind-forced near-inertial wave energy across spatio-temporal scales, thereby forming an internal wave continuum within three weeks. As a result, wave energy dissipation patterns are controlled by eddies and are substantially enhanced below the mixed layer. The efficiency of this process potentially explains why a phase lag between high-frequency and near-inertial wave energy was observed in eddy-poor regions but not in eddy-rich regions. Our findings highlight the importance of eddies in forming an internal wave continuum and in controlling upper ocean mixing patterns.

An Automatic Internal Wave Recognition Algorithm Based on CNN Applicable to an Ocean Data Buoy System

The application of internal wave recognition to the buoy system is of great significance to enhance the understanding of the ocean internal wave phenomenon and provide more accurate data and information support. This article proposes an automatic internal wave recognition algorithm based on convolutional neural networks (CNN), which is used in the tight-profile intelligent buoy system. The sea profile temperature data were collected using the Bailong buoy system in the Andaman Sea in 2018. The CNN network structure is applied to feature compression of ocean temperature profile data, reducing the input feature amount of the feature recognition network, thereby reducing the overall algorithm parameters and computational complexity. By adjusting the number of convolution kernels and the length of convolution steps, the original data features in the time domain and the space domain are compressed, respectively. The experimental results show that the identification accuracy and robustness of this method are clearly superior to those of other methods. Additionally, the parameter number and calculation amount of this algorithm are very tiny, which greatly improves the possibility of its deployment in the buoy system.

Estimating the Location of Acoustic Sources with Oceanic Internal Gravity Waves via Frequency-Difference Source Localization

Frequency-difference source localization (FDSL) is a relatively new method for locating sources emitting acoustic or electromagnetic waves from recorded data. Estimates of location are produced by matching a nonlinear function of the received data with the same nonlinear function of modeled data. This so-called matched-field approach depends on the accuracy of the modeled field. FDSL exhibits less sensitivity to model errors in some cases of interest. Fluctuations of the speed of sound in the ocean are affected by a wide variety of phenomena and internal gravity waves is one where their spectrum is usually well known. There is no analytical theory for predicting how these fluctuations of sound speed affect the accuracy of FDSL, leaving a numerical evaluation as the alternative for the focus of this paper. Experimental application of FDSL was previously made for an acoustic source of 100 Hz bandwidth surrounding 250 Hz for a 129 km section in the Philippine Sea with data recorded on a long vertical array [Geroski and Dowling, Long-range frequency-difference source localization in the Philippine Sea, J. Acoust. Soc. Am. 146 (2019) 4727–4739.]. The role of internal waves is quantified with a realistic FDSL simulation using models based on the parabolic and ray approximations of the linear acoustic wave equation. The modeled vertical variation of FDSL error is similar to observed while the modeled horizontal error is too small. The complicated software model is automatic but is computationally intensive, with our version needing about 1.5 mo to compute on a PC with 16 cpu cores.

Validating Finescale Parameterizations for the Eastern Arctic Ocean Internal Wave Field

AbstractIn the Arctic Ocean, vertical transport of heat by turbulent mixing is ultimately coupled to the sea‐ice cover, with immediate and far‐reaching impacts on the climate and ecosystem. Unfortunately, direct observations of mixing are difficult, expensive and sparse. Finescale Parameterization (FS) of turbulent energy dissipation rate (ɛ) allows for the quantification of turbulence from breaking internal waves using standard measurements, such as profiles of hydrography and velocity. While FS proved to be reliable in mid‐latitudes, the Arctic Ocean internal wave field is distinct in terms of composition and energy level, rendering the applicability of FS uncertain. To test FS in a wide range of eastern Arctic conditions, we compiled data from eight cruises. All profiles used to calculate FS were collocated with in‐situ measurements of ɛ obtained from microstructure profilers. FS was applied between 50 and 450 m below the surface. Results show a satisfactory performance of FS, with 84% of FS‐derived ɛ being within a factor of 5 to observations. This improved to 90% when using lower‐noise velocity profiles of lowered current meters instead of ship‐mounted current meters. In our data, FS performance is independent of the shear‐strain ratio (Rω) and internal wave field bandwidth (N/f), but there is evidence that highly stratified environments with large potential energy, low turbulence and substantially non‐white shear spectra are less suitable for FS. A widely used formulation of FS using only hydrography and a prescribed Rω = 7 results in 73% of FS estimates being within a factor of 5 to observations.

Higher-Order Benjamin–Ono Model for Ocean Internal Solitary Waves and Its Related Properties

In this study, the propagation of internal solitary waves in oceans at great depths was analyzed. Using multi-scale analysis and perturbation expansion, the basic equation is simplified to the classical Benjamin–Ono equation with variable coefficients. To better describe the propagation characteristics of solitary waves, we derived a higher-order variable-coefficient integral differential (Benjamin–Ono) equation. Subsequently, the bilinear form of the model was derived using Hirota’s bilinear method, and a multi-soliton solution was obtained. Based on the multi-soliton solution of the model, we further studied the interaction of the soliton, which led to the discovery of Mach reflection. Some conclusions were drawn, which are of potential value for further study of solitary waves in the ocean.

Detection of Ocean Internal Waves Based on Modified Deep Convolutional Generative Adversarial Network and WaveNet in Moderate Resolution Imaging Spectroradiometer Images

The generation and propagation of internal waves in the ocean are a common phenomenon that plays a pivotal role in the transport of mass, momentum, and energy, as well as in global climate change. Internal waves serve as a critical component of oceanic processes, contributing to the redistribution of heat and nutrients in the ocean, which, in turn, has implications for global climate regulation. However, the automatic identification of internal waves in oceanic regions from remote sensing images has presented a significant challenge. In this research paper, we address this challenge by designing a data augmentation approach grounded in a modified deep convolutional generative adversarial network (DCGAN) to enrich MODIS remote sensing image data for the automated detection of internal waves in the ocean. Utilizing t-distributed stochastic neighbor embedding (t-SNE) technology, we demonstrate that the feature distribution of the images produced by the modified DCGAN closely resembles that of the original images. By using t-SNE dimensionality reduction technology to map high-dimensional remote sensing data into a two-dimensional space, we can better understand, visualize, and analyze the quality of data generated by the modified DCGAN. The images generated by the modified DCGAN not only expand the dataset’s size but also exhibit diverse characteristics, enhancing the model’s generalization performance. Furthermore, we have developed a deep neural network named “WaveNet,” which incorporates a channel-wise attention mechanism to effectively handle complex remote sensing images, resulting in high classification accuracy and robustness. It is important to note that this study has limitations, such as the reliance on specific remote sensing data sources and the need for further validation across various oceanic regions. These limitations are essential to consider in the broader context of oceanic research and remote sensing applications. We initially pre-train WaveNet using the EuroSAT remote sensing dataset and subsequently employ it to identify internal waves in MODIS remote sensing images. Experiments show the highest average recognition accuracy achieved is an impressive 98.625%. When compared to traditional data augmentation training sets, utilizing the training set generated by the modified DCGAN leads to a 5.437% enhancement in WaveNet’s recognition rate.

Nonlinear Dynamic Analysis of Seafloor Pressure Time Series with and without Internal Waves

Ocean internal wave as a nonlinear process oscillates within ocean interior rather than the surface. Internal waves have significant influences on the transport of material, energy and even global climate. The generation, transmission and attenuation of internal wave play a significant role in the ocean dynamics owing to the exchange of the energy within each process. In this paper, reduced complexity of seafloor pressure time series induced by internal wave in the southern Taiwan bank has been investigated using nonlinear dynamic analysis. Cao method has been used to distinguish between the pressure signal with internal wave and the random signal without internal wave. The reconstructed phase spaces and correlation dimensions of the time series with and without internal wave are compared. Internal wave leads the seafloor pressure time series to produce a more regular phase space reconstruction and more low dimensional dynamics. Our study suggests that nonlinear dynamic analysis could be a valuable and promising method in real-time ocean internal wave detection.

Responses of Phytoplankton Communities to Internal Waves in Oligotrophic Oceans

AbstractUnderstanding the potential impacts of internal waves on phytoplankton communities in oligotrophic oceans remains an important research challenge. In this study, we elucidated the impact of internal waves on phytoplankton communities through a comprehensive 154‐hr time‐series of observations in the South China Sea (SCS). We identified distinctive variations in phytoplankton pigment biomass and composition across the upper, middle, and lower layers of the euphotic zone, which we attributed to the perturbations triggered by internal waves. Phytoplankton other than Prochlorococcus in the lower, nutrient‐replete layer likely benefitted from allochthonous nutrients introduced by internal waves, but their growth rates were constrained by light limitation, and their pigment biomass was held in check by microzooplankton grazing. In contrast, in the upper, nutrient‐depleted layer, the relative abundance of Prochlorococcus increased, likely because of the ammonium regenerated by zooplankton. The middle layer, characterized as the deep chlorophyll maximum layer, exhibited a dynamic equilibrium characterized by nutrient and light co‐limitation. This equilibrium resulted in high nitrate assimilation and growth by phytoplankton. The balancing of those rates by significant grazing losses maintained total chlorophyll a concentrations at a high level. Based on these findings, we proposed a three‐layer euphotic zone structure characterized by distinct physiological conditions, nutrient‐light dynamics, grazing pressure, and phytoplankton responses to internal waves. This three‐layer paradigm elucidated the intricate interplay between internal waves and phytoplankton communities and provided insights into the mechanisms that govern primary production and carbon cycling in oligotrophic oceanic ecosystems.