Deep-water Waves Research Articles

Momentum, energy and vorticity balances in deep-water surface gravity waves

The particle trajectories in irrotational, incompressible and inviscid deep-water surface gravity waves are open, leading to a net drift in the direction of wave propagation commonly referred to as the Stokes drift, which is responsible for catalysing surface wave-induced mixing in the ocean and transporting marine debris. A balance between phase-averaged momentum density, kinetic energy density and vorticity for irrotational, monochromatic and spatially periodic two-dimensional water waves is derived by working directly within the Lagrangian reference frame, which tracks particle trajectories as a function of their labels and time. This balance should be expected as all three of these quantities are conserved following particles in this system. Vorticity in particular is always conserved along particles in two-dimensional inviscid flow, and as such even in its absence it is the value of the vorticity that fundamentally sets the drift, which in the Lagrangian frame is identified as the phase-averaged momentum density of the system. A relationship between the drift and the geometric mean water level of particles is found at the surface, which highlights connections between the geometry and dynamics. Finally, an example of an initially quiescent fluid driven by a wavelike pressure disturbance is considered, showing how the net momentum and energy from the surface pressure disturbance transfer to the wave field, and recognizing the source of the mean Lagrangian drift as the net momentum required to generate an irrotational surface wave by any conservative force.

Hydrodynamic modulation instability triggered by a two-wave system.

The modulation instability (MI) is responsible for the disintegration of a regular nonlinear wave train and can lead to strong localizations in the form of rogue waves. This mechanism has been studied in a variety of nonlinear dispersive media, such as hydrodynamics, optics, plasma, mechanical systems, electric transmission lines, and Bose-Einstein condensates, while its impact on applied sciences is steadily growing. It is well-known that the classical MI dynamics can be triggered when a pair of small-amplitude sidebands are excited within a particular frequency range around the main peak frequency. That is, a three-wave system, consisting of the carrier wave together with a pair of unstable sidebands, is usually adopted to initiate the wave focusing process in a numerical or laboratory experiment. Breather solutions of the nonlinear Schrödinger equation (NLSE) revealed that MI can generate much more complex localized structures, beyond the three-wave system initialization approach or by means of a continuous spectrum. In this work, we report an experimental study for deep-water surface gravity waves asserting that a MI process can be triggered by a single unstable sideband only, and thus, initialized from a two-wave process when including the contribution of the peak frequency. The experimental data are validated against fully nonlinear hydrodynamic numerical wave tank simulations and show very good agreement. The long-term evolution of such unstable wave trains shows a distinct shift in the recurrent Fermi-Pasta-Ulam-Tsingou focusing cycles, which are captured by the NLSE and fully nonlinear hydrodynamic simulations with some distinctions.

Power absorption and flow-field characteristics analysis for oscillating coaxial twin-buoy wave energy converter by using CFD

The energy conversion capacity of wave energy conversion devices highly depends on the hydrodynamic characteristics of the energy-harvesting structure. To investigate the effect of hydrodynamic performance on the power conversion characteristics, a twin-buoy wave energy converter (WEC) was investigated by using a three-dimensional numerical wave pool based on the Computational Fluid Dynamics (CFD) method. Several factors are examined, including the elasticity coefficient of the anchor chain, the bottom configuration of the floating body, and the power take-off (PTO) damping coefficient. The heave displacement, heave velocity, and heave force of the converter are calculated under specific wave parameters, and the flow field cloud diagram during the heave motion is analyzed. The results indicate that a wave energy converter with a hemispherical floating body exhibits the best kinematic performance. The influence of the mechanical damping coefficient on the energy conversion performance of the device is studied. By appropriately reducing the mechanical damping coefficient, the energy capturing capability of the device can be increased to a certain extent. These findings can serve as a theoretical basis for the application of deep-water wave energy conversion in engineering and the optimization of future WEC designs.

Predicting heave and pitch motions of an FPSO using meta-learning

Real-time motion prediction is helpful in guaranteeing the operation stability of a Floating Production Storage Offloading (FPSO) unit. Recurrent neural networks (RNNs) are becoming feasible alternatives to numerical simulations for motion prediction as artificial intelligence develops rapidly. In this study, model-agnostic meta-learning (MAML) is combined with RNNs to deterministically predict the heave and pitch motions of a ship-shaped FPSO. This approach is motivated by the fact that MAML improves training efficiency without losing accuracy. The data came from a scaled model test conducted at Shanghai Jiao Tong University’s deepwater wave basin. Before introducing MAML, we verified that long short-term memory (LSTM) and gated recurrent unit (GRU) could accurately predict the heave and pitch of about 7.68 s into the future. With fewer learnable parameters than LSTM, GRU demonstrates slightly better accuracy. Therefore, this study focuses particularly on the combination of GRU and MAML. The parameters of MAML, including order of derivative, step size, number of adaption gradient updates, and batch size of the tasks, are evaluated systemically in terms of accuracy and training efficiency. With the assistance of MAML, GRU’s training efficiency for heave and pitch has significantly improved, increasing by approximately 65% and 55%, respectively. Meanwhile, the prediction error for both has decreased by about 10%. Notably, MAML’s performance is minimally affected by variations in incoming wave direction and sea state, as well as the randomness and temporal variability of the motion. MAML is a powerful tool that enables RNNs to achieve real-time prediction of FPSO motion.

Interaction Between Long Internal Waves and Free Surface Waves in Deep Water

Interaction Between Long Internal Waves and Free Surface Waves in Deep Water

Computational study on complex wave hydrodynamics of multidirectional extreme waves at fringing reef

Due to the special topography of coral reefs, most of the energy of incoming waves can be reduced through the wave breakings at the reef edge, thus playing a vital role in protecting the coastal regions behind coral reefs. In severe weather conditions, wind-generated waves tend to propagate in multiple directions in real ocean environment. The abrupt multidirectional wave-focusing mechanisms may result in the formation of huge waves in both deep and shallow waters, threatening the safety of coastal infrastructures and well-being of coastal communities at the reef islands. Therefore, it is crucial to thoroughly examine the complex hydrodynamics of multidirectional extreme waves at the fringing reefs. Previous studies on reef wave hydrodynamics predominantly focus on unidirectional waves. Only a small number of studies have considered the wave hydrodynamics of fringing reefs under multidirectional waves, let alone multidirectional extreme waves. To address the knowledge gap in previous studies, this study analyzes the complex hydrodynamic processes of multidirectional extreme waves at the fringing reef by applying a nonhydrostatic numerical wave model (NHWAVE). Influences of several key factors are analyzed, such as significant wave height, peak wave period, water depth, incident wave angle, and reef topography. It is expected that the research findings of this study will contribute to a deeper understanding of the hydrodynamics of multidirectional extreme waves at the fringing reef.

1D -FNO - Residual convolutional neural network autoregressive prediction model for deep water wave train evolution based on high-order spectral (HOS) data

Accurate prediction of deep-water wave dynamics is essential for ocean engineering, meteorology, and maritime safety. Traditional physical models, though effective, face significant computational and time challenges with complex nonlinear dynamics. This study introduces an autoregressive prediction model using a one-dimensional adaptive Fourier Neural Operator-Residual Convolutional Neural Network (1D-FNO-ResNet) to enhance prediction accuracy. We developed a high-precision single-step prediction model capable of capturing wave characteristics such as height and fluid particle velocity. A fine-tuning training strategy based on multi-step prediction errors was proposed for autoregressive evolution. Performance comparisons between the ResNet and FNO-ResNet models were made under various fine-tuning steps. A correction model was also introduced to control error accumulation, improving autoregressive prediction steps. The model's generalization performance was validated with modulated and random wave trains under different conditions, optimizing predictive stability and accuracy. This approach demonstrates efficient 1D deep-water wave evolution prediction, offering a novel deep learning solution for wave dynamics prediction in ocean engineering, supporting wave energy conversion device design and optimization.

Symmetric irrotational deep-water waves are traveling waves

In this paper, we show that a periodic solution to the irrotational two-dimensional deep-water wave problem with the horizontal velocity components at the surface, and wave profiles are symmetric, and periodic in the x-variable, necessarily defines a traveling wave. The proof relies on employing the Maximum principle for harmonic functions in unbounded domains and structural properties of the governing equations for nonlinear deep-water waves.

Numerical simulation of deep-water wave breaking using RANS: Comparison with experiments

Wave breaking is a multifaceted physical phenomenon that is not fully understood and remains challenging to model. An effective method for investigating wave breaking involves utilising the two-phase Reynolds-averaged Navier–Stokes (RANS) equations to directly simulate breaking waves. In this study, we apply a RANS model with an adaptively refined mesh to simulate breaking waves in deep water using the stabilised RANS model proposed by Larsen and Fuhrman. This approach enables a more efficient simulation of the physics of breaking waves compared to Direct Numerical Simulations, as it places less stringent demands on grid resolution. Our findings demonstrate that the RANS model compares well with deep water wave breaking experiments in terms of surface elevation. We also give estimates of the breaking strength parameter of our RANS simulations and compared them with the literature.

Effect of Fixed Bended Plate on Obliquely Incident Waves in Deep Water

Effect of Fixed Bended Plate on Obliquely Incident Waves in Deep Water

Soliton groups and extreme wave occurrence in simulated directional sea waves

The evolution of nonlinear wave groups that can be associated with long-lived soliton-type structures is analyzed, based on the data of numerical simulation of irregular deep-water gravity waves with spectra typical to the ocean and different directional spreading. A procedure of the windowed Inverse Scattering Transform, which reveals wave sequences related to envelope solitons of the nonlinear Schrödinger equation, is proposed and applied to the simulated two-dimensional surfaces. The soliton content of waves with different directional spreading is studied in order to estimate its dynamical role, including characteristic lifetimes. Statistical features of the solitonic part of the water surface are analyzed and compared with the wave field on average. It is shown that intense wave patterns that persist for tens of wave periods can emerge in stochastic fields of relatively long-crested waves. They correspond to regions of locally enhanced on average waves with reduced kurtosis. This eventually leads to realization of locally extreme wave conditions compared to the general background. Although intense soliton-like groups may be detected in short-crested irregular waves as well, they possess much shorter lateral sizes, quickly disperse, and do not influence the local statistical wave properties.

High-level Green–Naghdi model for large-amplitude internal waves in deep water

In this paper, a high-level Green–Naghdi (HLGN) model for large-amplitude internal waves in a two-layer fluid system, where the upper-fluid layer is of finite depth and the lower-fluid layer is of infinite depth, is developed under the rigid-lid free-surface approximation. The equations of the present HLGN model follow Euler's equations under the sole assumption that the horizontal and vertical velocity distributions along the vertical column are presented by known shape functions for each layer. The linear dispersion relations of the HLGN model for different levels are presented and compared with those obtained by other strongly nonlinear models for deep water, including the fully nonlinear models that include the dispersion effects $O(\mu )$ (where $\mu$ is the ratio of the upper-fluid layer depth to a typical wavelength) derived by Choi & Camassa (Phys. Rev. Lett., vol. 77, 1996, pp. 1759–1762) and $O(\mu ^2)$ derived by Debsarma et al. (J. Fluid Mech., vol. 654, 2010, pp. 281–303). It is shown that the HLGN model has a wider application range than other models. Solutions of travelling large-amplitude internal solitary waves in the absence and presence of background shear-current are then investigated by using the HLGN model. For the no-current cases, results obtained by the HLGN model show better agreement with Euler's solution on wave profile, velocity profile at the maximum interface displacement and wave speed compared with those obtained by other models. For the background shear-current cases, results obtained by the HLGN model also show good agreement with those obtained by solving the Dubreil-Jacotin–Long equation.

Nonlinear Fourier classification of 663 rogue waves measured in the Philippine Sea.

Rogue waves are sudden and extreme occurrences, with heights that exceed twice the significant wave height of their neighboring waves. The formation of rogue waves has been attributed to several possible mechanisms such as linear superposition of random waves, dispersive focusing, and modulational instability. Recently, nonlinear Fourier transforms (NFTs), which generalize the usual Fourier transform, have been leveraged to analyze oceanic rogue waves. Next to the usual linear Fourier modes, NFTs can additionally uncover nonlinear Fourier modes in time series that are usually hidden. However, so far only individual oceanic rogue waves have been analyzed using NFTs in the literature. Moreover, the completely different types of nonlinear Fourier modes have been observed in these studies. Exploiting twelve years of field measurement data from an ocean buoy, we apply the nonlinear Fourier transform (NFT) for the nonlinear Schrödinger equation (NLSE) (referred to NLSE-NFT) to a large dataset of measured rogue waves. While the NLSE-NFT has been used to analyze rogue waves before, this is the first time that it is systematically applied to a large real-world dataset of deep-water rogue waves. We categorize the measured rogue waves into four types based on the characteristics of the largest nonlinear mode: stable, small breather, large breather and (envelope) soliton. We find that all types can occur at a single site, and investigate which conditions are dominated by a single type at the measurement site. The one and two-dimensional Benjamin-Feir indices (BFIs) are employed to examine the four types of nonlinear spectra. Furthermore, we verify on a part of the data set that for the localized types, the largest nonlinear Fourier mode can be attributed directly to the rogue wave, and investigate the relation between the height of the rogue waves and that of the dominant nonlinear Fourier mode. While the dominant nonlinear Fourier mode in general only contributes a small fraction of the rogue wave, we find that soliton modes can contribute up to half of the rogue wave. Since the NLSE does not account for directional spreading, the classification is repeated for the first quartile with the lowest directional spreading for each type. Similar results are obtained.

Breaking Wave Statistics In Short-Crested Seas

The accurate prediction of loads to allow for a safe yet efficient design of coastal and offshore structures requires a thorough understanding of the distribution of crest heights. Although deep water waves are relatively well understood and accurately modelled by existing wave and crest height models, the same cannot be said for waves in intermediate and shallow waters (Tayfun & Alkhalidi, 2020). This work addresses this gap by analysing long, random records of laboratory-generated short-crested seas. The analysis focuses on the application of a novel method to identify waves that undergo significant nonlinear amplification and breaking. Following their identification, the associated energy transfers and dissipation are calculated per wave and probabilistically approximated. A modelling suite is proposed to describe the probability of wave breaking and associated dissipation in short-crested seas. This is then converted into a mixture model to recover crest height statistics. The success of the proposed approach is demonstrated through comparisons between model predictions and measurements.

A Study on the Scour Surrounding the Fixed Foundation of an Offshore Wind Turbine under Complex Waves, Tidal Currents, and Pile Vibration Conditions

Scouring around the fixed foundations of offshore wind turbines (OWTs) remains a pressing concern within the offshore wind industry, significantly jeopardizing OWT safety. Despite previous efforts, the current understanding of this phenomenon remains incomplete, as various factors related to offshore wind farms and OWT operations can exert intricate influences. To further explore this matter, this paper undertakes a new study to further understand the interplay between monopile vibrations, tidal currents, and sea waves, elucidating their combined impact on scour surrounding an OWT’s foundation. It was found that, besides pile vibration amplitude and frequency, the pile vibration direction can also notably influence the scour around the monopile. Specifically, the most pronounced scour mostly transpires when the monopile undergoes vibrations along the 135° direction relative to the inflow direction. The mildest scour mostly occurs when the pile vibrates along the 90° direction. Additionally, this study reveals that different types of waves, indicated by the ratio of water depth, H, to wavelength, L, also engender varying scour around the monopile. The influence of waves on scour increases with the decrease in the H/L ratio, implying that OWT foundations are more threatened by shallow water waves than by deep water waves.

Wake characteristics behind a tidal turbine with surface waves in turbulent flow analyzed with large-eddy simulation

To design tidal stream turbine arrays, turbine wakes need to be fully characterized to assess the adequate row spacing considering environmental factors such as onset turbulence, velocity shear, and surface waves. The role of waves on wake development varies depending on their characteristics, such as wavelength and amplitude, and needs to be carefully understood. Here large-eddy simulations are performed to analyze the instantaneous and time-averaged wake characteristics developed downstream of a tidal stream turbine for four wave-current conditions ranging from nearly deep to intermediate waves and compared to current-only results. The tip-vortex convection near the free surface is highly influenced by the waves. During the period of increased surface elevation, there is an upwards motion of the vortices which have merged into a single vortical structure during the wave trough period. Downwards vertical transport of tip vortices occurs after the pass of the wave crest, enabling vertical entrainment of momentum into the turbine wake region. Disk-averaged velocity deficit recovery increases for wave-current cases compared to the current-alone condition, with all intermediate wave cases showing a very similar wake recovery rate throughout the wake length. The nearly deep-water waves lead to a slightly slower recovery rate up to 12 turbine diameters (D) downstream while, after this distance, the recovery rate is the highest among the simulated wave conditions. The integration of the mean kinetic energy (MKE) budget in the region up to 12D shows there is a similar distribution in the balance between terms irrespective of the wave-current case. However, in the far-wake region covering from 12D to 20D, the nearly deep waves have a larger contribution for MKE replenishment from the transverse convection and turbulence transport terms, balanced with their vertical counterparts and the streamwise convection term. Analysis of the wake self-similarity shows that a Gaussian model description holds for the recovery rate and lateral profile of the wake of a tidal turbine operating in coupled wave-current conditions, with a similar rate of velocity recovery after 12D irrespective of the wave characteristics. Lateral wake spreading is smaller for the shortest wavelength, which relates to an increased contribution of transverse fluxes in the MKE budget. Published by the American Physical Society 2024

Standard log-capture differentials as performance metrics for deepwater wave power generation

As a support to performance assessment of deepwater wave energy converters, this work quantifies susceptibility of output power to short duration variations, independent of source wave statistics. Introduced as ς-metrics, the quantified measures may be a basis for technology comparisons, with possible convergence to a choice for implementation. The metrics further allow comparison between different versions of a specific converter technology, thereby helping in design revision and refinement.With power conversion represented in terms of a technology specific capture width function, the log-capture differential is defined relative to a hypothetical benchmark characterised by negligible power variability. Root-mean-square aggregation of the differential further leads to standard log-capture differentials as site independent ς-metrics for a capture-and-conversion mechanism.Study results are included in three phases. First, three established wave energy conversion technologies are analysed at respective power maximums, so as to illustrate applicability of log-capture differential and ς-metrics. Next, the metrics are analysed at fractional output power to realise converter performance at reduced variability. Finally for a given marine environment, ς-metrics are shown to be useful in projection of output power variability across technologies, together with the mean output power as fraction of rated maximum.

On elementary four-wave interactions in dispersive media

The cubic interactions in a discrete system of four weakly nonlinear waves propagating in a conservative dispersive medium are studied. By reducing the problem to a single ordinary differential equation governing the motion of a classical particle in a quartic potential, the complete explicit branches of solutions are presented, either steady, periodic, breather or pump, thus recovering or generalizing some already published results in hydrodynamics, nonlinear optics and plasma physics, and presenting some new ones. Various stability criteria are also formulated for steady equilibria. Theory is applied to deep-water gravity waves for which models of isolated quartets are described, including bidirectional standing waves and quadri-directional travelling waves, steady or not, resonant or not.

Interaction of soliton gases in deep-water surface gravity waves.

Soliton gases represent large random soliton ensembles in physical systems that display integrable dynamics at leading order. We report hydrodynamic experiments in which we investigate the interaction between two beams or jets of soliton gases having nearly identical amplitudes but opposite velocities of the same magnitude. The space-time evolution of the two interacting soliton gas jets is recorded in a 140-m-long water tank where the dynamics is described at leading order by the focusing one-dimensional nonlinear Schrödinger equation. Varying the relative initial velocity of the two species of soliton gas, we change their interaction strength and we measure the macroscopic soliton gas density and velocity changes due to the interaction. Our experimental results are found to be in good quantitative agreement with predictions of the spectral kinetic theory of soliton gas despite the presence of perturbative higher-order effects that break the integrability of the wave dynamics.

Nearshore wave buoy data from southeastern Australia for coastal research and management

Wind wave observations in shallow coastal waters are essential for calibrating, validating, and improving numerical wave models to predict sediment transport, shoreline change, and coastal hazards such as beach erosion and oceanic inundation. Although ocean buoys and satellites provide near-global coverage of deep-water wave conditions, shallow-water wave observations remain sparse and often inaccessible. Nearshore wave conditions may vary considerably alongshore due to coastline orientation and shape, bathymetry and islands. We present a growing dataset of in-situ wave buoy observations from shallow waters (<35 m) in southeast Australia that comprises over 7,000 days of measurements at 20 locations. The moored buoys measured wave conditions continuously for several months to multiple years, capturing ambient and storm conditions in diverse settings, including coastal hazard risk sites. The dataset includes tabulated time series of spectral and time-domain parameters describing wave height, period and direction at half-hourly temporal resolution. Buoy displacement and wave spectra data are also available for advanced applications. Summary plots and tables describing wave conditions measured at each location are provided.