Wind-wave Interaction Research Articles

A numerical analysis of wave type and steepness effects on cylindrical buoy dynamic characteristics

In the intricate realm of ocean-atmosphere dynamics, the sustained and stable operation of buoy observation systems under rough seas is crucial for ensuring marine resource security. This study employs a coupled Level-Set and Volume of Fluid (CLSVOF) method alongside the immersed boundary method. It investigates the physical processes of wind-wave interaction, the dynamic characteristics of cylindrical buoys, as well as the underlying mechanisms. The research highlights the significant impact of wave types (wind waves, swells) and wave steepness on the buoy's drift and oscillatory behavior. An increase in wave steepness and a shift from wind waves to swell markedly intensify both the buoy's drift and its oscillatory motions. Additionally, a thorough analysis of the loads on the buoy under varying maritime conditions is performed. By examining the vertical momentum equation on the windward and leeward sides, the secondary load cycle phenomenon is found to originate from collapse of the water column formed by the convergence of water flow behind the buoy. Wind waves are found to be less conducive to secondary load cycles compared to swell waves. Furthermore, the study investigates the relationship between the drag coefficient and the wave crest height on the windward side of the buoy, revealing a robust negative correlation between the two. This research not only illuminates the complex dynamics of wind-wave couplings but also offers theoretical insights for the optimization of buoy design.

Effects of Wind Speed on Size-Dependent Morphology and Composition of Sea Spray Aerosols.

Variable wind speeds over the ocean can have a significant impact on the formation mechanism and physical-chemical properties of sea spray aerosols (SSA), which in turn influence their climate-relevant impacts. Herein, for the first time, we investigate the effects of wind speed on size-dependent morphology and composition of individual nascent SSA generated from wind-wave interactions of natural seawater within a wind-wave channel as a function of size and their particle-to-particle variability. Filter-based thermal optical analysis, atomic force microscopy (AFM), AFM infrared spectroscopy (AFM-IR), and scanning electron microscopy (SEM) were employed in this regard. This study focuses on SSA with sizes within 0.04-1.8 μm generated at two wind speeds: 10 m/s, representing a wind lull scenario over the ocean, and 19 m/s, indicative of the wind speeds encountered in stormy conditions. Filter-based measurements revealed a reduction of the organic mass fraction as the wind speed increases. AFM imaging at 20% relative humidity of individual SSA identified six main morphologies: prism-like, rounded, core-shell, rod, rod inclusion core-shell, and aggregates. At 10 m/s, most SSA were rounded, while at 19 m/s, core-shells became predominant. Based on AFM-IR, rounded SSA at both wind speeds had similar composition, mainly composed of aliphatic and oxygenated species, whereas the shells of core-shells displayed more oxygenated organics at 19 m/s and more aliphatic organics at 10 m/s. Collectively, our observations can be attributed to the disruption of the sea surface microlayer film structure at higher wind speeds. The findings reveal a significant impact of wind speed on morphology and composition of SSA, which should be accounted for accurate assessment of their climate effects.

Large eddy simulation of wind turbulence over non-breaking and breaking waves

Wind-wave interaction affects the wind-wave fields and the combined loading on structures. This study aims to characterize turbulent airflow over non-breaking and breaking waves using a high-fidelity two-phase model in OpenFOAM. The volume of fluid method is utilized to model the complex air–water interface. Wind turbulence is considered via prescribing it at the inlet boundary. The model is validated against experimental data, and a numerical case study is conducted to simulate extreme wind and wave-plunging conditions. Results reveal that non-breaking waves induce wind turbulence and affect averaged wind velocity. For wave steepness exceeding 0.35, extreme wind forces amplify the turbulence by over 100% at wave crests. The amplified turbulence region extends to about 0.6λ in height. When waves plunge, an overturning jet propels the wind forward, generates a counterclockwise vortex, and enhances wind turbulence. Averaged wind speeds increase by over 20% above wave crests, with the enhanced region extending to a height of around 0.2λ for old waves. Maximum turbulence kinetic energy transiently increases by around 0.1Cp2 and maximum kinematic energy of wave-coherent velocity increases and subsequently decreases during wave plunging.

Effects of wind-wave-current-earthquake interaction on the wave height and hydrodynamic pressure based on CFD method

Wind-wave-current interaction is a phenomenon which is present in many practical engineering situations and vertical earthquakes may have significant influence on the hydrodynamic pressure of seawater in regions of active seismicity. This study is devoted to probe the wave height and hydrodynamic pressure of seawater caused by the joint wind, wave, current and earthquake action based on computational fluid dynamics (CFD) method, where a homogeneous multiphase model based on the noncompressible Reynolds Averaged Navier-Stokes (RANS) equation and the k-ε turbulence model implemented in ANSYS-Fluent code was used. First, a two-dimensional (2D) numerical flume excited by wind, wave, current and earthquake loadings was established through the secondary development of a user-defined function (UDF). Subsequent, effects of wind-wave interaction and wind-wave-current interaction on the wave height and hydrodynamic pressure of seawater were investigated. Finally, effects of wave-earthquake interaction and wave-current-earthquake interaction on the wave height and hydrodynamic pressure of seawater were investigated. The nonlinearity of seawater under joint wind, wave, current and earthquake action was also studied.

A dynamic wall modeling approach for large eddy simulation of offshore wind farms in realistic oceanic conditions

Due to the multitude of scales present in realistic oceanic conditions, resolving the surface stress is computationally intensive, motivating modeling approaches. In this work, a dynamic wave drag model is developed for large eddy simulation (LES) to quantify the effects of multiscale dynamically rough surfaces on the atmospheric boundary layer. The waves are vertically unresolved, and the total drag due to the horizontally resolved portion of the wave spectrum is computed through a superposition of the force from each mode. As LES can only resolve the horizontal wind–wave interactions to the filter scale Δ, the effects of the horizontally unresolved, subfilter waves are modeled by specifying a roughness length scale characterizing the unresolved wave energy spectrum. This subfilter roughness is set proportional to the subfilter root mean square of the wave height distribution, and the constant of proportionality is evaluated dynamically during the simulation based on the assumption that the total drag force at the wave surface is independent of the filter scale. The dynamic approach is used to simulate the airflow over a spectrum of moving waves, and the results are validated against high-fidelity phase-resolved simulations. The dynamic approach combined with the wave spectrum drag model is then used to study flow through a fixed-bottom offshore wind farm array, equivalent to an infinite farm, with each turbine represented using an actuator disk model. The dynamic model accurately adapts to the changing velocity field and accurately predicts the mean velocity profiles and power produced from the offshore wind farm. Furthermore, the effect of the wind–wave interactions on the mean velocity profiles, power production, and kinetic energy budget is quantified.

CMIP6 wave climate simulation in the European North East Atlantic Basin using WaveWatch III

Climate change is expected to have an impact on wind patterns, and therefore the generation of waves. Phase 6 of the Coupled Model Intercomparison Project (CMIP6), provides various realization of outputs integrated global coupled models for different centuries. Wind quality is a cornerstone for wave energy as it is the primary generation driver in any wave model. Therefore, proper quantification of wind wave interactions are key in the evaluation of future wave energy potential. In this study, a wave hindcast for the North-East Atlantic, using the WaveWatchIII model forced by CMIP6 winds is presented. The model uses a grid of 0.25° of spatial resolution, covering a longitude range of -21.0° to 10° (west to east) and a latitude range of 18° to 80° (south to north).The main objective of this work is to assess the quality of historical winds from all the CMIP6 wind data that are available under the first realization criteria (r1i1p1f1) at the time of this study. This leads to understanding limitations and proposing a selection method to choose the optimal wind dataset to force the wave model within the analyzed area.Thus, the optimal CMIP6 historical winds for the North-East Atlantic are used to create a 10 years hindcast(from 2003 to 2012). To further assess the suitability of the selected winds dataset for wave generation, results are compared with the ERA5 wave product. The available CMIP6 models show region-specific variations depending on the Regional Climate models used for their developments. The results show the impact of zonal and, meridional wind intensities, on wave characteristics in different regions over the domain.

On the development of a generalized atmospheric boundary layer velocity profile for offshore engineering applications considering wind–wave interaction

Within industrial standards, effects due to wind–wave interaction on the marine atmospheric boundary layer (ABL) may be included via the Charnock sea-surface roughness parameter for open-sea and near-coastal waters. This roughness parameter does not accommodate the wide variety of wave states possible, nor does it modify the ABL to include speed-up effects resulting from an undulating wave profile. In an attempt to provide an updated definition of the general ABL profile for offshore wind engineering applications, an experimental and numerical study is performed to assess the interaction of the ABL on a fixed wave geometry. By extracting the mean velocity field during the wind–wave interaction, bespoke values of the velocity profile power exponent and wind risk factor can be obtained. A range of wave heights and wave lengths are considered from which a Kriging surrogate model is trained to supply the relevant wind profile parameters depending on the wave state. These newly garnered parameters enable the practitioner to define a reference wind velocity, and then adjust the velocity profile characteristics to contain the influence of the wave. This approach provides a new inlet velocity condition that can be used within computational wind engineering investigations without the need to explicitly model the wave surface, as well as flexibility in specifying the underlying wave conditions. Application of the re-calibrated velocity profile shows that wind forces are significantly greater throughout an offshore wind turbine (OWT) swept blade area when large (H= 15 m) and small (H= 1.5 m) wave heights are compared.

Korteweg–De Vries–Burger Equation with Jeffreys’ Wind–Wave Interaction: Blow-Up and Breaking of Soliton-like Solutions in Finite Time

In this study, the evolution of surface water solitary waves under the action of Jeffreys’ wind–wave amplification mechanism in shallow water is analytically investigated. The analytic approach is essential for numerical investigations due to the scale of energy dissipation near coasts. Although many works have been conducted based on the Jeffreys’ approach, only some studies have been carried out on finite depth. We show that nonlinearity, dispersion, and anti-dissipation are the dominating phenomena, obeying an anti-diffusive and fully nonlinear Serre–Green–Naghdi (SGN) equation. Applying an appropriate perturbation method, the current research yields a Korteweg–de Vries–Burger-type equation (KdV-B), combining weak nonlinearity, dispersion, and anti-dissipation. This derivation is novel. We show that the continuous transfer of energy from wind to water results in the growth over time of the KdV-B soliton’s amplitude, velocity, acceleration, and energy, while its effective wavelength decreases. This phenomenon differs from the classical results of Jeffreys’ approach and is due to finite depth. In this study, it is shown that expansion and breaking occur in finite time. These times are calculated and expressed with respect to soliton- and wind-appropriateparameters and values. The obtained values are measurable in experimental facilities. A detailed analysis of the breaking time is conducted with regard to various criteria. By comparing these times to the experimental results, the validity of these criteria are examined.

Wave Age and Steepness Relation in Growing Wind Seas: A Case Study during Hurricane Matthew in 2016

For wind-wave interaction studies, from the literature, two criteria for the wind sea are used: one is the wave age and the other wave steepness. Analysis of pertinent datasets from a data buoy near the passage of Hurricane Matthew in 2016 indicates that = -15 + 1.3. Here is the wave age and is the wave steepness, where <em>C</em><sub>p</sub> is the phase speed of the peak wave, <em>U</em><sub>10</sub> is the wind speed at 10 m, <em>H</em><sub>s</sub> is the significant wave height, and <em>L</em><sub>p</sub> is the peak or dominant wave length. It is found that 87% of the variation between wave age and wave steepness can be explained by this relation. Application of this relation to estimate <em>U</em><sub>10</sub> and friction velocity from wave parameters are also presented for practical environmental and engineering use.

Sensitivity Analysis of Forecasting Performance for ST6 Parameterization in High-Resolution Wave Model Based on WAVEWATCH III

A regional wave forecasting system in East Asia, including the Korean Peninsula, was built based on WAVEWATCH III using offshore wind forecast data from the Global Data Assimilation Prediction System. The numerical simulations were performed on the sensitivity of the interaction between input wind and wave development. The forecasts for each condition were compared and verified with the observational data of marine meteorological buoys from 1 August to 30 September 2020. The sensitivity conditions were configured to have a specific range of variables related to the directional distribution of input winds (SINA0) and variables indicating the development of input wind–wave (CDFAC) in the ST6. The results were presented by calculating the mean error and root mean square error for all observation points. Overall, as the CDFAC increased, the mean error tended to decrease according to the forecast time and the root mean square error increased. Although the effect of SINA0 at the same CDFAC was insignificant, when SINA0 increased in sections where the significant wave height decreased rapidly, the significant wave height tended to decrease. In addition, the main variables that affect the physical process of wind–wave interaction should be considered to improve wave model forecasting performance and accuracy.

Wave Buoy Measurements at Short Fetches in the Black Sea Nearshore: Mixed Sea and Energy Fluxes

Wave buoy measurements were carried out near the northeastern Black Sea coast at the natural reserve Utrish in 2020–2021. In total, about 11 months of data records were collected during two stages of the experiment at 600 and 1500 m offshore and depths of 18 and 42 m. The measured waves propagate almost exclusively from the seaward directions. Generally, the waves do not follow the local wind directions, thus, implying a mixed sea state. Nevertheless, dimensionless wave heights and periods appears to be quite close to the previously established empirical laws for the wind-driven seas. The results of the wave turbulence theory are applied for estimates of spectral energy fluxes and their correspondence to the energy flux from the turbulent wind pulsations. These estimates are consistent with today’s understanding of wind–wave interaction. It is shown that the main fraction of the wind energy flux is sent to the direct Kolmogorov–Zakharov cascade to high wave frequencies and then dissipates in small amounts. Less than 1% of the wind energy flux is directed to the low frequency band (the so-called inverse Kolmogorov–Zakharov cascade), thus, providing wave energy growth.

Toward prediction of turbulent atmospheric flows over propagating oceanic waves via machine-learning augmented large-eddy simulation

Wind-wave interactions have important effects on the energy harvesting of offshore wind farms. High-fidelity large-eddy simulation (LES) is a powerful approach for investigating wind-wave interactions in turbulent oceanic environments. Due to the large scale of the flow domain and the high grid resolution required to resolve multi-scale flow motions, however, brute-force LES of wind-wave interactions is computationally very expensive. We propose augmenting brute-force LES via machine-learning data-driven modeling (ML-LES) to dramatically reduce the computational time required to obtain converged turbulence statistics when the brute-force approach is employed. Namely, we employ a convolutional neural network (CNN) autoencoder trained and validated with LES data sets to develop a highly efficient ML-LES approach for computing turbulence statistics from just a few snapshots of instantaneous LES flow fields. Our results demonstrate the accuracy and efficiency of ML-LES in predicting the mean velocity, velocity fluctuations, and turbulence kinetic energy in highly stretched computational grid systems required to carry out simulations in real-life oceanic environments.

Physics-statistics-based hybrid simulation scheme of coupled nonstationary hurricane wind and wave fields for long-span floating bridges

Abstract Long-span bridges with floating towers have recently drawn great attention from the engineering community. Due to their sensitivity to the aerodynamic and hydrodynamic loads during extreme storms, accurate and efficient simulation tools for hurricane winds and waves are needed for improved understanding of the complex dynamics of the fully coupled wind–wave–structure interaction system. Conventional simulation schemes usually generate winds and waves separately, and hence cannot capture the intense wind–wave interactions under hurricanes. In this study, a physics-statistics-based hybrid simulation scheme of nonstationary hurricane wind and wave fields is presented, where the winds and waves are coupled in both large and small scales. To simulate the large-scale winds and waves, a height-resolving hurricane wind model is coupled with a parametric hurricane wave model through a dependence between sea surface roughness and surface wind speed. In the small-scale simulations, the nonstationary wind fluctuations are statistically obtained by a Hilbert-wavelet-based scheme in which the target parameters (e.g. fluctuation intensity) are estimated based on the local sea state, while the nonstationary sea surface elevations are physically acquired by solving the governing equation of the nonlinear wave evolution under the action of winds. The simulation fidelity of the proposed physics-statistics-based hybrid scheme is demonstrated by generating the coupled nonstationary wind and wave fields approaching to a hypothetical long-span bridge with floating towers under a hurricane event.

Coastal flooding by wave, wind, tide interactions and related processes along the southern part of SW coast of India

Occurrences of coastal flash flooding events are not always associated with extreme events, which are often responsible for the high wave activity and ensuing flooding along coastal areas during the rough season. In the present study, the causative factors that led to a flash flood event at Valiyathura, Thiruvananthapuram, located along the southern part of the SW coast of India, during September 2018 have been investigated based on both field observations and numerical model studies. Numerical simulations reveal the presence of high period swells from the Southern Indian Ocean, leading to a rise in wave runup, coinciding with the Perigean Spring Tide (PST), along with the increased wind speed that sustained for more than 6 h, resulting in the piling up of nearshore waters and sudden flooding along the coast. This inference is well reflected in the beach morphology, particularly the sediment characteristics depicting the impact of coastal flooding. Further, field observations confirm that the nearshore wind–wave interaction prior to the event also contributed to the flooding. Thus, the study emphasizes the need for continuous monitoring of nearshore and other contributing factors such as waves, wind, tide, and beach morphological conditions for reliable predicting coastal flooding events and their probable impacts.

Wind–Wave Interaction for Strong Winds

Abstract In this paper, we revisit the problem of wind–wave interaction with emphasis on strong winds. For these events, it is assumed that nonlinearity is so large that the slope of the wind waves has reached a limiting steepness. Recent observations suggest that the drag decreases with wind in the strong wind speed regime. In this paper, we try to explain this. In the first step, we introduce a model for surface gravity waves and calculate explicitly the background roughness length from the original approach of Janssen. It is found that for young, steep wind sea, the background roughness length almost vanishes, giving a reduced drag. In addition, it is shown that for steep waves, the slowing down of the wind by waves is a nonlinear process; hence, the growth rate of the waves by wind depends in a nonlinear fashion on the wave spectrum. For strong winds, it is found that, as waves are typically steep, this nonlinear effect gives a further reduction of the wind input. As a consequence, in these extreme circumstances, the drag coefficient decreases with wind.

The Dynamics of the Oceanic Air-Sea Boundary Layer

This paper presents a model for the similarity structure of the velocity profiles in air and water in the wave boundary layer, which provides predictions in terms of two parameters, F and R, of all its important properties, including the Charnock parameter, the surface drift velocity and the condition for the cancellation of the surface Stokes velocity by the surface current. The parameter, F, arises from the fetch variability of the wave field, and the parameter, R, arises from the duration variability of the wave field. In the analysis two regimes emerge, namely the Ekman regime and the Hasselmann regime. In the Ekman regime, which occurs for R > ½(1 + F), there is a net loss of energy from the wave field to the deep ocean, and in the Hasselmann regime which occurs for R < ½(1 + F), there is a net gain of energy from the atmosphere to the wave field. The predictions are compared with observations from classical wave formulae, wind-wave studies, and also ROMS and SWAN modelling in the South Australian Basin. A general conclusion is that the condition, F = 1, which was used in the original inertial coupling model of Bye [1], is a good approximation for large scale theoretical ocean studies, and hence the wind-wave interaction is determined principally by R.

On wind–wave interaction phenomena at low Reynolds numbers

After decades of research efforts, wind–wave interaction mechanisms have been recognized as extremely elusive. The reason is the complex nature of the problem, which combines complex coupling mechanisms between turbulent wind and water waves with the presence of multiple governing parameters, such as the friction Reynolds number of the wind, the water depth and the wind fetch. As shown unequivocally here, the use of suitable flow settings allows us to reduce the complex problem of wind–wave interaction to its essential features, mainly as a function of the sole friction Reynolds number of the wind. The resulting numerical solution allows us to study the interactions between water and air layers with their own fluid properties, and to unveil very interesting features, such as an oblique wave pattern travelling upstream and a wave-induced Stokes sublayer. The latter is responsible for a drag reduction mechanism in the turbulent wind. Despite the simulated flow conditions being far from the intense events occurring at the ocean–atmosphere interface, the basic flow phenomena unveiled here may explain some experimental evidence in wind–wave problems. Among other things, the wave-induced Stokes sublayer may shed light on the large scatter of the drag coefficient data in field measurements where swell waves of arbitrary directions are often present. Hence the present results and the developed approach pave the way for the understanding and modelling of the surface fluxes at the ocean–atmosphere interface, which are of overwhelming importance for climate science.

A Sea Surface–Based Drag Model for Large-Eddy Simulation of Wind–Wave Interaction

Abstract Monin–Obukhov similarity theory (MOST) is a well-tested approach for specifying the fluxes when the roughness surfaces are homogeneous. For flow over waves (inhomogeneous surfaces), phase-averaged roughness length scales are often prescribed through models based on the wave characteristics and the wind speed. However, such approaches lack generalizability over different wave ages and steepnesses due to the reliance on model coefficients tuned to specific datasets. In this paper, a sea surface–based hydrodynamic drag model applicable to moving surfaces is developed to model the pressure-based surface drag felt by the wind due to the waves. The model is based on the surface gradient approach of Anderson and Meneveau applicable to stationary obstacles and extended here to the wind–wave problem. The wave drag model proposed specifies the hydrodynamic force based on the incoming momentum flux, wave phase speed, and the surface frontal area. The drag coefficient associated with the wind–wave momentum exchange is determined based on the wave steepness. The wave drag model is used to simulate turbulent airflow above a monochromatic wave train with different wave ages and wave steepnesses. The mean velocity profiles and model form stresses are validated with available laboratory-scale experimental data and show good agreement across a wide range of wave steepnesses and wave ages. The drag force is correlated with the wave surface gradient and out-of-phase with the wave height distribution by a factor of π/2 for the sinusoidal wave train considered. These results demonstrate that the current approach is sufficiently general over a wide parameter space compared to wave phase-averaged models with a minimal increase in computational cost. Significance Statement Understanding the physics of wind waves plays an important role in the context of numerous geophysical and engineering applications. A drag-based model is developed that characterizes the effect of the sea surface waves on the wind above. The model is validated with existing experimental datasets and is shown to be effective in predicting the average wind velocity and stress over waves with varied steepnesses and phase speeds. The ease of implementation and low computational cost of the model make it useful for studying turbulent atmospheric-scale flows over the sea surface important in offshore wind energy research as well as for modeling air–sea fluxes of momentum, heat, and mass.

A Dynamic Large-Scale Driving-Force to Control the Targeted Wind Speed in Large Eddy Simulations above Ocean Waves

We performed large eddy simulations to study micro-scale wind–wave interactions under undisturbed freestream conditions. We identified that standard approaches lead to wave-related disturbances at the top boundary. Therefore, we developed a numerical strategy to maintain an undisturbed wind speed at the top, while considering arbitrary waves at the bottom. In a broader context, the method is capable of controlling the wind speed at any height in the domain, and may also be used to enhance atmospheric simulations over land. The method comprises an evolution equation that controls the dynamic evolution of the large-scale driving force, representing the geostrophic forcing from the meso- to the micro-scales. In flat-bottom applications, this guided the reference freestream velocities towards a certain target; convergence to a steady state regime was favored and self-similarity was ensured. In wavy bottom applications considering the prescription of a monochromatic wave, we were able to maintain a quasi-steady wind speed close to the target on the freestream. The wave-induced disturbances were then investigated as functions of varying wave age conditions. We performed a systematic wave age variation study by varying the reference wind speed, and evaluated wave-induced disturbances in the velocity, normal, and shear stress profiles.

Evidence of Ocean Waves Signature in the Space–Time Turbulent Spectra of the Lower Marine Atmosphere Measured by a Scanning LiDAR

To achieve more accurate weather and climate forecasting, and propose efficient engineering solutions for exploiting offshore renewable energies, it is imperative to accurately describe the atmospheric turbulent flow in the offshore environment. The ocean’s dynamics raise specific challenges for the aforementioned applications, as they significantly alter the atmospheric flow through complex wind–wave interactions. These interactions are important in fairly common situations and notably in old-sea conditions, where ocean waves travel fast, under comparatively slow wind velocities. In the present study, a scanning LiDAR (sLiDAR) was deployed on the shore to study micro-scale wind–wave interactions by performing horizontal scans 18 m above the ocean, and as far as 2 km from the coast. In the proposed configuration, and in the test cases presented in old seas, the sLiDAR captures wave-induced disturbances propagating into the lower part of the marine atmospheric boundary layer. Based on measurements of high-resolution space–time maps of the Radial Wind Speed, an original two-dimensional spectral analysis of the space–time auto-correlation functions was performed. Unlike more conventional data-processing techniques, and as long as the waves travel sufficiently (∼twofold) faster than the mean wind at the measurement height, the upward transfer of motions from the waves to the wind can be clearly distinguished from the atmospheric turbulence in the wave-number–angular-frequency (k–w) turbulent spectra. These are the first space–time auto-correlation functions of the wind velocity fluctuations obtained at micro-scales above the ocean. The analyses demonstrate sLiDAR systems’ applicability in measuring k–w-dependent turbulent spectra in the coastal environment. The findings present new perspectives for the study of micro-scale wind–wave interactions.