Depth-limited Wave Research Articles

Low-Crested Structures In Front Of Rubble Mound Breakwaters

Climate adaptation of coastal structures has become more important due to climate change, resulting in sea level rise and increased wave loading for coastal structures with depth-limited wave conditions. If sea level rise causes wave loading that becomes too severe, one of the options is to reduce the wave loading before the waves reach the existing coastal structure (see for instance Van Gent, 2019, and Van Gent and Teng, 2023). This can be achieved by increasing the foreshore (e.g. sand nourishment) or by constructing a low-crested structure in front of the coastal structure. In this study the climate adaptation measure to add a submerged low-crested structure in front of an existing (emerged) coastal structure has been studied. Between the two structures, structure-induced wave set-up occurs. This structure-induced wave set-up has been studied based on wave flume tests. The effects of structure-induced wave set-up on wave transmission at the low-crested structures and the effects on wave overtopping at the emerged coastal structure were also measured and analyzed. To evaluate the performance of submerged low-crested structures Van Gent et al (2023) performed wave flume tests to examine wave transmission at various types of submerged low-crested structures, without an emerged structure behind the low-crested structures. For a submerged low-crested structure in front of an emerged coastal structure, the transmitted waves can be used as incident waves for estimates of wave overtopping at a rubble mound breakwater, using wave overtopping expressions described in Van Gent et al (2022). However, to verify whether the expressions for wave transmission and wave overtopping can be applied for submerged low-crested structures in front of rubble mound breakwaters, new physical model tests have been performed at Deltares. The new wave flume tests were performed with impermeable and permeable low-crested structures in front of impermeable and permeable emerged structures (see Figure 1 for permeable structure).

Submerged low-crested structures in front of coastal structures

Climate adaptation of coastal structures has become more important due to climate change, resulting in sea level rise and increased wave loading on coastal structures with depth-limited wave conditions. One of the climate adaptation measures to ensure that existing coastal structures continue to perform their function after unforeseen sea level rise, is to reduce the wave loading before the waves reach the existing coastal structure. This can be achieved by constructing a low-crested structure in front of the existing structure. Between the two structures, structure-induced wave set-up occurs. This structure-induced wave set-up has been studied based on wave flume tests. The effects of structure-induced wave set-up on wave transmission at the low-crested structures and the effects on wave overtopping at the emerged coastal structures were also measured and analyzed.The structure-induced wave set-up depends on the freeboard, wave steepness, and permeability of the low-crested structure. For configurations with impermeable low-crested structures, this wave-set-up does not depend on the distance between the two structures. Empirical expressions to estimate structure-induced wave set-up are derived for impermeable and permeable low-crested structures.The measurements indicate that the effect of structure-induced wave set-up on the wave transmission coefficients is negligibly small.The structure-induced wave set-up increases the wave overtopping discharges at the emerged coastal structure. This effect can be taken into account in wave overtopping estimates by reducing the freeboard with the structure-induced wave set-up.

Field observations and long short-term memory modeling of spectral wave evolution at living shorelines in Chesapeake Bay, USA

Living shorelines as a nature-based solution for climate change adaptation were constructed in many places around the world. The success of this type of projects requires long-term monitoring for adaptive management. The paper presents a novel framework leveraging scientific machine learning methods for accurate and rapid prediction of long-term hydrodynamic forcing impacting living shorelines using short-term measurements of water levels and wind waves in the largest estuary in the U.S. Different from existing data-driven wave prediction models focusing on significant wave heights, this study is focused on the prediction of wave energy spectra in shallow water using winds and tides as the input feature and short-term measurements of wave spectra and water depths as the label. Long Short-Term Memory (LSTM) models were developed using four-month wave measurements in the stormy seasons to predict integral wave parameters and energy spectra for multiple years. The developed models accurately predicted wave heights, peak periods, and energy spectra around the living shorelines, capturing complex wave dynamics, such as wave generation by wind, nonlinear wave-wave interactions, and depth-limited wave breaking in the shallow water of a large estuary. The validated models were then used to determine the long-term wave forcing impacting the living shorelines based on the modeled wave characteristics and spectra. Model results show that the surrogate models utilizing LSTM to predict wave spectra in the frequency domain enable long-term predictions of spectral wave evolution with a minimal computational cost. Our findings provide valuable insights into the efficacy of living shorelines in attenuating wave energy and demonstrate the utility of this approach in assessing the effectiveness of such living shoreline structures.

Observations of two-dimensional turbulence in the surfzone

Low-frequency, many-minute-period horizontal surfzone eddies are an important mechanism for the dispersion of material, transporting larvae, pollutants, sediment, and swimmers both across and along the nearshore. Previous numerical, laboratory, and field observations on alongshore uniform bathymetry with no or roughly uniform mean background flows suggest that the low-frequency eddies may be the result of a two-dimensional inverse energy cascade that transfers energy from relatively small spatial-scale vorticity injected by depth limited breaking waves to larger and larger spatial scales. Here, using remotely sensed high-spatial resolution estimates of currents, those results are extended to surfzones with strong complex mean circulation patterns [flows O(1 m/s)] owing to nonuniform bathymetry. Similar to previous results, wavenumber spectra and second-order structure functions calculated from the observations are consistent with a two-dimensional inverse energy cascade. The size of the largest eddies is shown to depend on the surfzone width and the spatial scales of the mean currents. Third-order structure functions also are consistent with an inverse cascade for spatial scales greater than ∼50 m. At smaller scales, the third-order structure functions suggest a mixture of inverse and forward cascades.

Unified depth-limited wave breaking detection and dissipation in fully nonlinear potential flow models

Unified depth-limited wave breaking detection and dissipation in fully nonlinear potential flow models

Observations of River Plume Mixing in the Surf Zone

Abstract We use salinity observations from drifters and moorings at the Quinault River mouth to investigate mixing and stratification in a surf-zone-trapped river plume. We quantify mixing based on the rate of change of salinity DS/Dt in the drifters’ quasi-Lagrangian reference frame. We estimate a constant value of the vertical eddy diffusivity of salt of Kz = (2.2 ± 0.6) × 10−3 m2 s−1, based on the relationship between vertically integrated DS/Dt and stratification, with values as high as 1 × 10−2 m2 s−1 when stratification is low. Mixing, quantified as DS/Dt, is directly correlated to surf-zone stratification, and is therefore modulated by changes in stratification caused by tidal variability in freshwater volume flux. High DS/Dt is observed when the near-surface stratification is high and salinity gradients are collocated with wave-breaking turbulence. We observe a transition from low stratification and low DS/Dt at low tidal stage to high stratification and high DS/Dt at high tidal stage. Observed wave-breaking turbulence does not change significantly with stratification, tidal stage, or offshore wave height; as a result, we observe no relationship between plume mixing and offshore wave height for the range of conditions sampled. Thus, plume mixing in the surf zone is altered by changes in stratification; these are due to tidal variability in freshwater flux from the river and not wave conditions, presumably because depth-limited wave breaking causes sufficient turbulence for mixing to occur during all observed conditions. Significance Statement River outflows are important sources of pollutants, sediment, and nutrients to the coastal ocean. Small rivers often meet large breaking waves in the surf zone close to shore, trapping river water and river-borne material near the beach. Such trapped material can influence coastal public health, beach morphology, and nearshore ecology. This study investigates how trapped fresh river water mixes with salty ocean water in the presence of large breaking waves by using high-resolution measurements of waves, salinity, and turbulence. We find that the surf zone is often fresh and stratified, which could have significant implications for the fate of riverine material. Wave breaking provides a constant source of turbulence, and the amount of mixing is limited by the degree of vertical salt stratification; more mixing occurs when stratification is higher.

In-situ measurements of energetic depth-limited wave loading

An extensive database of in situ measurements of wave impact pressure on the wall of a composite breakwater and associated explanatory variables (i.e., waves, wind, and water level) was collected in a particularly high-energy wave environment. Due to the bottom profile, which includes a wide mound of concrete blocks with a seaward edge that rises to the Lowest Astronomical Tide level over a large distance, most waves break before reaching the monitored barrier, making the data set representative of depth-limited wave loading. Maximum pressure is consistently found at the sensor position closest to the mean free surface, and statistically, a decrease of maximum pressure with the altitude is observed. Nevertheless, the database also shows a wide variety of vertical profiles of maximum pressure. A detailed analysis of the pressure signal shows that there are two impact classes associated with large pressure values. The first is mostly observed during stormy conditions with relatively slow pressure variations over time and a fairly uniform spatial distribution. The second class exhibits very limited pressure peaks in time and space and is most often observed during moderate sea states and high water levels. The pressure signals for each class agree well with the prediction of the PROVERB impact classification based on breakwater dimensions, water level, and wave parameters. Our dataset also shows the existence of impacts with simultaneous large pressure and rise time, which extends the range of the observed values of the pressure impulse already reported in-situ from 31000 Pa.s to 100000 Pa.s. For strong impacts that show regular upward propagation, the peak pressure propagation velocity was estimated and most values remain below 20 m/s. Finally, we performed a statistical analysis to relate the maximum pressure to a number of variables, including offshore wave parameters, water level, and wind parameters. The multiple linear regression model created confirms the dominant influence of wave height and the negligible influence of the wind on pressure maxima. The other wave parameters are still significant but of secondary importance. Finally, the upper bound of the 99% prediction interval predicts the maximum pressure given a set of wave and water level conditions with a good confidence level which is an encouraging result.

Wave-current interactions at the Tagus Estuary Mouth (Portugal) under storm wave conditions

This study investigates interactions between waves, water levels and currents at the mouth of the second largest estuary in Europe (the Tagus Estuary, Portugal) under storm waves, combining field observations and a three-dimensional fully coupled wave-current modelling system. Tidal-induced water depth variations substantially modulate waves over the ebb shoal. During energetic conditions, low tide levels promote depth-limited wave breaking and energy transfer towards higher harmonics (triad interactions), which reduces wave heights and periods. Furthermore, for a given water level, tidal currents also influence wave propagation and drive strong modulations over shallow regions characterized by cross-channel current gradients. Flood currents change the mean wave direction by about 10–15° and tend to focus the wave energy flux towards coastal regions of the southern margin compared to a run without currents (20 to 30% increase of wave heights), while ebb currents reduce the wave heights. In addition, model results suggest that the saturation level associated with wave and current conditions at shallow locations may be close to the threshold where waves start to dissipate by whitecapping. At the peak of the storm, waves become a main driver of the circulation at the mouth scale, even in the 45-m deep main channel. Wave breaking acceleration over the ebb-shoal locally increases flood currents by 50 to 300% and reduces ebb currents by 20 to 50%.

Statistical distribution of wave heights attenuation by entrained air bubbles in the surf zone

The paper focuses on the wave height distribution functions for long-term behavior of significant wave height, probability density function, Rayleigh distribution, depth-limited wave height distributions, etc. We have introduced the air bubble effects in the surf zone wave parameters, and the wave height distribution has been derived from the Rayleigh distribution. The calculated statistical wave height distributions by the present model indicate a good agreement with the field data than the Rayleigh distribution. Furthermore, the computed probability densities of the wave heights have been reduced considerably by the air bubble effects, making the wave heights lower than those determined by the Rayleigh distribution. It is also observed that the Rayleigh distribution overestimates large pressure amplitudes, but the present model and the Weibull distribution agree well with the collected field data. Finally, the proposed model performs less normalized root-mean-square error in predicting wave height statistics.

Evaluating the influence of slope limiters on nearshore wave simulation in a non-hydrostatic model

In a non-hydrostatic model utilizing a Godunov-type finite volume method, the numerical dissipation introduced by slope limiters can be a main source of total dissipation. The objective of this study is to look into how different slope limiters influence nearshore wave simulation with an emphasis on depth-limited wave breaking simulation in the open-source NHWAVE model. The minmod, superbee and van Leer limiter are analyzed. The performances of different slope limiters are correlated with their characteristics. The refinement of the grids do not necessarily help to improve the model accuracy. In the shock-capturing modeling of solitary waves, although superbee limiter introduces a small non-physical increase of wave energy before wave breaking, the overall dissipation is largest. The van Leer limiter does not introduce noticeable dissipation before breaking in a relatively short simulation time, and the energy dissipated is moderate of all three limiters. In the turbulent modeling of wave breaking, slope limiters do not introduce large numerical dissipation and the model accuracy is mainly influenced by the dissipation in wave propagation and generating turbulent dissipation at different orders of magnitude. The superbee limiter predicts the largest turbulent kinetic energy at breaking moment, which means superbee limiter tends to reproduce the most intensive breaking. It is consistent with the largest wave energy dissipation in the shock-capturing modeling.

Parametric Modeling of Depth-Limited Wave Spectra under Hurricane Conditions with Applications to Kinetic Umbrellas against Storm Surge Inundation

This paper presents a simplified parametric model for the estimation of depth-limited hurricane wave spectra, accounting for swell and wind-sea components, for coastal engineering applications. The model was evaluated against observations obtained from three shallow water sites in Florida during Hurricane David in September 1979. It was revealed that the parametric approach increases in accuracy with decreasing distance to the storm center and generally provides a conservative representation of the significant wave height, albeit overestimating the peak wave frequency. The model was subsequently adopted to evaluate the performance of tilted hyperbolic paraboloidal (hypar) shells (referred to as “kinetic umbrellas”) as an adaptable alternative to conventional floodwalls via smoothed particle hydrodynamics (SPH). The introduction of hypar geometry proved superior to conventional sloped barriers in reducing overtopping waves but decreases in effectiveness at levels of inundation greater than two-thirds the deployed height. Furthermore, umbrellas exhibiting larger geometrical warping were more capable at suppressing overtopping but must sustain larger base shear forces when subjected to irregular waves consistent with landfalling hurricanes.

Development of Depth-Limited Wave Boundary Layers over a Smooth Bottom

A theoretical and numerical study is carried out to investigate the transformation of the wave boundary layer from non-depth-limited (wave-like boundary layer) to depth-limited one (current-like boundary layer) over a smooth bottom. A long period of wave motion is not sufficient to induce depth-limited properties, although it has simply been assumed in various situations under long waves, such as tsunami and tidal currents. Four criteria are obtained theoretically for recognizing the inception of the depth-limited condition under waves. To validate the theoretical criteria, numerical simulation results using a turbulence model as well as laboratory experiment data are employed. In addition, typical field situations induced by tidal motion and tsunami are discussed to show the usefulness of the proposed criteria.

Influence of Mild Bottom Slopes on the Overtopping Flow over Mound Breakwaters under Depth-Limited Breaking Wave Conditions

The crest elevation of mound breakwaters is usually designed considering a tolerable mean wave overtopping discharge. However, pedestrian safety, characterized by the overtopping layer thickness (OLT) and the overtopping flow velocity (OFV), is becoming more relevant due to the reduction of the crest freeboards of coastal structures. Studies in the literature focusing on OLT and OFV do not consider the bottom slope effect, even if it has a remarkable impact on mound breakwater design under depth-limited breaking wave conditions. Therefore, this research focuses on the influence of the bottom slope on OLT and OFV exceeded by 2% of incoming waves, hc,2% and uc,2%. A total of 235 2D physical tests were conducted on conventional mound breakwaters with a single-layer Cubipod® and double-layer rock and cube armors with 2% and 4% bottom slopes. Neural networks were used to determine the optimum point to estimate wave characteristics for hc,2% and uc,2% calculation; that point was located at a distance from the model toe of three times the water depth at the toe (hs) of the structure. The influence of the bottom slope is studied using trained neural networks with fixed wave conditions in the wave generation zone; hc,2% slightly decreases and uc,2% increases as the gradient of the bottom slope increases.

Wave Forecasting in Shallow Water: A New Set of Growth Curves Depending on Bed Roughness

Forecasting relationships have been recognized as an important tool to be applied together, or not, with complete numerical modelling in order to reconstruct the wave field in coastal areas properly when the available wave data is limited. In recent years, the literature has offered several comprehensive sets of field experiments investigating the form of the asymptotic, depth-limited wind waves. This has made it possible to reformulate the original deep water equations, taking into account the effects of water depth, if wind waves are locally generated in shallow and confined basins. The present paper is an initial attempt to further contribute to the shallow water forecasting curves which are currently available, also considering the role on the wave generation of a variable equivalent bottom roughness. This can offer the possibility of applying shallow growth curves to a broad variety of contexts, for which bed composition and forms can be different. Simple numerical tests have been conducted to reproduce the fully developed conditions of wave motion with variable roughness values. To validate the new set of equations, they have been applied to a real shallow lake for which both experimental and numerical wave data is available. The comparison of the obtained results is very encouraging in proceeding with this approach.

Observations of mixing and transport on a steep beach

Surfzone mixing and transport on a sandy, steep (∼1/8 slope), reflective beach at Carmel River State Beach, California, are described for a range of wave and alongshore flow conditions. Depth-limited wave breaking occurred close to the shore due to the steepness of the beach, creating a narrow surf/swash zone (∼10 m wide). Fluorescent Rhodamine dye was released as a slug in the surfzone, and the temporal and spatial evolution was measured using in-situ dye sensors. Dye concentration measured as a function of time reveals sharp fronts that quickly decay resulting in narrow peaks near the dye release, which subsequently broaden and decrease in peak concentration with alongshore distance. The measurements indicate two stages of mixing and transport occur inside the surfzone on the steep beach. 1) In the near-field ( 50 m downstream from the dye release location), the mass transport was dominated by advection. The distance to the far-field is much shorter in the alongshore on a steep beach compared with a dissipative beach. Estimates of cross-shore and alongshore diffusion coefficients (κ x , κ y ) were found to be similar in magnitude within the surfzone. Outside the surfzone in the far-field, the results suggest that the mixing processes are independent of those inside the surfzone. The mixing and transport of material observed on this steep beach are found to be analogous to that previously measured on dissipative beaches, however the diffusion coefficients within and outside the surfzone were found to be smaller on this steep beach.

Breaking Wave Imaging Using Lidar and Sonar

The two-phase flow generated by breaking ocean waves plays a crucial role in various geophysical processes, including dissipation of wave energy and atmospheric gas exchange. This paper presents a technique to measure the two-phase flow generated by breaking waves at prototype scale. We have demonstrated the validity and potential of this technique in the Large Wave Flume (Grosser Wellenkanal, GWK) facility in Hanover, Germany. Actively breaking, depth-limited waves were measured using an array of three downward-looking lidars mounted above the water surface and an upward-looking multibeam sonar below. This novel setup enabled the characterization of the complete upper boundary (free water surface and splash-up) and seaward lower boundary (entrained cavity and bubble plume) of the breaking wave. We have quantified the migration of the lower boundary as the cavity and plume are entrained in the water column—penetrating toward the seabed, moving onshore with the passage of the wave crest, and then rising as it is slowly advected offshore. We have also estimated the overall composition of the splash, cavity, and plume as the breaking wave evolves over time. Our observations are consistent with results from previous small-scale laboratory experiments and the suitability of the technique for experimentation at prototype scale has been demonstrated.

ROCK ARMOR DAMAGE IN DEPTH-LIMITED BREAKING WAVE CONDITIONS

The armor layer of a mound breakwaters is usually designed with a formula derived from physical tests in non-breaking wave conditions; however, most rubble mound breakwaters are placed in the wave breaking zone where the highest waves break before reaching the structure. The hydraulic stability formulas developed for rock-armored breakwaters in non-breaking conditions are not completely valid to characterize the hydraulic stability of these structures under depth-limited wave attack. In this study, five series of 2D physical tests were carried out on a bottom slope m=1/50 to analyze the hydraulic stability of double-layer rock armored breakwaters in depth-limited breaking wave conditions. Measurements taken by 12 wave gauges placed along the wave flume were compared with estimations of Hm0, H2% and H1/10 obtained from numerical model SwanOne. The significant wave height, Hm0, estimated or measured at a distance 3hs from the toe of the structure was the best characteristic wave to relate armor damage with stability number. The six-power relationship between dimensionless armor damage and stability number, found in this study, explained more than 94% of the variance in the damage observations. This relationship is valid for conventional non-overtopping double-layer rock-armored breakwaters on bottom slope m=1/50 and depth-limited breaking wave conditions.

Variability of depth-limited waves in coral reef surf zones

Wave breaking and transformation on coral reef flats is an important process protecting tropical coastlines and regulating the energy regimes of coral reefs. However, the high hydraulic roughness, shallow water, and steep bathymetries of coral reefs may confound common surf zone assumptions, such as a depth-limited and saturated surf zone with a constant wave height to water depth ratio (γ). Here, we examine wave transformation across a coral reef flat, during three separate swell events, on both a time-averaged and a wave-by-wave basis. We use the relationship between significant wave height and water depth (γs) to examine the change in surf saturation across the reef flat and compare the measured wave height decay to results of modelled wave energy dissipation in the surf zone. Our results show that γs was not cross-reef constant and varied according to location on the reef flat and local water depth. On average, γs was greatest at the outer reef flat, near the reef crest, and progressively reduced towards the inner reef flat, near the reef lagoon. This was most pronounced in shallow water with large γs values (γs > 0.85) at the outer reef flat and small γs values (γs < 0.1) at the inner reef flat. This indicates that there is an increase in wave energy dissipation in shallow water, most likely due to increased breaker and bed frictional dissipation. The measured wave energy dissipation across the entire reef flat could, on average, be modelled accurately; however, this required location specific calibration of the free parameters, the wave friction factor (fw) and γ, and further suggests that there is no value for either parameter that is universally applicable to coral reef flats. Despite model calibration inaccuracies were still observed, primarily at the outer reef flat. These inaccuracies reflected the observed cross-reef variation of γ on the reef flat and potentially the limitations of random wave breaker dissipation models in complex surf zones. Our results have implications for the use of wave energy dissipation models in predicting breaker dissipation and subsequent benthic community change on coral reef flats, and suggest that careful consideration of the free parameters in such models (such as fw and γ) is required.

Field Observations and Modeling of Surfzone Sensible Heat Flux

AbstractSurfzone sensible heat flux (HS,SZ) obtained through direct eddy-covariance estimates was measured at four different sandy beach sites along Monterey Bay, California. The HS,SZ source region is estimated from a footprint probability distribution function (pdf) model and is only considered when at least 70% of the footprint pdf occupies the surfzone. The measured HS,SZ is 2 times the modeled interfacial sensible heat (HS,int) using COARE3.5. A formulation for estimating sensible heat flux from spray droplets (HS,spray) generated during depth-limited wave breaking is developed. The sea-spray generation function for droplet radii ranging over 0.1 &lt; ro &lt; 1000 μm is based on self-similar spectra of spray droplets measured from the surfzone forced by the average depth-limited breaking wave dissipation across the surfzone. However, it is shown that the size of the spume droplets that contribute to HS,spray is limited owing to the relatively short residence time in air as the droplets fall to the sea surface during wave breaking. The addition of the surfzone-modeled HS,spray to the COARE3.5 HS,int gives values similar to the observed surfzone HS,SZ, highlighting the importance of depth-limited wave-breaking processes to sensible heat flux. Measured HS,SZ values are an order of magnitude larger than simultaneous open ocean observations.

Increased Aerodynamic Roughness Owing to Surfzone Foam

AbstractDrag coefficients Cd obtained through direct eddy covariance estimates of the wind stress were observed at four different sandy beaches with dissipative surfzones along the coastline of Monterey Bay, California. The measured surfzone Cd (~2 × 10−3) is twice as large as open-ocean estimates and consistent with recent estimates of Cd over the surfzone and shoaling region. Owing to the heterogeneous nature of the near shore consisting of nonbreaking shoaling waves and breaking surfzone waves, the surfzone wind stress source region is estimated from the footprint probability distribution derived for stable and unstable atmospheric conditions. An empirical model developed for estimating the Cd for open-ocean foam coverage dependent on wind speed is modified for foam coverage owing to depth-limited wave breaking within the surfzone. A modified empirical Cd model for surfzone foam predicts similar values as the measured Cd and provides an alternative mechanism to describe roughness.