Depth inversion in shallow water based on nonlinear properties of shoaling periodic waves

Two depth inversion algorithms (DIA) applicable to coastal waters are developed, calibrated, and validated based on results of computations of periodic waves shoaling over mild slopes, in a two-dimensional numerical wave tank based on fully nonlinear potential flow (FNPF) theory. In actual field situations, these algorithms would be used to predict the cross-shore depth variation h based on sets of values of wave celerity c and length L, and either wave height H or left–right asymmetry s₂/s₁, simultaneously measured at a number of locations in the direction of wave propagation, e.g., using video or radar remote sensing techniques. In these DIAs, an empirical relationship, calibrated for a series of computations in the numerical wave tank, is used to express c as a function of relative depth k_oh and deep water steepness k_oH_o. To carry out depth inversion, wave period is first predicted as the mean of observed L/c values, and H_o is then predicted, either based on observed H or s₂/s₁ values. The celerity relationship is finally inverted to predict depth h. The algorithms are validated by applying them to results of computations for cases with more complex bottom topography and different incident waves than in the original calibration computations. In all cases, root-mean-square (rms)-errors for the depth predictions are found to be less than a few percent, whereas depth predictions based on the linear dispersion relationship—which is still the basis for many state-of-the-art DIAs—have rms-errors 5 to 10 times larger.     

Profiles of fully developed (Airy) waves in different water depths

Airy waves have a sinusoidal profile in deep water that can be modeled by a time series at any point x and time t, given by η(x,t) = (H_o/2) cos[2πx/L_o − 2πt/T_w], where H_o is the deepwater height, L_o is the deepwater wavelength, and T_w is the wave period. However, as these waves approach the shore they change in form and dimension so that this equation becomes invalid. A method is presented to reconstruct the wave profile showing the correct wavelength, wave height, wave shape, and displacement of the water surface with respect to the still water level for any water depth.     

Calibration and modification of energy dissipation models for irregular wave breaking

This study is undertaken to recalibrate eight existing energy dissipation models and find out the suitable models, which can be used to compute H_rms for a wide range of experimental conditions. The examination shows that the coefficients in the existing models are not the optimal values for a wide range of experimental conditions. Using the new calibrated coefficients, all existing models can be used for computing H_rms and the model of Battjes, J.A., Stive, M.J.F. [1985. Calibration and verification of a dissipation model for random breaking waves. Journal of Geophysical Research 90 (C5), 9159–9167] gives the best predictions. The existing models are also modified by changing the breaker height formulas in the dissipation models. The accuracy of most existing models is improved significantly by using the suitable breaker height formula.     

Experimental study of unidirectional irregular wave slamming on the three-dimensional structure in the splash zone

The experimental investigation of unidirectional random wave slamming on the three-dimensional structure in the splash zone is presented. The experiment is conducted in the marine environment channel in the State Key Laboratory of Coastal and Offshore Engineering, Dalian University of Technology. The test wave is unidirectional irregular wave. The experiments are carried out with perpendicular random waves (β=0°) and oblique random waves (β=15°, 30°, 45°), the significant wave heights H_1/3 ranging from 7.5 to 20 cm with 2.5 cm increment, the peak wave periods T_p ranging from 0.75 to 2.0 s with 0.25 s increment, and the clearance of the model with respect to the significant wave height s/H_1/3 ranging from 0.0 to 0.5 with 0.1 increment. The statistical analysis results of different test cases are presented. The statistical distribution characteristics of the perpendicular irregular wave impact pressures are compared with that of the oblique irregular wave on the underside of the structure. The effect of the wave direction β on the wave impact forces on the underside of the structure is determined. The relation between the impact forces and the parameters such as the significant wave height, the relative structure width and the relative clearance of the structure is also discussed.     

Evaluation of a high-resolution wave hindcast model SWAN for the West Mediterranean basin

This study aims to present an evaluation and implementation of a high-resolution SWAN wind wave hindcast model forced by the CFSR wind fields in the west Mediterranean basin, taking into account the recent developments in wave modelling as the new source terms package ST6. For this purpose, the SWAN model was calibrated based on one-year wave observations of Azeffoune buoy (Algerian coast) and validated against eleven wave buoys measurements through the West Mediterranean basin. For the calibration process, we focused on the whitecapping dissipation coefficient C_ds and on the exponential wind wave growth and whitecapping dissipation source terms. The statistical error analysis of the calibration results led to conclude that the SWAN model calibration corrected the underestimation of the significant wave height hindcasts in the default mode and improved its accuracy in the West Mediterranean basin. The exponential wind wave growth of Komen et al (1984) and the whitecapping dissipation source terms of Janssen (1991) with C_ds = 1.0 have been thus recommended for the western Mediterranean basin. The comparison of the simulation results obtained using this calibrated parameters against eleven measurement buoys showed a high performance of the calibrated SWAN model with an average scatter index of 30% for the significant wave heights and 19% for the mean wave period. This calibrated SWAN model will constitute a practical wave hindcast model with high spatial resolution (˜3 km) and high accuracy in the Algerian basin, which will allow us to proceed to a finer mesh size using the SWAN nested grid system in this area.     

Effective fetch and non-linear four-wave interactions during wave growth in slanting fetch conditions

Wave growth in slanting fetch (with wind blowing obliquely off a coast) is investigated with 7 years worth of routine wave measurements in Lake IJssel in The Netherlands and with the SWAN wave model. Two aspects are considered in particular for this case: the validity of the concept of effective fetch and the role of the non-linear four-wave interactions. For slanting and parallel fetch conditions, we found some significant deviations from the effective fetch assumption, leading to 20–35% mismatch in either the peak period T_p or the significant wave height H_m0 respectively. However, the effect of discrepancies between various widely accepted wave growth formulas turned out to be even more important. The wave directions during slanting fetch are significantly ‘steered’ by the coastline, especially in the first kilometre(s) off the coast. The role of the non-linear four-wave interactions is investigated by running the SWAN (version 40.41) wave model with three different quadruplet formulations. Exact quadruplet methods (Xnl) yielded relatively strong wave steering, despite the four-wave interactions being relatively weak. Application of Xnl did not lead to better overall agreement with measurements — improvements for the mean wave period T_m01 were offset by some deterioration for the wave height H_m0.     

Improved representation of breaking wave energy dissipation in parametric wave transformation models

An improved formulation to describe breaking wave energy dissipation is presented and incorporated into a previous parametric cross-shore wave transformation model [Baldock, T.E., Holmes, P., Bunker, S., Van Weert, P., 1998. Cross-shore hydrodynamics within an unsaturated surf zone. Coastal Engineering 34, 173–196]. The new formulation accounts for a term in the bore dissipation equation neglected in some previous modelling, but which is shown to be important in the inner surf zone. The only free model parameter remains the choice of γ, the ratio of wave height to water depth at initial breaking, and a well-established standard parameter is used for all model runs. The proposed model is compared to three sets of experimental data and a previous version of the model which was extensively calibrated against field and laboratory data. The model is also compared to the widely used model presented by Thornton and Guza (1983) [Thornton, E.B., Guza, R.T., 1983. Transformation of wave height distribution. Journal of Geophysical Research 88 (No.C10), 5925–5938].     

A morphodynamic model to simulate the seasonal closure of tidal inlets

Seasonally open tidal inlets usually occur in microtidal, wave-dominated coastal environments where strong seasonal variations of streamflow and wave climate are experienced. These inlets are closed to the ocean for a number of months every year due to the formation of sand bars across their entrances. The annual closure of these inlets inhibits ocean access for boats and could also cause deterioration of water quality in the estuary/lagoon connected to the inlet. As these estuaries/lagoons are commonly used as harbours or recreational facilities there is increased interest in keeping the inlets permanently open. A process-based numerical model capable of simulating inlet closure is invaluable in terms of identifying the natural processes governing inlet closure. As a further step, this type of model could also be used to determine the effect of any proposed engineering solutions to keep the inlet open on the adjacent beaches. A morphodynamic model capable of simulating the seasonal closure of inlets, which includes both longshore (LST) and cross-shore transport (CST) processes, was developed in this study. Application of the model to two idealised scenarios indicated that cross-shore processes govern inlet behaviour when LST rates were low. The Dean's criterion [Dean, R.G., 1973. Heuristic models of sand transport in the surf zone. Proc. Conf. on Eng. Dynamics in the Surf Zone, Sydney, pp. 208–214.] for on–offshore transport was employed to show that, for small offshore wave incidence angles, onshore transport aided inlet closure when the offshore wave steepness (H_o/L_o) was less than the critical wave steepness (H_o/L_o)_crit, while offshore transport helped to keep the inlet open when (H_o/L_o) was greater than (H_o/L_o)_crit. LST was found to be the dominant process leading to inlet closure when (H_o/L_o) was much larger than (H_o/L_o)_crit or when the offshore wave incidence angle was large.     

Wave groupiness and spectral bandwidth as relevant parameters for the performance assessment of wave energy converters

To date the estimation of long-term wave energy production at a given deployment site has commonly been limited to a consideration of the significant wave height H_s and mean energy period T_e. This paper addresses the sensitivity of power production from wave energy converters to the wave groupiness and spectral bandwidth of sea states. Linear and non-linear systems are implemented to simulate the response of converters equipped with realistic power take-off devices in real sea states. It is shown in particular that, when the converters are not much sensitive to wave directionality, the bandwidth characteristic is appropriate to complete the set of overall wave parameters describing the sea state for the purpose of estimating wave energy production.     

Prediction of significant wave height using regressive support vector machines

Wave parameters prediction is an important issue in coastal and offshore engineering. In this literature, several models and methods are introduced. In the recent years, the well-known soft computing approaches, such as artificial neural networks, fuzzy and adaptive neuro-fuzzy inference systems and etc., have been known as novel methods to form intelligent systems, these approaches has also been used to predict wave parameters, as well. It is not a long time that support vector machine (SVM) is introduced as a strong machine learning and data mining tool. In this paper, it is used to predict significant wave height (H_s). The data set used in this study comprises wave wind data gathered from deep water locations in Lake Michigan. Current wind speed (u) and those belonging up to six previous hours are given as input variables, while the significant wave height is the output parameter. The SVM results are compared with those of artificial neural networks, multi-layer perceptron (MLP) and radial basis function (RBF) models. The results show that SVM can be successfully used for prediction of H_s. Furthermore, comparisons indicate that the error statistics of SVM model marginally outperforms ANN even with much less computational time required.     

Wave model predictions in the Black Sea: Sensitivity to wind fields

This paper evaluates the impact of using different wind field products on the performance of the third generation wave model SWAN in the Black Sea and its capability for predicting both normal and extreme wave conditions during 1996. Wind data were obtained from NCEP CFSR, NASA MERRA, JRA-25, ECMWF Operational, ECMWF ERA40, and ECMWF ERA-Interim. Wave data were obtained in 1996 at three locations in the Black Sea within the NATO TU-WAVES project. The quality of wind fields was assessed by comparing them with satellite data. These wind data were used as forcing fields for the generation of wind waves. Time series of predicted significant wave height (H_mo), mean wave period (T_m02), and mean wave direction (DIR) were compared with observations at three offshore buoys in the Black Sea and its performance was quantified in terms of statistical parameters. In addition, wave model performance in terms of significant wave height was also assessed by comparing them against satellite data.The main scope of this work is the impact of the different available wind field products on the wave hindcast performance. In addition, the sensitivity of wave model forecasts due to variations in spatial and temporal resolutions of the wind field products was investigated. Finally, the impact of using various wind field products on predicting extreme wave events was analyzed by focussing on storm peaks and on an individual storm event in October 1996. The numerical results revealed that the CFSR winds are more suitable in comparison with the others for modelling both normal and extreme events in the Black Sea. The results also show that wave model output is critically sensitive to the choice of the wind field product, such that the quality of the wind fields is reflected in the quality of the wave predictions. A finer wind spatial resolution leads to an improvement of the wave model predictions, while a finer temporal resolution in the wind fields generally does not significantly improve agreement between observed and simulated wave data.     

Evaluation of two WAM white capping parameterizations using parallel unstructured SWAN with application to the Northern Gulf of Mexico, USA

The performance of two well accepted formulations for white capping and wind input of third generation wave models, viz., WAM-3 and WAM-4, were investigated using parallel unstructured SWAN (PunSWAN). Several alternative formulations were also considered to evaluate the effects of higher order steepness and wave number terms in white capping formulations. Distinct model configurations were calibrated and validated against available in situ measurements from the Gulf of Mexico. The results showed that some of the in situ calibrated models outperform the saturation level calibrated models in reproducing the idealized wave growth curves. The simulation results also revealed that increasing the power of the steepness term can enhance the accuracy of significant wave height (H_s), at the expense of a higher bias for large waves. It also has negative effects on mean wave period (T_a) and peak wave period (T_p). It is also demonstrated that the use of the quadratic wave number term in the WAM-3 formulation, instead of the existing linear term, ameliorates the T_a underestimation; however, it results in the model being unable to reach any saturation level. In addition, unlike H_s and T_p, it has been shown that T_a is sensitive to the use of the higher order WAM-4 formulation, and the bias is decreased over a wide range of wave periods. However, it also increases the scatter index (SI) of simulated T_a. It is concluded that the use of the WAM-4 wind input formulation in conjunction with the WAM-3 dissipation form, is the most successful case in reproducing idealized wave growth curves while avoiding T_a underestimation of WAM-3 and a potential spurious bimodal spectrum of WAM-4; consequently, this designates another perspective to improve the overall performance of third generation wave models.     

Examination of wind-wave interaction source term in WAVEWATCH III with tropical cyclone wind forcing

Results of drag coefficient(CD) from field observations and laboratory wave tank experiments indicate that the operational wave model can overestimate wind energy input under high wind conditions. The wind-wave interaction source term in WAVEWATCH Ⅲ has been modified to examine its behavior with tropical cyclone wind forcing. Using high resolution wind input,numerical experiments under idealized wind field and tropical cyclone Bonnie(1998) were designed to evaluate performance of the modified models. Both experiments indicate that the modified models with reduced CD significantly decrease wind energy input into the wave model and then simulate lower significant wave height(SWH) than the original model. However,the effects on spatial distribution of SWH,mean wavelength,mean wave direction,and directional wave spectra are insignificant. Due to the reduced wind energy input,the idealized experiment shows that the modified models simulate lower SWH than the original model in all four quadrants. The decrease in the front quadrants is significantly larger than that in the rear quadrants;it is larger under higher winds than lower winds. The realistic experiment on tropical cyclone Bonnie shows that the modified model with the various downward trends of CD in high winds creates a simulation that agrees best with scanning radar altimeter observations.     

Interannual variability and predictability of summertime significant wave heights in the western north pacific

We propose and validate a linear regression model which enables us to predict the summer (June–August) mean of the monthly 90th percentile of significant wave heights (H₉₀) in the western North Pacific (WNP). The most prevailing interannual variability of H⁹⁰ is identified by applying an Empirical Orthogonal Function analysis to H⁹⁰ obtained from the ERA-40 wave reanalysis as well as from the optimally interpolated TOPEX/Poseidon (OITP) wave data. It is found that the increase of H⁹⁰ is correlated with cyclonic circulation in the WNP which links with warm SST anomalies in the Niño-3.4 region. We adopt zonal wind anomaly averaged over the region 5°N–15°N, 130°E–160°E (U^10N) as a predictor of the first principal component (PC1) of H⁹⁰, since U^10N is closely correlated with the PC1 of H⁹⁰. It is revealed that regression models obtained from two different wave datasets are nearly identical. The predictability of the regression model is assessed in terms of the reduction of the root-mean-square (rms) errors between H⁹⁰ and the reconstructed data. The predictor is found to be successful in reducing the rms errors by up to 40% for the ERA-40 wave reanalysis and by up to 70% for the OITP wave data within the latitudinal band 10°N–25°N, though rms errors exceeding 0.3 m still remain, particularly in the East China Sea.     

A new predictive formula for inception of regular wave breaking

Efforts are made to enhance the predictive formula for the inception of wave breaking. To achieve success, the existing formulas are extensively reviewed. They are categorized into four types, i.e., the McCowan type, the Miche type, the Goda type and the Munk type. The inherent relations among the different types are then exploited. The differences among each formula within a group are also discussed. Four representative formulas from the different types are chosen to compare with the measured data for a total number of 1193 cases reported in literatures. It is shown that Goda's and Ostendorf and Madsen's formulas are advantageous in general among the selected ones. Goda's formula, however, is found to be inaccurate as the beach slope becomes steeper than 1/10. Ostendorf and Madsen's formula is fairly good even for cases of very steep slopes, but its accuracy for the cases of ordinary slopes is not as good as Goda's. A new predictive formula for the inception of wave breaking is proposed. The unique index, defined by ψ′_b = (1.21 − 3.30λ_b)(1.48 − 0.54γ_b)ψ_b, where ψ_b = gH_b/C_b², H_b is the breaking wave height, C_b is the breaking wave celerity, λ_b is the breaking wave steepness, γ_b is the relative breaking wave height, and g is the gravity acceleration, is introduced. The incipient condition of wave breaking is then given by ψ′_b = 0.69. This formula is a significant improvement to the existing ones in terms of the accuracy. In addition, it is a local relation. Further verification shows that the proposed formula performs similarly well when applied to the field and to the waves over permeable bed.     

Wave modelling in the vicinity of submerged breakwaters

This paper describes a simple method for modelling wave breaking over submerged structures, with the view of using such modelling approach in a coastal area morphodynamic modelling system.A dominant mechanism for dissipating wave energy over a submerged breakwater is depth-limited wave breaking. Available models for energy dissipation due to wave breaking are developed for beaches (gentle slopes) and require further modifications to model wave breaking over submerged breakwaters.In this paper, wave breaking is split into two parts, namely: 1) depth-limited breaking modelled using Battjes and Janssen's (1978) theory [Battjes, J.A. and Jannsen, J.P.F.M. (1978). Energy loss and setup due to breaking of random waves. Proceedings of the 16th Int. Conf. Coast. Eng., Hamburg, Germany, pp. 569-587.] and 2) steepness limited breaking modelled using an integrated form of the Hasselmann's whitecapping dissipation term, commonly used in fully spectral wind–wave models. The parameter γ₂, governing the maximum wave height at incipient breaking (H_max = γ₂d) is used as calibration factor to tune numerical model results to selected laboratory measurements. It is found that γ₂ varies mainly with the relative submergence depth (ratio of submergence depth at breakwater crest to significant wave height), and a simple relationship is proposed. It is shown that the transmission coefficients obtained using this approach compare favourably with those calculated using published empirical expressions.     

Use of nautical radar as a wave monitoring instrument

Common marine X-Band radars can be used as a sensor to survey ocean wave fields. The wave field images provided by the radars are sampled and analysed by a wave monitoring system (called WaMoS II) developed by the German research institute GKSS. This measuring system can be mounted on a ship, on offshore stations or at coastal locations. The measurement is based on the backscatter of microwaves from the ocean surface, which is visible as ‘sea clutter' on the radar screen. From this observable sea clutter, a numerical analysis is carried out. The unambiguous directional wave spectrum, the surface currents and sea state parameters such as wave periods, wave lengths, and wave directions can be derived. To provide absolute wave heights, the response of the nautical radar must be calibrated. Similar to the wave height estimations for Synthetic Aperture Radars, the so-called ‘Signal to Noise Ratio' leads to the determination of the significant wave height (H_S). In this paper, WaMoS II results are compared with directional buoy data to show the capabilities of nautical microwave radars for sea state measurements.     

Wave height parameter for damage description of rubble-mound breakwaters

In this paper it will be shown that the wave height parameter H₅₀, defined as the average wave height of the 50 highest waves reaching a rubble-mound breakwater in its useful life, can describe the effect of the wave height on the history of the armor damage caused by the wave climate during the structure's usable life.Using Thompson and Shuttler (Thompson, D.M., Shuttler, R.M., 1975. Riprap design for wind wave attack: A laboratory study on random waves. HRS Wallingford, Report 61, UK) data it will be shown that H₅₀ is the wave parameter that best represents the damage evolution with the number of waves in a sea state. Using this H₅₀ parameter, formulae as van der Meer (van der Meer, J.W., 1988. Rock slopes and gravel beaches under wave attack. PhD Thesis. Technical University of Delft) and Losada and Giménez-Curto (Losada, M.A., Gimenez–Curto, L.A., 1979. The joint effect of the wave height and period on the stability of rubble mound breakwaters using Iribarren's number. Coastal Engineering, 3, 77–96) are transformed into sea-state damage evolution formulae. Using these H₅₀-transformed formulae for regular and irregular sea states it will be shown how damage predictions are independent of the sea state wave height distribution.To check the capability of these H₅₀-formulae to predict damage evolution of succession of sea states with different wave height distributions, some stability tests with regular and irregular waves have been carried out. After analysing the experimental results, it will be shown how H₅₀-formulae can predict the observed damage independently of the sea state wave height distribution or the succession of sea states.     

Numerical modelling of erosion and accretion of plane sloping beaches at different scales

The paper focuses on the numerical simulation of erosion of plane sloping beaches by irregular wave attack in three wave flumes of different scales. One of the prime objectives of the tests was to provide a consistent data set for the improvement of numerical beach profile models. A practical application of this research with wave attack on plane sloping beaches is the erosion of the plane beaches after nourishment. Three models (CROSMOR, UNIBEST-TC and DELFT3D) have been used to simulate the flume experimental results focusing on the wave height distribution and the morphological development (erosion and deposition) along the beach profiles. Overall, the model predictions for wave heights show consistent results. Generally, the computed wave heights (H_rms and H_1/3) are within 10% to 15% of the measured values for all tests (under-prediction of the largest wave heights close to the shore). The three models can simulate the beach erosion of the wave flume tests (erosive tests) reasonably well using default values of the sand transport parameters. The model performance for the accretive tests is less good than that for the erosive tests. A practical field application of this research is the erosion of nourished beaches, as these beaches generally have rather plane beach slopes immediately after nourishment. Various graphs are given to estimate the beach erosion of nourished beaches.     

Measured and modeled wave overtopping on a natural beach

The rate of wave overtopping of a barrier beach is measured and modeled. Unique rate of wave overtopping field data are obtained from the measure of the Carmel River, California, lagoon filling during a time when the lagoon is closed-off with no river inflow. Volume changes are based on measured lagoon height changes applied to a measured hypsometric curve. Wave heights and periods are obtained from directional wave spectra data in 15 m fronting the beach. Beach morphology was measured by GPS walking surveys. Three empirical overtopping models by Van der Meer and Janssen (1995), Hedges and Reis (1998) and Pullen et al. (2007) with differing parameterizations on wave height, period and beach slope and calibrated using extensive laboratory data obtained over plane, impermeable beaches are applied in a quasi-2D manner and compared with the field observations. Three overtopping events are considered when morphology data were available less than 2 weeks prior to the event. The models are tuned to fit the data using a reduction factor to account for beach permeability, berm characteristics, non-normal wave incidence and surface roughness influence. In addition, the run-up model by Stockdon et al. (2006) based on field data is examined and found to underestimate run-up as the calculated values were too small to predict any of the observed overtopping. The three overtopping models performed similarly well with values of 0.72–0.87 for the two narrow-banded wave cases, with an average reduction factor of 0.78. The European model (Pullen et. al., 2007) performed best overall and in particular for the case of the broad-banded, double peaked wave spectrum.