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Fang, K.; Wang, H.; Sun, J.; Zhang, J., and Liu, Z., 2020. Including wave diffraction in XBeach: Model extension and validation. Journal of Coastal Research, 36(1), 116–127. Coconut Creek (Florida), ISSN 0749-0208.
An extended version of the XBeach model is presented in this paper to improve the accuracy of predicted wave and current fields and sandy beach evolution in scenarios in which wave diffraction takes effect. Following the approach successfully implemented in the Simulating Waves Nearshore (SWAN) spectral wave model, a diffraction parameter is introduced into the wave action balance equation in XBeach to take wave diffraction into account. An alternative wave breaking index and the formulation for the critical velocity are also tested in the extended model to achieve better model–data agreement. The extended model is validated by comparing the simulation results with those of physical tests for fixed and movable sea beds. The computed wave height, wave-induced current field, and beach evolution are found to be in satisfactory agreement with the measurements, and the numerical modeling capacity of the extended model to represent wave diffraction effect is thus demonstrated.
INTRODUCTION
The erosion of sandy coast is quite common around the world. The most direct causes of erosive processes are the increase in wave energy along the coast and decrease in sediment supply, leading to an unbalanced sediment budget. Dealing with coastal erosion has become extremely challenging. For example, the observed global trend in coastal erosion will be enhanced by sea level rise and more frequent extreme events under a changing climate (Lorenzo et al., 2018). In the meantime, some shoreline protection structures have failed in their purpose and intensified erosion processes (Pranzini et al., 2018; Rangel-Buitrago, Williams, and Anfuso, 2018) because of the lack of systematic theoretical guidance. Over the years, large numbers of numerical models have been developed, differing in complexity and hence (operational) applicability to predict the evolution of sandy beach, e.g., Mike21 (Moghaddam et al., 2018), Surfacewater Modeling System (Kuiry, Ding, and Wang, 2009), Delft3D (Chaichitehrani, 2018), and XBeach (Roelvink et al., 2009). Among these models, the process-based and open-source model XBeach, developed by Deltares, together with the United Nations Educational, Scientific and Cultural Organization–International Institute for Hydraulic and Environmental Engineering and Delft University of Technology, has become increasingly popular and has been widely used for a large variety of beach types in various coastal environments (De Santiago et al., 2017; McCall et al., 2010; Roelvink et al., 2018; Suh, Kim, and Kim, 2017; Van Dongeren et al., 2016; among others).
Despite XBeach's increased use, further development of the model is needed to improve or extend its adaptability in various application scenarios. For instance, McCall et al. (2010) and Elsayed and Oumeraci (2017) found XBeach substantially overestimates erosion volumes, deposition volumes, and consequently washover volumes, and they proposed methods to overcome this deficit. The simulations of Stockdon et al. (2014) and Palmsten and Splinter (2016) show that despite accurate predictions of the morphodynamics of dissipative sandy beaches, the XBeach model does not correctly simulate the individual contributions of setup, infragravity, and incident-band swash to wave run-up. Roelvink et al. (2018) thus proposed an improved numerical scheme and a different approach for simulating the propagation of directionally spread short wave groups to achieve better predictions of grouping of the short waves and the resulting infragravity waves. In addition, XBeach contains numerous parameters and formulations in its hydrodynamic and morphological modules, and careful adjustment or even the use of alternative formulations is usually needed to minimize data–model mismatch (e.g., Do et al., 2018; Elsayed and Oumeraci, 2017; Splinter, Kearney, and Turner, 2018; Van Geer et al., 2015).
Wave diffraction is quite common in the nearshore region, e.g., in the presence of a breakwater, headland, offshore island (Goda, 2010), irregular bathymetries such as a circular shoal (Dalrymple et al., 1989), or other regions of localized energy dissipation (Dalrymple, Kirby, and Hwang, 1984). Wave diffraction makes the waves and wave-induced currents more complicated, which influences the bed evolution (Nam and Larson, 2010; Pérez, 2011; Tang et al., 2017). However, the present form of the XBeach model (in surf-beat mode or stationary mode) uses the spectral model to calculate wave fields, failing to represent short wave diffraction appropriately (Caminada, 2018). This lack of capacity might yield further uncertainties once XBeach is used in scenarios in which diffraction takes effect, as already noted by XBeach developers (e.g., Caminada, 2018; McCall, 2008). In this context, the objective of the present work is to include wave diffraction in the XBeach surf-beat mode and validate the extension by comparing the computed hydrodynamics and bed evolution with physical experiments.
METHODS
This section details the procedure of introducing a diffraction parameter into the wave action balance module of XBeach and alternative options for the wave breaking index and critical velocity formulation. Additional details of the governing equations and numerical schemes of XBeach are referred to in references (e.g., De Santiago et al., 2017; McCall et al., 2010; Roelvink et al., 2009; Van Dongeren et al., 2016).
Wave Action Balance Equation of XBeach
To obtain the wave field, the surf-beat mode of XBeach solves a time-dependent version of the wave action balance equation, which reads:
[Image Omitted. See PDF.]
where, wave action A = A(x,y,t,θ) is calculated as:
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Here, θ is the wave incident angle relative to the x-axis, and E is the wave energy density represented as a function of geophysical location (x, y), time (t), and direction (θ). σ represents the intrinsic wave frequency obtained from the linear dispersion relation. The x-axis and y-axis correspond to the shore-normal and shore-parallel directions, respectively. The right side of Equation (1) is the dissipation term, which consists of three short wave dissipation processes (i.e. wave breaking, bottom friction, and vegetation). XBeach also employs a roller model to represent the momentum stored at the surface after breaking, which is included in the wave dissipation terms. The wave action propagation speeds in x, y, and directional space without considering wave–current interaction are given by:
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where, cg is the group velocity, k is the wave number, and h is the local water depth.
Application of a Diffraction Parameter in XBeach
The phase-decoupled refraction–diffraction approximation (Holthuijsen, Herman, and Booij, 2003) derived from the mild slope equation has been successfully incorporated in the Simulating Waves Nearshore (SWAN) model (Booij, Ris, and Holthuijsen, 1999). The wave action balance equation in XBeach is primarily the same as that in SWAN; this approach is thus followed, and the wave action propagation speeds in Equation (3) are replaced by:
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where,
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The total energy Ew is calculated by integrating over the wave directional bins as:
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As seen, the diffraction effect is mainly embodied in the calculation propagation speed in θ-dimension cθ.
Numerical Implementation
XBeach uses a curvilinear coordinate for which the s-axis is always oriented toward the coast, approximately perpendicular to the coastline, and the n-axis is alongshore. The corresponding grid resolutions are denoted Δs and Δn (Deltares, 2015). The second-order finite difference scheme is used to approximate the numerator of Equation (5) along the s direction, which reads:
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where, indices i,j refer to the grid index in the s and n directions, respectively. The expression for the n direction is almost identical and will not be repeated here for simplicity. As argued by Holthuijsen, Herman, and Booij (2003), a convolution filter is needed to smooth the wave field before computing the diffraction parameter, which reads:
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where, the superscript m indicates the iteration number and is set to 3 as a default value. Equations (7) and (8) are primarily the same as those in Holthuijsen, Herman, and Booij (2003), but they are implemented on the variable grid size system used in XBeach.
Wave Breaking
In XBeach, a breaker coefficient γ is needed to initiate wave breaking once local wave height exceeds a maximum value Hmax, which is defined as:
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where, Hrms represents root-mean-square wave height. In XBeach, γ = 0.55 is specified as a default value. Numerical tests demonstrate that adjusting this parameter with given breaking models cannot satisfactorily predict wave height for the Large-Scale Sediment Transport Facility (LSTF) data (see “Results”). To obtain better agreement with the measurements, an alternative breaker formulation (Grasmeijer and Ruessink, 2003; Nam et al., 2009; Tang et al., 2017), which takes the contribution from local bed slope to wave breaking into account, is used to replace the default Equation (9) (see Appendix A).
Critical Velocity for Sediment Transport
The calculation of critical velocity is a key procedure in XBeach, because it determines whether the sediment is in motion and then significantly influences the bed evolution. Using two available equations provided in XBeach, i.e. the Soulsby–van Rijn equations (Soulsby, 1997; van Rijn, 1984) and the Van Thiel–van Rijn equations (van Rijn, 2007; Van Thiel de Vries, 2009), numerical experiments show that sediment incipient motion is rare for LSTF experiments (see the next section), and the computed bed evolution deviates largely from the measurements. An alternative formulation proposed by Cao, Kong, and Jiao (2003), based on hundreds of physical flume tests for sand incipient motion in a wave–current coexistence environment, is thus used to determine the critical velocity (see Appendix B). The default option of the Van Thiel–van Rijn equations (van Rijn, 2007; Van Thiel de Vries, 2009) in XBeach is then used to calculate the equilibrium sediment concentration.
RESULTS
In this section, a series of numerical simulations including unidirectionally and directionally spread irregular waves propagating over fixed and movable beds using the extended XBeach model are carried out. The computed results are compared against the experimental data and numerical results without considering wave diffraction.
Waves Propagating through a Breakwater Gap
Yu et al. (2000) carried out a series of experiments to study combined refraction and diffraction of waves through a breakwater gap. Three tests with various combinations of gap width and wave parameters are chosen for comparison, as shown in Table 1. The layout of the wave basin and wave gauges is shown in Figure 1. The constant water depth is 0.4 m, and there are two 0.35-m-wide breakwaters with rounded tips close to the wavemaker side in the wave basin. The Joint North Sea Wave Project spectrum is used in the experiments. The incident significant wave height Hs,i is 0.05 m with peak spectral period Tp = 1.20 seconds and peak enhancement factor γ = 4.
Table 1
Test conditions for the physical experiment test (Yu et al., 2000).
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Figure 1
Layout of the wave basin: (a) incident principal wave direction θ0 = 90°; (b) incident principal wave direction θ0 = 45° (Yu et al., 2000).
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A fixed grid size with Δs=Δn=0.196 m and Δθ=10° is used in the simulation. The directional spreading parameter (the power of cosine law; see Goda, 2010, and Deltares, 2015, for details) is set to s = 100,000 to represent a unidirectional incident wave. The contour maps of computed significant wave height H (normalized by the incident wave height Hs,i) with and without wave diffraction are given in Figure 2. Further comparison with the experimental data along three transects parallel to the x-axis after the breakwaters is given in Figure 3.
Figure 2
The normalized significant wave height from the simulations with diffraction (a) and without diffraction (b) for Test 1 (B= 2L), Test 2 (B= 4L), and Test 3 (B = 4L). x = 0 represents the gap center (arrow = wave direction, dashed line = transect locations used for comparison in Figure 3).
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Figure 3
Comparison of computed wave height with observations of Yu et al. (2000) along transects for Test 1, Test 2, and Test 3 (circle = measured, solid line = with diffraction, dashed line = without diffraction).
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It is seen from Figure 2 for waves propagating through the breakwater gap that the extended model predicts a more reasonable wave field for unidirectional incident wave cases in Tests 1 and 2, while the original XBeach model presents almost zero values of wave height in the shadow zone because of the lack of wave diffraction. The improvement is clearly seen in Figure 3, where the computed results from the extended model agree well with the experimental data, while those that do not consider wave diffraction deviate from the measurements. Wave diffraction becomes weak for directionally spread irregular waves, because some wave components directly penetrate into the shadow zone with little effect of diffraction induced by the breakwater; thus, the improvement should not be so obvious. This expectation is confirmed from the simulation results for Test 3, where the difference between computed wave height with and without considering wave diffraction is marginal.
Waves Propagating over a Submerged Elliptical Mound
It is well known that a region of localized energy dissipation leads to wave diffraction (Dalrymple, Kirby, and Hwang, 1984). An irregular bathymetry is commonly encountered in modeling sandy beach evolution, for example, when deploying submerged breakwaters or reefs and artificial sandbars in beach nourishment projects or in the presence of naturally formed sandbars incised by rip channels. The second test is thus to simulate the physical experiment for the unidirectional incident wave with a frequency distribution based on a Texel, Marson, And Arsole spectrum for propagation over an elliptical mound as conducted by Vincent and Briggs (1989). The water depth in the wave flume is 0.457 m. An elliptical shoal with a major radius of 3.96 m (13 ft), a minor radius of 3.05 m (10 ft), and a maximum height of 0.3048 m (1 ft) was placed in the flume center. Test U1, with incident significant wave height Hs,i=0.0775 m, peak spectral period Tp=1.3 seconds, and peak enhancement parameter γ=2, is simulated. Δs=Δn=0.2 m and Δθ= 2° are used in the simulation.
Figure 4a,b presents the spatial distribution of computed significant wave height H (normalized by incident value Hs,i) with and without wave diffraction, respectively. Figure 4c compares the computed and measured wave height along a transect 12.2 m (40 ft) from the incident boundary. As seen from this figure, the extended XBeach model again shows great improvements, demonstrating the necessity of taking wave diffraction into account for the case considered.
Figure 4
The normalized significant wave height from simulation with diffraction (a) and without diffraction (b) for the experiment of Vincent and Briggs (1989), and the comparison between computed wave height and experimental data along transect 4 (c). Dashed line in (a) and (b) denotes the location of transect 4.
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LSTF Data
High-quality data sets have been collected in physical experiments conducted in the LSTF basin at the U.S. Army Engineer Research and Development Center in Vicksburg (Gravens and Wang, 2007; Gravens et al., 2006), and these data have been widely used to validate beach evolution models (Nam et al., 2009; Tang et al., 2017). Test T1C1 from the experiment for sandy beach evolution behind a detached breakwater is employed here for verification. The configuration of the LSTF basin and the location of the detached breakwater are shown in Figure 5. The wave conditions and currents are sampled at 13 cross-shore transects from profile Y14 to profile Y34. Three wave gauges (#11, #12, and #13) are located 18.43 m seaward of the initial shoreline to measure incident waves.
Figure 5
Detached breakwater layout within LSTF for Test T1C1 (Gravens and Wang, 2007).
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The computational domain is discretized into Δs = Δn = 0.2 m and Δθ = 10°. A composite incident significant wave height of 0.227 m and peak spectral period of 1.457 seconds are set as model inputs. A TMA spectrum is simply activated by setting keyword tma = 1 in XBeach with the parameter values γ = 3.3 and s= 25. In addition, the wave propagates at a 6.5° angle from north to the beach. The dimensionless bed friction coefficient is determined by setting the Manning coefficient of n = 0.015. The main parameter settings for this case are listed in Table 2.
Table 2
Main parameter settings for the LSTF test in XBeach.
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Figure 6 compares the computed wave fields with diffraction and without diffraction. The contour lines in this figure show that the extended model predicts a more reasonable wave height distribution, especially around the breakwater tips and in the shadow zone. More wave energy penetrates the shadow zone after incorporating wave diffraction in XBeach; however, as mentioned earlier, the quantitative difference is marginal (on the order of centimeters) because of the directional spreading of wave energy. The comparison of wave height along transects in Figure 8 show this trend, while Figure 7 compares the computed wave-induced current field including diffraction with the measurements. The computed wave-induced current field, including the velocity magnitude and direction, compares well with the experimental data.
Figure 6
Comparison between the computed wave field with diffraction and the one without diffraction for LSTF Test T1C1 (solid line = with diffraction, dashed line without diffraction).
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Figure 7
Comparison between computed and measured nearshore currents for LSTF Test T1C1 (black vector = measured result, red vector = simulation results with diffraction).
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Figure 8
Comparison between computed and measured significant wave heights for LSTF Test T1C1: (a) Y18, (b) Y20, (c) Y22, (d) Y24, (e) Y26, and (f) Y30.
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Figures 8 to 10 show in detail the comparison between simulated and measured wave height, cross-shore current, and longshore current, respectively, at six profiles. It seems that including wave diffraction improves the prediction accuracy of longshore currents, while the cross-shore currents change little. The comparison between the simulated bed level with diffraction and the measurements after 185 minutes of wave action is plotted in Figure 11. Because of the short simulation duration, the differences in bed evolution with and without considering wave diffraction are indistinguishable, and only the predictions from the extended model are presented. The model correctly predicts the formation of a salient, and the overall agreements between the computed bed evolution and the measurements are satisfactory. The computed results in Figures 6 to 11 are comparable to those simulated by Nam et al. (2009) and Tang et al. (2017).
Figure 9
Comparison between computed and measured cross-shore currents for LSTF Test T1C1: (a) Y18, (b) Y20, (c) Y22, (d) Y24, (e) Y26, and (f) Y30.
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Figure 10
Comparison between computed and measured alongshore currents for LSTF Test T1C1: (a) Y18, (b) Y20, (c) Y22, (d) Y24, (e) Y26, and (f) Y30.
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Figure 11
Comparison between computed and measured beach evolution after 185 min of wave action for LSTF Test T1C1. All units are in meters (solid line = simulation results with diffraction, dashed line = measurements).
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Offshore Breakwater Test Case
Nicholson et al. (1997) set up a numerical test involving sandy beach evolution in the vicinity of a shore-parallel breakwater on a plane-sloping beach with a slope of 1:50 to compare the performance of five coastal area morphodynamic models. Lesser et al. (2004) subsequently used the same configuration to test the DELFT3D model. This test is used herein to further investigate the effect of wave diffraction on beach evolution. The breakwater is 300 m long and 220 m from the shoreline. The median diameter of grains is d50 = 0.25 mm. The incident wave is unidirectional, with a height of Hrms = 2.0 m and a peak spectral period of Tp = 8.0 seconds. A spatial grid size with Δs = Δn = 10 m and Δθ = 5° is used in the simulation. Other parameters use the default values according to the report of Nicholson et al. (1997). The computed bathymetry and flow field after 72 hours of wave action are given in Figure 12.
Figure 12
The computed bathymetries and flow field using XBeach with diffraction (a) and without diffraction (b) after 72 hours of wave action for the offshore breakwater test reported in Nicholson et al. (1997).
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As has been shown in context, the effect of including wave diffraction is more pronounced for unidirectional incident waves and/or long-duration simulations. The comparison presented in Figure 12 illustrates this conclusion. For this test, the wave-induced current field and bathymetry change with and without considering waves showing significant discrepancies. Without wave diffraction, the vortex mainly concentrates near breakwater tips, and the relatively calm shadow zone allows more sediment deposition.
DISCUSSION
The present extension to the surf-beat mode of XBeach is valid for the stationary mode of XBeach, which uses the quasi-state wave action balance equations. In contrast, the nonhydrostatic mode of XBeach intrinsically considers wave diffraction because of its phase-resolving nature. The nonhydrostatic wave module in the latest version of XBeach works well for coastal wave hydrodynamic simulations, but the coupled simulation with sediment transport and beach evolution is still under development. In addition, it requires expensive computation efforts.
The comparisons shown in the present study illustrate that the extended XBeach model offers reasonable improvements over the original model for the cases in which wave diffraction takes effect. However, there are still some uncertainties when adding the diffraction term in the XBeach model. For example, the value of iteration number m for the convolution filter needs to be tested to find an optimum value, as suggested by Holthuijsen, Herman, and Booij (2003). Otherwise, large m will smear the computed wave fields, while small m might induce numerical instabilities for some cases. Moreover, grid resolutions (Δs and Δn) and angular resolution (Δθ) affect simulation results (Kim, Jho, and Yoon, 2017), which remain for further study.
It is a common approach to choose different formulations of wave hydrodynamics and sediment transport to achieve better agreement for a specific simulation of sandy beach evolution. An alternative wave breaking index and alternative critical velocity formulation have been used in this paper (Appendices A and B) for the same purpose. However, it is undeniable that testing the performance of different formulas in XBeach is worth further investigation.
CONCLUSIONS
By introducing a diffraction parameter into the wave action balance equations, wave diffraction has been successfully implemented in the spectral wave model SWAN. The study follows a similar approach and presents an extended version of the XBeach model with improved numerical modeling capacity to predict more accurate hydrodynamics and beach evolution in scenarios in which wave diffraction takes effect. An alternative wave breaking index and alternative critical velocity formulation have also been tested in the extended model.
The comparisons with the fixed bed physical tests for waves propagating through a breakwater gap and over a submerged hump show that the extended model can obtain more accurate wave field predictions. With the increase in the directional spreading parameter, the improvement becomes more apparent; this observation suggests profound implications in the design stage of beach nourishment projects in which unidirectional waves are typically used as the incident wave condition. Because of the scarcity of physical experiments for mobile bed evolution under the effects of wave diffraction in the literature, the comparisons are only made for computed wave height, nearshore current, and bed changes from the extended XBeach with LSTF data. Reasonable agreements between the simulation results and the measurements are found, while the improvement over the original version of XBeach is found to be marginal because of wave directional spreading and short simulation duration. Further evaluation of the improvement and performance for different wave breaker indices and sediment transport formulations needs the support of detailed physical experiments.
Additional Notes
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Corresponding author: [email protected]ACKNOWLEDGMENTS
The authors acknowledge support from the National Key Research and Development Program (Grant No. 2017YFC1404200), the National Natural Science Foundation of China (Grant Nos. 51579034, 51779022, and 51809053), the Ocean Engineering Joint Research Center of the Dalian University of Technology and the University of Western Australia (Grant No. LY1902), and the Fundamental Research Funds for the Central Universities (Grant No. DUT18ZD214). We also thank the research team of the XBeach model for providing open-source code.
APPENDIX A: ALTERNATIVE BREAKER FORMULATION
According to Nam et al. (2009) and Tang et al. (2017), the following expression is used to determine γ:
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where, . is the local bottom slope. Then, the limiting wave steepness for maximum wave height takes the form (Grasmeijer and Ruessink, 2003):
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where, L is the wave length and k is the wave number.
APPENDIX B: OPTIONAL FORMULATION TO CALCULATE CRITICAL VELOCITY FOR SEDIMENT TRANSPORT
In Cao, Kong, and Jiao (2003), the critical velocity is determined as:
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with
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and
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where, d50 is the median diameter of grains (in meters), υ is the kinematic viscosity of water, vmg is the velocity magnitude (in meters per second), g is the gravity acceleration (in square meters per second), and ρs and ρ are the densities of sediment grains and water, respectively (in kilograms percubic meter). fcw is the bottom friction factor because of wave and current, and uorb is the orbital velocity (in meters per second) calculated from linear wave theory:
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where, Trep refers to representative wave period or periods defined as the zeroth moment of the spectrum divided by the first moment of the spectrum:
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where, S(f) is the frequency spectrum and f is the frequency. fc and fw in Equation (B3) are calculated in accordance with van Rijn (1993) and Swart (1974), respectively, as:
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where, ks is a user-defined current-related effective roughness height with a default of 0.0025 m. Âδ is the semiorbital excursion at the bed: Âδ = uorbTrep/2π. Wave-related roughness ks,w is obtained from the estimated ripple height in a range of 0.01 to 0.1 m.
Copyright Allen Press Inc. Jan 2020