OMAE Verification of a 3rd generation FEM spectral wave model for shallow and deep water applications. Copyright 2004 by ASME

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1 Proceedings of OMAE 3rd International Conference on Offshore Mechanics and Arctic Engineering June -9,, Hamburg,Germany OMAE-9 Abstract Verification of a 3rd generation FEM spectral wave model for shallow and deep water applications. Aron Roland Ulrich Zanke Tai-Wen Hsu University of Technology University of Technology National Cheng Kung University Darmstadt, Germany Darmstadt, Germany Tainan, Taiwan, R.O.C Shan-Hwei Ou Jian-Ming Liau Swun-Kwang Wang National Cheng Kung University National Cheng Kung University National Cheng Kung University Tainan, Taiwan, R.O.C Tainan, Taiwan, R.O.C Tainan, Taiwan, R.O.C This paper shows some results of the work currently carried out to improve the wave forecasting and hindcasting in oceanic and coastal regions. A new spectral wave model with a flexible numerical scheme using triangular elements to describe the model domain was developed by Hsu et al. (). This new spectral wave model called WWM (Wind Wave Model) is feasible for the spectral wave modeling of irregular coastlines and complicated bathymetries because of its numerical scheme. The Wave Action Equation is solved with the aid of the Fractional Step Method (Yanenko, 97). The Integration in the spatial space is carried out with the Taylor-Galerkin Method and the terms describing depth and current induced refraction are integrated with the aid of Leonard s (979) TVD Ultimate Quickest scheme, which was already introduced in the WWIII (H. Tolman, 99) for the same purpose. In three applications the wave model was verified against in-situ spectral measurements of directional and non-directional wave buoys. The results show that the new spectral wave model is capable of hindcasting the wave climate with a comparable accuracy like the SWAN model (Ris et al., 99), though with a better efficiency since fewer nodes are necessary to resolve the model domain and the boundary conditions adequately. Introduction Numerical simulation of the sea state has become an important tool in scientific and engineering applications during the last 5 years. The spectral wave models that are actually available are designated for certain applications and only in that field they are optimally applicable. This is due to the chosen numerical method and/or due to the mathematical formulation of the sources and sink terms in the wave action equation. The further improvement of wave forecasting and hindcasting methods is of fundamental scientific interest and it is of great importance for the sustainable planning and development of our coastal regions and off-shore structures (e.g. oil platforms) or for the secure ship routing. In the past years since the development of the well known SWAN model, spectral wave models have been increasingly applied for coastal studies where also the morphodynamic evolution plays an important role (e.g. Witting et al. ()). Complicated environments and boundary conditions encountered e.g. in tidal estuaries are difficult to discretize with structured grids. In certain cases cumbersome nested calculation runs have to be carried out in order to have a proper representation of the nature. Most of the available spectral wave models use structured meshes (WWIII, WAM (WAMDI Group) and SWAN, Ris et al.). The methods today available on unstructured meshes incorporate the method of characteristics Benoit (99, TOMAWAC), Ardhuin (, CREST) or a first order finite volume scheme Sorensen et al. (; MIKE) to calculate the wave energy advection in spatial space. The weakness of the characteristics approach is that in inhomogeneous media (e.g. tidal estuaries) the characteristic fields changes with time and must be permanently recalculated, which makes the numerical scheme computationally expensive. The first order upwind scheme in MIKE may generate numerical diffusion which becomes significant for the case when swell waves propagate over larger distances within the model domain. The WWM uses like the SWAN model higher order numerical schemes for the various advection effects. The flexibility of the Fractional Step approach makes it possible to use multiple time steps for the integration of the wave action equation. The utilization of unstructured grids makes it easier to resolve the model area with a better accuracy but with fewer nodes than with structured meshes. This must be seen especially in the context to the ongoing advance in the understanding of the nonlinear processes in deep and shallow water and the development of more sophisticated theories and algorithms to evaluate the nonlinear energy transfer, not only in homogenous (Hasselmann, 9, Zakharov, 9)) moreover in inhomogeneous media (Rasmussen (99), Stiassnie ()). New developments in this context lead to more complicated theories and algorithms for the calculation of the nonlinear energy fluxes within the wave spectrum e.g. Webb-Resio-Tracy Method (WRT; van Vledder ). These transfer integrals must be evaluated at every time step and grid point. Efficient discretization of the model domain with unstructured meshes reduces the computational demand for these source terms significantly and makes a more sophisticated model formulation possible. Copyright by ASME

2 The newly developed model has been used in different environments; at the Sargasso Sea (Atlantic Ocean), the Baltic Sea and the Haringvliet estuary (The Netherlands). The results have been compared to the results of the SWAN model for the same cases. The source functions describing the effects of wind input and white capping dissipation have been replaced in both models. For the wind input function the formulation of Makin & Stam () and for the white capping dissipation function the formulation of Alves & Banner () has been incorporated into the wave models. The reason for the modification of the spectral balance is that the dissipation function suggested by Komen et al. (9) following the theoretical approach of Hasselmann (97) leads to an underestimation of low frequency energy and an overestimation of high-frequency energy which results in an erroneous spectral shape and an underestimation of the average period (Rogers et al. (), Roland et al. (, 5)).Moreover Holthuisen et al. () and Hurdle & van Vledder have showed that the actual formulation that is used in the SWAN model may lead to erroneous dissipation rates in the case of mixed sea conditions (windsea and swell). Van der Westhuysen et al. (5) have shown that the dissipation function of Alves & Banner reduces this effect and that it also leads to better convergence rates and better estimation of the average period. Model formulation The conservation equation describing the advection and refraction of wave action due to depths and currents can be written for Cartesian coordinates as follows. Stot N+ ( cxn) + ( cyn) + ( cσn) + ( cθn) = () t x y σ θ σ Equation : Wave action equation where N=N(t,x,y,σ,θ) is the wave action density spectrum; t is the time; c x and c y are the wave propagation velocities in x and y space, respectively; c σ and c θ are the wave propagation velocities in σ and θ space, respectively; σ is the relative frequency and θ is the wave direction. S tot is the source function including the energy input due to wind (S in ), the nonlinear interaction in deep and shallow water (S nl and S nl3 ), the energy dissipation in deep and shallow water due to whitecapping and wave breaking (S ds and S br ), the energy dissipation due to bottom friction (S bf ) and nonlinear interactions between waves and the sea floor (S bg, Braggscattering.) Stotal = Sin + Snl+ Sds + Snl3 + Sbr + Sbf + Sbg () The nonlinear terms Snl and Snl3 have been evaluated with the Discrete Interaction Approximation (DIA, Hasselmann et al. (9)) and the Lumped Triad Approximation (LTA, Elderberky (999)) respectively. Bragg Scatering is not accounted in the latest version but it will be considered in the future following the approach of Ardhuin & Herbers (). Wave diffraction is already implemented on the foundation of the approach by Holthuijsen et al. (5) and under testing. In the investigated simulations these effects have been neglected. The wind input function (eq. 3) is defined following Makin & Stam, on the foundation of the work of Makin & Kudryavtsev, 999. n c c ρ a p S in( σ, θ ) = min AW, mβ m c ρ w u * u * cos( θ θw) cos( θ θw) σ S(, ) (3) σ θ c p The dissipation function (eq. ) is based on the work of Alves & Banner p / m n B( σ ) α k Sds(, ) = Cds S σθ σ ( σθ, ) Br α PM k α = E k tot π 3 ( ) = ( ),, (, ) g (, σ θ = σ σ θ σ θ σ θ ) B B d B k C S p =, B( ) < B σ r p p B( σ ) p = + tanh, B( ) B σ r > B r In the above equations S (σ, θ) is the variance density spectrum [m²/hz], S in (σ, θ) [m²] and S ds (σ, θ) [m²] are the rate of change in variance density due to the influence of wind and dissipative processes respectively. B (σ, θ) is the saturation spectrum, ρ a [kg/m³] and ρ w [kg/m³] are the wind and the water density, u * [m/s] and c p [m/s] is the wind induced friction velocity and the phase velocity of the certain wave component. θ and θ W are the wave direction and the wind direction respectively and σ is the Doppler shifted (by current in the water column) relative frequency, k [/m] and k [/m] are the certain wave number and the average wave number of the wave spectrum, α [-] and α PM =.57E-3 [-] are the average steepness and the Pierson- Moskowitz steepness. Br, m β, m c, n c, p, m, n, are parameters (for details see Alves & Banner and Makin & Stam. A W controls the wave attenuation of waves that are running faster than the wind. This parameter was set to zero in this study so that this effect was not accounted in this study. We will consider it in future studies. In the following we will refer to the default source term formulation in the SWAN model as C-I (combination I) and to the new source term formulation as C-III (combination III). In the latest version of the code the friction velocity u * is estimated with the sea drag parameterization by Makin 3, which incorporates the wind-over-waves coupling theory (WOWC). This parameterization is based on a resistance law, which relates sea drag to the sea state. In the version used for this study the friction law by Wu (9) was used throughout. The dissipation formulation for bottom friction is based on the empirical JONSWAP model by Hasselmann et al. (973) with a constant dissipation coefficient of -.7. () Copyright by ASME

3 Verification cases For the verification of the new wave model for deep water and intermediate water conditions we have investigated different wind situations at the Baltic Sea and the Sargasso Sea with the SWAN model and the WWM. At the Baltic Sea we used for the validation of the models data from two WAVERIDER buoys, which are operated by the Institute for Coastal Research, Gestacht (IfK, GKSS) and located nearby the island of Rügen. Assimilated wind fields have been obtained from the Local Model (LM, German Meteorological Service (DWD)), which have a spatial resolution of.5 and a time increment of one hour. For the case of the Sargasso Sea we used the dataset from the ONR Suite which are also forecasting results with a time step of one hour and a spatial resolution of.5. For the Haringvliet case we used stationary wind velocity over the whole domain. The bathymetry for the region around Rügen was obtained from the Baltic Sea Research Institute, Warnemünde (IOW). For the remaining area the foundation for bathymetry was the ETOPO- dataset. The boundary conditions for the Sargasso Sea case were taken from The ONR Test bed for Coastal and Oceanic Wave Models, Ris et al. (3). The bathymetry data for the model in the region of the Sargasso Sea is based on the ETOPO- dataset courtesy of the NGDC (NOAA). The wind data used courtesy of V.J. Cardone from Oceanweather Inc. and wave data used courtesy of Dr. R.E. Jensen from the US Army Engineer Research and Development Center (ERDC), USA. The wave data for the Haringvliet case was taken from the The ONR Test bed for Coastal and Oceanic Wave Models. Baltic Sea The Baltic Sea is a nearly enclosed basin located at the northeastern part of Europe. The area where the measurement buoys are moored is close to the island of Rügen in the western part of the Baltic Sea. Two different storm events have been investigated in this study. Both wind events are resulting from depressions which approached the area of interest from northwestern and north-eastern directions respectively. Both events have similar wind velocities but the second event has a longer duration and the greatest fetch lengths. The wave heights during this event exceed 3 m at the location of buoy and are the biggest measured waves in this study for the Baltic Sea case. The two buoys are moored within a water depth of m and m respectively (see fig. ). The buoy is nearly unaffected by shallow water processes, at the location of the buoy bottom friction becomes important. For the SWAN model it was necessary to utilize a nested grid in order to describe the depth distribution and the coastline curvature properly. The coarse grid for the SWAN simulations ( nodes, 77 active) has a resolution of.5 (approx. 7km) which is the same like the atmospheric model LM (Local Model) of the German Meteorological Service (DWD) from which the atmospheric boundary conditions were obtained. The nested grid ( nodes, 377 active), which covers the region around the island of Rügen where the buoys are located, has a resolution of.. For WWM a grid was used with a resolution from. up to. (59 nodes) in the region of interest. The spectral resolution was set in both models to 3 direction increments and 3 frequency increments with a frequency bandwidth from.5-.hz. Fig. : Bathymetry of the Baltic Sea and computational mesh of the WWM. The boundaries of the coarse mesh and the nested mesh for the SWAN computation are indicated by the magenta frames. The comparison of the integral wave parameters with the buoy measurement showed, that the wave models have the ability to adequately hindcast the significant wave height, though they underestimate the average period (see fig.), especially during event II. One reason for the underestimation of the average period is the influence of the cut-off frequency of the buoy on the integral wave parameters. Here the data from the Waverider buoys has a cut-off frequency of.5 Hz and the wave simulations have usually a cut-off frequency of Hz. The influence of the cut-off frequency on the integral wave parameters is plotted in fig. with dashed lines. Especially the average period is sensitive to the choice of the frequency range in which the integral wave parameters are estimated, whereas the significant wave height is only slightly influenced by the choice of the cut-off frequency. This effect is especially pronounced in growing and young wind-sea spectra, where a huge part of the wave energy is located in the high frequency band. To similar conclusions came already Dykes et al. (). Another reason is the overestimation of high-frequency energy and underestimation of low frequency energy (see fig. and fig.3) with C-I. The implementation of source term combination like suggested by Makin & Stam (C-III) reduced the overestimation of high-frequency energy and also the underestimation in the low frequency region. 3 Copyright by ASME

4 Fig. : Simulation results for the Baltic Sea case at the location of buoy I with the SWAN default source term formulation. The dashed lines demonstrate the effect of the cut-off frequency on the integral wave parameter of the wave model Measurement at the location of buoy WWM with SWAN default source term formulation C- WWM Makin & Stam + Alves & Banner C Measurement at the location of buoy SWAN default source term formulation C- SWAN Makin & Stam + Alves & Banner C Fig. 3: Measured and hindcasted -d wave spectrum at the location of buoy at the -. : PM. Fig. : Differences between measured and hindcasted variance density as a function of time and frequency at the location of buoy during the north-east storm. Sargasso Sea The results at the Baltic Sea with the implemented source term combination have been promising and therefore our ambition was to investigate a case with different climatic conditions in order to prove our results from the Baltic Sea. We chose the Sargasso Sea. In this area swell advection over great distances and wave growth and dissipation plays an important role for the evolution of the wave spectra. Moreover the The ONR Test bed for Coastal and Oceanic Wave Models summarizes several weather situation in this region with buoy measurements with the purpose to verify the already existent wave models on these datasets. The first event that was investigated is the Halloween storm, an extra-tropical depression occurred in autumn 99. The second event was Hurricane Felix, occurred in August 995. In fig. 5 the computational mesh of the WWM is presented. For the SWAN model we have used a mesh which covers the same area with a spatial resolution of.5 (9 Nodes, 799 Active). For the WWM a mesh was used with a minimum mesh size about.5 and a maximum of.35 (59 Nodes). The calculation time step was set to min for both models. The directional resolution was set to 7 increments and the frequency resolution was set to 35. The frequency bandwidth was set from.. Hz. The chosen directional resolution was necessary to resolve the swell wave field adequately and to avoid the Garden-Sprinkler effect. The GSE effect occurs in both models for a directional resolution of. In fig. the measurement of the buoys and the simulation results of both models for C-I and C-III are presented. The wave models underestimate the average period with C-I significantly. Especially when the wave height reaches its peak value the underestimation becomes much stronger than for the Baltic Sea case. Copyright by ASME

5 Fig. 5: Bathymetry of the Sargasso Sea, finite element mesh and locations of the NOAA buoys. Measurements from buoys,, (, ) are only accessible during Halloween Storm ( Hurricane Felix ). The underestimation of the average period is reduced significantly and the results fit much better to the measurements than with C-I. Especially at buoy 9 and where the significant wave height was underestimated with the default formulation there is a significant improvement of the hindcast results with C-III. The influence of the cut-off frequency on the wave results is also illustrated in fig., though this cannot explain the underestimation of the average period for C-I. The statistical analysis of the simulation results are presented in the tables below. The integral wave parameters of the wave models are recalculated on the basis of the cut-off frequency of the NDBC buoys (.35 Hz) and compared with the measurements (see fig.5). The Bias reduces significantly for the significant wave height and the average period while the correlation coefficient remains mainly unaffected. The analysis of the average wave directions shows a reduction of bias and rmserrors with C-III. The newly implemented source term formulation improves the model performance in the Sargasso Sea case significantly. Fig. : Measured and simulated wave height during Halloween storm. The vertical lines separate the measurements from each buoy in the plot. Blue and pink lines show the influence of the cut-off frequency of the buoy on the integral wave results of the simulations. 5 Copyright by ASME

6 HS TM MDIR WWM C- BIAS [m] RMS [m] SCI [-] R² [-].7... BIAS [s] RMS [s] SCI [-] R² [-]..3.. BIAS [ ] MEA [ ] RMS [ ] Table : Statistical analysis for the Halloween storm case of all measurements (HS: Significant wave height, TM: Zero-down crossing period, MDIR: average wave direction, SCI: Scatter Index, MAE: Mean average Error, RMS: Root mean square error, R²: Correlation coefficient) from Roland et al. (). HS TM MDIR WWM C- BIAS [m] RMS [m] SCI [-].... R² [-] BIAS [s] RMS [s] SCI [-] R² [-] BIAS [ ] MAE [ ] RMS [ ] Table : Statistical analysis for the Hurricane Felix case of all measurements Roland et al. (). Haringvliet estuary (The Netherlands) The Haringvliet estuary lies in the south-west of the Netherlands. It is a shallow estuary of the Rhine river. This test case is used in the ONR Testbed for the verification of wave models for wave transformation over a complex bathymetry and wave reformation due to local wind. During the investigated weather situation the waves are entering the model domain mainly from north-western direction with a wave height around 3.5m and an average period around.5s. One part of the wave energy that enters the model domain is dissipated due to depth induced wave breaking over a shallow shoal called Hinderplaat, behind this shoal the sea state is influenced by the local wind sea and the part of the incoming wave energy which is not dissipated due to wave breaking. In the area below the Hinderplaat the water depth is greater and the waves can propagate deep into the estuary. In this problem depth induced wave breaking and near-resonant triad wave-wave interactions dominate the spectral evolution. The boundary condition at the open boundary is defined through the -d directional wave spectra given at Location. For model setup was same for both models. The frequency bandwidth is set from.5. Hz and has a logarithmical resolution of 3 increments. The directional resolution is set to. Measurements of one dimensional wave spectra are available at eight measurement locations at four different times during a flood period with varying wind velocities from 3 (nautical convention). The boundary conditions are summarized in tab.3. Tidal currents have been neglected for this case and the water level is assumed to be constant over the whole domain. Fig. 7: Bathymetry, buoy positions and mesh for the Haringvliet estuary. Day..9 Case Nr. Time [hh:mm] Water Level [m] Wind Velocity [m/s] Wind Direction [ ] : : :.7. 3 : Table 3: Boundary conditions for the Haringvliet estuary simulations. Wind direction in nautical convention. The results show that both wave models produce reasonable results in the shallow water case. Before the Hinderplaat the wave models produces the wave spectrum well. Behind the zone of strong wave dissipation due to depth induces wave breaking the wave models reproduce the spectral shape fairly well. The models tend to underestimate the measured wave energy behind the Hinderplaat (buoy 5) at low water level (case ) significantly. The smallest deviations between Copyright by ASME

7 measured and hindcasted spectra are for case when the water level is maximal. This leads to the conclusions that strong nonlinearities in the spectral evolution are not well represented in the wave models. Especially at higher frequencies the wave models generates second harmonics in the inner estuary which are too strong. The overestimation of the secondary peaks may be due to the LTA, which is only derived for the case of horizontal bottom and unidirectional sea state (see Dingemans, 99). Beside these two weaknesses both models give a good estimation of the sea state for the Haringvliet estuary. Conclusions The newly developed wind wave model WWM was applied in three different environments. The results have been compared with the SWAN model and measurements from wave buoys. Both models gave a reasonable hindcast of the sea state in deep water conditions. However in shallow water applications the models produce the measured wave spectra fairly well. The advance of the WWM is the flexibility of the utilized numerical scheme, which reduces the needed amount of nodes to discretize the model domain. This must be seen especially in the context to the ongoing developments in the nonlinear wave-wave interaction processes in shallow water (Rasmussen (99)) and inhomogeneous media (Stiassnie (). The alternative source term formulation with its parameterization like suggested by Makin & Stam improved the results of the simulations significantly, especially at the places where wind input and white capping dissipation govern the sea state. For the Baltic Sea case both models reproduce the wave spectra well. Nevertheless there is still an underprediction of wave energy in the vicinity of the spectral peak with the new suggested source term formulation. This may result from the DIA for the estimation of the nonlinear energy transfer in deep water because it transfers to much energy to higher and lower frequency and distorts so the wave spectrum (see e.g. van Vledder (). For the Sargasso Sea case the results are encouraging, especially concerning the hindcast of the average period. The impact of the new spectral balance in deep water at the Haringvliet estuary on the results was naturally weak as in this region strong nonlinearities due to shallow water processes govern the sea state. In the wave regeneration zone of the inner estuary the new source term formulations gives better results for certain locations but a general trend can not be seen. The influence of tidal currents and water level variations induced by tides, wind and waves should be accounted for future studies. The coupling of the wave model with a hydrodynamic numerical model, through water levels, currents and radiation stresses in a permanent feed-back can clarify the influence of currents and water level variations on the evolution of the wave spectra. nonlinearities on the evolution of the wave spectra in shallow inhomogeneous environments. This approach was never investigated; however a critical review is given by Dingemans. The contributions by Stiassnie, Rasmussen and Willebrand, which extends the theory of nonlinear resonant and near-resonant wave-wave interactions to inhomogeneous media, are important contributions for the further development of spectral wave models in the near shore zone and should be practically investigated in the framework of a numerical wave model. The wave breaking formulation in the SWAN model is believed to be formulated according to Battjes & Janssen (97). However in the SWAN model the wave breaking parameter for the estimation of the maximum local wave height is treated as a constant whereas in the referenced publication the adapted Miche criterion is used which depends on the shallowness parameter k*h, with h the local water depth (see Dingemans). Dingemans showed in his review that this simplification has considerable impact on the dissipation rates due to depth induced wave breaking. The wave breaking formulation should be revised in both models and revalidated against a considerable amount of measurements. The dissipation process of plunging breakers should be investigated in detail. Plunging breaker wave dissipation cannot be described with the bore type dissipation model. Meza et al., showed in laboratory experiments that plunging breaker generates new waves in the region below the spectral peak. This effect is neglected in the actual model formulation and should be investigated in future applications. According to Groeneweg et al. () the Lumped Triad Approximations is one main reason for the overestimation of the higher harmonics in shallow waters. Our results underline their findings. We found out that the models are very sensitive to the choice of the parameterization of the LTA approach in the shallow water regions and especially in the vicinity of the wave breaking zone. The results presented here are obtained with a value of.5 for the free tuning parameter (α) in the LTA. Dingemans has summarized the weaknesses of the LTA approach, which seem to be considerable. Thus a revision of this source term is one of the important steps towards more reliable wave forecasts in shallow waters. Acknowledgments The authors like to thank the anonymous reviewers for their fruitful comments. The wave action equation (eq. ) is derived with the assumption that waves are traveling through homogeneous media (Komen et al. (99)). This is not the case in tidal environments. Willebrand (973) has extended the linear dispersion relation and the conservation part of (eq. ) to account for the effect of 7 Copyright by ASME

8 Wave Spectra at location "" P Wave Spectra at location "5" P Wave Spectra at location "" P Wave Spectra at location "" P Wave Spectra at location "3" Wave Spectra at location "" P-3 P Wave Spectra at location "7" P Wave Spectra at location "" P- Fig. : Case, Waterlevel =.3m Copyright by ASME

9 Wave Spectra at location "" P Wave Spectra at location "5" P Wave Spectra at location "" P Wave Spectra at location "" P- Wave Spectra at location "3" P Wave Spectra at location "7" P Wave Spectra at location "" P Wave Spectra at location "" P Fig. 9: Case, Waterlevel =.9m 9 Copyright by ASME

10 Wave Spectra at location "" P Wave Spectra at location "5" P Wave Spectra at location "" P Wave Spectra at location "" P Wave Spectra at location "3" P Wave Spectra at location "7" P Wave Spectra at location "" P Wave Spectra at location "" P Fig. : Case 3, Waterlevel =.7m Copyright by ASME

11 Wave Spectra at location "" P Wave Spectra at location "5" P Wave Spectra at location "" P Wave Spectra at location "" P Wave Spectra at location "3" P Wave Spectra at location "7" P Wave Spectra at location "" P Wave Spectra at location "" P Fig. : Case, Water level =.m Copyright by ASME

12 References: Alves, G.M.J.H., and Banner, M.L., 3. Impact of a saturation-dependent dissipation source function on operational hindcasts of wind-waves in the Australian region. Atmosh. And Ocean system. Vol. No. pp Ardhuin, F., and T. H. C. Herbers, : Bragg scattering of random surface gravity waves by irregular sea bed topography. J. Fluid Mech., 5, 33.,, and W. C. O Reilly, : A hybrid Eulerian Lagrangian model for spectral wave evolution with application to bottom friction on the continental shelf. J. Phys. Oceanogr., 3, 9 5., T. G. Drake, and T. H. C. Herbers, : Observations of wave generated vortex ripples on the North Carolina continental shelf. J. Geophys. Res., 7, 33, doi:.9/jc9., W. C. O Reilly, T. H. C. Herbers, and P. F. Jessen, 3: Swell transformation across the continental shelf. Part I: Attenuation and directional broadening. J. Phys. Oceanogr., 33, , W. C. O Reilly, T. H. C. Herbers, and P. F. Jessen, 3: Swell transformation across the continental shelf. 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