WAVEWATCH-III is a community wave modelling framework that includes the latest scientific advances in the field of wind-wave modelling and dynamics. The WAVEWATCH-III third-generation wave model was developed at the US National Centers for Environmental Prediction (NOAA/NCEP). In this study the version 6.07 is employed .

WAVEWATCH-III computes surface gravity wave propagation solving the discrete phase spectral action density balance equation for directional wavenumber spectra and accounts for the main physical processes influencing the propagation of waves as sinks or sources of wave energy: 1∂∂tN(k)+∂∂x⋅x˙N(k)+∂∂θθ˙N(k)+∂∂kk˙N(k)=Sσ,2x˙=(cg+u),3θ˙=-1k∂σ∂h∂h∂m-k⋅∂u∂m,4k˙=-∂σ∂h∂h∂s-k⋅∂u∂s, where N(k)=F(k)/σ(k) is the wave action as a function of k, which is the wave number vector, and F(k) is the directional wavenumber spectrum; ω(k) is the frequency according to the dispersion relationship; h is the water depth; cg=∂σ/∂k is the group velocity; u is the surface current vector; s is a coordinate in the θ direction; and m is a coordinate perpendicular to s . The second term on the left side of Eq. () is the advection, the third term is the refraction, and the fourth term is direct forcing by topography and current variations. The implicit assumption of this equation is that properties of medium (water depth and current) and the wave field itself vary on temporal and spatial scales that are much larger than the variation scales of a single wave. The third-order accurate numerical scheme is presently used to describe wave propagation in combination with the total variance diminishing limiter .

The source terms on the right side of Eq. () are integrated in time using a dynamically adjusted time-stepping algorithm, which concentrates computational efforts in conditions with rapid spectral changes. In deep waters, the dominant source terms are wave growth due to wind action Sin, nonlinear wave–wave interactions Snl, and whitecapping Sds. Presently, nonlinear wave–wave interactions are modelled using the discrete interaction approximation , while input and dissipation source terms are based on . Proceeding into shallow waters, additional source terms must be included as bottom friction Sbot and depth-induced breaking Sdb. In this study, the parameterization of bottom friction for sandy bottoms is employed as later calibrated by in field measurements . Depth-induced breaking is modelled following the formulation of .

CROCO (Coastal and Regional Ocean COmmunity model, 10.5281/zenodo.7415343, ) is a new oceanic modelling system built upon the ROMS-AGRIF model . It solves finite-difference approximations of the Reynolds-averaged Navier–Stokes (RANS) equations on a horizontal free-surface Arakawa C grid and vertical stretched terrain-following coordinates with a split-explicit time-stepping algorithm. CROCO has a flexible structure that allows choosing several numerical schemes and parameterizations. Here we use the model with the hydrostatic and Boussinesq approximations, the WENO5 scheme for horizontal and vertical advection, the generic length scale scheme for vertical mixing based on the k–ϵ turbulent closure , and the parameterization of sub-grid-scale processes in the bottom boundary layer considering the combined wave–current drag as in . Momentum, scalar advection, and diffusive processes are represented using transport equations.

For the phase-averaged wave–current interactions, the following equations are solved in Eulerian framework and Cartesian coordinates using hydrostatic, Boussinesq, and incompressible assumptions: 5∇⋅vL=0,∂u∂t+∇⋅(vLu)-fvL=-∂ϕc∂x+us∂u∂x+vs∂v∂x+Fu+Du+Fuw,∂v∂t+∇⋅(vLv)+fuL=-∂ϕc∂y+us∂u∂y+vs∂v∂y+Fv+Dv+Fvw,∂ϕc∂z+ρgρ0=vs⋅∂v∂z, where vL=(uL,vL,wL) is the phase-averaged Lagrangian velocity, v=(u,v,w) is the phase-averaged Eulerian velocity, and vs=(us,vs,ws) is the 3D Stokes velocity. The phase-averaged Lagrangian velocity is calculated such that vL=v+vs. Du and Dv are diffusive terms including wave-enhanced drag and mixing. Fu and Fv are forcing terms (in the present study we only consider winds at the free surface), while Fuw and Fvw are wave-induced forcing terms. ρ0 is the reference density, g is the gravity acceleration, and f is the Coriolis parameter. ϕc=ϕ+ϕ^ is related to the fluid pressure, where ϕ is the dynamic pressure calculated such that ϕ=Pρ0 (with P the total pressure) and ϕ^ is the Bernoulli head due to waves.

In the v1.1 of CROCO, the wave-induced terms in the ocean dynamic equations (Stokes drift, Bernoulli head, bottom wave orbital velocity, wave-to-ocean energy flux) are computed from their monochromatic approximations, which are introduced in the next subsections. In the present study, we have implemented the exchanges of these fields computed from the spectral wave model. These implementations are now included in CROCO v1.2. The formulation of the wave-induced terms in these two CROCO versions is detailed below.

In this study, CROCO and WAVEWATCH-III are coupled as presented in Fig. . Instantaneous hydrodynamic fields are exchanged between both models every coupling time step (Δtc=60 s) thanks to the OASIS coupler , according to a similar procedure to that in , , and . The choice of exchanged variables, coupling frequency, and grid interpolation options is managed through a text file read by OASIS (named namcouple). CROCO provides the sea surface height and surface flow velocity to WAVEWATCH-III, which are used in the wave model to compute depth- and current-induced wave refraction. In CROCO v1.1, WAVEWATCH-III provides only three mean wave parameters based on the integration of the wave spectrum (Fig.  left panel, blue labels): the significant wave height (Hs), the mean wave period (Tm01), and the mean wave direction (θw), which are used by CROCO to compute wave-induced terms in the hydrodynamic equations, including horizontal and vertical vortex force, wave-induced pressure, wave-induced tracer diffusivity, non-conservative wave dissipation, non-conservative wave accelerations for currents, and wave-enhanced vertical mixing (Eq. ). In the new v1.2, we have implemented the additional exchanges of the mean wavelength (LM), near-bottom wave orbital velocity (UBR), magnitude and direction of the surface Stokes drift (USS), Bernoulli head pressure (BHD), and wave-to-ocean energy flux (FOC) from WAVEWATCH-III to CROCO (Fig.  right panel, red labels). These wave-induced terms are already computed from the full spectrum in the wave model. They are here exchanged through the coupler and used in CROCO wave-averaged equations instead of their monochromatic approximations, allowing a wider range of applications. The additional exchanges are coded in the cpl_prism_get.F routine of CROCO, which manages all the possibly received variables from OASIS-MCT. The use of these spectrally computed terms in the wave-averaged equations of CROCO instead of their monochromatic approximation is activated with a CPP preprocessing key (as other options in CROCO) named OW_COUPLING_FULL that can be defined in the cppdefs.h file and which impacts the choice of terms used in the mrl_wci.F (wave-averaged equations) and bbl.F (bottom boundary layer) routines. These modifications have been available in CROCO since the v1.2 release (10.5281/zenodo.7415343, ). Both the monochromatic and full coupling procedures remain accessible for comparison in different configurations, and users can easily switch between them by activating or deactivating the OW_COUPLING_FULL preprocessing key.

Each numerical experiment performed in this study considers a rectangular computational domain delimited by the points 50.316° N, 1.156° E, and 50.083° N, 1.781° E, respectively, at the NW and SE corners (Fig. ). The same computational grid is adopted for both the wave and the circulation models, but the use of different grids is possible. The horizontal mesh has a spatial resolution of 100m, and the bathymetry from HOMONIM is interpolated over this grid. A total of 20 sigma layers are used for the vertical discretization. They are uniformly distributed between the seabed and the free surface, resulting in a maximum layer thickness offshore of about 2 m and a minimum layer thickness onshore of about 5 cm.

The wave model uses 32 frequencies (0.04–0.7 s-1) and 24 directions, leading to a directional resolution of 15°. Bi-dimensional (frequency and direction) full wave spectra from the HOMERE database are used to interpolate wave forcing at the deep-water open boundaries. It is worth noting that the use of full spectra at the boundaries is the recommended practice in modelling spectral waves, particularly when investigating wave–current interactions, as suggested by and .

Along these open boundaries, tidal forcing is interpolated from the high-resolution (250 m) PREVIMER atlas MANE focused on the east part of the English Channel, which includes 37 harmonic constituents . A cold start is imposed as an initial condition, and no stratification due to temperature and salinity gradients is considered. To assess the independence of the hydrodynamic results used to validate the model from the relative distance between the forcing boundaries and the measurement sites, we conducted an additional numerical simulation with an extended computational grid, effectively placing the measurement location further from the open boundaries. The results of this supplementary simulation align closely with the outputs of the simulation which solely employed tidal forcing. This congruence has been assessed in terms of circulation patterns, time series of near-bottom currents, and vertical profiles.

Wind forcing is provided by NCEP throughout hourly wind speeds at 10 m above mean sea level from the CFSR reanalysis at 0.312° horizontal resolution . This resolution is much coarser than the numerical model resolution, and this may have an impact on the simulation results. It is important to acknowledge this limitation and consider it when interpreting the results. Wind stresses at the free surface are computed according to the formulation of , which estimates the wind drag for sea surface wind stress as a function of wind speed while accounting for a constant Charnock coefficient. This approach parameterizes the impact of the sea state on the drag through the wind speed only and does not consider an eventual variation of the drag coefficient for various sea states for a given wind speed. Since the effect of waves on the wind drag have been proven to have substantial effect on free-surface currents in similar macro-tidal settings , we performed a sensitivity experiment in which the surface wind stress was computed by WAVEWATCH-III (two terms) and input into CROCO, therefore accounting for the impact of varying sea states on the wind drag coefficient. This sensitivity analysis showed that while wave effects may alter wind stresses at the free surface, their impact on vertical velocity profiles is negligible under the investigated metocean conditions. This finding reinforces the robustness of the present modelling approach in forcing conditions similar to those presently investigated. Furthermore, no water and heat fluxes from the atmosphere are considered.

Hydrodynamic motions were not computed for depths smaller than 1m while wave-induced forcing terms were activated for depths greater than 2m. This is necessary due to the low resolution of the employed bathymetry, whose interpolation resulted in steep seabed gradients within the bay, leading to numerical instabilities in shallow waters. The observed mass median diameter of sediment particles equal to 200µm is used to compute the effective Nikuradse roughness length employed in the parameterizations of bottom friction (CROCO using , and WAVEWATCH-III using , ST4 package).

To understand the impact on the macro-tidal hydrodynamics of BoS of each metocean forcing (tide, wind, and waves) and each additional spectral field exchanged compared to its monochromatic approximation, a total of nine final numerical experiments have been performed. These simulations are detailed in Table . A simulation forced by only tidal levels and currents (CRX) was initially performed to be used as baseline case, providing a pure tidal vertical current profile. To assess the impact on the vertical profile of wind forcing alone, a second simulation with tidal and wind forcing (WND) was performed but without accounting for waves. Waves are then considered in all subsequent simulations. The first simulation uses the CROCO v1.1 configuration with computation of wave-induced terms from their monochromatic approximation. The five following simulations each include a different additional spectral wave field exchange instead of its monochromatic approximation (mean wavelength LM, Bernoulli head pressure BHD, wave-to-ocean energy flux FOC, surface Stokes drift USS, and near-bottom wave orbital velocity UBR). The last simulation, named full coupling, uses all the spectral-wave-induced terms included in CROCO v1.2.

Sea states were simulated for a time period ranging from 1 to 6 August 2008 (Fig. ). During this period, an energetic wind event (fresh to strong breeze based on Beaufort's wind scale with moderate to large waves) occurred on 2 August with a wind velocity magnitude at 10 m above the sea surface reaching 10.5 m s-1 between 17:00 and 19:00 GMT+2 The root-mean-square significant wave height (Hrms) computed by WAVEWATCH-III turns out to be smaller than the observed one (top panel of Fig. ), with a maximum value around 1.05m instead of the observed 1.40m, leading to an underestimation of about 25 % by the model. By contrast, simulated wave heights, periods, and directions are close to those predicted by the HOMERE hindcast (Fig. ). For the entire time series, the correlation coefficient r2 is 0.86, while RMSE is 0.23 m. This partly validates our numerical configuration. Since the HOMERE hindcast is alike in wind forcing, physical parameterizations, and computational grid size, it is consistent to obtain similar results. Moreover, this hindcast was used to force the spectral wave model at these boundaries. Slight differences between the two model predictions are thus supposedly due to the higher spatial resolution employed in the present study (100 m), which results in a better description of wave shoaling and refraction.

Waves are deviated from their direction of propagation by the surface currents e.g.. In the study area, the flood flow is oriented towards the east-northeast while the ebb flow goes to the west-southwest (Fig. ). Due to the tide asymmetry often observed in the English Channel, the ebb current is less intense than the flood current with a mean value of 0.45 and 0.6 m s-1, respectively, during spring tide . Although the current magnitude is relatively weak, a change in the wave direction is observed with a slight modulation following the tidal phase, as shown in Fig.  (bottom panel) at the ADCP location, and this matches very well (r2=0.99) with the predictions of the HOMERE hindcast.

The circulation within and off the bay is mainly controlled by semi-diurnal tidal currents which are oriented to the west-southwest during ebbs (Fig. , top panel) and east-northeast during floods (Fig. , bottom panel). The model well reproduces the observed tidal asymmetry predicting modelled peak surface velocity magnitudes larger than 2 ms-1 during flood and smaller than 1.5 ms-1 during ebb in the simulated period.

Figure  shows the comparisons of our results with the ADCP data in terms of sea surface height. The hydrodynamic model well captures the macro-tidal range off the BoS (about 8 m for the simulated time period), and excellent scores are obtained (r2=0.99 and RMSE = 0.29) for all runs. Similar results are obtained with the standalone CROCO model forced only with tides (CRX), the simulation including wind forcing (WND), and the simulation accounting for waves and their interactions with levels and currents (2WC), showing that winds and waves do not significantly affect water levels at the ADCP mooring station, which are mainly controlled by tides.

Figure  shows the comparisons of our model results with the AQDP measurements in terms of current magnitude and direction at 1 m above the bottom (m a.b.). The modelled current magnitude is correctly predicted in line with the values recorded by the AQDP during the different phases of the tidal cycle. Particularly well captured is the tidal current reversal. This is confirmed by near-bottom current directions that are shown to be fairly replicated. A small phase shift (around 10 min) between measured and modelled currents is present. This can be seen by comparing the time at which the velocity peak occurs during tidal floods. Flood peaks are reached slightly later in the model than observed in the measurements, as if the model would predict longer flood phases than observed. This phase shift could be due to bottom friction effects associated with the presence of widespread bedforms of very different sizes e.g., ranging from small-scale ripples to large-scale sand waves, whose impact on hydrodynamics cannot be properly modelled using the horizontal resolution employed in the present study. Indeed, current measurements have been collected in a field of sand dunes with an average crest distance of about 425 m, which cannot be properly represented with the 100 m grid resolution. Winds and waves are shown to not significantly affect near-bottom currents which are clearly dominated by tides. Similar results have indeed been obtained with the standalone CROCO model forced only with tides (CRX), considering also the wind forcing (WND), and accounting for wave–current-level interactions (2WC).

Figure  shows the comparisons of model predictions with the ADCP measurements in terms of current magnitude and direction at 1 m below the free surface (1 m b.f.s.). The modelled current magnitude is correctly predicted during most of the phases of the tidal cycle, particularly at tidal current reversal and especially during ebb phases. Also, current directions are fairly replicated. The phase shift between measured and modelled currents is present also at the free surface when comparing modelled currents with the ADCP measurements. It is worth noting that peak flood velocities delay is not coincident with passing storms, thus suggesting that winds and waves are not responsible for the phase shift and adding up to the likelihood that this is due to the presence of sub-grid-scale bedforms. Results obtained by the standalone CROCO model forced only with tides (CRX) are modified by wind stresses (WND), resulting in an acceleration and a deceleration of surface currents, respectively, at tidal floods and ebbs due to winds blowing towards the east-northeast. This wind-driven modulation is pronounced during storms. Wave effects on currents tend to smooth this modulation (2WC) by means of wave–current interaction mechanisms, as explained in the following section.

Tests of model results (2WC) against in situ measurements through the water column in terms of eastward (Fig. ) and northward (Fig. ) current velocity components at the ADCP location further validate our modelling. Overall, the fair matching between measured and modelled current velocity components during the different phases of the tidal cycle and within the entire water column proves the reliability of the hydrodynamic model. However, an overestimation (underestimation) of the northward tidal current velocity computed by the model is observed during ebb (flood) peaks, particularly around the low tide slack water event, from the surface to the bottom.

To assess the importance of exchanging spectral wave fields instead of computing them from monochromatic approximations, we compared measured vertical profiles of currents extracted at noteworthy time instants within various simulations listed in Table . Figure  shows two vertical current profiles associated with storm waves and calm sea states. Measured (squares) and modelled (continuous lines) time series of current velocities (red) and root-mean-square wave heights (blue) are compared in the top panel (a). The middle panel (b) analogously compares current and wave directions. Vertical profiles were extracted at two time instants characterized by the occurrence of waves coming from west with Hrms, respectively, larger and smaller than 1 m. In the case of storm waves (bottom-left panel c), wind-driven stresses are shown to modify the logarithmic tidal velocity profile accelerating the current towards the free surface and decelerating it towards the bottom, as reported in former studies e.g.. However, this velocity increase seems too strong in view of the data with an overestimate of about 5 cm s-1 at the surface, which represents about 18.5 % of the measured surface velocity. Wave–current coupling considering terms computed with their monochromatic approximations (CROCO v1.1, 2WC) produced a realistic surface flow which now fits the measurements very well. The smoothing of the wind-induced profile is due to waves because of the wave–current interaction mechanism described in , which is activated by waves moving in a direction similar to that of the current and thus opposite to the wind stress (Fig. c). The adding of the full wave spectral terms (CROCO v1.2) results in a light smoothing of the vertical profile throughout the entire water column, but the overall impact on model predictions is minor (Fig. , 2WF). In the case of calm sea states (bottom-right panel d), wind and wave effects are negligible and do not significantly modify the logarithmic tidal velocity profile.

Figure  shows two vertical current profiles associated with ∼10 ms-1 winds. Measured (squares) and modelled (continuous lines) time series of current velocities (red) and wind speeds (blue) are compared in the top panel (a). The middle panel (b) analogously compares current and wind directions. Note that modelled wind speeds and directions from the CFSR global reanalysis data set are tested against the records of a Met Office mooring station in the English Channel just offshore the study area (50.4° N, 0.0° E) from the Copernicus Marine In-Situ Near Real Time Observations of the Atlantic Iberian Biscay Irish Ocean . Vertical profiles were extracted at two time instants characterized by the occurrence of winds coming from west, contrasting ebb flows, and favouring flood flows. In the case of winds blowing in a direction which is opposite to that of tidal currents (bottom-left panel c), wind-driven stresses are shown to modify the logarithmic tidal velocity profile (black curve) by slowing the flow over a depth of 10 m due to wind resistance of the flow, which results in a reduction of the surface velocity magnitude of about 8 cm s-1 (or 25 % of the pure tidal velocity). CROCO v1.2 coupling results in a current profile in between the logarithmic tidal profile and that of CROCO v1.1. It is closer to that of the simulation only influenced by wind stresses with a lower decrease in the close-to-surface velocity and a slight increase in the close-to-bottom velocity. The profile modelled with CROCO v1.2, while still showing significant biases with observations, is, however, the best fitting with measurements throughout the entire water column compared to other simulations (Fig. , 2WF). Waves accelerate the wind-induced current of about 3 cm s-1 at the surface due to an angle between wave and current directions of propagation slightly larger than 90°. Indeed, the wave energy spectrum at this time shows a sea state with two peak frequencies (around 0.15 and 0.25 Hz), and thus the use of spectral forcing terms improves the accuracy of the results (Fig. ; RMSE ∼1.8 cm s-1, r2=0.99). Differences between CROCO v1.1 and v1.2 are mainly related to the near-bed wave orbital velocity, which is increased by a factor of about 1.5 at this time (Fig. ), leading to a reduction in the bottom stress (Eq. 9) due to the weak value of the near-bottom flow velocity (about 20 cm s-1). This reduces the flow intensity across the entire water column, as described by . In the case of winds blowing in the same direction of tidal currents (bottom-right panel d), winds and waves only slightly affect the tidal logarithmic profile, leading to less significant differences. Indeed, the wind forcing magnitude at 14:30 and at 23:00 is similar, but its impact on current is minor at 23:00 as the instantaneous surface velocity magnitude is more than twice as high as at 14:30 (60 cm s-1 versus 25 cm s-1). As in the previous case (Fig. ), wind stresses accelerate the tidal profile at the free surface since wind is following the current, while waves tend to smooth it.

The velocity magnitude on the water column is reduced by the use of wave spectral forcing terms because of the increase in the near-bed orbital velocity (about 2 cm s-1) that causes an enhancement of the bottom stress. It is also worth noting that, in both situations, maximum current magnitude in the middle of the water column is not captured by the model. Better fits are obtained near the surface than in the middle of the water column. At 14:30, it appears that vertical profiles are smoother than observations. This is likely due to an overestimation of the vertical viscosity coefficient employed in the RANS modelling. At 23:00, many breakings occur due to opposite wave and current directions of propagation and also due to a wind velocity causing whitecaps. The wave-induced turbulence is transmitted thanks to a surface term (Eqs. –). This could be the origin of discrepancies since the mixing is not propagated in the water column but just located at the surface. Otherwise, at 14:30 and at 23:00, breakings make the measurements difficult due to the flow aeration. Thus, a part of the differences can also be caused by the measurement techniques and site conditions. Overall, stronger effects due to the newly exchanged spectral terms are predicted during storm conditions when incoming sea-state spectra are likely broad or multi-modal.

This section is devoted to the assessment of the impact of spectral versus monochromatic computation of each newly exchanged wave-induced term on the hydrodynamics. Six simulations were performed adding only the exchange of the spectral fields one by one through the coupling interface and then comparing the results in terms of vertical current profiles. Figure  shows these comparisons at time instants when the tidal current is affected by wind forcing (also Fig. ). The profiles predicted by the simulations including only the effect of the spectral mean wavelength (2WC+LM) or the effect of the spectral Bernoulli head (2WC+BHD) match the results of CROCO v1.1 (2WC), indicating that considering these two spectral quantities rather than their monochromatic counterparts does not significantly affect the local hydrodynamics in our configuration. Simulations including only the spectral wave energy dissipation due to breaking (2WC+FOC), the Stokes drift (2WC+USS), or the near-bottom orbital velocity (2WC+UBR) present interesting deviations from that of CROCO v1.1 (2WC). The main effect is caused by the use of spectral near-bed velocity which induces a decrease in the bottom shear stress, leading to an increase in the velocity magnitude in between 1 and 1.5 cm s-1 across the entire water column. The addition of the spectral Stokes drift at the surface and its distribution over the depth according to Eq. () slightly changes the vertical shape of the current due to the vortex force, which redistributes the momentum. The velocity magnitude is also altered by the Stokes drift contribution to the vertical advection. Differently, the use of the wave-to-ocean energy flux from the spectral wave model affects the first 5 m below the sea level. This flux represents the wave-breaking contribution to the circulation, which is, in such intermediate depths, associated with whitecapping. It is important to note that in the CROCOv1.1 implementation the wave-to-ocean energy flux is only computed for depth-induced wave breaking, which does not apply here. Consequently, the difference observed here between 2WC and 2WC+FOC shows the added value of including the representation of whitecapping dissipation. With the breaking acceleration force being used for computing and the surface boundary condition for mixing, it is shown to reduce the flow velocity near the surface.

These newly exchanged spectral terms correct the solution predicted by coupling using their monochromatic counterparts when the sea-state spectrum is far from having most of the energy concentrated in one frequency. Figure  shows the spectra associated with vertical profiles of Figs.  and . As expected, the two-peak spectrum (Fig. a, c) corresponds to the vertical profile when wind forcing opposes the tidal current (left panel of Fig. ), which is strongly affected by the spectral versus monochromatic computation of wave-induced terms. Conversely, the one-peak spectrum (Fig. b, d) corresponds to the vertical profile when the contributions of the spectral terms are minor (right panel of Fig. ), as expected.

Differences between the spectral and monochromatic Stokes drifts computed at the ADCP location are shown in Fig.  for the entire simulation time. It can be seen that the free-surface (<0.1 ms-1) and depth-averaged (< 0.02 ms-1) values of the Stokes drift are mainly governed by the occurrence of storms, with higher and longer waves associated with larger Stokes velocities (∼0.09 ms-1). The surface Stokes velocity is 2 to 6 times stronger than its barotropic counterpart (2WC and 2WF cases), showing a modulation in time of the vertical shape of the Stokes velocity with the largest changes for strong winds and high waves. The magnitude of the Stokes drift computed by the spectral wave model (2WC, red and magenta lines in Fig. ) turns out to be smaller than its monochromatic counterpart (2WF, blue and cyan lines in Fig. ) on 2 August 2008 at 14:30, while spectral and monochromatic values are higher than monochromatic ones on 2 August 2008 at 23:00. Despite the fact that spectral and monochromatic values remain close (only about 1 cm s-1 differences), the impact of the spectral Stokes velocity at the surface on the overall current profile is not negligible, especially at 14:30 as shown in Fig. 14.

Differently from the Stokes drift, the near-bottom wave orbital velocity is strongly modulated by the time evolution of the macro-tidal sea-level range (Fig. ). The near-bed wave orbital velocity is increased during low tide, while a reduction in its intensity is observed during high tide (2WC and 2WF cases). This is primarily due to the fact that in shallower waters the action of waves close to the seabed is enhanced, especially in macro-tidal settings where water depths are of the same order of tidal ranges as at the ADCP location. The modulation according to the tidal phase is more intense for the spectral velocity with an amplification up to 30 %.

Differences between the Stokes drift predicted by the spectral model when coupled within the CROCO v1.2 framework (2WF) and its monochromatic approximation computed when coupled within the CROCO v1.1 framework (2WC) are shown for two time instants in the top and bottom panels of Fig. . The spectral Stokes drift is accelerated for depth smaller than 5 m, which corresponds to the nearshore and estuarine areas. This acceleration is mainly driven by the water depth and bottom morphology rather than by wind effects because similar patterns are observed at 14:30 and at 23:30. However, the 100 m spatial resolution of the employed computational grid smooths the bathymetry, and thus some key processes such as the depth-induced refraction of waves (e.g. ) are likely misrepresented. This low resolution is not appropriate for studying the nearshore dynamics. Off the BoS, a higher spectral-surface Stokes velocity is found with respect to its monochromatic value at the time when the bi-frequency spectrum is observed (14:30). By contrast, at 23:00 the spectral velocity is smaller than the monochromatic one, but the difference is negligible, in line with the observed mono-frequency spectrum. Generally these differences are rather homogeneous across the entire computational domain.

The near-bed wave orbital velocity computed by WAVEWATCH-III coupled to CROCO v1.2 (UBRs from run 2WF) and its monochromatic approximation computed by CROCO v1.1 (UBRm from run 2WC) are compared for two time instants (Fig. ). These results indicate that an increase in spectral near-bottom wave orbital velocity with respect to the monochromatic values is present offshore and mainly depends on the nature of offshore incoming wave spectra and wind forcing. Conversely, and differently from the Stokes drift, bathymetric effects in shallower waters lead to a decrease in spectral wave orbital velocity that reduces to the value predicted by the Airy theory or attains even a smaller value. Also in the case of the wave orbital velocity, the larger differences between spectral and monochromatic values are observed in shallower waters within the bay under both forcing conditions. This confirms that these differences are not only associated with the nature of the forcing spectra, but also with the wave propagation in shallow waters. As mentioned above, the limitations arising from the low horizontal resolution of our computational grid and the omission of wave effects on currents in depths below 2 m hinder our ability to accurately represent wave processes in shallow waters. Consequently, crucial processes influencing near-bottom orbital velocity in the surf zone may not be faithfully captured and need additional dedicated measurements and an increased resolution in this area for a comprehensive investigation.