1 Introduction

Wind-wave-current coupling is of growing interest since it has long been recognized that these interactions control the momentum and energy exchange between the atmosphere and the ocean, and need to be better understood and resolved. The currents are driven by surface wind stresses that in turn are a function of the sea state. On the other hand, the sea state depends on the wind stress and the currents. These are complicated feedback mechanisms which can be modelled by coupling hydrodynamic and wave models which to date have been developed separately.

Model coupling can be achieved at various levels of complexity. A complete review of wind-wave-current interactions processes can be found in Jonsson (1990) and more recently in Cavaleri et al. (2012). The present work focuses on wave and current modifications due to interactions of waves with surface currents, and the wind speed correction due to a stability parameter that depends on air-sea temperature differences (Tolman 2002), and to the surface drag coefficient for currents which takes into account the wave effects.

Wind generated waves are affected when they interact with currents since wave characteristics such as wavelength, amplitude, frequency, and direction are modified due to the Doppler shift. For the wave frequency, the shift can be written as follows:

$$ \omega =\sigma +\mathbf{K}\bullet {\mathbf{U}}_{\mathrm{c}} $$
(1)

where ω is the absolute wave frequency, K is the wavenumber vector, U c is the surface current velocity vector, and σ is the wave intrinsic frequency (Phillips 1977). The degree to which the Doppler shift modifies the surface waves depends on the current speed relative to the wave propagation speed, which means that slow propagating waves are most affected by currents. High winds and strong currents can lead to wave merging, thus, creating single exceptionally large waves (rogue waves), waves travelling in the area of strong opposite currents may become short and steep (being potentially hazardous for navigation), and wave blocking or breaking may occur in the case of strong opposite currents against waves.

The difference between sea surface temperature (SST) and air temperature affects the stability of the lower atmosphere and thus the wind velocity structure. Tolman (2002) formulated a stability correction by replacing the wind speed with an effective wind speed so that the wave growth reproduces Kahma and Calkoen (1992) stable and unstable wave growth curves.

The atmospheric wind stress is the main driving force for the hydrodynamic currents that are produced by internal turbulent stresses. The wind stress depends on the wind speed and sea roughness that arises from surface waves; thus, ocean current energy input depends on surface waves. The surface wind stress, τ, is parameterized following the bulk formula:

$$ \tau ={\rho}_a{C}_D\left|{\mathbf{U}}_{10}-{\mathbf{U}}_{\mathrm{c}}\right|\left({\mathbf{U}}_{10}-{\mathbf{U}}_{\mathrm{c}}\right) $$
(2)

where ρ a is the air density, C D is the surface drag coefficient, U 10 is the atmospheric 10 m wind speed, and U c is the surface current. Uncertainties in wind stress calculations arise from wind model resolution and depend on wind speed and direction and from the specification of the drag coefficient, which quantifies how much the surface winds are slowed down because of surface roughness due to waves. Several authors have proposed different parameterizations for the drag coefficient usually based on local observations and the function of wind speed and air-sea temperature difference (Wu 1980, 1982; Hellerman and Rosenstein 1983; Large 2006). However, for more realistic computations, a wave model should provide the estimate of the drag as a function of the wave spectrum.

In this first step approach to couple wind waves and currents, all three abovementioned processes, consisting in feeding the wave model with sea surface temperature and surface currents computed by the hydrodynamic model and returning a neutral wind drag coefficient to the latter, were selected in order to develop the model coupling between a hydrodynamic model and a wave model forced by winds from an atmospheric analysis and forecasting model. The paper concentrates on the process of momentum exchange between wind waves and currents in the Mediterranean Sea. For a full coupling between waves and currents, the hydrodynamic equations should make use of Stokes Drift velocities, sea-state-dependent momentum flux, a parameterization of wave-induced vertical mixing, and include coastal radiation stresses or vortex force term in the hydrodynamic model equations (Mellor 2003, 2008, 2011; McWilliams et al. 2004; Rascle 2007; Ardhuin et al. 2008; Bennis et al. 2011; Uchiyama et al. 2010; Kumar et al. 2012; Michaud et al. 2012; Breivik et al. 2014, 2015, Alari et al. 2016, Staneva et al. 2017); however, this is out of the scope of the development of the present work. By excluding these processes, surface wave effects are not accounted in the ocean model primitive equations; moreover, breaking waves effect in enhancing turbulence is not performed nor wave absorbed (or released) stress is directly accounted.

The coupled modelling system is composed of the ocean circulation model NEMO (Madec et al. 1998; Madec 2008), the third-generation wave model WaveWatchIII (WW3, Tolman 2009), and the ECMWF analysis wind forcing. NEMO and WW3 are implemented in the Mediterranean Sea with 1/16° horizontal resolution (Fig. 1). The ECMWF forcing is considered at 1/4° resolution for 10 m wind velocities and 2 m air temperatures. The models are two-way coupled by exchanging the following fields at hourly intervals: the sea surface currents and air-sea temperature differences from NEMO to WW3 for the wave refraction and the wind speed stability parameter calculation, respectively. The neutral drag coefficient, computed by WW3, is then passed to NEMO which determines the turbulent part of the surface wind stress following Large and Yeager (2004) and Large (2006), as explained in Appendix 3. Such coupling is fairly new for semi-enclosed seas such as the Mediterranean Sea, and impacts need to be quantified in order to improve current and wave forecasting.

Fig. 1
figure 1

Mediterranean model domain, bathymetry [m] and river positions (red circles)

Full size image

The effect of wave-dependent drag coefficient was analyzed in Mastenbroek et al. (1993) showing improvements by modelling surge heights in the North Sea when the wave-dependent drag is taken into account instead of a standard Smith and Banke (1975) stress relation. Other works in the Baltic Sea (Alari et al. 2016) and the North Sea (Staneva et al. 2017) show a coupling approach previously implemented by Breivik et al. (2014, 2015) consisting in coupling the wave model to the circulation model modified in order to include the Stokes-Coriolis effect and both the sea-state-dependent momentum and energy fluxes. These works show a pronounced effect due to wave coupling on the vertical temperature distribution and on meso-scale events as well as an improved skill in the predicted sea level and currents during storm events. Moreover, in the Mediterranean basin, Lionello et al. (2003) investigated the importance to couple the wave and ocean models with an atmospheric model analyzing air-sea interface fields on a short time scale range for regional meteorological prediction.

The performance of the wave model in the present work is evaluated by comparing numerical results with buoy measurements and altimeter data for satellite significant wave height, mean, and peak wave periods. The performance of the hydrodynamic model is assessed by comparing sea surface currents with coastal buoy observations, model SST with satellite estimates, and vertical temperature and salinity model profiles with ARGO float observations.

The paper is organized as follows. Section 2 presents the model system and the coupling strategy. Section 3 illustrates the numerical experiment design and setup. Section 4 shows the numerical model results and comparison with observations, and in Sect. 5, the conclusions are given.

2 The model system

The model system presented in this work is a two-way coupled hydrodynamic circulation model with a third-generation spectral wind wave model as described in the following sections.

2.1 The atmospheric forcing fields

ECMWF 6-hourly operational analysis fields were used to force both wave and hydrodynamic models for the 5-year experimental time period 2009–2013. The ECMWF system that produced the atmospheric forcing fields is the IFS (Integrated Forecast System, http://www.ecmwf.int/en/forecasts/documentation-and-support/changes-ecmwf-model) at 1/4° horizontal resolution and 91 vertical levels.

The meteorological fields of interest were as follows: meridional and zonal components of the velocity field at a 10-m height, air temperature at 2 m, dew point temperature at 2 m, mean sea level atmospheric pressure, and cloud cover. All fields were linearly interpolated to the hydrodynamic and wave model time steps of 600 s.

2.2 The wave model: WaveWatchIII (WW3)

The wave model used in the simulations is the third-generation spectral WaveWatchIII model version 3.14 (Tolman 2009), hereafter denoted as WW3. The model solves the wave action balance equation written for a Cartesian grid as follows:

$$ \frac{\partial }{\partial t}N+{\nabla}_x\bullet \dot{\boldsymbol{x}}N+\frac{\partial }{\partial k}\dot{k}N+\frac{\partial }{\partial \theta}\dot{\theta}N=\frac{S}{\sigma } $$
(3)

where N(k, θ; x, t) is the wave action density spectrum defined as the variance density spectrum divided by the intrinsic frequency σ, θ is the wave direction, k is the wavenumber, x = (x,y) is the position vector, \( \dot{\boldsymbol{x}} \) is the wave group velocity vector, t is time, and S(k, θ; x, t) represents the net effect of source and sink terms.

Equation 3 describes the evolution, in a slowly varying depth domain, of a 2D ocean wave spectrum where individual spectral components satisfy the linear wave theory locally.

The source function S on the right-hand side of Eq. 3, is a superposition of wind input source term S in representing the momentum and energy transfer from air to ocean waves, the wave dissipation due to white-capping S ds , and the nonlinear transfer by resonant four-wave interactions S nl :

$$ S={S}_{in}+{S}_{ds}+{S}_{nl} $$
(4)

In our application, WW3 was implemented following WAM Cycle4 model physics (Gunther et al. 1993). Wind input and dissipation terms are based on Janssen’s quasi-linear theory of wind-wave generation (Janssen 1989, 1991): the surface waves extract momentum from the air flow and therefore the stress in the surface layer depends both on the wind speed and the wave-induced stress. The dissipation source term was based on Hasselmann’s (1974) white-capping theory according to Komen et al. (1984). The nonlinear wave-wave interaction was modelled using the discrete interaction approximation (DIA, Hasselmann et al. 1985, Hasselmann and Hasselmann 1985).

When waves interact with surface currents provided by the hydrodynamic model, the propagation velocity in the various phase spaces can be written as follows:

$$ \dot{\boldsymbol{x}}={\boldsymbol{c}}_g+{\boldsymbol{U}}_c $$
(5)
$$ \dot{k}=-\left(\frac{\partial }{\partial d}\sigma \right)\left(\frac{\partial }{\partial s}d\right)-\mathbf{K}\bullet \frac{\partial }{\partial s}{\boldsymbol{U}}_c $$
(6)
$$ \dot{\theta}=-\frac{1}{k}\left[\left(\frac{\partial }{\partial d}\sigma \right)\left(\frac{\partial }{\partial m}d\right)-\boldsymbol{K}\bullet \frac{\partial }{\partial m}{\boldsymbol{U}}_c\right] $$
(7)

where c g is the wave propagation velocity vector, d is water depth, s and m are the directions, respectively, along and perpendicular to the wave direction.

The wave model was implemented in the Mediterranean Sea (Fig. 1) considering closed boundaries in the Atlantic Sea, which is a fairly good approximation for the Mediterranean Sea. This is because the occasional propagation of Atlantic swell through the Gibraltar Straits only affects the western part of the Alboran Sea.

2.3 The hydrodynamic model: NEMO

The NEMO model version 3.4 (Nucleus for European Modelling of the Ocean, Madec 2008) was used as the hydrodynamic component of our coupled system. The NEMO code solves the primitive equations (derived assuming the hydrostatic and the incompressible approximations), and here, we used the linear free surface formulation solved by the time-splitting technique; thus, the external gravity waves are explicitly resolved. Additionally, the atmospheric pressure effect was introduced in the model dynamics (Oddo et al. (2014) describes the NEMO implementation with time-splitting and atmospheric pressure effect in the Mediterranean Sea). Figure 1 shows the bathymetry, the river positions (red circles), and the model domain, which extends into the Atlantic Ocean. Lateral boundary conditions in the Atlantic are open and nested into the monthly mean climatological fields computed from 10-year daily output of the 1/4° global model (Drevillon et al. 2008); the nesting details are given in Oddo et al. (2009). Seven rivers are considered as volume inputs: Ebro, Rhone, Po, Vjose, Seman, Bojana, and Nile, and the Dardanelles Strait is closed but is considered as net volume input through a river-like parameterization.

The NEMO configuration parameters can be found in Appendix 2. The advection scheme for active tracers, temperature and salinity, is a mixed upstream/MUSCL (Monotonic Upwind Scheme for Conservation Laws, Van Leer 1979), originally implemented by Estubier and Lévy (2000) and modified by Oddo et al. (2009). The vertical diffusion and viscosity terms are a function of the Richardson number as parameterized by Pacanowsky and Philander (1981).

The model interactively computes air-surface fluxes of momentum, mass, and heat. The bulk formulae implemented are described in Pettenuzzo et al. (2010) and are currently used in the Mediterranean operational system (Tonani et al. 2015). A detailed description of other specific features of the model implementation can be found in Oddo et al. (2009, 2014).

2.4 Model coupling

The coupling between wave and circulation models is achieved through an hourly exchange of instantaneous fields of sea surface currents from NEMO to WW3, air-sea temperature differences computed by NEMO are used for including their effects on the wave generation, and the neutral drag coefficient evaluated from WW3 is passed to NEMO. Surface currents and sea surface temperature from NEMO are at 1.5-m depth, and no additional interpolation is carried out for the coupling as both models share exactly the same computational grid. A sketch of the coupling mechanism is represented in Fig. 2.

Fig. 2
figure 2

The coupling mechanism between WW3 and NEMO. When the simulation starts, models exchange information after the first time step (10 min), then communicate every hour. NEMO sends air-sea temperature differences (ΔT) and current fields (U,V) to WW3, while WW3 sends the neutral drag coefficient (C Dn ) to NEMO

Full size image

In the first part of the coupling, the effects of surface currents on waves are taken into account as specified in Eqs. 5–7.

The second part of the coupling is related to the model underestimation of deep-ocean wave growth and the effects of stability on the growth rate of waves as identified by Kahma and Calkoen (1992), which should be explicitly included in the parameterization of the source terms. Instead of correcting the source terms directly, Tolman (2002) proposes using a wind speed correction. The air-sea temperature difference is used to evaluate a stability parameter, ST, which is written as follows:

$$ ST=\left(\frac{h_g}{U_h^2}\frac{T_a-{T}_s}{T_0}\right) $$
(8)

where U h and T a are the wind speed and air temperature at height h, and T s and T 0 are the surface and reference temperature, respectively.

An effective wind speed, U e , is used:

$$ {\mathbf{U}}_{\mathrm{e}}={\mathbf{U}}_{10}{\left(\frac{c_0}{1\pm {c}_1\mathit{\tanh}\left[\pm {c}_2\left(ST-{ST}_0\right)\right]}\right)}^{1/2} $$
(9)

where U 10 is the wind speed at 10 m derived from the ECMWF model output. Values of c 0, c 1, c 2, and ST 0 are set, respectively, to 1.4, 0.1, 150, and −0.01 default values, according to Tolman (2002), and the plus or minus sign is the same as (ST-ST 0).

The third part of the coupling consists in the transfer of the wind neutral drag coefficient from the wave model to the hydrodynamics. In the wave model, the neutral wind drag coefficient is computed by the WW3 model formulation based on the quasi-linear theory of wind-wave generation developed by Janssen (1989, 1991) and based on Miles (1957). According to this theory the neutral drag coefficient (C Dn ) depends on the effective roughness length and thus on the sea state through the wave-induced stress estimated from the wave spectra.

$$ {C}_{Dn}={\left(\frac{\kappa }{\mathit{\log}\left({z}_u/{z}_0\right)}\right)}^2 $$
(10)

where z u is the height at which the wind is specified (10 m in the present work), κ = 0.4 is the von Karman constant, and z 0 is the roughness length modified by the wave-supported stress τ w as follows:

$$ {z}_0=\frac{\alpha_0\tau }{\sqrt{1-{\tau}_w/\tau }} $$
(11)

where α 0 is set to 0.01. The neutral drag coefficient (Eq. 10) is transferred to the circulation model where it is used to estimate the turbulent drag coefficient following Large and Yeager (2004) and Large (2006), as described in Appendix 3. The full wind stress for the currents is evaluated using Eq. 2 with the drag coefficient given by Eq. 23.

3 Numerical experiment design and validation data sets

Three numerical experiments were carried out for the period January 2009 to December 2013, and a shorter experiment (year 2010) was also performed to evaluate model coupling without wind corrections. Both wave and current models were defined on the 1/16° (about 6–7 km) grid, and the NEMO vertical resolution was defined by 72 unevenly spaced z-levels using partial cells. For the wave model, the spectral discretization was achieved through 30 frequency bins ranging from 0.05 Hz (corresponding to a period of 20s) to 0.79 Hz (corresponding to a period of about 1.25 s) and 24 equally distributed directional bins (15° directional increment). The numerical experiments are set as follows and listed in Table 1:

  • EXP1: WW3 uncoupled. The wave model is a standalone model that does not use sea surface currents and temperature derived from the circulation model. This means that no wave-current interactions (refraction of waves due to currents) take place, and no wind correction due to air-sea temperature differences is included.

  • EXP2: NEMO uncoupled. The hydrodynamic model is uncoupled, and the turbulent drag coefficient is calculated using the Hellerman and Rosenstein (1983) formulation.

  • EXP3: WW3-NEMO coupled. The two models are two-way coupled by hourly exchanging parameters as described in Sect. 2.4 and depicted in Fig. 2.

  • EXP4: WW3-NEMO coupled as in EXP3 but WW3 does not receive air-sea temperature difference fields from NEMO, which means that no wind correction is performed.

Table 1 List of numerical experiments carried out and simulation period
Full size table

3.1 Observational data sets for validation

Four sets of data were used to evaluate the accuracy of the model results, i.e., moored buoys in situ measurements, Jason-2 along track significant wave height, satellite-derived SST, and ARGO profiles. Figure 3 shows the location of the data sets in the study period 2009–2013.

Fig. 3
figure 3

Networks of measuring stations, satellite tracks, and ARGO profile locations used to validate numerical results. Numbered circles represent coastal buoys (Appendix 1, Table 9) measuring wave and current data (data originators: Puertos de L’Estado, ES (cyan); ISPRA, IT (green); HCMR, GR (purple), and CSIC, ES (orange)); buoy numbers are listed in Table 9. Red dots indicate the altimetry dataset measuring wave characteristics. Black dots indicate the ARGO profiler positions

Full size image

The first source of data consists of daily averages of in situ observations derived from a fixed buoy network (http://calval.bo.ingv.it/). Figure 3 shows the spatial buoy locations, and their name, position, and corresponding networks are listed in Table 9 (Appendix 1).

The second set of data is composed of satellite altimeter-derived wave heights from OSTM/Jason-2. The recommended calibrated significant wave height data (with corrections applied to the altimeter 1 Hz estimated values) were used.

The satellite sea surface temperature daily gap-free maps (L4) are used at 1/16° horizontal resolution over the Mediterranean Sea (Buongiorno Nardelli et al. 2013). These data were made available through the CMEMS (Copernicus Marine Environment Monitoring Service) catalogue (http://marine.copernicus.eu/web/69-interactive-catalogue.php).

The fourth dataset is composed of vertical profiles of temperature and salinity measured by ARGO profiling floats integrated by the CMEMS In Situ Thematic Assembly Group (CMEMS INSITU-TAC), quality checked, and made available through the CMEMS catalogue.

4 Results and discussion

The ability of the models to represent the observations was evaluated by standard statistics such as: Bias, root mean square error (RMSE), normalized standard deviation (STDN), and correlation coefficient (R):

$$ Bias=\frac{1}{N}{\sum}_{i=1}^N\left({M}_i-{O}_i\right) $$
(12)
$$ RMSE=\sqrt{\frac{1}{N}{\sum}_{i=1}^N{\left({M}_i-{O}_i\right)}^2} $$
(13)
$$ STDN=\frac{\sigma_M}{\sigma_O}=\sqrt{\frac{\sum_{i=1}^N{\left({M}_i-\overset{-}{M}\right)}^2}{\sum_{i=1}^N{\left({O}_i-\overset{-}{O}\right)}^2}} $$
(14)
$$ R=\frac{\sum_{i=1}^N\left({M}_i-\overset{-}{M}\right)\left({O}_i-\overset{-}{O}\right)}{\sqrt{\frac{1}{N}{\sum}_{i=1}^N{\left({M}_i-\overset{-}{M}\right)}^2{\left({O}_i-\overset{-}{O}\right)}^2}} $$
(15)

where M represents the model fields, O the observations, N is the number of data, and the overbar indicates the time mean value over the whole period.

In order to compare statistics for different experiments, bootstrap 95% confidence intervals (CI) were evaluated for the Mean, Bias, and RMSE metrics, being this approach applicable to any sample distribution.

This section is organized into three parts. The first describes the wave model comparison with buoys and altimeter data, highlighting the impact of the coupled system on waves. The second part shows the circulation model comparison with buoys, satellite, and ARGO measurements in order to assess the model performance in both uncoupled and coupled modes. The third one presents the effects of coupling on both waves and currents considering a short time scale analysis.

4.1 Impact of the coupled system on waves

Results of WW3 uncoupled (EXP1) and coupled (EXP3) experiments are compared to buoy measurements by daily averaged wave fields for the entire experimental period (2009–2013).

Figure 4a–f shows scatter plots of buoy measurements versus the uncoupled (left) and coupled (right) numerical significant wave height (Fig. 4a, b), mean period (Fig. 4c, d), and peak period (Fig. 4e, f). The regression lines of the data from the buoys versus uncoupled and coupled model results (dashed red and blue lines, respectively) are also plotted in order to represent the distance from the best-fit (1:1) line (dashed black line). Different colors and markers correspond to different buoy networks.

Fig. 4
figure 4

Scatter plot comparing buoy measurements with uncoupled (left, EXP1) and coupled (right, EXP3) model results in terms of: significant wave height HS (a, b), mean period TM (c, d), and peak period TP (e, f). Black dashed lines represent the best-fit (1:1); red and blue dashed lines are the buoy-uncoupled and buoy-coupled model data fit, respectively

Full size image

The comparison of significant wave heights (Fig. 4a, b) shows that there is a relatively good agreement between measurements and the model output. In particular, the coupled model significant wave heights (Fig. 4a) correlate better with in situ observations than the uncoupled model estimates (Fig. 4b). In general, the model results underestimate the data from the buoys, particularly the largest wave heights, and similar outcomes have already been highlighted by Ardhuin et al. (2007) and Korres et al. (2011). In the period considered, the maximum daily average wave height was always lower than 6 m, and 95% of wave heights came between 0.2 and 2 m for both measured and modelled data, with an average value of 0.8 m for measurements and 0.7 m for model results. These results are in agreement with other works carried out in the Mediterranean Sea for different periods and using different wave models by Cavaleri and Bertotti (2003) and Korres et al. (2011).

The comparison of mean and peak wave periods is represented in Fig. 4c, d and e, f, respectively, showing a larger scatter with respect to the significant wave height data. Coupled model values correlate slightly more with observations than the uncoupled model. As already found by Korres et al. (2011) for uncoupled model simulations, the predicted mean period is lower than the measured one and presents a lower performance compared to the significant wave height. Mean period values are included between 2 and 10 s with an average value of 4.1 s for the model and 4.3 s for the measurements. Peak period values range between 2 and 12 s and the simulated values, characterized by an average value of 5 s, underestimate the measured average value of 5.5 s.

Table 2 summarizes the main statistics (Mean, Bias, RMSE, and their bootstrap 95% CI, STDN, and R) of significant wave height (HS), mean period (TM), and peak period (TP) for the buoy measurements. In all cases, the coupled model reproduces observations better than the uncoupled model. The coupled model significant wave height and mean period biases decrease by about 20% with respect to the uncoupled model, while the analysis of the CI shows that the peak period results to be not affected by the coupling. Considering the model significant wave heights, these are characterized by lower standard deviation than the observations (STDN < 1) and a high correlation (95%). Both mean and peak model periods are more dispersed than the measurements (STDN > 1) with a correlation of about 75% for the mean period and 85% for the peak period.

Table 2 Statistics evaluated by comparing buoy measurements and model results in terms of wave height (HS), mean period (TM), and peak period (TP) for uncoupled (EXP1) and coupled (EXP3) wave models
Full size table

In order to check the spatial pattern of the errors, Fig. 5 shows significant wave height RMSE (yellow to red triangles) and Bias (green to blue circles) of the coupled model results. Some regionalization of the errors is evident: the model performs better in the Aegean Sea, but not in the Western Ionian Sea. However, the Adriatic, Tyrrhenian, and Ligurian seas have small error values as well as on the Southern Spanish coast.

Fig. 5
figure 5

Significant wave height RMSE (triangles) and Bias (circles) from the buoy data evaluated for the coupled model (EXP3). Colors represent the metric magnitude (left: RMSE from yellow to red; right: Bias from green to blue)

Full size image

Daily averaged significant wave height time series of the Valencia buoy (Puertos del Estado network), where the statistics are more robust due to the long measurement time series, are shown in Fig. 6a–e for the years 2009 to 2013 (black line) with corresponding uncoupled (red dashed line) and coupled (blue dashed line) model results. Both experiments correctly reproduce the significant wave height variability and on average underestimate measurements (negative Bias for all the years and experiments). Averaging over the entire simulated period, the coupled model system presents a lower Bias (−11%) and RMSE (−5%) compared to the uncoupled experiment.

Fig. 6
figure 6

Daily averages of significant wave height for the years: 2009 (a), 2010 (b), 2011 (c), 2012 (d), and 2013 (e) at the Valencia buoy. Black lines represent buoy data, red dashed line corresponds to WW3 uncoupled model (EXP1), and blue dashed line refers to WW3-NEMO coupled model (EXP3). Bias and RMSE statistics are listed in the figure with the corresponding CI for each year

Full size image

Considering that the comparison with the coastal buoy data could be affected by the coastline and shallow water bathymetry representation, by the relatively coarse resolution models, the significant wave heights were also assessed by comparing them with Jason-2 observations (Fig. 7a–h and Tables 3 and 4).

Fig. 7
figure 7

Scatter plots of significant wave height (HS) comparison between Jason2 satellite data and numerical results for 2010 to 2013. Left panels (a, c, e, g) represent WW3 uncoupled model results (EXP1); right plots (b, d, f, h) show the coupled model (EXP3) significant wave height results. Dot colors refer to the data probability density; black dashed line represents the best-fit (1:1) line; solid red and blue lines show the satellite-uncoupled and coupled model data fit, respectively

Full size image
Table 3 Statistics evaluated by comparing satellite significant wave height (HS) with uncoupled model results (EXP1) for years 2010 to 2013
Full size table
Table 4 Statistics evaluated by comparing satellite significant wave height (HS) with coupled model results (EXP3) for years 2010 to 2013 and coupled model results without Tolman (2002) wind correction (EXP4) for year 2010
Full size table

Figure 7a–h shows how the model results fit the altimeter data from 2010 to 2013. For all the years, the coupled model performs better than the uncoupled WW3. The main statistics for EXP1 and EXP3 are listed in Tables 3 and 4, respectively, and show that the coupled model decreases the Bias by about 25%.

The impact of wave-current interaction without considering wind speed correction was also evaluated by performing a 1-year (2010) integration of the system where the Tolman (2002) correction in Eq. 9 was not used (EXP4). Thus, only the exchange of currents and wind drag between NEMO to WW3 was kept. Model results were compared to satellite altimeter significant wave height and are summarized in Table 4 (last column) showing that this simple current-wave interaction coupling slightly improves the RMSE but worsens the Bias. This result is in agreement with previous studies by Galanis et al. (2012) probably due to the still coarse resolution of the models and the winds used in both studies. In a semi-enclosed basin like the Mediterranean Sea, where the wind-sea strongly determines the shape of the wave spectrum, better improvements in the wave field predictions could be achieved by using higher resolution winds. This is evidenced by Ardhuin et al. (2007) who demonstrate that different forcing winds produce larger wave differences than using different wave models.

4.2 Impact of the coupled system on the hydrodynamic fields

The effects of waves on the hydrodynamics are only due to the usage of the turbulent wind drag coefficient shown in Eq. 23. Figure 8a, b shows maps of the neutral drag coefficient estimated by WW3 and the turbulent drag coefficient estimated by NEMO (in EXP3), respectively, following the methodology explained in Appendix 3. Figure 8c shows the turbulent surface drag coefficient as parameterized by the uncoupled NEMO model using the Hellerman and Rosenstein (1983) formulation (in EXP2). The two turbulent drag coefficient formulations (Fig. 8b, c) show a similar pattern. The differences between the coupled and uncoupled drag coefficient, shown in Fig. 8d as a percentage difference, are concentrated in the coastal areas especially of the Aegean and the Liguro-Provençal basin. Overall, the changes are of the order of 10% and are mainly attributed to the changed Cd provided by the WW3 model. All the maps in Fig. 8 are evaluated as time averages over the entire 5-year period.

Fig. 8
figure 8

Maps of drag coefficient averaged over the entire time period (2009–2013). a Neutral drag coefficient evaluated by WW3. b Turbulent drag coefficient evaluated by NEMO in coupled mode EXP3. c Hellerman and Rosenstein (1983) turbulent drag coefficient in uncoupled NEMO EXP2. d Percentage difference between b and c

Full size image

In order to evaluate the impact of the coupled system on the circulation model, we compared model surface currents with coastal buoy measurements; the sea surface temperature was validated using satellite data, and vertical profiles of temperature and salinity were compared to ARGO floating measurements.

Table 5 summarizes the main statistics derived from the comparison of model results and buoy surface current amplitude, showing that a slight improvement is achieved when coupling with waves (lower Bias and RMSE): model predictions generally underestimate measurements and have lower standard deviations; however, the CI show that the differences between the two systems are negligible (note that in this case the limits of the CI correspond since these differ at the fourth decimal).

Table 5 Statistics evaluated by comparing current amplitude measurements from buoy and model results in the uncoupled (EXP2) and coupled (EXP3) circulation model
Full size table

Tables 6 and 7 evaluate the Bias and RMSE of the single zonal and meridional components of the surface velocities, respectively, thus giving an estimate of the impact on current directions. Again, the improvements in the coupled model system are too small to be significant and metrics CI are in some cases larger than the improvements.

Table 6 Bias, RMSE, and confidence interval (CI) values evaluated by comparing zonal surface velocities at coastal buoys listed in Table 9 (Appendix 1) for uncoupled (EXP2) and coupled (EXP3) model systems
Full size table
Table 7 Bias, RMSE, and confidence interval (CI) values evaluated by comparing meridional surface velocities at coastal buoys listed in Table 9 (Appendix 1) for uncoupled (EXP2) and coupled (EXP3) model systems
Full size table

Figure 9 shows the 5-year mean SST from EXP2 and EXP3 compared with satellite derived SST. The model overestimates satellite SST in the spring and summer seasons showing a positive Bias of about 0.5 °C. However, it is still confined within one standard deviation and there is no substantial difference between the results of the two experiments.

Fig. 9
figure 9

Time series of climatological sea surface temperature (SST). Black line: satellite retrieved SST; dashed red line: NEMO uncoupled results (EXP2); dashed blue line: NEMO coupled (EXP3); shaded grey area: standard deviation of satellite data. The figure also lists values of mean SST

Full size image

Table 8 summarizes the Mean, Bias, RMSE, their confidence intervals (CI), and the correlation coefficient R metrics for both uncoupled (EXP2) and coupled (EXP3) model systems for temperature and salinity profiles compared with ARGO and integrated along the first 500 m. As for the SST, there are no significant differences between the two experiments.

Table 8 Statistics evaluated by comparing temperature and salinity vertical profile measurements and model results from uncoupled (EXP2) and coupled (EXP3) circulation models
Full size table

We conclude that the effects of the different wave-induced turbulent wind drag coefficients have a small, if nonexistent, impact on the quality of the simulated current, temperature, and salinity fields when considering space or time averaged fields. Possible impacts on restricted area at short time scale are presented in the following section.

4.3 Wave-current coupling at short time scale

The performance of the system was evaluated considering an event of particularly high wind speed and assessing the role of the wave effects on coastal circulation and temperature as well as the effect that the circulation and temperature exert on the waves. The selected event, which is considered interesting in terms of both atmospheric conditions/extreme events and observational data availability, took place on the 26th December 2010 in the Balearic Sea with maximum wind speed (U10) higher that 20 m/s close to the Spanish coastal area of Tarragona with a maximum daily average velocity of about 15 m/s as represented in Fig. 10a. The turbulent drag coefficient evaluated in the selected area by the coupled and uncoupled systems for the considered day are represented in Fig. 10b, c showing that when the coupling is activated, the drag coefficient has a spatial pattern similar to the wind field with higher values in the area affected by the storm.

Fig. 10
figure 10

Daily averaged maps of the Balearic Sea on the 26th December 2010 for a wind speed at 10 m [m/s], b turbulent drag coefficient in coupled system (EXP3), and c turbulent drag coefficient in uncoupled system (EXP2)

Full size image

The modelled significant wave height, surface temperature, and currents have been compared to in situ observations at the Tarragona and Dragonera buoys (buoy nos. 6 and 4 in Table 9 and Fig. 3) for 1 week: from the 24th December 2010 (2 days before the selected event) to the 30th December 2010 (4 days after the event).

Hourly time series of significant wave height are presented in Fig. 11a, b for the Tarragona and the Dragonera buoy, respectively, showing an improved skill of the coupled system resulting intensified by the interaction with the hydrodynamic fields with respect to the uncoupled one.

Fig. 11
figure 11

Time series at Tarragona (a, c, e) and Dragonera (b, d, f) coastal buoys for the period 26–30 December 2010 for hourly significant wave height [m] (a, b), daily surface temperature [°C] (c, d), and daily surface current velocity [m/s] (e, f). Black lines represent measurements; blue lines correspond to coupled system results; red lines show uncoupled system results

Full size image

The comparison of the modelled surface temperature against daily observations at the Tarragona buoy is presented in Fig. 11c where the coupled model shows a good agreement with the measurements with respect to the uncoupled one, which overestimates the buoy surface temperature. Figure 11d illustrates the surface temperature at the Dragonera buoy showing that the uncoupled model underestimates the observations, while the coupled one provides more reliable predictions of the measurements especially in correspondence of the storm event.

Figure 11e, f shows that the coupled model overestimates the measured surface currents at the Tarragona and Dragonera buoys, but it better represents the evolution of the measured velocity after the extreme event with respect to the uncoupled model, which is less affected by the increased wind speed.

The short time scale analysis for the selected event demonstrates that both wave and hydrodynamic fields are affected by the coupling, resulting in an enhanced skill, particularly of the modelled significant wave heights and, to a lesser extent, of the surface temperature and current fields.

5 Conclusions

A coupled wave-current numerical model system was developed and validated against observations for a 5-year period between 2009 and 2013. The coupled model was implemented in the Mediterranean Sea using NEMO and WW3 model codes. Fields were exchanged hourly between the two components while atmospheric forcing fields from ECMWF were interpolated to the single model time steps.

The coupling consists of feeding the wave model with sea surface temperature and surface currents computed by the hydrodynamic model and returning a neutral wind drag coefficient to the latter, which then computes a turbulent wind drag coefficient used in the momentum surface boundary condition.

One major conclusion drawn from the various results presented in the work is that the wave model is impacted by this kind of wind wave–hydrodynamic model coupling, while the hydrodynamics changes are negligible at large space and time scales becoming more evident when considering the coupling impacts during storm events.

Both the uncoupled and coupled wave models perform well in reproducing in situ as well as satellite measured wave parameters, and the coupled system improves the significant wave height simulation values with respect to the uncoupled system. The results also highlight that the enhanced performance of the coupled model is mainly achieved by better representing effect of air-sea temperature differences impacting the wave growth, while surface currents lead to only a minor improvement. This might be due to the low resolution of the hydrodynamic and wave models and in particular to the scarce resolution of the coastal areas geometry and bathymetry.

The wave-induced turbulent wind drag correction and the traditional Hellerman and Rosenstein (1983) formulations differ significantly only in the coastal areas and even there only by 10–20%. The impact of the coupling on the simulated hydrodynamic fields is thus much lower, and most of the time is negligible. Evidence of the effect of wave dependent surface stress on the computation of coastal currents and temperature is shown on a short time scale analysis, while, on the typical time scale of ocean circulation, the main effect is expected because of a wave-dependent ocean mixing, which is not included in this study.

We probably need to wait for a stronger coupling between waves and currents before an improvement in the hydrodynamics will be evident. Future work in fact will consider the wave dissipated energy (as provided by the wave model) in the vertical mixing of the water column and will add the Stokes drift velocity in the momentum and tracer equations.

The present work suggests that a two-way coupled model could improve the prediction of wave characteristics, in particular the significant wave height, for both open ocean and coastal areas.