The circulation model used in this study is the General Estuarine Transport Model (GETM, Burchard and Bolding, 2002). The nested-grid model setup for the German Bight has a horizontal resolution of 1 km and 21 σ layers (Fig. 1) (Stanev et al., 2011). GETM uses the k-ε turbulence closure to solve for the turbulent kinetic energy k and its dissipation rate ε. The data for temperature, salinity, velocity and sea surface elevation at the open boundary are obtained from the coarser resolution (approximately 5 km and 21 σ layers) North Sea–Baltic Sea GETM model configuration (Staneva et al., 2009). The sea surface elevation at the open boundary of the outer (North Sea–Baltic Sea) model was prescribed using 13 tidal constituents obtained from the satellite altimetry via OSU Tidal Inversion Software (Egbert and Erofeeva, 2002). Both models were forced by atmospheric fluxes computed from bulk aerodynamic formulas. These formulas used model-simulated sea surface temperature, 2 m air temperature, relative humidity and 10 m winds from atmospheric analysis data. This information was derived from the COSMO-EU regional model operated by the German Weather Service (DWD, Deutscher Wetter Dienst) with a horizontal resolution of 7 km. River run-off data were provided by the German Federal Maritime and Hydrographic Agency (BSH, Bundesamt für Seeschifffahrt und Hydrographie).

Ocean surface waves are described by the two-dimensional wave action density spectrum N(σ, θ, φ, λ, t) as a function of the relative angular frequency σ, wave direction θ, latitude φ, longitude λ and time t. The appropriate tool to solve the balance equation is the advanced third-generation spectral wave model WAM (WAMDI group, 1988; ECMWF, 2014). The use of the wave action density spectrum N is required if currents are taken into account. In that case, the action density is conserved, in contrast to the energy density, which is normally used in the absence of time-dependent water depths and currents. The action density spectrum is defined as the energy density spectrum E(σ, θ, φ, λ, t) divided by σ observed in a frame moving with the ocean current velocity (Whitham, 1974; Komen et al., 1994): N(σ,θ)=E(σ,θ)σ. The wave action balance equation in Cartesian coordinates is given as ∂N∂t+(cg+U)∇xyN+∂cσN∂σ+∂cθN∂θ=Swind+Snl4+Swc+Sbot+Sbrσ. The first term on the left side of Eq. (2) represents the local rate of change of wave-energy density and the second term describes the propagation of wave energy in two-dimensional geographical space, where cg is the group velocity vector and U is the corresponding current vector. The third term of the equation denotes the shifting of the relative frequency due to possible variations in depth and current (with propagation velocity cσ in σ space). The last term on the left side of the equation represents depth-induced and current-induced refraction (with the propagation velocity cθ in θ space). The term S=S(σ, θ, φ, λ, t) on the right side of Eq. (2) is the net source term expressed in terms of the action density. It is the sum of a number of source terms representing the effects of wave generation by wind (Swind) quadruplet nonlinear wave–wave interactions (Snl4), dissipation due to white capping (Swc), bottom friction (Sbot) and wave breaking (Sbr). The current version of the third-generation wave model WAM Cycle 4.5.4 is an update of the former Cycle 4, which is described in detail in Komen et al. (1994) and Günther et al. (1992). The basic physics and numerics are maintained in the new release. The source function integration scheme is provided by Hersbach and Janssen (1999), and the updated source terms of Bidlot et al. (2007) and Janssen (2008) are incorporated. Depth-induced wave breaking (Battjes and Janssen, 1978) is included as an additional source function. The depth and/or current fields can be non-stationary. The wave models have the same resolution, and the model uses the same bathymetry and wind forcing as the GETM model. The boundary values of the North Sea model are taken from the operational regional wave model of the DWD, while the boundary values for the German Bight are taken from the North Sea model. The wave models run in shallow water mode, including depth refraction and wave breaking, and calculate the two-dimensional energy density spectrum at the active model grid points in the frequency and direction space. The solution of the WAM transport equation is provided for 24 directional bands at 15∘ each with the first direction being 7.5∘, measured clockwise with respect to true north, and 30 frequencies logarithmically spaced from 0.042 to 0.66 Hz at intervals of Δf/f=0.1.

The implementation of the depth-dependent equations of the mean currents u(x, z, t) in the presence of waves follows Mellor (2011). The momentum equation for an incompressible fluid is du / dt = F -∇δp, where F is the sum of external forces (Coriolis, gravity, friction) and ∇δP is the pressure gradient, which includes the influence of wave motion on the mean current. Within the radiation stress formulation of Mellor (2011), the prognostic velocity u is related to the Eulerian wave-averaged velocity. Using linear wave theory and accounting for the second-order terms of the wave height, the equation of motion is ∂u∂t=F-u⋅∂u∂x-∂∂x⋅S, where the angle brackets denote averaging over the wave period, and S is the radiation stress tensor: S=E⋅cfcgk⊗kk2+δsinh⁡2kh+2khsinh⁡2kD+2kD-δcosh⁡2kh-14sinh⁡2kD, where E=1 / 16gHs is the wave energy, k is the wave vector, and h=D(1+ξ) is the local depth of layer ξ. Thus, the divergence of the radiation stress is the only force related to waves in the momentum equations. The equation for kinetic energy, which is derived from the momentum equation by multiplication with the velocity vector, is ∂Ekin∂t=F⋅u-u⋅∂Ekin∂x-∂∂x⋅S⋅u, where the gradients in wave energy (i.e. dissipation due to wave breaking) may lead to increased surface elevation (wave set-up).

The wave state information required to account for the divergence of the radiation stress in the GETM momentum equations is provided by WAM. The dissipation source functions (wave breaking, white capping and bottom dissipation) estimated by the wave model WAM are also used in the turbulence module of GOTM (General Ocean Turbulence Model). These data are used to specify the boundary conditions for the dissipation of the turbulent kinetic energy and the vorticity due to wave breaking and bottom friction (Pleskachevsky et al., 2011). Following Moghimi et al. (2013), an enhanced bottom roughness length z0b is computed as a function of the base roughness z0 and wave properties (e.g. the bottom orbital velocity of the waves) according to Styles and Glenn (2000). This allows accounting for the generated turbulence at the bottom due to the non-resolved oscillating wave motion. In the two-way coupling experiments, the GETM model provides the water level and ambient current for WAM.

The coupling between GETM and WAM is performed via the coupler OASIS3-MCT: Ocean, Atmosphere, Sea, Ice, and Soil model at the European Centre for Research and Advanced Training in Scientific Computation Software (Valcke et al., 2013). The name OASIS3-MCT is a combination of OASIS3 (Ocean, Atmosphere, Sea, Ice, and Soil model coupler version 3) at the European Centre for Research and Advanced Training in Scientific Computation (CERFACS) and MCT (Model Coupling Toolkit), which was developed by Argonne National Laboratory in the USA. The details of the properties and use of OASIS3 can be found in Valcke (2013). The exchange time between models is 5 min. This small coupling time step is a major advantage for modelling fast-moving storms compared to off-line (without using a coupler) coupled models, as in Staneva et al. (2016), where hourly wave fields are used in GETM.

This study is focussed on the period during the winter storm Xaver that occurred on the 5 and 6 December 2013, which caused flooding and serious damage to the southern North Sea coastal areas (Dangendorf et al., 2016). During 4 to 7 December, the storm depression Xaver moved from the south of Iceland over the Faroe Islands to Norway and southern Sweden and further over the Baltic Sea to Lithuania, Latvia and Estonia. It reached its lowest sea level pressure on the 5 December at 18:00 UTC over Norway (approximately 970 hPa, Figs. 2 and 3). Over the German Bight, the arrival of Xaver coincided with high tides; therefore, an extreme weather warning was given to the coastal areas of north-western Germany due to high tides and wind gusts greater than 130 km h-1 (Deutschländer et al., 2013). The extremely high water level and waves triggered sand displacement on the barrier islands and erosion of dunes in the Wadden Sea region. The German Weather Service reported the storm to be worse or similar to the North Sea flood of 1962, in which 340 people lost their lives in Hamburg, saying that improvements in sea defences since that time would withstand the storm surge (Deutschländer et al., 2013; Lamb and Frydendahl, 1991).

For the control simulation (CTRL run), GETM is run as a fully three-dimensional baroclinic model without coupling with the wave model. The effects of using different coupling methods are studied by comparing the two-way fully coupled GETM–WAM model simulation (FULL run) with the one-way coupled model, in which the circulation model obtains information from WAM (one-way coupling). We denote this experiment as FORCED run. Additionally, we run the circulation model GETM as a two-dimensional barotropic model (2-D run). In the final experiment, we exclude the river runoff forcing (NORIV run). The list of experiments is given in Table 1.

In this section, we analyse the wave model performance during Xaver using the FULL experiment. The significant wave heights (Hs) from the model simulations are in good agreement with the measured values. As can been seen in the time-series graph for the Elbe (top) and Westerland (bottom) stations, the measured Hs was greater than 7.5 m during 2–8 December 2013 (Fig. 4). The peak Hs during the storm is reached earlier in the model simulations compared to the observations (Fig. 4b, d). This could be due to the DWD wind data (see also Wahle et al., 2016). In addition, the maximum of statistical wave height simulated by the model for the two locations (Fig. 4a, c) occurs earlier than that of the measurements, which is due to the shifted maximum of the DWD wind forecasts. The standard deviation between the model and the measurements is 0.16 m for the Elbe station and 0.12 m for the Westerland station. The correlation coefficients between the WAM simulations and measurements are greater than 0.9 for all stations, and the normalized RMS error is relatively low (between 0.09 and 0.16 m). For the analyses of the wave model performance, including different statistical parameters computed during the extreme event for all available German Bight stations, we refer to Staneva et al. (2016).

The wave spectra at the FINO1 and Elbe BSH buoy stations are given in Fig. 5 for the study period. The wave spectra from the model simulations (Fig. 5a, c) are in good agreement with the spectra from the observations (Fig. 5a, c). The time variability of the spectral energy is accurately reproduced by the model, and the energy around the peak is similar in the observations and simulations; however, the model patterns are smoother than the observed patterns.

In addition to the in situ measurements, the satellite altimetry data provide a unique opportunity to evaluate both the temporal and spatial variability simulated in the model along its ground track at the time of the overflight of the German Bight, lasting approximately 38 s (see Fig. 6a, b). The modelled Hs varies along the satellite ground tracks between 1.2 and 1.9 m during calm conditions on 3 December 2013 at 18:00 UTC (Fig. 6a), while during Xaver Hs varies between 6.3 and 9.4 m (6 December 2013 at 04:00 UTC, Fig. 6b). The spatial distribution of Hs (Fig. 6c, d) is in good agreement with the satellite data in both cases. The latitudinal distribution of Hs simulated by the wave model (green dots) is smoother than that of the satellite data. This can be explained by different post-processing of the satellite data of the significant wave height and by the statistical nature of its estimate by the model. For calm conditions (Fig. 6c), Hs is slightly underestimated (approximately 15 cm) in the coastal area and overestimated (approximately 20 cm) in the open German Bight. During Xaver, the model slightly overestimates the satellite data in open areas (20–30 cm). These results are consistent with the results of Fenoglio-Marc et al. (2015), who compared the SARAL data with the DWD wave simulations.

In this section, we demonstrate the performance of the hydrodynamic model to simulate the mean sea level and present statistics obtained for the study period. Detailed statistical analyses of the model comparisons with measurements for the area of German Bight are quantified by Staneva et al. (2016), where the coupled model performance is shown to be in a good agreement with observations, not only during the calm conditions but also during storm events. Therefore, we only provide new examples of model-data validations, including satellite data that have not been used in previous studies.

The geographic representation of the bias between the model simulations and all available tide gauge data shows that the bias for most tide gauge stations is within ±0.1 m (Fig. 7). Exceptions are found in some coastal tide gauge data stations in very shallow areas. This can be attributed to the relatively coarse spatial resolution (1 km) and smoother model bathymetry in shallow coastal waters. Storm surges are estimated by subtracting from the simulations and tide gauge observations the ocean tide estimated using the T_TIDE routine (Pawlowicz et al., 2002). The comparisons between individual simulations are only marginally affected by tidal simulation errors because the simulations share the same systematic tidal errors. Estimating the surge component, the direct influence of tidal simulation errors in overtides is minimized because the surge signals from observations and model runs are derived by subtracting an individual estimate of the tidal signal for each dataset.

From the comparison between the surge model and satellite data (Fig. 8), it can be concluded that the model results are in good agreement with the observations. This holds for calm conditions (3 December 2013), when the surge was weak (less than 10 cm offshore and up to 25 cm near the coastal area, Fig. 8c), as well as during Xaver on 6 December 2013, when the surge reached almost 3 m. The statistics from the comparisons between the observations and experiments are presented in Table 2. The coupling between circulation and waves significantly improves the surge predictions; when the effects of the interactions with waves are considered, both the bias and the RMSE are substantially reduced (see Table 2).

The temporal evolution of the water level for the Helgoland tide gauge data (see Fig. 1 for its location) is shown in Fig. 9. The consistency between the model simulations from the CTRL and FULL runs is very good during normal meteorological conditions; however, during the storm, the water level simulated by the stand-alone circulation model is approximately 30 cm lower than the data from the Helgoland tidal gauge station. When the wave-induced processes are considered, the simulated sea level (FULL run) approaches the observations. Including wave–current interactions improves the root mean square error and the correlation coefficient between the tide gauge data and the simulated sea level over the German Bight (Table 2).

The surge height reaches approximately 2.5 m during Xaver, with its maximum at low water. During Xaver, two surge maxima (Smax⁡1 and Smax⁡2 as the green line in Fig. 9) are observed. Fenoglio et al. (2015) described the first surge maximum as a wind-induced maximum. They found that at Aberdeen and Lowestoft stations, the surge derived from the tide gauge records had only one maximum, reaching the eastern North Sea coastal areas (anticlockwise propagation) approximately 10 h later than Lowestoft (easternmost UK coast), causing the second storm surge maximum detected by the measurements in the German Bight. As shown by Staneva et al. (2016), the wave-induced mechanisms contribute to a persistent increase of the surge after the first maximum (with slight overestimation after the second peak). At the two maxima, the observed water level at the Helgoland tide gauge is in better agreement with the coupled model (FULL run: black line) than the CTRL-simulated water level. The two maxima are underestimated by the stand-alone circulation model (CTRL: red line), especially at high water, when the surge difference between the model results and the measurements is approximately 30 cm for the first peak and more than 40 cm for the second peak (Fig. 9).

In this section, we analyse the role of wave–current interactions in the storm surge model and demonstrate the sensitivity to one-way vs. two-way coupling. Figure 10 shows the time series of the water level (black line) and the storm surge (red line) for six stations (see Fig. 1 for their locations) together with the differences in the surge between the FULL and CTRL runs (FULL–CTRL, green line) and the differences between the FULL and FORCED runs (FULL–FORCED, blue line). The surge during the extreme exceeds 2 m in the open-ocean stations and increases to 2.8 m near the coastal stations. The two storm surge maxima during Xaver (described in Sect. 4) are seen at the near-coastal stations ST1–4, whereas at ST6 (in the Elbe Estuary) the surge remains high, even in the period between the two maxima. The coupling with waves leads to a persistent increase in the surge, especially after the occurrence of the first maximum (Smax⁡1). The difference in the simulated surge between the FULL and CTRL runs (green line) reaches a maximum during the first peak of the surge and is substantial during the following 2 days. For the Hörnum station (ST3), the increase in the surge due to coupling with waves exceeds 35 % compared to the CTRL data (Fig. 10c). At the north-easternmost station (ST4), the surge difference between the FULL and CTRL runs is greater than 70 cm, which results in a contribution of the wave–current interaction processes of greater than 40 %. For the deeper open-water station (ST5, Fig. 10f), the maximum contribution is approximately 30 cm, a 25 % increase in the surge. The differences between the FORCED and FULL runs are relatively small (less than 4 % of the total for all stations; see the blue line). However, for the shallower Elbe station (ST6, Fig. 10e), the effects of two-way coupling compared to the FORCED run (one-way coupling) are important. Staneva et al. (2016) provided a summary of improved model performance with respect to the prediction of sea level, which is the main variable considered below in the analysis of extreme surges in the German Bight. The quantification of the performance shows that in a large number of coastal locations, both the RMS difference and the bias between the model estimates and observations are significantly reduced because of the improved representation of physics. Only in very few near-coastal tide gauge stations does the coupling not lead to improvements, which might be due to the insufficient resolution of the near-coastal processes in very shallow water regions.

To provide an illustration of the coastal impact caused by Xaver, we analyse the horizontal patterns of the maximum storm surge (Fig. 11) over the four tidal periods T1–T4 (as specified in Fig. 9). During the second peak (T3), the surge exceeds 2.8 m over the whole German Bight coast (Fig. 11c); the storm surge near Elbe is greater than 3 m. During the period of the first surge peak (T2, Fig. 11b), the maximum occurs in the Sylt–Römo Bight area (above 2.8 m) and along the Elbe and Weser estuaries (approximately 2.5 m). Over the whole German Bight, the simulated surge exceeds 1.5 m. In the period of relatively calm conditions before the storm (T1), the surge is relatively low (Fig. 11a, less than 30 cm). A decrease in the surge towards the north-western German Bight is simulated during T4 (Fig. 11d). The intensification of the storm surge from the open sea towards the coastal area is consistent with the specific atmospheric conditions during Xaver (Fig. 3).

To better understand the impact of wave–current interactions on the surge simulations, we also analyse the horizontal patterns of the maximum differences in the storm surge between the coupled model (FULL run) and the stand-alone GETM (CTRL run). The maximum differences for each grid point are estimated over the four tidal periods (T1–T4, Fig. 12). The patterns show that the differences between the FULL and CTRL runs during the first surge maximum are more noticeable in the very shallow North Frisian Wadden Sea. The maximum surge simulated by the fully coupled model exceeds that of the CTRL run by approximately 60 cm along the Sylt–Römo Bight during T2. The enhancement of the surge in the coastal area (see Fig. 11b) may be due to the nonlinear interaction between circulation and waves (the contribution of the wave–current interaction to the increase of the surge is greater than 25 %) along the German Bight coastal region (Fig. 12a). For T3, the maximum surge difference (approximately 55 cm) is concentrated along the Elbe river; however, the increase in the surge due to wave-induced processes exceeds 40 cm along the entire German Bight coast. During the second Xaver peak, the radiation stress contributes to a rise in the sea level, which is directed towards the Elbe–Weser river area. During the first peak (T1), the differences between FULL and CTRL are more pronounced near the North Frisian Wadden Sea. The computed maximum surge differences are higher during T2 than during T3. For T4 (Fig. 12d), the maximum difference of approximately 15 cm occurs for the east Frisian coast towards the Elbe river area, whereas in the north-eastern area, the wave-induced processes do not contribute much to the mean sea level and the surge simulations of the FULL runs are similar to the CTRL run. The horizontal distribution of the patterns of Fig. 12 demonstrates the good consistency with the meteorological situation (Fig. 3). The effects of wave-induced forcing during the storm are also noticeable in the open North Sea (maximum surge differences are approximately 30 cm Fig. 12b, c) due to the dominant role of the radiation stress; even in the deeper areas, the differences between the FULL and CTRL surge estimates are greater than 20 %. Although the wave heights are much higher in the open sea, the water there is much deeper; thus, the differences in sea level between the FULL and CTRL runs are relatively small.

Depth-averaged two-dimensional flow models are widely applied in storm surge simulation and have been assumed to meet the requirements of operational forecasts. They are also widely used in many scientific studies. However, to study the flow characteristics of storm surges, the use of only barotropic models is insufficient, especially in large estuaries. The flows in the surface and bottom layers are usually quite different, so depth-averaged two-dimensional models cannot sufficiently depict the flow structure. Furthermore, storm surge models do not account for baroclinic processes, such as density-driven changes in water masses, which are important in estuarine environments.

The changes in the sea level due to temperature for Dutch coastal areas have been studied by Tsimplis et al. (2006). Dangendorf et al. (2013) showed that laterally forced steric variation and baroclinic processes are important at decadal scales, while atmospheric forcing causes the annual variability in the sea level. Chen et al. (2014) studied the role of remote baroclinic and local steric effects in the interannual sea level variability and found that a three-dimensional model that considers the temperature and salinity can more accurately simulate the changes in the water level related to the North Atlantic Oscillation (NAO). In these models, more realistic open boundary conditions (than in the barotropic models) are used to account for the dynamics of heat and salt. We quantify the benefit of using a fully three-dimensional model that also considers temperature and salinity to simulate the sea level during extremes.

The surge differences between the FULL and 2-D runs are much larger during Xaver (T2, Fig. 13b) than during calm conditions (T1, Fig. 13a). For T2, the maximum difference increases eastward from 2 to 5 cm at the western boundary of the German Bight to more than 80 cm along the North Frisian Wadden Sea coast and near the Elbe and Weser estuaries. The surge differences decrease to 30 cm during the second peak of Xaver. After the storm, the three-dimensional effects contribute to an increase in sea level in the direction of the Elbe Estuary (Fig. 13d). These effects can exceed 25 % of the sea level increase, compared to the 2-D model simulations (Fig. 14). For the Elbe area, the 2-D model underestimates the mean sea level by approximately 1 m. This could cause significant underestimation of the sea level predictions of the barotropic models. For T4, the impact of baroclinicity is localized along the south-eastern coastline (Fig. 13d). The differences between FULL and NORIV (Fig. 14, blue line) are negligible at ST3, while at ST6 they are approximately 15 cm in the vicinity of the Elbe Estuary during the storm Xaver. When analysing the impact of baroclinicity on sea level model results, we use the barotropic hydrodynamic model that is not coupled to the wave model since our aim is to demonstrate only those effects. Introducing wave–circulation coupled processes (as demonstrated in the previous sections) to the barotropic model can reduce the differences between this model and the FULL run.