Timevarying storm surges on Lorentz’s Wadden Sea networks
Abstract
We present an idealized network model for storm surges in the Wadden Sea, specifically including a timedependent wind forcing (wind speed and direction). This extends the classical work by H.A. Lorentz who only considered the equilibrium response to a steady wind forcing. The solutions obtained in the frequency domain for the linearized shallowwater equations in a channel are combined in an algebraic system for the network. The velocity scale that is used for the linearized friction coefficient is determined iteratively. The hindcast of the storm surge of 5 December 2013 produces credible timevarying results. The effects of storm and basin parameters on the peak surge elevation are the subject of a sensitivity analysis. The formulation in the frequency domain reveals which modes in the external forcing lead to the largest surge response at coastal stations. There appears to be a minimum storm duration, of about 3–4 h, that is required for a surge to attain its maximum elevation. The influence of the water levels at the North Sea inlets on the Wadden Sea surges decreases towards the shore. In contrast, the wind shearing generates its largest response near the shore, where the fetch length is at its maximum.
Keywords
Shallowwater equations Storm surges Idealized models Networks Wadden Sea1 Introduction
Storm surges, the raised water levels induced by strong winds in coastal areas, pose a serious hazard of flooding and of loss of life and property. This is amplified by trends such as a growing population pressure, sea level rise, and increasing storminess projections due to climate change (Pugh 1987). Weather systems act upon the water by means of the atmosphericpressure differences and the wind stresses acting on the free surface. The surge dynamics is further influenced by tides, nonlinear tidesurge interactions, wave dynamics, bed interactions, and the physiographic features of the coastal area—see, e.g., Pugh (1987) for general information on storm surges.
The report of State Committee for the Zuiderzee (1926) provides a pioneering idealization of the physiographic complexity of a tidal basin. The extensive flats of the Wadden Sea are separated by deep channels originating at the inlets, also visible in Fig. 1a. Since most of the water flows along these tidal channels, the State Committee for the Zuiderzeerepresented the western Wadden Sea as a network of equivalent channels having depth and width uniform over their length—specifically commented upon in Section 2.1. Whereas originally this simplified approach aimed to overcome limitations of computing, idealized basins have been used until recently for investigating phenomena in coastal dynamics, such as tides (e.g., Hill and Souza 2006; Alebregtse et al. 2013; Alebregtse and de Swart 2014; Alebregtse and de Swart 2016), storm surges (Stroband and Wijngaarde 1977), and tidesurge interactions (Prandle and Wolf 1978). Their enduring advantage over the models retaining the full complexity of physics and topography—for example Zijl et al. (2013), DuranMatute et al. (2014), and DuranMatute et al. (2016)—lies in their computational efficiency. Idealized models do reproduce key physical processes at a limited computational cost, and provide accurate results (both in a quantitative and qualitative sense), for example usable for extensive sensitivity analyses against geometrical and physical modelling parameters driving the system’s response—as presented in this article.
Further, in order to compute the onedimensional flow inside the individual channels, the State Committee for the Zuiderzee (1926) linearized the shallowwater equations averaged over the channels’ cross sections. To this end, the seabed friction was parametrized through a novel procedure based on energyequivalence arguments, known as Lorentz linearization, described in Section 2.2.2.
Unfortunately, for the lack of adequate processing power at the beginning of the twentieth century, the tide and stormsurge simulations could only be performed separately. In particular, the stormsurge simulations consisted of calculating the steadystate water levels in equilibrium with an extreme wind having fixed speed and direction. Therefore, the State Committee for the Zuiderzee could not identify that both motion and storage of the surge water inside the Wadden Sea are modulated by the temporal variability of the wind field, as well as by the fluxes across the tidal inlets—see, for example, the reanalyses of Lipari et al. (2008), Lipari and van Vledder (2009), and DuranMatute et al. (2016). This implies that, in a semienclosed basin, the most severe surges are not necessarily generated by the storms with the highest wind speed. Timevarying storms can cause higher surges than steadystate storms do, even when the peak wind speed is the same (Lipari et al. 2008).
Jallah and Bakker (1994) already coded a computerized transcription of the model and algorithms of the State Committee for the Zuiderzee. Here, inspired by Lorentz’s seminal studies of the Wadden Sea surges, and drawing from the work of Chen et al. (2015, 2016), we have developed a new idealized network model allowing for timevarying external forcing, namely the storm surge elevation in the outer sea and the windstress field. The model is based on the linearized shallowwater equations applied to the flow in the network channels. Unlike the State Committee for the Zuiderzee, the equations are cast in the frequency domain after Fourier transformation of both input and output variables and the equations. Because of the linearity, the superposition of the solutions of the individual modes gives the unsteady solution in the time domain.
With this tool, we aim to provide insights on the transient behavior of storm surges within tidal basins. We will specifically investigate how basin characteristics, storm characteristics, and different forcing mechanisms affect the transient behavior of storm surges in tidal basins.
The model, the forcings, and the outline of the solution method are presented in Section 2. The simulations presented in Section 3 deal with the hindcast of the storm surge of 5 December 2013, the sensitivity analysis of the peak surge on the coast to selected basin and storm parameters, and a modal analysis of the surge response to the external forcing. Finally, Sections 4 and 5 contain the discussion and the conclusions.
2 Methods
2.1 Network geometry
Characteristics of the tidal network and storm network
Tidal  Storm  Storm network  

network  network  with Zuiderzee  
Number of nodes  
Internal  31  10  11 
Open sea  4  5  5 
On coast  0  0  1 
Total  35  15  17 
Number of channels  
Total  69  33  36 
Max. ratio of \(h/b\)  0.03  0.01  0.01 
2.2 Hydrodynamic model
2.2.1 Governing equations
2.2.2 Lorentz’s linearization of the bottom friction
In their surge simulations, the State Committee for the Zuiderzee used a constant Chézy coefficient \(\chi = 50\:\:\mathrm {m}^{1/2}\:\:\mathrm {s}^{1}\), whereas we use a constant Manning coefficient \(\mu = 0.0242\:\:\mathrm {s}\:\:\mathrm {m}^{1/3}\). This corresponds to coastal waters with characteristic grain sizes of \(D_{50}= 0.2 \;\text {mm}\) and D_{90} = 0.5 mm with a typical depth of 5 \(\mathrm {m}\) (Barua 2017). These conditions are typically found in tidal basins, such as in the Borndiep basin near Ameland (van Straaten 1954), and in the basin of Spiekeroog (Flemming and Ziegler 1995).
2.2.3 Boundary conditions
2.2.4 Wind forcing
2.3 Outline of solution method
Drawing from the work of Chen et al. (2015, 2016), a temporal Fourier transform is applied to the governing (1) and (2), which are solved for the individual modes in the frequency domain. The superposition of the individual solutions then gives the solution to the full problem in the time domain.
The full solutions of each mode are given in Appendix A.2.
2.3.1 Network
The solution for an entire network (see Fig. 1b,c) is obtained by solving the linear system of the channel equations and the corresponding boundary conditions at the nodes. First, for each mode, the flow solution for the network is derived. The summation of the \(2M + 1\) networkwide solutions then gives the evolution over the simulation period of all flow quantities at the nodes. The distribution of elevations and velocities inside each channel are obtained from summing up the expressions (20) and (21) in Appendix A.2.
Finally, the determination of the velocity scale \(u_{j, \text {peak}}\) for the bottom friction coefficient in formula (7) implies that the above procedure is nested in a loop: starting from an estimated guess and using an underrelaxation procedure, the velocity scale (u_{j,peak}) is adjusted iteratively until it corresponds to the actual solution (u_{j}). We require that the residual R between \(u_{j, \text {peak}}^{2}\) and \({u_{j}^{2}}\) should not exceed \(10^{3}\); with \(R^{2} = (u_{j, \text {peak}}^{2}{u_{j}^{2}})\) and threshold \(R<10^{3}\).
2.3.2 Simulations

The storm of 5 December 2013, known as Sinterklaas storm or Xavier storm (Watermanagementcentrum Nederland 2013). We have considered the measured windvelocity and direction at the station of Vlieland, located as in Fig. 1a. The time series are shown in Fig. 2a. The wind direction during the storm was almost NWly. In our simulations, both the magnitude and direction are unsteady;

An artificial episode with a schematized wind stress pattern (Fig.2b), for evaluating the influence that the storm rampup time \(T_{\text {ramp}}\), storm duration \(T_{\text {event}}\), and wind direction \(\theta \) have on the freesurface setup and setdown at the coast. The rampup time \(T_{\text {ramp}}\) is the duration of the rampup stage (same as rampdown). The duration of the storm event \(T_{\text {event}}\) is defined as the period between halfway the rampup and halfway the rampdown of the storm, and should be larger or equal to the rampup time (i.e., \(T_{\text {event}} \geq T_{\text {ramp}}\)). The rampup time can be varied (in the range \(0 \leq T_{\text {ramp}} \leq T_{\text {event}}\)) without changing the total “amount” of wind stress experienced by the system, as illustrated by the blue shaded areas. The peak wind stress is 1.25 N m^{− 2}, and the direction is fixed.
3 Results
3.1 Hindcast
To gain confidence in our model, we first present a hindcast of the 2013 Sinterklaas storm here. We will check both the qualitative and the quantitative performance of our model, by requiring that the simulated water levels \(\zeta _{modelled}\) do not show large phase lags compared to the measured water levels \(\zeta _{measured}\) and that the maximum water level during the surge lies within the 20% error range. The storm surge of 5 December 2013 has been simulated on both networks of Fig. 1b,c by applying the measured time signals of Fig. 2a as a timevarying and spatially uniform wind field over the basin. At the opensea nodes, the time series of elevation at Den Helder, Vlieland haven, and Wierumergronden have been applied, which include tide. The physical parameters are those given in Section 2. The numerical parameters are \(T_{\text {simul}}=\) 10 days and \(M=\) 127. As a result, the period of fastest resolved oscillations is approximately 1.9 \(\mathrm {h}\), sufficiently close to the interval of 1 h of the hydrometeo data.
The wind stress measured, during the 2013 Sinterklaas storm, at the KNMI measurement location at Vlieland (KNMI 2016) is used as spatially uniform wind forcing. This simulation uses measurement data from six measurement locations in and near the Dutch Wadden Sea (see the circle in Fig. 1), obtained from Rijkswaterstaat (2016). The time series of the measurement locations, Den Oever (DO), Kornwerderzand (KO), and Harlingen (HL), were used to assess the model performance. The time series of the measurement locations, Den Helder (DH), Vlieland Haven (VH), and Wierumergronden (WG), were used as boundary conditions for the model. For every tidal inlet, the nearest measurement location was used. For the Marsdiep, Vlie, and Frisian inlet, this means using the measurement data from the nearby measurement location. For the Eierland inlet, this means using the measurements from Vlieland Haven, and for the Borndiep inlet, this means using the data from Wierumergronden. The modelling error introduced by not having direct data for these two inlets is minor, due to the limited importance of these two inlets on the elevation deeper in the Wadden Sea (see Fig. 6).
Using the denser tidal network of Fig. 1b (Fig. 3, top) leads to slightly better quantitative agreement during the surge, than the coarser storm network of Fig. 1c (Fig. 3, bottom). This is clearly observable for the surge at stations Den Oever (DO) and Kornwerderzand (KZ), since the dots in the top panel are closer to the perfect agreement line. The agreement at the station of Harlingen is possibly affected by the fact that the networks do not allow flow towards the eastern Wadden Sea. Also note that the network nodes that can be associated to each stations are not exactly the same in either network. Before and after the surge (when the water levels are lower), the storm network (Fig. 3, bottom) performs better than the tidal network (Fig. 3, top). There is a reasonably good qualitative agreement between the modelled water levels \(\zeta _{modelled}\) and the measured ones \(\zeta _{measured}\) and the error in peak elevation. Nonetheless, the performance of the model is qualitatively correct and quantitatively acceptable at all three stations, remarkably in the lack of any ad hoc calibration.
Having considered that the tidal network gives only slightly better results, at the cost of accounting for double the number of channels, we proceed to use the storm network in the remainder of our analysis.
3.2 Sensitivity of the peak elevations to basin and storm parameters
3.2.1 Basin parameters
The sea level rise has been imposed by uniformly increasing the depth of the network channels. Its influence, shown in the left panel, results in an almost linear increase of elevations by the coast. Interestingly, the increase of the peak elevations by the coast is slightly smaller than the assigned sea level rise, which indicates that the combined effect cannot be reduced to the addition of surge and sea level rise.
Next, the middle panel shows the peak elevations against different southward dam displacements from its real position. Moving the location of the Afsluitdijk southwards increases both the wind’s fetch length and the basin’s water storage. The network has been extended by adding channels representing the Zuiderzee (as shown in Fig. 1c, in blue). A larger basin size increases the peak elevations at the fictional dam location because of the increased fetch, while reducing those at Den Oever and Kornwerderzand (at the real dam position) because of the basin’s wider extent. The peak elevations at Harlingen, further away from the bay entrance, are less sensitive to the repositioning of the dam.
Finally, the right panel shows the influence of the Manning’s roughness coefficient, \(\mu \), with a single value applied to the whole network. Higher roughness coefficient leads to lower elevations, since increased friction hampers the inflow of surge water into the network. The range of roughness coefficients ranges from unrealistically smooth (to identify resonance peaks); (i.e., \(\mu = 0.006\:\mathrm {s}\:\mathrm {m}^{1/3}\) and \(\mu = 0.012\:\mathrm {s}\:\mathrm {m}^{1/3}\)) through earth channels (clean and straight: μ = 0.018 s m^{− 1/3} and winding: \(0.024\:\mathrm {s}\:\mathrm {m}^{1/3}\)) to rubbly (μ = 0.03 s m^{− 1/3}), stony (μ = 0.036 s m^{− 1/3}), and cobblebottomed (μ = 0.042 s m^{− 1/3}) channels (Chow 1959). The range of the roughness coefficients used here is the same as that in the frequency response analysis of Section 3.3.1. Within the range of realistic parameters (i.e., \(\mu = 0.018\:\mathrm {s}\:\mathrm {m}^{1/3}\) to μ = 0.042 s m^{− 1/3}) the impact of the uncertainty of the roughness coefficient, on the surge level at the coast is below 0.5 m. It should be noted that different estimates can be obtained when the roughness has a spatial distribution of its own.
3.2.2 Storm parameters
The parameters defining the artificial storm profile of Fig. 2b have been varied to highlight some influences on the peak elevations. These parameters are the wind direction 𝜃, the peak duration \(T_{\text {event}}\), and the rampup time \(T_{\text {ramp}}\). Unlike in Section 3.2.1, here we have isolated the effect of wind stress from the other forcing in our model (i.e., we take the elevation signal at the inlets equal to 0). This is also a pragmatic step, since we do not know the elevation at the inlets during the artificial wind event. This approach is justified by the linearity of our model; the solution to the fully forced model is the superposition of the individual solutions of the separate forcings, except for the frictional iteration.
3.3 Frequency response analysis
The formulation of a timevarying process in the frequency domain makes it possible to dissect the causal relationship between external forcings and resulting flow fields into the constituting individual modes. To this end, we consider as many forcing scenarios as there are modes: in the mth scenario, the mth mode in the forcing has unit amplitude, while all other modes are 0. Again, as in Section 3.2.2, this is motivated by the model’s linearity. The corresponding solution highlights the degree of sensitivity of the overall response to that unimodal unit forcing. Therefore, we can identify the modes in the forcings (external hydrography, wind shear) that are conducive to higher responses (elevations) at any selected location (the coastal stations).
The forcing factors considered separately are the elevations assigned at the network open boundaries (the tidal inlets), and the direction of the wind stress vector, these simulations have been carried out on the storm network.
3.3.1 Forcing of elevations at the tidal inlets
Higher friction coefficients (μ = 0.033 s m^{− 1/3} and \(\mu = 0.042\:\mathrm {s}\:\mathrm {m}^{1/3}\)) result in a lowering of the response for all cases. This is due to the increased friction holding back the flow of water and reducing the response at the measuring stations further in the basin. Lower friction coefficients (μ = 0.015 s m^{− 1/3} and \(\mu = 0.006\:\mathrm {s}\:\mathrm {m}^{1/3}\)) lead to strong variations in the frequency response at the observation locations. There are clear response peaks around \(\omega = 3 \times 10^{4}\ \mathrm {rad\ s}^{1}\); these are observed at all three locations for forcings originating at one of three inlets (Marsdiep inlet, Eierland inlet, and Vlie inlet). In practice, this maximum response suggests that the basin experiences resonance due to waves of this precise frequency. For the two other inlets (Borndiep inlet and Frisian inlet), resonance peaks can be observed at different frequencies. We also observe that the effect of the friction coefficient is small for modes with a low angular frequency. It is at these low angular frequencies that the power of a storm is concentrated (gray bars in Fig. 6), since they are lowfrequency events (i.e., nonrecurring) in our simulation window. So, the effect of changes in roughness coefficient will be limited for the storm surge.
The range of the roughness coefficients used here is the same as that in the sensitivity analysis for the roughness coefficient in Section 3.2.1.
3.3.2 Wind direction
4 Discussion
The networkbased idealized model for the Wadden Sea described in this study appears to be wellsuited for gaining insight in the behavior of unsteady storm surges in a semienclosed tidal basin. The hindcast of the storm surge of 5 December 2013 presented in Section 3.1 agrees with measurements within 25% of magnitude and with small phase errors at the peak time, all the simplifications of its construction and settings notwithstanding. Here, we will discuss the model assumptions and the factors that affect storm surges.
4.1 Critique of the model assumptions
One of the model simplifications is the linearized form of the shallowwater equations. Neglecting nonlinear dynamics implies the neglect of tidesurge interactions, the relevance of which has been noted by Spencer et al. (2015) in their study of the same 2013 storm surge on the English coast: “Storm surge impacts are not simply linearly related to maximum elevation but rely on more complex, nonlinear interactions between tidesurge condition.” Nonetheless, there is reasonable agreement between measured and simulated elevations for the 5 December 2013 storm surge.
Furthermore, Prandle and Wolf (1978) signal that on the Thames estuary the dominant interaction mechanism between tides and storm surges is nonlinear (quadratic) friction. Instead, our model implements a channelwise, timeinvariant, linearized parametrization of bottomfriction using the peak velocity as scale. Also, this friction coefficient overestimates bottom friction before and after the storm, when the actual velocities are smaller. Overcoming this limitation is already the focus of ongoing research (Roos et al. 2017).
Along the same lines, Horsburgh and Wilson (2007) explained the surge clustering at the time of rising tide mathematically as the consequence of a tidal phase shift combined with the modulation of surge generation due to water depth. Another assumption of linear dynamics that water elevations are small in comparison to the water depth becomes less realistic under storm conditions.
Here, we should notice that Lorentz’s networks do not allow the water to flow further into the eastern Wadden Sea, which naturally occurs and can be an important factor in the basinwide surge dynamics, hence of which storms actually generate severe surges (Lipari et al. 2008 and DuranMatute et al. 2016). The importance of this is not reflected in our model results: we found that the two easternmost inlets only have a limited effect on the rest of the basin (see Fig. 6).
4.2 Determinants of the surge elevations
Storm surges in tidal basins are caused by a local wind effect, a setup of water at the tidal inlets (e.g., as an indirect wind effect), and atmospheric pressure effects (neglected in this study). Our linear model is capable of simulating storm surges in tidal basins and, by linearity, allows us to study the effect of the different forcing mechanisms separately. Both the wind and elevation in the tidal inlets show a stronger response at lower frequencies (and thus longer periods), at locations Den Oever, Kornwerderzand, and Harlingen in the Wadden Sea. The elevation at the Marsdiep and Vlie inlets (the two dominant inlets) shows peak responses around \(\omega = 3 \times 10^{4} \text { rad s}^{1}\), with lowering responses at lower frequencies (for \(\mu = 0.0242\) s m^{− 1/3}). The wind response shows distinct peak values at frequencies in the range of \(\omega = 0 \text { rad s}^{1}\) to ω = 4 × 10^{− 4} rad s^{− 1}. At higher frequencies, the response is significantly lower. Waves with a long period are only present in storms with a sufficiently long duration. Therefore, storm surges in tidal basins require a minimum duration before they attain their maximum elevation.
We studied this minimum duration by focusing only on the locally generated storm surge (i.e., the local wind effect). We found that a minimum duration in the order or several hours (3–4 h) is required for the wind stress in the domain to generate its maximum surge. The largest setup is obtained for winds from the NW; and the largest setdown, for winds from the SE.
The minimum duration required for reaching the peak surge is caused by the noninstantaneous adaptation of the water to changes in the forcing (e.g., a sudden increase of wind stress, or an increase of water in the inlets). It is important to realize that storms in real life never have a constant wind stress, but rather a timevarying wind stress (as can be seen in Fig. 2 for the 2013 Sinterklaas storm). Therefore, during a storm, the system constantly adjusts to the evolving timedependent forcing instead of aiming at steadystate equilibrium.
The influence of the elevations imposed at the tidal inlets on the surge at the coast can be ascribed to two factors: the inlet size and the distance from the tidal inlet, given the network. The narrower Eierland Inlet has a trailing influence on the water elevations at the output stations Den Oever and Harlingen, regardless of the proximity to it. Then, the response generated by the conditions set at the tidal inlets fades with the distance from them. For example, the Texel Inlet influences Den Oever more than elsewhere, and likewise for the Vlie Inlet and Harlingen. Furthermore, there is a small response at Den Oever, Kornwerderzand, and Harlingen to water level variations in the Borndiep Inlet and Frisian Inlet. The wind forcing shows an opposite relationship with the distance from the boundary shoreward because of the fetch length. Locations with a shorter fetch length (when considering a NW storm), such as Den Oever and Harlingen, show smaller surge response to a unit wind stress than locations with a larger fetch length like Kornwerderzand.
5 Conclusion
We have developed a new idealized network model for storm surges in the Wadden Sea, inspired by the model of the State Committee for the Zuiderzee (1926), and extended with a timedependent wind stress (both in magnitude and direction). We probed the validity of our approach by simulating the 5 December 2013 Sinterklaas storm, leading to sufficient confidence in our model. The effect of basin characteristics on the transient behavior of storm surges was studied using our full model (i.e., including wind forcing over the domain and elevation signals forced at the inlets). The effect of storm characteristics on storm surges was studied using only wind forcing.
To study the effect of basin characteristics on storm surges, we investigated the impact of changes in sea level, basin extension, and friction coefficient. We found that sea level rise, expressed in deepened channels, results in an increase of the surge height, slightly less than the amount of sealevel rise imposed. The effect of a basin extension, up to the historic situation before the closure of the Zuiderzee, was found to be that the surge height increases at the back of the basin. In the middle of the basin, near the presentday location of the Afsluitdijk, we found lower elevation. The sensitivity of the model to the roughness coefficient showed that an increase in bottom roughness results in a lower surge and vice versa.
The effect of storm characteristics on the transient behavior of storm surges was studied by varying the rampup time, duration, and wind direction for an artificial wind event. We found that the effect of the rampup time is limited, whereas a minimum duration is required for a surge to attain its maximum level. The wind direction has a clear effect on the setup, winds from a NWly direction result in the highest surge, followed by winds from a Nly and Wly direction. A setdown is obtained when the wind direction is from a Ely direction to a Sly direction, with a maximum setdown for winds from a SEly direction.
The effect of different forcing mechanisms on the transient behavior of storm surges was studied by dissecting the solution of our model. This has been done for both the wind forcing and the elevation signal that is forced at the tidal inlets. For the elevation signal, we found that the response in the basin is affected by the distance from the inlet, and by the size of the inlet. Larger inlets have a larger impact on the basin, and inlets have a larger impact on the part of the basin that is close to them. The wind forcing was shown to have the largest effect near the coastal boundary of our basin. There the fetch is the largest, resulting in the largest setup of water against the coast.
Notes
Acknowledgements
The authors thank H.E. de Swart for his feedback and knowledge of the work of the State Committee for the Zuiderzee. J.R. de Graaf, J.C. Hazeleger, and M.D. Krol are thanked for their contribution to the sensitivity analysis. The remarks of the anonymous reviewers stimulated the achievement of valuable insights.
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