For this study, the three-dimensional hydrodynamic numerical model UnTRIM2 (Casulli, 2008) is used, which solves the three-dimensional shallow-water equations and the three-dimensional transport equation for salt, suspended sediment and heat on an orthogonal unstructured grid (Casulli and Walters, 2000). A special feature of UnTRIM2 compared to its predecessor, UnTRIM, is the subgrid technology, which allows a high resolution of the topography independently of the computational grid (Casulli, 2008). The computational grid cells can be wet, partially wet or dry, allowing for a precise mass balance and realistic wetting and drying (Sehili et al., 2014), which is important for a realistic representation of the large intertidal areas in the German Bight. The variation of the surface drag coefficient with wind speed is parameterised according to Smith and Banke (1975). The generation of wind waves as well as sediment and heat transport are not calculated in the model setup used for this study in order to reduce computational effort.

The regional model we use is very similar to the model used by Rasquin et al. (2020). The model domain covers the German Bight from Terschelling in the Netherlands to Hvide Sande in Denmark including the estuaries of the Elbe, Weser and Ems rivers with their main tributaries (Fig. 1). The model boundary is defined by the dike line and thus cannot be overflowed. The resolution of the computational grid ranges from 5 km at the open boundary to about 100 m in the coastal area and the estuaries. In the Wadden Sea region and the estuaries, a higher subgrid resolution of about 10 to 50 m in the finest part is used. The model has vertically fixed layers with a resolution of 1 m up to a depth of 27.5 m and a resolution of 10 m below this depth. The topography data of the year 2010 implemented into the model were generated in the EasyGSH-DB project (Sievers et al., 2020).

The atmospheric forcing over the model domain (wind field at 10 m height and surface pressure) is derived from COSMO-REA6 (Bollmeyer et al., 2015). The data are generated and made available by the Hans Ertel Centre of the University of Bonn in cooperation with Germany's National Meteorological Service (Deutscher Wetterdienst, DWD) (Bollmeyer et al., 2015). Salinity is set to a constant value of 33 psu at the open boundary (based on Klein et al., 2016) and 0.4 psu (based on Michael Bergemann, personal communication, 2009) at the upstream boundary of the Elbe estuary.

The German Bight model is used to simulate a spring–neap cycle in July 2013 with a constant river discharge at the upstream boundary of the Elbe estuary of 600 m3s-1. The weir in Geesthacht is open in the model during the simulated period. The selected period and discharge are chosen to estimate changes in average conditions without extreme events.

Water levels at the seaward open boundary of the German Bight model are provided by the Dutch Continental Shelf Model (DCSMv6FM) (Zijl, 2014), a 2-D hydrodynamic model which covers the northwestern European shelf and which is a further development of DCSMv6 (Zijl et al., 2013, 2015) (Fig. 1). DCSMv6FM uses the flexible mesh technique D-Flow FM (D-Flow Flexible Mesh) (Kernkamp et al., 2011) based on the classical unstructured grid concept. At the seaward open boundary, the DCSMv6FM model is forced by the amplitudes and phases of the 22 main diurnal and semidiurnal constituents, which are derived by interpolation from the data set generated by the GOT00.2 global ocean tide model (Ray, 1999). Sixteen additional partial tides are adopted from FES2012 (Carrère et al., 2013). As for the smaller regional model, the atmospheric forcing for DCSMv6FM is derived from COSMO-REA6 (Hans Ertel Centre for Weather Research; Bollmeyer et al., 2015).

SLR is added at the open boundary of the German Bight model; therefore, SLR-induced changes in tidal dynamics seaward of the German Bight are neglected. Ideally, SLR would be added at the boundary of the shelf model to consider changes in tidal dynamics in the continental shelf seaward of the German Bight model boundary. However, this approach is not suitable in our case, since the resolution of DCSMv6FM is insufficient for estimating SLR-induced changes (Rasquin et al., 2020).

Since the focus of our study is on the Elbe estuary, a brief validation of the model for this specific region is presented below. Further analysis of the model's performance can be found in Rasquin et al. (2020). To compare the simulation results with observations, we simulated seven spring–neap cycles between January and April 2013 with measured river discharge provided by the Federal Waterways and Shipping Agency (WSV, 2022). A comparison of water levels between model results and observations at 39 gauges in the model domain for this period reveals a mean RMSE of 16.4 cm, a mean bias of 7.3 cm and a mean skill score after Willmott et al. (1985) of 0.993. Figure 2 shows a visual comparison between the simulation result and the observation of the water level at the stations of Cuxhaven (mouth of the estuary) and St Pauli (close to the port of Hamburg) for a short period of time. It can be seen that the phase and shape of the vertical tide are well reproduced by the model, but tidal low water is slightly higher in the simulation results compared to the observations. A similar display for an entire spring–neap cycle is shown in Fig. A2 Appendix A. It shows that there are no distinctive differences in performance during the different phases of the illustrated spring–neap cycle.

Figure 3 shows the mean tidal range (TR) and mean tidal mean water level (MW) along the estuary for the seven spring–neap cycles, calculated for both the simulation results and observational data at stations along the estuary (WSV, 2022). It demonstrates that the model is able to reproduce the characteristic development of TR along the Elbe estuary, with a strong increase starting at around kilometre 60, a maximum reached around kilometre 115 and a decrease further upstream. As a result of the too high tidal low water, the model underestimates TR by 10–20 cm and overestimates MW by around 10 cm. The RMSE data for the mean TR, MW, tidal high water (HW) and tidal low water (LW) for the gauges Cuxhaven, St Pauli and the mean of 15 gauges in the Elbe estuary are shown in Table 1.

To investigate the effect of SLR and potential corresponding tidal flat growth, several scenarios are simulated and analysed. Two SLR scenarios of 55 and 110 cm were simulated by adding SLR at the open boundary of the German Bight model. According to the IPCC Sixth Assessment Report (AR6), mean global SLR in 2100 compared to the reference period 1995–2014 will be in the likely range between 0.43 and 1.01 m for the intermediate- to high-emission scenarios (SSP2-4.5, SSP3-7.0 and SSP5-8.5) (Fox-Kemper et al., 2021). Our selected SLR scenarios are close to the median of the intermediate scenario and close to the upper range of the high-emission scenario for 2100. Projected values for regional SLR in the southeastern North Sea region (Delfzijl, Cuxhaven, Esbjerg) are within a range of ±20 cm of the median of the projected global mean SLR until 2100 (Fox-Kemper et al., 2021; Garner et al., 2021).

As mentioned in the Introduction, there are uncertainties and difficulties in quantifying to what extent the tidal flats in the German Bight will be able to keep up with future accelerated SLR. The amount of tidal flat accretion can strongly differ between the tidal basins of the German Bight and not only depends on future SLR acceleration and magnitude but also on sediment availability and meteorological conditions. To gain a better understanding of the possible effects of potential SLR and tidal flat growth in the Elbe estuary, we analyse a range of 0 %, 50 % and 100 % tidal flat growth with SLR. In these scenarios, all tidal flat areas in the model domain are uniformly elevated by the same amount, which is a highly simplified assumption.

The scenarios with tidal flat growth are further differentiated firstly by elevating the tidal flats in the German Bight and in the mouth section of the Elbe estuary (Scenario A) and secondly by elevating the tidal flats in the German Bight, the mouth section and also the lower section of the estuary (Scenario B) (see Fig. 4). The purpose of this differentiation inside the estuary is to gain a better understanding of the estuarine system in the context of SLR and tidal flat elevation. Table 2 summarises the scenarios simulated with the German Bight model. The letters “a” and “b” in the names stand for Scenarios A and B, while “tf” stands for “tidal flats”.

In Fig. 7, tidal range (TR) is visualised for the different simulated scenarios along the profile of the estuary. Other parameters of the vertical tide (high water (HW), low water (LW), mean water (MW)) can be found in Appendix A (Fig. A1). We focus on the results of the scenarios with 110 cm SLR to gain a better system understanding. The results of the scenarios with SLR of 55 cm are not shown here in detail but are included to determine whether the changes due to SLR are in principle similar for a different SLR scenario. As apparent in Fig. 7, the mean tidal range (TR) in the Elbe estuary increases in the upstream direction to reach a maximum in the Hamburg section and decrease subsequently. The simulation result for the scenario with SLR of 110 cm (slr110) reveals an increased tidal range (TR) due to lower LW and higher HW relative to the applied SLR of 110 cm. TR is increased by up to around 15 cm in the mouth section and by around 10 cm in the lower section. Upstream of the Hamburg section the increase intensifies, mainly due to a lower LW. If tidal flats are elevated by 50 % with SLR in scenario A (slr110tf55a), no notable changes in TR are visible compared to scenario slr110. If, however, tidal flats in the German Bight and the mouth section of the estuary are fully (100 %) elevated with SLR (slr110tf110a), TR is damped relative to scenario slr110. In contrast, an additional elevation of the tidal flats in the entire Elbe estuary (slr110tf110b and slr110tf55b) strongly increases TR relative to scenario slr110, especially in the lower section.

For all scenarios, the maximum value of TR along the estuary is reached in the Hamburg section. Table 3 lists the changes in maximum TR relative to the reference condition (maximum TR=3.87 m) for all simulated scenarios with SLR 110 cm as well as SLR 55 cm. Maximum TR increases by 6.5 cm for a SLR of 55 cm and by 12.5 cm for an SLR of 110 cm, which is about 11 %–12 % of the respective SLR. Both SLR scenarios with 100 % tidal flat elevation in Scenario A (slr55tf55a and slr110tf110a) show an increase in maximum TR that is less than with SLR alone. In both SLR scenarios with 100 % tidal flat elevation in Scenario B (slr55tf55b and slr110tf110b) the increase in maximum TR is greater than in the scenarios without tidal flat elevation.

Our estimated value for the estuary's convergence length (La) in the reference condition is 46.5 km. This value lies in the same order of magnitude as the values estimated by Dronkers (2017) (42 km) and Savenije et al. (2008) (30 km) for the Elbe estuary. Scenario slr110 results in a significant decrease in La and therefore a stronger convergence of the Elbe estuary relative to the reference condition. In scenario slr110tf110a, upstream convergence weakens relative to slr110, which results in an La close to the reference condition.

A change in La is the result of differing changes in the cross-sectional area (A) along the estuary in the upstream direction, which are due to regional dissimilarity in cross-sectional geometry. As discussed in Friedrichs et al. (1990), changes in intertidal storage capacity, cross-sectional flow area and channel width due to SLR are strongly dependent on the gradient of the estuary banks. The model results correspondingly show a stronger increase in the cross-sectional flow area in the mouth section of the Elbe estuary where tidal flat areas (meaning low topographic gradient) are larger compared to other sections. It can therefore be assumed that the stronger convergence of A as a result of SLR is due to the decrease in the amount of relative intertidal area further upstream in the estuary (Fig. 10). This effect is sketched in Fig. 11. Tidal flat elevation decreases A regionally and seems to significantly counteract SLR-induced changes in convergence in scenario slr110tf110a but not in the other scenarios.

In the reference scenario, we derive a mean hydraulic depth averaged over the entire estuary (up to the weir in Geesthacht) of 7.0 m for ht (including intertidal area) and 9.2 m for hc (excluding intertidal area). In comparison, Savenije et al. (2008) listed a mean depth at MW of 7.0 m, increasing to 9.0 m further upstream for the Elbe estuary, and Dronkers (2016) listed a time-averaged channel depth of 10.0 m for the Elbe estuary. However, it is not clear how these numbers were derived.

Our simulation results for the SLR scenarios might be unexpected and counter-intuitive since they show that SLR of 110 cm does not, in general, cause an increase in mean hydraulic depth along the estuary. To the contrary, the mean hydraulic depth shows varying changes and even a decrease in some parts of the estuary for slr110 relative to the reference scenario. These differing changes in mean hydraulic depth along the estuary are caused by the varying topographic gradients of the control volumes. A decrease in hc due to SLR can be explained by an effect where the shallow areas next to the previous channels become part of the enlarged channels (Friedrichs et al., 1990); due to an elevated LW, some of the areas next to the channel that previously belonged to the intertidal zone can develop into subtidal areas and thus become part of the channel (Fig. 12). Because of the relatively small water depth in this new part of the channel, the hydraulic depth averaged over the channel cross-section can decrease. Moreover, elevated HW can cause the shallow previously supratidal areas to become intertidal areas (see Fig. 12), which explains the decrease in ht. However, if tidal flats are elevated with SLR in the model, this results in a regional increase in mean hydraulic depth relative to slr110 and relative to the reference condition in the Elbe estuary. Tidal flat elevation counteracts the abovementioned effect where shallow areas become part of the subtidal and intertidal cross-sections and this overall results in a greater mean hydraulic depth due to SLR in these scenarios.

For the reference condition, we analyse a mean ϙSINT of 0.4 for the entire estuary, which is slightly lower than the value of 0.5 derived by Dronkers (2016) for the Elbe estuary and in the range of 0.412 to 0 (decreasing in the upstream direction) given by Savenije et al. (2008). Note that Dronkers (2016) and Savenije et al. (2008) used a different form (ratio of width at HW to width at LW) and that these numbers are converted for comparability.

According to our simulation results, an SLR of 110 cm causes regionally, strongly and widely scattered changes in ϙSint along the estuary with a decrease in some control volumes and an increase in others. Tidal flat elevation counteracts these changes regionally. The varying changes along the estuary can be explained by the differing topographic gradients. In general, SLR can either cause an increase or decrease inSINT, or it causes no change at all, depending on the local topographic gradient above LW and a potential change in TR (see Fig. 12). An increase in SINT can result from previously supratidal areas (above old HW) becoming part of the SINT due to SLR (Dronkers, 2016), and/or it can be caused by an increased TR. A decrease in SINT can occur in tidal systems that are contained by dikes or restricted by a strong-gradient topography. In such cases, the size of previous SINT that becomes subtidal area (SLW) can be larger as a result of SLR than the size of a previously supratidal area that becomes SINT due to SLR (Dronkers, 2016) (see Fig. 12). However, a decrease in SINT can also be caused by a decrease in TR.

The effects of the previously discussed changes in geometric parameters on tidal dynamics act simultaneously and can therefore counteract, outweigh or enhance each other in the resulting impact on TR. However, we want to point out correlations between the detected changes in geometry and in TR to find explanatory approaches for the latter.

The simulation results show an increased TR in the estuary in scenario slr110 relative to the reference condition. The analysis of the upstream convergence of the cross-sectional flow area (A) accordingly shows a significant increase in convergence in scenario slr110 relative to the reference condition. We assume this is the main reason for the increased TR in slr110, as the gradual convergence in width and depth causes an amplification of the tidal wave according to Green's law (1837) (Sect. 2.3.3). Averaged over the different sections, a decrease in ϙSINT in the mouth section and an increase further upstream are detected. These could contribute to or counteract the increase in TR respectively (Sect. 2.3.5). According to our analysis, SLR of 110 cm does not cause a general increase in mean hydraulic depth, but causes strongly varying changes along the estuary. Averaged over the sections, mean hydraulic depth remains approximately unchanged relative to the reference condition for the largest part of the estuary. In the Hamburg section and the upper section, a greater mean hydraulic depth is detected. The hydrodynamics in this upper part of the estuary are not fully dominated by the tide but are also highly influenced by discharge. We assume the strong increase in TR in these sections to be caused by an increase in tidal influence relative to discharge influence.

In scenario slr110tf55a, TR is the same as in slr110. Accordingly, an analysis of convergence shows no significant changes compared to slr110. In contrast, if the tidal flats in Scenario A are elevated by 100 % with SLR (slr110tf110a), TR decreases relative to slr110 along the entire estuary. Our analysis accordingly shows a significant decrease in convergence for slr110tf110a compared to slr110, which might be the main reason for the decrease in TR. On average, the mouth section is characterised by an increase in both relative intertidal area and mean hydraulic depth, the impacts of which might counteract each other.

In scenario slr110tf55b, TR strongly increases relative to SLR alone (scenario slr110) in the entire estuary, while in scenario slr110tf110b a decrease can be seen in the mouth section and an increase is observed further upstream. In both scenarios, a significant change in convergence relative to slr110 cannot be detected with a significance level of α=0.1. The average of ϙSINT in the lower section shows a decrease for scenario slr110tf110b compared to scenario slr110 which might partly explain the increase in TR. However, a notable decrease in slr110tf55b cannot be seen. Therefore, we assume that in the scenarios slr110tf55b and slr110tf110b the main reason for the increase in TR is the change in mean hydraulic depth, which increases as the tidal flats are elevated. In contrast to the scenarios where the tidal flats are only elevated in the mouth of the estuary, mean hydraulic depth increases in a much larger part of the estuary and is therefore most likely to have a stronger effect on TR. As mentioned in Sect. 2.3.4, changes in water depth influence frictional damping of a tidal wave due to energy dissipation, and they can also push a system closer to or further away from resonance. Whether the increase in TR due to increased mean hydraulic depth is mainly caused by a decrease in frictional damping or by a shift towards resonance is a question that needs further investigation.

Seiffert and Hesser (2014) simulated the effects of 80 cm SLR without topographic changes in the Elbe estuary and found an increase in TR, which is in accordance with our results. Jordan et al. (2021) did not focus on the estuaries when investigating the effects of SLR and tidal flat elevation on tidal dynamics in the Wadden Sea. Nevertheless, their results show a slight increase in M2 amplitude in the Elbe estuary due to SLR of 80 cm without tidal flat elevation. They also show a much stronger increase when the tidal flats are elevated in the entire estuary, which qualitatively corresponds to our results. Du et al. (2018) analysed tidal response to SLR in different types of idealised estuaries and for different realistic estuaries in the USA. The authors point out the relevance of length, convergence and lateral bathymetry of estuaries for the resulting changes due to SLR. In contrast to our study, they tried to find explanatory approaches for the changes in realistic estuaries by matching their geometric characteristics to the different types of geometry of idealised estuaries (e.g. differences in length, convergence and cross-sectional bathymetric gradient), but they did not analyse the changes in geometry due to SLR. Without further evaluating, they mention the possible change in convergence characteristics under SLR conditions and spatially variable tidal response due to spatially variable lateral geometry (e.g. size of intertidal area).

In our study, SLR-induced changes in tidal dynamics seaward of the German Bight model are neglected. Previous research by Jordan et al. (2021) shows large-scale changes in M2 amplitude in the North Sea due to SLR. Relating the results of Jordan et al. (2021) to our model boundary, we neglect changes in the M2 amplitude that are less than ±2 cm. We assume that this has no bearing on the key results of our study, which aims to improve system understanding of the changes in the Elbe estuary induced by SLR and tidal flat growth.

To reduce computational effort, the generation of wind waves as well as sediment and heat transport is not included in our model setup. Hence, potential changes in sediment dynamics, e.g. changes in the ETM (estuarine turbidity maximum) and their potential effect on tidal dynamics, are neglected. Furthermore, our investigation does not include potential future changes in river discharge into the Elbe estuary, as the discharge in the model is kept constant (600 m3s-1).

We selected the SLR scenario of 110 cm with corresponding hypothetical tidal flat elevation scenarios which we analysed in detail. For scenarios with 55 cm SLR, we found qualitatively similar changes in maximum TR and therefore assume similar alterations in estuarine geometry. However, to ensure that our results are in principle applicable to other scenarios than an SLR of 110 cm, it would be necessary to simulate a range of several SLR scenarios with their corresponding tidal flat growth scenarios and analyse the changes in tidal dynamics and estuarine geometry for each of them.