Sea-level rise induced amplification of coastal protection design heights

Coastal protection design heights typically consider the superimposed effects of tides, surges, waves, and relative sea-level rise (SLR), neglecting non-linear feedbacks between these forcing factors. Here, we use hydrodynamic modelling and multivariate statistics to show that shallow coastal areas are extremely sensitive to changing non-linear interactions between individual components caused by SLR. As sea-level increases, the depth-limitation of waves relaxes, resulting in waves with larger periods, greater amplitudes, and higher run-up; moreover, depth and frictional changes affect tide, surge, and wave characteristics, altering the relative importance of other risk factors. Consequently, sea-level driven changes in wave characteristics, and to a lesser extent, tides, amplify the resulting design heights by an average of 48–56%, relative to design changes caused by SLR alone. Since many of the world’s most vulnerable coastlines are impacted by depth-limited waves, our results suggest that the overall influence of SLR may be greatly underestimated in many regions.

Nonlinear effects are assessed using a storm tide model which combines the coupled effects of tides, surge, and wind waves with an average horizontal resolution of 1 km. The model is forced with the 20 th Century Reanalysis meteorological data 16 and local SLR 17 from 1970-2013, and was previously validated against observational data 13 (see also Fig. 1s). We next apply the same meteorological forcing to three SLR scenarios: the median of estimates by 2100 associated with RCP4.5, RCP8.5 18 and a high-end scenario based on RCP8.5 19 (hereafter RCP8.5HE) (see Methods) ( Fig. 1c). Coastal design heights are estimated from common run-up formulas (see ref. 20) using storm tides and waves as input (see Methods). All relevant storm tide and wave combinations at the ARI 100 are obtained introducing a novel application of bivariate design height assessment based on copulas (e.g. ref. 21), considering that storm tide and wave magnitudes are partially correlated (see Methods). As shown in Fig. 2a, the multivariate copula analysis yields an ensemble of events, each instance of which describes a particular design height resulting from combinations of storm tides and waves that together have a 0.01 annual probability of occurrence (i.e. the ARI 100 ). From the "elbow shaped" contour line, we choose the most-damaging ARI 100 event, i.e. the one causing the largest wave run-up height. These coastal design heights (red circles in Fig. 2a) are the most-likely source of dike-failure 20 .

Results
Water level changes. Our simulations show that storm tide levels outpace SLR at most locations ( Fig. 3a-c).
Under RCP4.5 (Fig. 1c), storm tides at the ARI 100 increase by up to 0.67 m, exceeding the expected SLR (0.54 m) by up to 0.13 m. Similar excess values of 0.14 and 0.17 m are observed under RCP8.5 (SLR = 0.71 m) and RCP8.5HE (SLR = 1.74 m) scenarios, respectively. The changes are spatially heterogeneous and non-linear increases tend to be the least in areas adjacent to large tidal streams, i.e. where depths are larger (see Figs 1b and 3a-c). Changes in storm tides can be explained by a reduction of both the effective friction and shallow water effects. In linearized tidal equations, the friction term is inversely proportional to depth (e.g. ref. 20  increases without geomorphic adjustment (our assumption), shallow water areas will increasingly be subject to less frictional damping and tidal deformation 13 . Wave-current interaction (at the surface and bed) changes as the water level increases, also altering energy dissipation in the sea. This decreases the observed damping and depth dependence of tides during storm events, and helps explain why tidal ranges increase ( Fig. 2s) (see also ref. 13). The increase in tidal range counteracts the meteorological component of the storm tide (i.e., surge), which is observed to decrease by 1.8% (RCP4.5), 2.3% (RCP8.5), and 5.1% (RCP8.5HE) on average (not shown here) as sea-level rises. Storm surge is subject to the same constraints as tides, and reduced bottom friction or increased depth can amplify the surge in estuaries (e.g. ref. 3); in the North Sea, however, the decreased bottom friction appears to be counteracted by the lessened effectiveness of surface wind stress. Basically, the same wind forcing (surface stress) is less effective at dragging water and produces a smaller surge when water becomes deeper.
Overall, tidal changes outweigh surge changes (Fig. 4a). This effect is illustrated on the western side of the Wadden Sea island Pellworm, where the mean tide range over all considered storm events (mean storm tide range) increases by 0.32 m (RCP4.5), 0.4 m (RCP8.5), and 0.77 m (RCP8.5HE) (Fig. 2s), a phenomenon consistent with observational evidence in this region (e.g. refs 14 and 22). Although the amplification is more pronounced in the tidal low water levels (Fig. 2s), the tidal high water levels (which are relevant to design) of almost all scenarios significantly increase by a factor of up to 1.24 faster than when SLR is considered alone (see Fig. 4a and Table 1s). Interestingly, the predicted sensitivity of tides to an incremental increase in sea-level diminishes when depths become large; as a result, in the extreme RCP8.5HE scenario, depth-induced decreases in surge outweigh increases in high water. These observations again highlight the complexity on non-linear interactions.
Wave changes. Simulations suggest that waves and wave run-up height are much more sensitive to SLR than tides or surge ( Fig. 3d-f), for the shallow bathymetry considered here. At exposed, westward oriented locations, wave magnitudes at the ARI 100 exceed modern conditions by up to 0.25 (RCP4.5), 0.33 (RCP8.5), and 0.78 m (RCP8.5HE) (Fig. 3d-f). Positive increases are observed even in the more protected areas on the lee-side of barrier Islands.
The observed amplification in wave run-up is driven by a predicted decrease in wave breaking away from the coast (see Fig. 5a). Depth-induced breaking occurs when waves propagate into very shallow areas, and the wave height can no longer be supported by the water depth. An empirically based criterion states that wave breaking occurs at an average breaker parameter γ (wave height to depth ratio) of ~0.78 (here only used to explain general mechanisms), but significant variation is observed with wave conditions and the bathymetry 23 . As sea-level increases, fewer waves exceed this criterion. Hence, for a given wave period, larger waves impinge on coastal defenses, to a degree related to the percent SLR. Moreover, increased depth allows longer period waves to reach the coast, due to decreased non-linearity and wave steepness at low frequencies. Both effects increase the amount of energy impinging on coastal defenses, and produce a shift in both the significant wave height and its period (see Fig. 5b   Waves at the coast are smallest during low water due to the filtering effect, remain low during ebb and flood due to wave-current interaction effects, but are elevated during low current, high water periods (Fig. 3s). As SLR continues, the natural bathymetric protection caused by shallow water reduces and coastal structures will increasingly observe more open-ocean like wave conditions. This effect is also evident in our modeled dependency between storm tides and waves; typically, a larger storm tide results in larger wave magnitudes, up to a 'saturation' depth at which waves are no-longer filtered. As sea-level increases, the same saturation depth is reached for smaller storm tides. Therefore, the observed Kendall's rank correlation coefficient between waves and storm-tides decreases from 0.4 (HIS) to 0.37 (RCP4.5), 0.36 (RCP8.5), and 0.32 (RCP8.5HE). In effect, waves and storm tides become more independent with SLR, subtly affecting the bivariate statistics.
Design height changes. The joint, coupled influence of changing storm tides and waves on design heights is highlighted in Fig. 3g-i. Along exposed westward oriented coastlines, changes in design heights clearly exceed the simultaneous SLR, with values ranging from 56 to 122 cm (RCP4.5), 73 to 157 (RCP8.5), and 181 to 357 cm (RCP8.5HE). Over the entire region, predicted changes in design heights exceed SLR by an average of 48-56% (slightly decreasing with larger SLR scenarios) ( Fig. 4s and Table 1s). At exposed locations such as Pellworm Island, the change in design height is more than doubled relative to SLR alone (Fig. 4c, Fig. 4s, and Table 1s).

Discussion
Our results have broad implications for coastal impact studies: until recently, climate change studies have largely focused on the effects of SLR 8,18 and/or changing storm tracks 7,9 . However, our analysis shows that in shallow coastal areas, wave heights, tides, and surges are strongly correlated with SLR, and with each other. To obtain design heights and assess risk, these non-linearities need to be considered by coupled, multivariate assessment such as the copula approach used here. Since many of the densely populated and highly vulnerable world's open coasts are fringed by shallow shelfs, present-day coastal bathymetry likely provides significant protection against wave attack. If SLR is not accompanied by morphodynamic adjustment (e.g. increased deposition), the feedback effects highlighted here are likely to occur, and coastlines worldwide will be more subject to open-ocean conditions. An example is the US Gulf Coast around Louisiana and Texas, where loss of wetlands and intertidal areas is having a significant impact on surge risk 24 . Simple allowances from uncertain SLR projections 8 , as adopted by the Fifth Assessment Report of the IPCC 5 , therefore tend to underestimate the impact of future SLR on the required coastal design heights at many locations around the world. The investigations are based on a number of assumptions introducing uncertainties in the potential feedback of design heights to SLR. This includes but is not limited to the wave run-up formula we used (see e.g. ref. 25 uncertainties in future SLR which directly affect future coastal extremes 28 . Furthermore, the results presented here are for the specific case study of an impermeable 1:6 slope dike and similar assessments need to be adapted to other locations. However, the general processes identified here-changing storm tides and wave heights-are endemic worldwide. Many locations such as the South China Sea or the Arctic are, as the German North Sea Coast, relatively shallow, semi-enclosed ocean basins that are subject to significant waves and altered tides. To plan for the future and protect vital coastal areas, current best-practice needs to be revised to include the effects of non-linear, but coupled changes in risk. The effects, as shown by the large spatial variability observed in Fig. 3, depend on the complex effects of the bathymetry on waves, surges, and tides, and may be further influenced by changes in the morphodynamic equilibrium condition and anthropogenic interventions (e.g., beach nourishment). Nonetheless, the precautionary principle suggests that the approach used here -numerical modeling and multivariate analysis under the assumption of constant depth changes everywhere-will provide significantly better design heights and protection than current methodologies. using historical observations 17 and RCP projections 18,19 for present-day and future conditions by 2100 added on to the observed SLR, respectively. Waves and storm tides for all scenarios are produced by applying continuous wind and pressure fields from reanalysis data for the years 1970-2013 16 . This time period was chosen due to the availability of high quality in-situ data but also to include the 1976 storm tide, the largest event ever recorded in many places of the German Bight. Model runs were output every 10 minutes. For every scenario, the 99.7 th percentile water level exceedances are then estimated at ~500 m increments along the ~470 km North Sea coastline in Schleswig Holstein.

SLR projection.
We consider the median of three different SLR projections by 2100 associated with the RCP4.5 18 , RCP8.5 18 , and RCP8.5HE 19 (high end). All projections represent the effects of thermosteric and halosteric density changes, the response of the ocean to wind and pressure forcing, changes in ocean mass (Greenland and Antarctic ice sheets, glaciers, and groundwater), and glacial isostatic adjustment (see the Fifth Assessment Report of the IPCC 8 for details). In addition, the RCP8.5HE projection includes rapid ice melt in the Antarctic, a plausible but more extreme sea level rise scenario 29  Extreme value statistics. Extreme value statistics are used to infer magnitudes of both storm tides and waves at specific ARI's. We employ the Peak Over Threshold (POT) method 4,30 and fit the following generalized Pareto distribution (GPD) to a ranked list of independent events exceeding a specified threshold of simulated high water peaks, where c is the location (threshold) parameter, b is the scale parameter, a is the shape parameter and the threshold of exceedances is x. The parameters are estimated using the Maximum Likelihood Estimation (MLE) method 4 , with the threshold level of 99.7 th percentile yielding consistent and stable results in the German Bight 30 . A declustering scheme based on the extremal index 4 ensured that data were independent. Wave heights are described using the GPD but also a range of other common distribution functions including the Lognormal, Normal (Gaussian), Exponential, Weibull, and Generalized Extreme Value (GEV) distribution (see e.g. ref. 4). The best fitting distribution is assessed by calculating the minimum RMSE between the theoretical and empirical wave distributions.
The two univariate marginal distributions of storm tides and waves where then applied to assess their joint magnitudes and frequencies. We used Archimedean Copulas to describe the dependence between the two marginal distributions 31 , and hence the bivariate ARI's. Specifically, for each SLR scenario and coastal grid point, we first obtain coincident samples of peak storm tides and wave heights in a window that is ± 120 minutes from the predicted high tide (Fig. 4s). The marginal distributions for storm tides (Fig. 2b) and waves (Fig. 2c) are obtained using univariate analysis. The dependence of storm tides and waves in our modelled data sets are then assessed using Kendall's rank correlation. The correlation coefficient then becomes an input parameter in our copula analysis. Next, three types of copulas (Gumbel-, Clayton-, and Frank Copula) are evaluated and the model with the minimum RMSE between the parametric and the empirical copula 32 is retained to estimate bivariate ARI's. To qualitatively assess whether results are reasonable, 10,000 random events are generated from the parametric copula and the marginal distributions and compared to the numerical model data for consistency (see e.g. ref. 21 for an example).
Next, bivariate contour lines for the ARI 100 are calculated, resulting in a family of possible combinations which have the same recurrence interval. For example, in Fig. 2a, a small storm tide (300 cm) with large waves (185 cm) has the same historical ARI as a large storm tide (450 cm) with small waves (~100 cm; see HIS case). Wave run-up height (see next section) is then calculated for each ARI 100 event (elbow shaped contour line in Fig. 2a). The maximum overall height (i.e. storm tide plus wave run-up height) is assumed to be design relevant (see the red dot in Fig. 2a) and differences between the HIS and scenario runs indicate SLR induced changes in design heights.

Run-up.
Dikes are constructed to withstand the impact of extreme water levels and waves. Potential dike failures result from several mechanisms, including overflow induced by elevated water levels and dike breaching caused by wave overtopping. In practice, dikes are built to withstand the wave run-up height R u,2% , the vertically measured distance which is exceeded by 2% of all incident waves 20 . Along the German North Sea coast, we assume all dikes consist of smooth embankments with a 1:6 slope (see Fig. 5a), following current recommended best practice. However, in reality the dike slopes slightly vary along different coastal stretches, and would need to be considered in a site-specific assessment.
Though extensively studied, uncertainty and bias is still found when wave run-up formulas are compared to physical test results (see e.g. ref. 20; and references therein). Here, we focus on relative wave run-up height changes from different SLR scenarios to minimize the effect of these uncertainties. Our assessment is based on a formula provided in ref. 20 describing the wave run-up on smooth and straight slopes (assuming all waves to attack perpendicular and in relatively deep water at the dike toe but without any wave breaking in front of the dike 20  where α is the outer dike slope and the deep water wave length L 0 is given by m 0 2 where g is the gravitational acceleration and the modal wave period T m as obtained from the scenario and HIS runs.