How Sea Level Rise May Hit You Through the Backdoor: Changing Extreme Water Levels in Shallow Coastal Lagoons

Due to their choked geometry, coastal lagoons can attenuate extreme water levels compared to the open sea. However, this protective property is expected to decrease due to sea‐level rise. By studying idealized lagoons in a non‐dimensional parameter space, this study describes non‐linear interactions between tides, storm surges, freshwater fluxes into the lagoon, and sea‐level rise. The non‐dimensional numbers include lagoon geometry and forcing scales. The main objective is to provide an overview of potentially affected lagoons and to highlight the importance of attenuation changes due to sea‐level rise. Tidal and storm surge induced maximum water levels inside lagoons rise faster than sea‐level rise for most of the parameter space. Maximum water levels due to freshwater fluxes rise slower than sea‐level rise for strongly choked lagoons. For compound events, the response between rising faster or slower than sea‐level rise depends strongly on the lagoon geometry.

• Tides, storm surges, freshwater fluxes, and sea-level rise interact non-linearly on the water level in choked coastal lagoons • Maximum water levels inside choked lagoons can increase faster than sea-level rise due to non-linear attenuation changes • Maximum water levels due to freshwater fluxes increase slower than sea-level rise

Supporting Information:
Supporting Information may be found in the online version of this article. 10.1029/2023GL105512 2 of 11 to 1.02 m for the median by 2100, depending on anthropogenic greenhouse gas emissions as well as the response of the ocean and the cryosphere in the coming decades (see e.g., Fox-Kemper et al., 2021).SLR will shift the frequency distribution of water levels to higher base levels, increasing both the magnitude and frequency of EWLs (Wahl et al., 2017), as well as the associated impacts (e.g., Hinkel et al., 2014) such as chronic flooding (Hague et al., 2023).It will further alter tides and waves in shallow coastal waters (Haigh et al., 2020;Horsburgh & Wilson, 2007;Idier et al., 2019;Melet et al., 2018).In estuaries, SLR will change the attenuation of tides, storm surges, and water level extremes (Holleman & Stacey, 2014;Khojasteh et al., 2021).Therefore, it will impact morphodynamics and coastal flooding (e.g., Ding et al., 2013;Hague et al., 2023;Talke & Jay, 2020).Already today, high tides alone can cause minor floods due to risen the mean sea level (Pareja-Roman et al., 2023).
Coastal lagoons are a particularly vulnerable type of estuary to climate change (Newton et al., 2014).Approximately 32,000 lagoons (Carrasco et al., 2016) are reported to occupy 13% of the world's coastline (Carter & Woodroffe, 1994).A lagoon can be defined as "an inland water body, usually oriented parallel to the coast, separated from the ocean by a barrier, connected to the ocean by one or more restricted inlets, and having depths which seldom exceed a couple of meters" (Kjerfve, 1994).Due to their restricted inlets, water exchange is limited, reducing tidal amplitudes (e.g., Hill, 1994;Stigebrandt, 1980) and maximum water levels during an EWL event, the so-called "choking" effect.SLR will increase the tidal amplitudes in coastal lagoons as well as tidal currents (Carrasco et al., 2016(Carrasco et al., , 2018;;Passeri et al., 2016), therefore increasing the probability of coastal flooding (De Leo et al., 2022).Due to the increased mean water depth, the water transport increases and wave attenuation decreases.This is also true for EWLs, for example, tide/surges (Atkinson et al., 2013;Smith et al., 2010).It is shown that in lagoons, the response of maximum water levels to SLR is non-linear and site-specific (Atkinson et al., 2013;Bilskie et al., 2014;Orton et al., 2020;Smith et al., 2010).Since many impact studies assume linearity of the different components of total water level (e.g., Kirezci et al., 2020;Rueda et al., 2017;Vousdoukas et al., 2018), a lack of non-linear increases of maximum water levels with SLR can underestimate potential impacts, which could literally hit coastal communities through the backdoor, that is, through currently well-protected shallow, coastal lagoons.
Therefore, this paper aims to provide a general overview of the non-linear interactions for coastal lagoons and highlight their importance when assessing maximum water level changes with SLR.We consider not only tides and storm surges, but also the effects of river discharge.We will use an idealized box model to mimic real lagoons in the Gulf of Mexico and the Baltic Sea.

Idealized Box Model
To study the general impact of SLR on EWLs in shallow, coastal lagoons, we employ a widely used, idealized box model of a lagoon connected to the ocean via a narrow and shallow inlet (Hill, 1994;Keulegan, 1967;MacMahan et al., 2014;Morel et al., 2022;Stigebrandt, 1980), see also Figure S1 in Supporting Information S1.The momentum equation reads as a balance between sea level gradient, friction, and wind stress: where L is the length of the inlet, u is the depth averaged mean horizontal velocity, k is a non-dimensional friction parameter, which can be itself a function of water depth, for example, the law of the wall, τ is the wind stress, ρ 0 = 1,000 kg/m 3 , and the total water depth is D = H + (η 0 + η)/2, where H is the depth of the inlet, η 0 is the water level of the open sea and η is the water level of the lagoon.Note that wind stress is not considered in Stigebrandt (1980) or Hill (1994).The key assumptions are that the wavelength of the forcing is much larger than the inlet length and that the flow is always subcritical  (  √ ) (Stigebrandt, 1980).The continuity equation reads as: 10.1029/2023GL105512 3 of 11 where A is the area of the lagoon, W is the width of the inlet, and Q r is the total freshwater input to the lagoon (river runoff plus precipitation).Combining Equations 1 and 2 yields: where sgn denotes the sign function and depends on the argument X, whether the water level inside the lagoon is rising or falling.For τ = 0, Equation 3 can be written in a non-dimensional form (Stigebrandt, 1980): The choking parameter P and the non-dimensional filling rate due to freshwater input S read as  are the non-dimensional elevations scaled with the vertical length scale a.The scales T and a are taken from the dominant forcing, for example, tides.Therefore, T is usually the tidal period, and a is the tidal amplitude.The wave amplitude to water depth ratio, a/H, is an additional non-dimensional parameter of the problem.However, a/H is also included in P. For P < 5 choking is important and EWLs are damped, whereas for P > 5 choking becomes less important (Hill, 1994).In this study, Equation 3 and its non-dimensional form Equation 4 are integrated with a fourth-order Runge-Kutta scheme.Equation 3 can be evaluated for n different inlets at the same time with where the index i counts the inlets and its different properties.In this way, multiple inlets and depth variations across the inlet can be accounted for.Therefore, we can resolve complicated cross-sections of real inlet bathymetries, which often exhibit dredged shipping ways.All simulation are run for 20 tidal cycles.

Parameter Space
We will study the parameter space spanned by the choking number P, a/H, and S. Since P summarizes both the geometry and the forcing, the interpretation can be twofold: either one assumes fixed forcing scales a and T, from which one can estimate which lagoons may be vulnerable to such forcing.Or one can assume a fixed lagoon geometry and estimate to which forcing scales the lagoon may be vulnerable to.We will use the former interpretation and will use idealized, fixed forcings (scales based on realistic values), see the next section.Note that we only consider wind stress forcing for the validation in Section 3.1 and not in the parameter space studies.

Idealized Forcing
We force the model with sinusoidal tides of amplitude a and period T, Gaussian shaped storm surges (amplitude a surge , timescale αT), and precipitation plus river runoff in the form of Gaussian shaped freshwater pulses (amplitude Q amp , timescale βT) with background supply Q 0 , 10.1029/2023GL105512 4 of 11 For simplicity, we prescribe that the maximum of the storm surge and freshwater pulses are aligned with the high tide, t 0 = 17.25 T which is not necessarily realistic (Familkhalili et al., 2020).The tidal period is set to 24 hr since tides in the Gulf of Mexico are diurnal.The surge amplitude is chosen to be twice the tidal amplitude a: a surge = 2a since the tidal amplitudes in the Gulf of Mexico are small  () = 1 m.The timescale for the storm surge is chosen with αT = 6 hr which corresponds approximately to a 24-hr event which is realistic for both the Gulf of Mexico (Familkhalili et al., 2020) and the Baltic Sea (Kiesel et al., 2023).The freshwater pulse amplitude Q amp is chosen with 1,000 m 3 s −1 , which is a value in the range of the 90th percentile discharge, see Table 1.For comparison, the discharge during Hurricane Harvey in 2017 for the Trinity River entering Trinity Bay was ≈3,500 m 3 s −1 (Yao et al., 2022)).The event lengths were chosen with the timescale αT = 72 hr which mimics the discharge time series of Harvey and other extreme cases (Yao et al., 2022).We consider SLR = 2/3 a. Effects of wave-setup (e.g., Lentz & Raubenheimer, 1999) are not considered.

Quantification of Non-Linear Effects Due To Sea-Level Rise
To analyze the effect of SLR on the maximum water level, we use the Normalized Non-linearity index (NNL, Bilskie et al., 2014), where η 2 is the maximum water level inside the lagoon for the simulation with SLR, and η 1 is the maximum water level inside the lagoon for the simulation without SLR.A NNL index of zero indicates that non-linear effects of SLR are not important.NNL ≠ 0 indicates that maximum water levels will accelerate (NNL > 0) or decelerate (NNL < 0) compared to the new mean sea level.However, the new total maximum water level would always increase by (1 + NNL) times SLR.

Validation of the Box Model
We evaluated the idealized box model for lagoons along the Texas Gulf coast and the German Baltic Sea coast, see  et al., 2017).River discharge data are obtained from the Gulf of Mexico Coastal Ocean Observing System (GCOOS, https://gcoos.org) and EMODnet.The box model results are compared to observed GESLA gauges from inside the lagoons.EWL events above the 99.75th percentile (peak over threshold) are evaluated for each lagoon.The simulation period begins 3 weeks before the maximum EWL of the outside gauge and ends 1 week later.The first week of the simulation is discarded as a spin-up before evaluating the performance of the box model.We compare the maximum water level inside the lagoon with the observed one, the mean bias, root mean square error, and the correlation of the 3-week water level time series, see Table 1.Overall, the idealized model can reproduce the sea level variability inside the lagoons to a reasonable degree, Table 1, see also Figures S4-S12 in Supporting Information S1.However, the maximum water levels inside many lagoons are often underestimated.We believe that one reason is the complicated interconnections within the Gulf of Mexico lagoon systems which we do not describe well with our box model assumption.Nevertheless, the idealized box approach can provide an overview of the conceptual effects of SLR on EWLs.Since we resolve the inlets, each of the n subsections of the inlet possesses its own choking number P i .To find a representative choking number for the whole lagoon, one can either sum the individual choking numbers or compute it using the mean depth of the inlet (e.g., Lopes et al., 2022), see Text S4 in Supporting Information S1.Both approaches yield different values.Therefore, we list both values as a range of choking numbers in Table 1.
To account for additional uncertainties, we have extended the ranges of the non-dimensional parameters by 10%.

The Present as Reference
To study the water level changes due to SLR, we first have to set a reference for the NNL (Section 2.4, Bilskie et al., 2014), which is done in the following.The tidal amplitude  (max( η0) = 1) is significantly dampened inside the lagoons for most of the parameter space,     1 , see Figure 1a.For P < 5, the amplitude inside the lagoon is less than 70% of the amplitude in the open sea.For even smaller choking numbers, P < 2, the amplitude drops to 40%.Due to the asymmetry of water depth during a tidal cycle, the mean water level in the lagoon is higher than in the open ocean (e.g., Hill, 1994).For a storm surge (  max( η0) = surge∕ = 2 and α = 1/4), the attenuation of the surge occurs mainly for P < 2, Figure 1b.The maximum water level for a surge with tidal forcing  (max( η0) = 3) is also damped for small choking numbers, P < 2, or P ≥ 2 and small a/H, Figure 1c.Non-intuitively, the maximum water level in the lagoon becomes smaller for smaller a/H (greater depths H, constant a).This is due to the interconnection between the two non-dimensional parameters.If H becomes larger, other geometric variables or the friction must adjust to keep P constant, for example, the inlet becomes longer, narrower, or more frictional at constant surface area A. During a tide/surge event, a competition between the increased filling rate due to the larger cross section versus the reduced length of the event compared to a single surge due to the ebb phases determines whether the maximum water level is higher or lower compared to a linear response, Figure 1d.Freshwater supply to the lagoon can increase the mean water level inside the lagoon relative to the open ocean.However, only at very low choking numbers the inlet is choked enough that the freshwater cannot immediately flow out to sea, leading to a mean water level gradient, Figure 1e.When tides are present, the mean water level inside the lagoon increases as freshwater is prevented from leaving the lagoon (Morel et al., 2022), Figure 1f.A freshwater pulse increases the water level inside the lagoon, Figure 1g, but only again for small choking numbers P < 1.Consider ing a compound event of all the above processes, that is, a storm surge at high tide, coinciding with a freshwater pulse with background freshwater supply, Figure 1h, the response is a combination of the individual components.Non-linear effects are important for this case, Figure 1i.Depending on the geometry, either the freshwater may more easily leave the lagoon due to the tide and surge (well-choked lagoons, P < 0.3, shallow inlets, a/H > 0.5) or the freshwater helps the tide-surge to propagate more easily into the lagoon (rest of the parameter space).1.

The Future With Sea-Level Rise
In the following, we compare how the maximum water levels in lagoons change with SLR based on the geometries of the present time.With SLR, the tidal amplitude inside the lagoons will increase compared to today, Figure 2a.Maximum water levels during storm surges will increase faster than solely SLR (NNL > 0) for lagoons with P < 3.This effect is greater for shallow inlets (large a/H), Figure 2b.For P ≥ 3, the NNL is ≈0, meaning that SLR is additive.These lagoons do not attenuate in their present geometry, compare Figure 1b.For a surge at high tide, maximum water levels will also increase more than only SLR, affecting most of the parameter space with maximum increases of 0.4 SLR, Figure 2c.With SLR, the mean water level due to freshwater supply will rise slower than SLR (NNL < 0), Figure 2d.The additional cross-sectional area allows the freshwater to flow out of the lagoon.This effect is limited to well-choked lagoons P < 1.Similarly, the maximum water levels due to a freshwater pulse rise slower than SLR, Figure 2e.For a compound event, Figure 2f, maximum water levels rise slower than SLR for lagoons with P < 0.9.For larger choking numbers, the maximum water levels rise faster than SLR.This depicts the competition between the fluvial forcing of freshwater, which can more easily flow out of the lagoon with SLR, versus the oceanic forcing, which can more easily flow into the lagoon with SLR.
The lagoons considered in the Gulf of Mexico and along the German Baltic Sea coast (Table 1) fall into the parameter space that will experience mostly weak increases in maximum water levels with SLR, see the gray boxes in Figure 2. In particular, the maximum water levels for storm surges during high tides (Figure 2c) show an increase for many of these lagoons, which is also true for the compound event considered.Storm surges are most important for the micro-tidal Baltic Sea.For this case, the results emphasize that the Saaler Bodden lagoon lies in the parameter space where a great increase of the maximum water level with SLR can be expected.

Interpretation and Sensitivity of the Results
We have shown that maximum water levels for tides, surges, and tide/surge events increase faster than SLR.In contrast, water levels due to constant freshwater supply or freshwater pulses increase slower than SLR.Both effects are also true for other time scales (α, β) and higher/lower amplitudes (a surge , Q amp ).The respective patterns and NNL indices change slightly compared to Figure 2 (Text S5 in Supporting Information S1), while the main point of this study remains unchanged: EWLs in lagoons change non-linearly with SLR, which should be considered when evaluating future EWL events and flood risks.
Since sea-level rise increases the water depth, it effectively changes the geometry of the lagoons.Due to our static geometry approach, the new choking number P and the new a/H can be easily identified by adding the SLR scenario to the present water depth H, yielding: see also the Text S5.5 and Figure S19 in Supporting Information S1.Therefore, the NNL index describes the maximum water level differences between future and present lagoon geometries (summarized by the non-dimensional numbers) for similar forcing.The NNL itself is a non-linear function of SLR.Hence, it can only give a general idea of how lagoon EWLs change with SLR.

Limitations
Our approach of using an idealized lagoon model has limitations and simplifies the dynamics of the water exchange.The assumption of a static geometry allows the study of the parameter space, but in reality, the geometry and the friction change with water depth; thus water level changes.
A main assumption is that the flow is always subcritical  (  √ ) .However, for a/H → 1 this assumption may be violated during the tidal cycle, when water depths become very shallow.In this case, the neglected advection term in the momentum equation would become important and a hydraulic control would establish (see also Stigebrandt, 1980).However, for the considered surges and compound events, we expect the errors to be small.On short timescales, for example, during an EWL event, the surface area of the lagoon may increase significantly (e.g., Holleman & Stacey, 2014), the inlet may widen, or new inlets may form.The former may decrease whereas the latter may increase the maximum water levels in the lagoon.These processes could easily be added to the simulations, but these are beyond the scope of providing a first overview of the parameter space.On long timescales, the morphology of the lagoon and inlets may change with SLR (Donatelli et al., 2018), which would alter the water exchange and therefore the influence of SLR.For example, barrier islands may migrate or erode with SLR (Carrasco et al., 2016;Ranasinghe et al., 2013).In addition, inlets may be deepened, opened, or closed (De Leo et al., 2022).
We have neglected the effect of wind waves.Wave-setup (e.g., Lentz & Raubenheimer, 1999), wind-wave induced residual transport, wind-wave induced bottom friction, and wind-wave-current interaction can be important for lagoon dynamics, also for extreme conditions (e.g., Mao & Xia, 2018).
It should be noted that the non-dimensional form of the water level evolution Equation 4 can exclude non-linear effects between different tidal constituents, for example, the spring-neap cycle (MacMahan et al., 2014).The results of this study on the interaction between tides and surges show no significant differences between the dimensional and non-dimensional forms.
Differently shaped and timed forcings as well as observed time series should be considered in future studies as the lengths and heights of surges or freshwater pulses change the non-linear results.

Responses of Real Lagoons to Sea-Level Rise
Due to the limitations of the idealized model and the various scales of the forcing to which real lagoons are exposed, the non-dimensional numbers listed in Table 1 are only an estimate.For each EWL event and during an event, the forcing scales, the geometry and thus the non-dimensional parameters change.
In addition, wind stress and internal water level processes inside the lagoons are neglected in the SLR assessment of this study, for example, reflections off infrastructure (Pareja-Roman et al., 2023).Additionally, water level changes due to wind stress are important for most lagoons, as the comparison with observations improved when wind stress was included, shown in Text S3 in Supporting Information S1.However, wind stress would change the non-dimensional form of Equation 3, making the choking number wind stress dependent or adding new inter-dependencies.
Furthermore, the most extreme events that the considered real lagoons experience in reality are beyond the scale of the forcings we consider.For such events, the geometry of real lagoons changes significantly and the assumption of a static geometry would be violated.
Nevertheless, based on our results, it is possible to identify potential locations of lagoons within the parameter space.Additionally, the estimates provide insight into how EWLs in lagoons can be affected by SLR.All considered real lagoons are located in the parameter space, where SLR will change future lagoon EWLs (mostly increasing more than just SLR).
Since SLR could decrease EWLs along the open coast (Arns et al., 2020), the increase of EWLs inside the lagoons could be partially compensated.Overall, detailed modeling studies are needed to provide insights and answers on how specific coasts will respond to SLR, especially how the most extreme events will change.Furthermore, the competition between oceanic forcing (tides, surges) versus fluvial forcing (river discharge) should be explored in more detail.The parameter space presented can help identify which lagoons will be more affected by SLR than similar lagoons nearby.

Summary and Conclusions
As sea levels continue to rise, the impact on coastal lagoons is becoming increasingly apparent.We show that there are several non-linear interactions between tides, storm surges, freshwater supply, sea-level rise, and geometry at play.These effects can have a large impact on the maximum water levels in these lagoons, with many real lagoons falling into a parameter space where maximum water levels increase more than just SLR.Maximum lagoon water levels during storm surges and surges at high tides increase faster than SLR alone.In contrast, water levels due freshwater input, rise slower than SLR.For compounding events (tide, surge, and freshwater pulse), both can be true, depending on the lagoon geometry.Unfortunately, these non-linear effects on maximum water levels are often not considered in impact studies, which can lead to under-or overestimation of flooding and its associated consequences.To address this issue, non-linear effects need to be included in future impact studies.Regional high-resolution modeling studies will be necessary to more accurately estimate these changes for EWL events with SLR.Without these efforts, future floods may hit communities through the backdoor: the lagoons.

Figure 1 .
Figure 1.Responses of the lagoon water level    as a function of the choking parameter P and the ratio of vertical length scale to the water depth of the inlet a/H, and S: (a) Attenuation of the tidal amplitude and the mean water level due to tidal water depth asymmetry (gray contours).(b) Attenuation of a storm surge.(c) Attenuation of a surge at high tide.(d) Deviation from the assumption that the water levels due to tide and surge add up linearly.(e) Mean lagoon water level with constant freshwater supply.(f) Same as (e), but with tides.(g) Maximum water level due to a freshwater pulse (Q 0 = 0).(h) Compound event.(i) Deviation from the assumption that water levels due to tide, surge, freshwater supply, and freshwater pulse add linearly.The dashed line indicates P = 5.The rectangles indicate the parameter ranges of the lagoons in Table1.

Figure 2 .
Figure 2. Responses of the lagoon sea level    to sea-level rise (SLR): (a) Increase in tidal amplitude and decrease in mean water level difference to the open sea (gray contours).(b) Normalized Non-linearity index (NNL, Bilskie et al., 2014) for a storm surge.(c) NNL for a surge at high tide.(d) NNL of the lagoon mean water level for constant freshwater supply and tides.(e) NNL of a freshwater pulse.(f) NNL of a compound event.The dashed line indicates P = 5.The rectangles indicate the parameter ranges of the lagoons of Table1.

Table 1
Overview of Lagoons and Gauges Used to Validate the Idealized Box Model, See Figure S2 in Supporting Information S1 for the Locations and Figures S4-S12 for the Validation in Supporting Information S1 Kiesel et al. (2023)ngth scale a is set to the tidal amplitude (sum of the greatest constituents: M2, S2, N2, K1, O1) and the time scale is set to T = 24 hr (approximately diurnal tides) for all Gulf of Mexico lagoons.For the Baltic Sea lagoons, a corresponds to the 99.75th percentile (see TableS2in Supporting Information S1) of the water level distribution and T = 24 hr (a typical length of an EWL eventKiesel et al. (2023)).The bias and RMSE in the last column refer to the 3-week time series comparison.

Table 1 ;
Saha et al., 2014)2016;Haigh et al., 2022).Fo, 2001b)agoon, we set up the geometry of the idealized box model to mimic the real lagoons.The inlet cross sections are extracted from the U.S. Coastal Relief Model for the central and western Gulf of Mexico(NOAA National Geophysical Data Center, 2001a, 2001b)and the European Marine Observation and Data Network (EMODnet, https://emodnet.ec.europa.eu),seealsoTextS3 in Supporting Information S1.We prescribe the open ocean water level with gauges from the Global Extreme Sea Level Analysis data set (GESLAWoodworth et al., 2016;Haigh et al., 2022), which summarizes gauges from the National Oceanic and Atmospheric Administration (NOAA) and the United States Geological Survey (USGS) for the Gulf of Mexico, freshwater supply by river discharge plus precipitation, and wind stress.Atmospheric data are prescribed from the NCEP Climate Forecast System Reanalysis (CFSR,Saha et al., 2010), the NCEP Coupled Forecast System Model Version 2 (CFSv2Saha et al., 2014), and the HARMONIE V1 dataset (https://apps.ecmwf.int/datasets/data/uerra/) from the project 'Uncertainties in Ensembles of Regional ReAnalyses' (UERRA, https://www.uerra.eu,Ridal