## Introduction

Coral reefs effectively dissipate incident wave energy through a combination of breaking and friction1 and fundamentally regulate how ocean waves interact with reef-fringed shorelines2. This ‘natural breakwater’ effect has enabled dense development on coastal margins and complete urbanisation on a number of atoll islands that are now highly vulnerable to impacts of future sea-level rise (SLR)3,4. Small and low-lying atoll islands are prone to flooding from spring tides5, storm surge6 and wave overtopping7, with marine inundation predicted to become increasingly costly in the near future3,4,8. At present sea-level (SL), wave overtopping occurs every 2–3 years8, with destructive flooding events every 10–30 years7,9,10 allowing sufficient recovery time for agriculture, fresh-water resources and infrastructure. Global SL is predicted to rise by 0.44 ± 0.17 m (RCP2.6) to 0.74 ± 0.23 m (RPC8.5) by 210011, with a continued increase predicted for several centuries regardless of emission mitigation efforts12,13. SLR is a serious climate change hazard that has already impacted some reef-island systems14 and is predicted to severely impact all reef-fringed and atoll island shorelines in the coming decades15,16, affecting an estimated global population of 197 million people4. Unmitigated, SLR will increase the frequency and magnitude of wave-driven flooding events8, eventually reaching a threshold when compounding events prevent recovery and compromise permanent habitation on coastal margins or entire islands17. Atoll-populations (>500,000 people across five nations) have limited land available for in-country relocation and may become climate change refugees when local adaptation efforts are exceeded16,18. However, future responses are unlikely to be uniform as there is considerable heterogeneity in island physical structure and energy exposure19. To assist local adaptation and timelines for future overtopping, there is a pressing need to resolve how the physical drivers of wave-driven flooding on reef coastlines vary spatially due to differences in wave climate exposure and island morphology. Here, we present a new model of wave overtopping that can be applied to predict SL and (or) wave height thresholds for overtopping on reef-fringed coastlines globally. The model is applied to a number of different atoll islands to understand how future inundation trajectories vary according to local morphology, wave climate exposure and potential geomorphic feedbacks.

Overtopping occurs when wave runup surges over the beach crest and across the island surface (Fig. 1a). Wave runup is a product of interactions between incident wave energy and reef morphology, with overtopping determined by the capacity of a shoreline to accommodate runup surges. Theoretically, runup levels and the occurrence of overtopping are, therefore, a predictable function of incident wave energy and first-order morphology characteristics that control dissipation. However, predicting runup and overtopping on reef coastlines is significantly complicated by a complex suite of nonlinear wave transformation processes that increase reef flat water level (setup) and produce energetic low-frequency (infragravity) wave surges20. Wave breaking on coral reefs creates a setup of mean water level across the reef flat21 and the dissipation of incident wave groups drives a transfer of energy to long-period infragravity wave motions that surge across the reef flat22. Wave setup and infragravity waves are dynamically influenced by incident wave conditions and reef morphology and both significantly influence runup and overtopping on reef coastlines20,23. It is, therefore, essential that the nonlinear interactions between incident wave dissipation and these dynamic surf-zone processes are represented accurately when investigating runup and wave-driven flooding on reef coastlines8,22.

Previous attempts to understand the physical drivers of future wave overtopping on reef coastlines focus on single case study islands8,17,24,25 that are not globally applicable, neglect overtopping magnitudes26 or use simplified hydrodynamic models that do not resolve nonlinear surf-zone processes17,27,28. Our research advances previous work by applying a comprehensively evaluated physics-based numerical model20,29 to directly simulate wave overtopping motions resulting from incident wave dissipation and nonlinear surf-zone processes. Our approach builds on previous studies that utilised a wide range of idealised bathymetries and wave conditions to investigate the impacts of SLR on reef coastlines26,27,28,30. Consequently, we present the first analysis of wave overtopping that is applicable to reef coastlines globally, under present and future SLs by representing 60,000 unique morphology, wave energy and SL scenarios.

## Results

### The reef-island overtopping threshold

Of the 60,000 unique combinations of reef depth (hr, below still water level), reef width (Wr), significant incident wave height (Hs), wave period (Ts) and island elevation (Zi, above still water level) used in this analysis (Methods; Supplementary Table 1), wave overtopping occurred on 31.7% of the simulations, with maximum wave overwash depth (how) at the beach crest ranging between 0.01 and 2.5 m (Fig. 1a). For simulations where overtopping was identified (Methods; Supplementary Fig. 1; Supplementary Data 1), a deterministic relationship was established between how and each forcing variable (Hs, hr, Wr, Zi), when all other factors were consistent (Fig. 1). Modelled how increased proportional to Hs and hr (Fig. 1b, e), but decreased with increasing Zi and Wr (Fig. 1b–e).

Subsequent analysis identified two new dimensionless parameters that collectively control the occurrence of wave overtopping on atoll islands (Fig. 2). Relative reef width (RRW) (Wr/hr) quantifies the dissipative capacity of a reef and relative island elevation (Zi/Hs) represents the potential for an island to withstand overtopping during a given event. RRW decreases with increasing SL and therefore represents how reefs will become less effective at dissipating incident wave energy, resulting in elevated runup levels at the shoreline. Low-relative island elevation (RIE) values (<0.6) indicate that an island is likely to be overtopped by a given event, unless RRW is high and the reef is extremely dissipative. At present SL, overtopping occurs because high-wave events coincide with high tide, resulting in a low-RIE value. As SL increases, the associated decrease in Zi will lower RIE and therefore the incident wave energy required for overtopping will decrease. Across the spectrum of forcing and morphology conditions used in this analysis (Supplementary Table 1), RRW and RIE present a clear boundary that determines whether overtopping occurs on a reef-fringed shoreline (Fig. 2), which is defined by the reef-island overtopping threshold (RIOT; Eq. (1)), a simple function of island morphology and incident wave height:

$$\frac{{Z_{\mathrm i}}}{{H_{\mathrm s}}} = 2.8\left( {\frac{{W_{\mathrm r}}}{{h_{\mathrm r}}}} \right)^{ - 0.29}$$
(1)

Modelled overwash magnitudes (Supplementary Data 2) were tested against RRW and RIE combinations from 23 published wave runup, overtopping and flooding events across five atolls and showed good agreement (Table 1). Elevated runup events (without overtopping) associated with either moderate waves at spring high tide (SHT), or large waves during a neap tide, are appropriately located above the overtopping threshold line (Fig. 2). Five events in the Marshall Islands that were associated with nuisance level overwash and minimal damage are all positioned below the overtopping threshold7,23, with modelled inundation depths averaging 0.12 ± 0.01 m (Table 1). RRW and RIE combinations from fifteen significant island flooding events from the Marshall Islands7,8,23, Taaku Atoll31 and the Maldives9 are located well below the inundation threshold, with modelled how = 0.24 ± 0.02 m, on average (Table 1). Of note, these significant island inundation events all occurred during high tide and were driven by either local storm activity or long-period swell with wave heights between 2 and 5 m (Table 1). Wave period is shown to have some influence on overwash magnitude (Fig. 1), but this is of a secondary importance compared to other input variables (Supplementary Fig. 2) and was, therefore, not included in the RIOT formula (Methods).

### Future overtopping trajectories with SLR

The RIOT provides a powerful tool to explore and predict the vulnerability trajectory of different atoll islands to SLR, by adjusting Zi and hr magnitudes in accordance with how SLR will increase reef depth and decrease island elevation (Fig. 3). Trajectory curves are presented for a number of studied islands (from Table 1) using reef specific morphology characteristics and wave climate statistics for each islands 50 percentile (p50), 99 percentile (p99) and extreme significant wave height (Methods; Supplementary Table 2). Trajectory profiles highlight how SLR will exacerbate overtopping on atoll islands by compromising wave dissipation capacity (decreasing RRW), while also decreasing the magnitude of energy required for overtopping. It is important to note that the trajectories and thresholds presented here are based on simulations of nonlinear wave transformation on simplified reef geometries and do not represent the site-specific complexity of each individual reef-island system. This enables a first-order assessment of overtopping vulnerability across a wide spectrum of reef systems but does not represent site-specific variations in reef flat or beach-face topography and roughness. Therefore, the following assessment is appropriate for understanding trends and trajectories in reef-coast vulnerability and for identifying sites where higher resolution investigations are required.

Most islands do not experience overtopping at present SL when exposed to their respective p50 or p99 wave heights, although overtopping does occur on some lower elevation islands during p99 events at SHT (Fig. 3a, b). Significant overtopping is predicted to occur on the lower elevation islands of Nukutoa (Takuu), Fares-Maathoda and Majuro South under modal (p50) wave heights at SLR = 1 m. However, higher islands, including Fatato and Roi-Namur remain resilient to overtopping by mean waves at SLR = 1 m. All islands will experience overtopping or potential flooding under p99 wave conditions with 1 m SLR, with major flooding predicted for each island during extreme wave events.

Overtopping trajectories indicate that an increase in reef depth associated with small steps in SL drives a substantial decrease in RRW and rapidly shifts island vulnerability towards the threshold for overtopping (Fig. 3). Once the threshold for overtopping is exceeded, the vulnerability trajectory for each island accelerates (drops rapidly) as RIE becomes the primary variable that determines the magnitude of potential flooding (Fig. 3). Significantly, our findings highlight that the capacity for islands to withstand SLR without wave-driven flooding is variable and strongly dependent on the antecedent morphology and wave climate of each island. In general, shallow reefs present a more rapid trajectory towards the overtopping threshold compared to deeper reefs, which is driven by a proportionally greater increase in wave energy at the shoreline. It is also evident that island elevation above SHT has a critical influence on future island vulnerability, however the timing and magnitude of future wave overtopping on islands of similar elevation will be controlled by differences in wave climate exposure and reef dissipation characteristics (Fig. 3).

Our results highlight that there is significant variation in the potential for atoll islands to withstand wave overtopping under future SLR scenarios, under the assumption that there is no adjustment to reef or island morphology. Predicted SLR thresholds for overtopping during modal wave conditions (p50 wave height specific to each reef) ranged between 0.1 m (Takuu) and 1.22 m (Fatato) above present SHT (Fig. 4a). This equates to a temporal range in the onset of regular overtopping occurring in the next few decades or beyond 2100. These results indicate that higher islands with a narrow reef (Fatato) are more resilient to flooding than low islands with a wide reef (Taaku). Differences between extreme and mean wave heights within a local wave climate also exert an important control on inundation vulnerability. For example, although Roi-Namur exhibits a high SLR threshold for inundation under modal wave conditions, it is one of the most vulnerable islands during energetic events because of its exposure to comparatively large 99.9 percentile (3.7 m) and extreme (6.2 m) waves (Supplementary Data 2; Methods). A key impact of SLR in the context of flood exposure on reef coastlines is that less energy will be required for overtopping, but more energy will be available at the shoreline. All the islands examined (Table 1) require Hs ≥ 1.4 m for overtopping at contemporary SHT, with Hs > 2.8 m required for overtopping on Fatato and Roi-Namur (Fig. 4b). The predicted threshold wave height for overtopping decreases by 69% (on average) for SLR = 1 m at SHT, with the greatest reduction noted for Takuu (100%; SL inundated), Fares (86%) and Majuro (71%), compared to Roi-Namur (54%) and Fatato (47%).

### Overtopping thresholds and geomorphic feedbacks

Numerical modelling simulations and the subsequent analysis of future overtopping trajectories were undertaken based on the conservative assumptions that island morphology and reef structure remain static over decadal timescales. Such assumptions suggests that model outputs yield worst-case overtopping scenarios by omitting two important morphological feedbacks that have significant potential to mitigate the flood susceptibility of islands. First, vertical accretion of the outer reef flat in response to new accommodation space for coral growth, which in some areas is predicted to exceed rates of SLR associated with RPC6.0 (7.4 ± 2.8 mm y−1)32. Using Eq. (1) to quantify SLR thresholds for overtopping with theoretical keep-up (Δhr = 0) and catch-up reef (Δhr = 0.5ΔSL) responses (Methods) identifies that future reef growth will delay the onset of flooding by raising SLR thresholds for overtopping by an average of 0.11 ± 0.01 m for keep-up reefs and 0.08 ± 0.01 m for catch-up reef responses (Fig. 4c). Significantly, future reef accretion has some potential to delay the impacts of SLR on atoll islands but has limited capacity to prevent overtopping, especially during higher energy events (Fig. 4). Reef accretion is effective because it limits the decrease in RRW but it cannot protect islands from overtopping because RIE still decreases with SLR. Therefore, wave overtopping will still occur on islands with a keep-up reef, but the magnitude of potential flooding will be more manageable.

Second, island morphodynamic feedbacks will likely occur as SLR allows wave overwash processes to transport sediment from the beach-face to the island surface, resulting in an increase in island elevation and lagoonward migration33. We do not consider how potential adjustments to island shoreline morphology will change future overtopping thresholds, but we do apply the RIOT formula to calculate how much vertical accretion is required to prevent overtopping, assuming no change in reef width. Eq. (1) was used to predict the magnitude of vertical island adjustment required to prevent overtopping at different SLs, wave events and reef responses (Fig. 4c–e). Once the SLR threshold for overtopping is reached, vertical accretion of the island shoreline will need to exceed the rate of SLR by an average of at least 20% (ΔZi = 1.2ΔSL) to maintain a shoreline elevation at equilibrium with the overtopping threshold associated with a give-up reef response (Fig. 4c–e). This proportionally higher increase in Zi is required to compensate for the decrease in hr that allows higher waves to reach the shoreline. Accordingly, island accretion requirements with a keep-up reef response are equal to the magnitude of SLR. Since many of the islands in Table 1 currently have some capacity to prevent overwash under moderate waves at low magnitudes of SLR, the average vertical island accretion required to prevent overtopping with 1 m of SLR and p99 waves is 0.82 m, notably less than the magnitude of SLR. However, the magnitude of adaptation varies widely, with Takuu and Fares-Maathoda both needing 1.3–1.5 m of vertical island accretion to prevent overtopping at SLR = 1 m (Fig. 4c).

### Implications for atoll island vulnerability

Our results present valuable new insight on the physical drivers of wave inundation on coral reef coastlines and provide a widely applicable tool for predicting thresholds for wave overtopping driven by extreme events or future SLR. Our results indicate that the lowest islands examined will suffer from chronic wave overtopping with small increases in SL (0.1–0.4 m) that are predicted to occur before 210011, assuming no eco-morphodynamic adaptation to reef or island structure. We also identify characteristics that promote resilience to inundation with SLR, allowing some islands to potentially remain habitable beyond 2100. Consequently, our research reveals that the impacts of SL rise on atoll islands will be highly variable across different reef systems and highlights the importance of antecedent morphology in determining future vulnerability trajectories. We resolve the nonlinear balance between boundary forcing and dissipation variables that collectively determine the occurrence of wave overtopping on reef-fringed coastlines by deriving an idealised formula to quantify thresholds for wave-driven flooding that is applicable to a wide range of reef-island settings. The RIOT can be applied in coastal and climate change management practise to identify inundation-prone areas and locations with high natural resilience, based on easily measured morphological characteristics. Such knowledge enables a strategic and site-specific foundation to plan for and mitigate the impacts of future flooding events.

## Methods

### Numerical model

A previously evaluated fully nonlinear numerical model was used to simulate wave shoaling, transformation, runup, overtopping and inundation for this research. The open-source Green–Naghdi solver34 provides a fully nonlinear, but weakly dispersive solution for the Boussinesq equations that is applicable to large amplitude waves interacting with complex morphologies, such as coral reefs. A shock-capturing nonlinear shallow water solution is applied in areas of wave breaking and runup to ensure areas of rapid dissipation are handled with stability and accuracy34,35. The phase-resolving model we apply here has been comprehensively evaluated for accurately simulating wave dissipation, infragravity wave behaviour, wave setup, runup and overtopping on coral reef environments, using field data20 and wave flume29 experiments. The documented open-source model is available online36, with existing literature providing more information on the numerical scheme34 and its application to simulate waves on coral reefs20,25.

### Numerical simulations

A total of 60,000 unique simulations were undertaken for this analysis (Supplementary Table 1). These combinations comprise 60 boundary conditions (10 Hs variations and 6 Ts variations) and 1000 morphology variations (10 Wr variations, 10 hr variations and 10 Zi variations) that collectively represent a global spectrum of atoll islands and energy settings. Consistent values were used for beach slope (0.1) and reef slope (0.14) because these variables are known to have a deterministic but secondary influence on wave runup elevation26,27. Friction is known to vary on coral reefs, depending on ecological and sedimentary characteristics. We account for friction using an implicit quadratic bottom friction equation with a coefficient of Cf = 0.04, representative of a moderately rough reef environment20. We do not account for variability in roughness by adjusting Cf because this has a deterministic influence on runup and is of secondary importance compared to changes in water depth26. The contribution of possible future reductions in friction associated with declining structural reef complexity to overtopping scenarios is a subject recommended for future study. To represent wave breaking in the numerical model, a threshold free-surface slope was used to split between dispersive and dissipative equations, with a slope threshold of 1 based on previous work20. Modelled wave conditions were generated with a JONSWAP spectrum, using γ = 3.3 and each simulation was run for 35 min of wave activity to ensure that a saturated wave field developed on the reef platform, allowing a number of wave groups to potentially overwash the island. Waves released at the offshore boundary propagated across a 60 m deep shelf for 200 m before interacting with the moderately steep reef slope, reef flat and island (Supplementary Fig. 1). To maintain consistency with previous model calibration simulations20, a uniform Δx of 1 m was used to discretise the model domain. The island profile was designed to ensure that maximum elevation (input Zi value) remained constant for 50 m before a leeward beach slope decreased island elevation to zero (Supplementary Fig. 1). This leeward section was sufficient for overwashed water to pool without reflecting back onto the ocean reef flat. All simulations for this research were computed on the NeSI Pan cluster, through the University of Auckland.

### Data analysis

Wave overwash was identified by extracting time-series data at 10 Hz at a location on the island berm positioned 70 m landward of the beach toe (Supplementary Fig. 1). Time-series data after 512 s was used to calculate mean and maximum overwash depth for each simulation. Overwash outputs for each simulation are presented in Supplementary Data 1, alongside input conditions. To establish a representative value of overwash depth for each RRW and RIE combination, the top third of all values for maximum overwash depth that had the same RRW and RIE combination were averaged (Supplementary Data 2). The same data points were used to calculate standard error, which are also presented in Supplementary Data 2, along with the number of points used to calculate averages (n). When calculating RRW, a minimum reef depth of 0.1 was used to avoid dividing by zero for dry reef scenarios.

The influence of wave period on overtopping is not accounted for in the RIOT formula. Wave period is known to exert some influence on runup magnitude and the relationship between Ts and overwash depth was investigated for this analysis. Based on our numerical outputs, very short period waves (6 s) result in reduced overwash magnitudes compared to moderate (8–12 s) and long (14–16 s) period waves. Overall, the influence of wave period on overwash occurrence and magnitude was of a secondary order compared to wave height and island elevation, which became the focus of subsequent analysis (Supplementary Fig. 2).

### Wave climate statistics

Hourly wave climate data for 30 years (1980–2009), from the CSIRO Wave Watch 3 hindcast database37, were used to calculate wave climate statistics for Fatato (179°E; −8.5°N), Roi-Namur (167°E; 9.4°N), Majuro (171°E; 6.9°N), Takuu (158°E; −4.4°N), and Fares-Maathoda (72.8°E; 0°N). Time-series wave data were filtered to only include wave events where peak wave direction was travelling within 90 degrees of shore-normal to the island. Directionally filtered wave heights were used to calculate 50, 99, 99.9 and 99.99 percentile wave heights, and extreme wave heights, calculated as the average of the top 10 wave heights over the 30 years. Wave height statistics for each example island were used to calculate inundation trajectories and future SL thresholds for wave inundation (Supplementary Table 2).

### SL thresholds

SLR thresholds for wave inundation were calculated for each island and wave statistic by incrementally adjusting Zi and hr at small (0.001 m) steps that represent increases in SL. Thresholds were identified by iterating the increase in SL until the calculated Hs threshold for overtopping (using Eq. (1)) was less than the wave height statistic being examined. SLR thresholds were calculated for three reef response morphologies, allowing us to quantify how effective future reef accretion may be at mitigating SLR impacts. To represent no change in reef structure (give-up response), the increase in reef depth (Δhr) was equal to the increase in SL (ΔSL). To represent a catch-up reef response, where the rate of reef accretion is less than the rate of SLR, the increase in reef depth was equal to half the increase in SL at each increment. To represent a keep-up reef response, where carbonate accumulation processes allow a reef to accrete vertically at the same rate as SLR, the change in reef depth was zero. Given the significant uncertainty and regional variability regarding the rate and magnitude of future SLR12, we chose not to convert our identified SLR thresholds into temporal dates.

### Limitations

Simulations used to develop the RIOT were undertaken using a fully nonlinear Boussinesq-type model that has been comprehensively evaluated for simulating wave processes on coral reefs. However, there are limitations associated with numerical simulations and data synthesis that need to be understood before Eq. (1) can be applied in practise. First, using a one-dimensional across-reef profile omits the influence of wave refraction that is important on some reefs. Second, using an idealised island and reef morphology in each simulation omits the influence of micro-topography on wave processes on the reef flat and beach-face. Accordingly, our findings are based on a simplified geometry that represents the broad morphological characteristics of a reef. Third, the hydrodynamic model does not account for porosity, sediment transport or morphodynamic change which means our inundation values likely present a worst-case scenario. Fourth, we do not represent variations in friction, reef slope and beach slope that are known to influence runup on coral reef shorelines because we consider these to be second-order variables. Fifth, the location of the threshold curve that defines whether inundation occurs or not for a given combination of RIE and RRW was defined visually and accounts for 99.2% of all combinations associated with overtopping or inundation.

Despite these limitations, the placement of RRW and RIE combinations for 23 known inundation, overtopping and runup events (Fig. 2), provides confidence that these limitations are acceptable on many atoll settings. The motivation for this analysis was to develop a first-order method for rapid assessment of island vulnerability to overtopping, which requires focusing on key variables and omitting local factors such as reef slope and friction that have a secondary influence. Therefore, the RIOT is most applicable for wide-scale and general assessments of island vulnerability and can identify areas where more specific investigations can be made, if required.

### Code availability

The Green–Naghdi solver used in this research is part of the open-source Basilisk software package. The documented source code is available online (http://basilisk.fr/src/green-naghdi.h) and the software is freely available to install and use. Linux shell scripts that were used to pre-process model runs and Matlab codes that were used to process model outputs are available from the corresponding author (e.beetham@auckland.ac.nz) on request.