# Probabilistic structure of events controlling the after-storm recovery of coastal dunes

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Contributed by Ignacio Rodriguez-Iturbe, November 11, 2020 (sent for review June 29, 2020; reviewed by Carlo Camporeale and A. Brad Murray)

## Significance

Coastal dunes help stabilize barrier islands by reducing the frequency of overwashes and promoting vertical accretion. In doing so, they build unique habitats and protect coastal infrastructure. The after-storm recovery of coastal dunes thus underpins the short-term resiliency of sandy coasts to storm impacts and sea-level rise. Here, we show that this recovery is controlled by the size and frequency of random high-water events too small to overtop a mature dune, but large enough to prevent the development of small embryo dunes. Using data from several locations around the world, we find that the statistical properties of high-water events change surprisingly little and do not depend on tidal range and wave regime.

## Abstract

Coastal dunes protect beach communities and ecosystems from rising seas and storm flooding and influence the stability of barrier islands by preventing overwashes and limiting barrier migration. Therefore, the degree of dune recovery after a large storm provides a simple measure of the short-term resiliency (and potential long-term vulnerability) of barrier islands to external stresses. Dune recovery is modulated by low-intensity/high-frequency high-water events (HWEs), which remain poorly understood compared to the low-frequency extreme events eroding mature dunes and dominating the short-term socio-economic impacts on coastal communities. Here, we define HWEs and analyze their probabilistic structure using time series of still-water level and deep-water wave data from multiple locations around the world. We find that HWEs overtopping the beach can be modeled as a marked Poisson process with exponentially distributed sizes or marks and have a mean size that varies surprisingly little with location. This homogeneity of global HWEs is related to the distribution of the extreme values of a wave-runup parameter,

The statistics of extreme events has been well studied in hydrology using the generalized extreme value distribution (1). A particularly relevant statistical model consists of a Poisson process for the occurrence of an exceedance of a high threshold and a generalized Pareto (GP) distribution for the size of the event (termed “Poisson–GP model”). This model underpins the peak-over-threshold method (1, 2) and has been used to analyze the statistics of extreme wave heights (2, 3) and to simulate extreme total water levels in the Pacific West Coast to complement the scarce observational data on events with return periods above 10 y (4).

In the context of dune erosion, the focus on extreme events (4, 5)—usually referred to as events with return periods in the 10- to 100-y range—is understandable, as they can overtop mature dunes and lead to an overwash, where most of the dunes and the plant ecosystem supported by them are eroded away. However, extreme events don’t control the short and midterm dune dynamics, as dunes have plenty of time to recover before the next one. In contrast, high-water events (HWEs) with much shorter return periods (of the order of few months or less; Fig. 1*A*) can affect the after-storm recovery process by eroding small embryo-dunes, preventing plant colonization of washover fans (i.e., low-lying regions created during an overwash) (Fig. 1*B*).

Remote-sensing data of dune recovery in low barrier islands suggest a sigmoid growth curve with a slower initial recovery and a final saturation to a maximum dune size (6, 7). However, is the slow initial growth a result of intrinsic limitations in the dune-building process or evidence of the erosional effects of HWEs? After all, storm intensity and frequency have been reported to hinder embryo-dune development (8), and initial dune growth can be fast in locations with little impacts of HWEs, such as on the high-elevation backshore of the coast of Oregon (9).

The potential for disruption of low-intensity HWEs becomes evident when superimposing calculated return periods of potentially dune-overtopping HWEs (obtained from data in the Virginia barrier islands; see *Calculation of HWEs*) to theoretical predictions of the undisturbed after-storm dune-recovery curve (Fig. 1*C*). Recent numerical simulations (10) and dune-growth data (9) suggest that the undisturbed dune recovery follows an exponential saturation curve of the form *B*), HWEs flooding the Virginia barrier islands (e.g., Fig. 1*A*) could overtop embryo-dunes for about 4 y after an overwash, potentially leading to widespread erosion and slowing down dune recovery. Once dunes reach about 1 to 1.5 m; they seem to enter a safer period of a few decades where most HWEs don’t reach the dune crest and the overtopping probability is low (although the dune base and front can be partially eroded). Mature dunes can then survive for about 100 y before an extreme event leads to another overwash (Fig. 1*C*).

The competition between dune formation and water-driven erosion has important implications for the stability of barrier islands, with low-intensity HWEs preventing dune recovery and potentially keeping the barrier in a low-elevation, highly vulnerable state (11). In fact, the resiliency of barrier islands can be partially defined by the degree of after-storm dune recovery and, thus, by the frequency and intensity of HWEs (11).

Here, we use the peak-over-threshold method to define HWEs and describe their probabilistic structure. Our analysis suggests that HWEs can be modeled as a marked Poisson process with exponentially distributed sizes. In a companion paper (12), we use this result to derive and analytically solve a stochastic model of barrier-elevation dynamics.

## Calculation of HWEs

Following the peak-over-threshold method, we define HWEs as clusters of the daily maximum of total water levels (*A* and *B*).

The total water level η at the shoreline is defined as the sum of the still-water level *Materials and Methods* for details). Wave runup was found to scale as *Materials and Methods*).

We calculate η from time series of still-water levels and deep-water wave data collected by buoys in 12 locations on the North Atlantic, North and South Pacific, and the Adriatic Sea (Fig. 2*A*; see *SI Appendix*, Table S1 and Fig. S1 for further details). These locations represent mostly wave-dominated sandy beaches with a relatively wide range of tidal and wave conditions (Fig. 2*B*). In what follows, we use local values of beach slopes (*SI Appendix*, Table S2 and Fig. S2).

At each location, HWEs are identified as the discrete set of clusters of daily maximum total water levels *A*).

## Results

### Statistical Properties of HWEs.

The frequency and intensity of HWEs at each site are described by the probability distributions of interarrival times

For high enough elevations (see below for a proper definition), we find that both probability distribution functions are consistent with exponential distributions in all sites analyzed (Fig. 3; see *SI Appendix*, *SI Methods* and Figs. S3 and S4 for the result of statistical tests). Consecutive HWEs as defined before may thus be considered statistically independent and can be represented by a Poisson-GP model with null shape parameter (2).

As a consequence, the change in the frequency of HWEs with elevation can be approximated as*A*) and corresponds to an annualized frequency **1** and the constancy of

### Distribution Parameters.

At each location, the statistical properties of HWEs, modeled as a marked Poisson process with exponentially distributed sizes, are summarized by Eq. **1** in terms of two parameters: the reference elevation

Interestingly, the reference elevation *SI Appendix*, Fig. S5), which suggests a causal relation between HWEs and beach morphodynamics. In this interpretation,

For simplicity, and based on this interpretation of

### Effects of External Drivers on the Distribution Parameters.

Tides and waves are the main drivers of HWEs, and their effect on the distribution parameters (*Materials and Methods* and Eq. **2**). We find that the reference beach elevation *A*). In contrast, the mean intensity of HWEs flooding the beach (*B*). In fact, *B*).

### Homogeneity of Global HWEs.

Assuming **1**). The relatively constancy of *B*), for the locations analyzed thus suggests an interesting homogeneity of global HWEs in natural beaches.

The origin of both the little variation of the mean size of HWEs and the exponential distribution of their size can be traced back to the distribution of the extreme values of the runup parameter **3**) leading to the extreme runups behind HWEs (Fig. 7). Indeed, the distribution of the daily maximum of *A* and *B*). This exponential tail becomes even more pronounced when considering only values of the runup parameter *C* and *D*). Therefore, by filtering out the bulk of the distribution (which drives the mean

The scaling and distribution properties of the extreme values of

## Conclusions

Here, we defined and analyzed the statistical properties of global HWEs flooding beaches around the world. These events potentially control after-storm dune recovery, wave-driven nuisance flooding, and even barrier-island dynamics. We find that a particular frequency of HWEs defines a reference elevation or threshold that can be identified as a characteristic beach elevation. This frequency, of about one event per month, suggests an interesting connection between traditional cross-shore beach morphodynamics and the stochastic properties of HWEs. We also find that HWEs overtopping this characteristic beach elevation belong to a general class of extreme events described by a Poisson-GP model, i.e., a marked Poisson process with exponentially distributed sizes or marks. Furthermore, their mean intensity changes little with location and does not seem to depend on the average wave height or wave runup. Our results thus imply a homogeneity of HWEs affecting global natural beaches. Although we currently lack a simple explanation for this homogeneity, it seems to be related to the distribution of the extreme values of a runup parameter

## Materials and Methods

### Calculation of the Total Water Level.

The total water level η at the shoreline is defined as the sum of the still-water level **2** can be directly measured by tide gauge, *SI Appendix*, *SI Methods* and Fig. S2).

We used available hourly data of significant wave height *A*), to generate a time series of total water levels

The time series

### Calculation of the Tidal Amplitude.

The tidal amplitude was calculated following the National Oceanic and Atmospheric Administration (NOAA) guidelines as half of the mean tidal range, which is defined as the difference between the mean high and mean low water levels. The mean high/low water levels were extracted from the time series of verified tide

## Data Availability.

Data have been deposited in Texas Data Repository Dataverse at https://doi.org/10.18738/T8/HVKLND.

## Acknowledgments

T.R., I.R.-I., and O.D.V. were supported by the Texas A&M Engineering Experiment Station. K.A.R. was supported by a fellowship from the Hagler Institute at Texas A&M University.

## Footnotes

↵

^{1}T.R. and K.A.R contributed equally to this work.- ↵
^{2}To whom correspondence may be addressed. Email: oduranvinent{at}tamu.edu or irodriguez{at}ocen.tamu.edu.

Author contributions: T.R., K.A.R., I.R.-I., and O.D.V. designed research, performed research, analyzed data, and wrote the paper.

Reviewers: C.C., Polytechnic University of Turin; and A.B.M., Duke University.

The authors declare no competing interest.

This article contains supporting information online at https://www.pnas.org/lookup/suppl/doi:10.1073/pnas.2013254118/-/DCSupplemental.

Published under the PNAS license.

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