# Stochastic dynamics of barrier island elevation

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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

Barrier islands sustain important coastal ecosystems and are our first line of defense from storm impacts. However, they are threatened by sea level rise and lack of sand supply and could undergo rapid deterioration. The uncertainty in barrier response is amplified by the complexity of the processes involved and their stochastic nature. Here we find that barrier stochastic behavior can be described by a stochastic equation that suggests the presence of a tipping point in barrier response to external drivers. The solution of this equation allows the quantitative estimation of after-storm recovery, the rate of barrier landward migration, the effectiveness of dune protection in reducing back-barrier flooding, and the effects of potential interventions to accelerate after-storm recovery.

## Abstract

Barrier islands are ubiquitous coastal features that create low-energy environments where salt marshes, oyster reefs, and mangroves can develop and survive external stresses. Barrier systems also protect interior coastal communities from storm surges and wave-driven erosion. These functions depend on the existence of a slowly migrating, vertically stable barrier, a condition tied to the frequency of storm-driven overwashes and thus barrier elevation during the storm impact. The balance between erosional and accretional processes behind barrier dynamics is stochastic in nature and cannot be properly understood with traditional continuous models. Here we develop a master equation describing the stochastic dynamics of the probability density function (PDF) of barrier elevation at a point. The dynamics are controlled by two dimensionless numbers relating the average intensity and frequency of high-water events (HWEs) to the maximum dune height and dune formation time, which are in turn a function of the rate of sea level rise, sand availability, and stress of the plant ecosystem anchoring dune formation. Depending on the control parameters, the transient solution converges toward a high-elevation barrier, a low-elevation barrier, or a mixed, bimodal, state. We find the average after-storm recovery time—a relaxation time characterizing barrier’s resiliency to storm impacts—changes rapidly with the control parameters, suggesting a tipping point in barrier response to external drivers. We finally derive explicit expressions for the overwash probability and average overwash frequency and transport rate characterizing the landward migration of barriers.

Barrier island elevation is determined by the competition between the formation of vegetated dunes or foredunes (the highest natural feature on a barrier) and random water-driven erosional events (Fig. 1*A*). Vegetated dunes form when plants trap wind-blown sand and their growth thus depends on the establishment of a dune-building plant ecosystem, the availability of fine sand, and the presence of a dry beach (1). Fast-growing dunes can recover before the next storm or high-water event (HWE) hits the island, in which case islands will tend to have well-developed dunes, resist storm impacts, migrate slowly (if at all), and support a rich ecosystem and/or human development. In contrast, slow-growing dunes can be frequently eroded, which keeps island elevation low and prone to frequent overwash, resulting in rapid landward migration and low biodiversity. These two extreme cases can be associated with high-elevation and low-elevation barrier states, respectively (2) (Fig. 1*A*).

HWEs—defined by clusters of total water levels above a given threshold elevation—can be divided into two broad groups based on the relation between maximum total water level, beach elevation, and the height of mature dunes (3). The first group consists of interannual high-intensity events (e.g., large storms) overtopping and potentially eroding mature dunes. The second group consists of intraannual low-intensity events flooding the beach, which can disrupt after-storm dune recovery when barrier elevation is low (Fig. 1*A*).

Recent measurements of the stochastic properties of intraannual HWEs in several locations around the globe, reported in a companion study in PNAS (3), show they can be modeled as a marked Poisson process with exponentially distributed marks. The mark of a HWE is defined as the maximum water level above the beach during the duration of the event and characterizes its size and intensity. This result opens the way for a probabilistic model of the temporal evolution of the barrier/dune elevation, along the lines of the stochastic model of soil moisture dynamics (4). The knowledge of the transient probability distribution function of barrier elevation allows the calculation of after-storm recovery times, overwash probability and frequency, and the average overwash transport rate driving the landward migration of barriers.

## Master Equation for Barrier Elevation

In general, the barrier elevation z at a point *B*) and its temporal evolution has to be calculated with complex eco-morphodynamic models (1, 2). In what follows, we propose several approximations to reduce this complex two-dimensional problem to a point (zero-dimensional) description.

Following ref. 1, we assume dunes form at a given cross-shore location *B*). For simplicity, we consider dune elevation relative to a washover fan (i.e., a bare low-elevation area), such that the condition

Numerical simulations (1) and field measurements (5) show dune growth in the absence of wave overtopping has a characteristic time *Discussion and Conclusions*.

The along-shore profile of barrier elevation is assumed to consist of different random realizations of the stochastic dune elevation **1**), and random erosional HWEs decreasing dune size by an amount

The stochastic erosional dynamics can be included into a Chapman–Kolmogorov forward equation for the evolution of the probability density function (PDF) of barrier elevation *Materials and Methods*). After rescaling dune height by H and time by **1**), and the term

Integrating Eq. **3** over ξ and using the normalization condition

### Control Parameters.

The evolution of the probability density function

## Solution, Interpretation, and Derived Quantities

### Transient Solution.

Eq. **3** is solved using the method of characteristics, where the curve

The transient PDF for *Materials and Methods*.

For

### Steady-State Solution.

Taking **3**, we arrive at the steady-state solution

### Intermodal Transition Times and Barrier Elevation Regimes.

In the parameter subset where the steady-state PDF exhibits bimodality (

Following ref. 8, **9** into the definition of

The behavior of

### After-Storm Recovery Time and Tipping Point in Barrier Response.

The mean excursion time

The characteristics of the tipping point can be derived analytically for **17** can be approximated by

The emergence of strongly time-separated modes (*Discussion and Conclusions*.

### Average Overwash Frequency and Transport Rate.

In addition to describing the natural variability of barrier elevation, the transient PDF **4**). This allows the calculation of the average overwash frequency

The average overwash frequency also quantifies the exchange of sediments between the beach and the back barrier leading to the landward migration of barriers. Indeed, as shown in *Materials and Methods*, the expected value of the transport rate

### Overwash Probability during Dune Recovery and Effects of Potential Interventions.

A final metric to study the vulnerability of dune recovery and the effectiveness of potential interventions is the probability **6**) and

As expected, *A*). Furthermore, *B*). Interpreting initial dune size as the degree of intervention in the system,

For **17**. The relation

## Discussion and Conclusions

Our results complement those of a previous process-based model (2) focused on the controls of after-storm vegetation dynamics. Together, they propose two complementing mechanisms for the slowdown in dune growth (Fig. 1*A*): first, the slow recovery of vegetation (2), which is required for dune formation and was assumed to depend on wind-driven sand accretion in the absence of plants (2), and second, direct water-driven erosion of small vegetated dunes (proto-dune), which we find can lead to a low-elevation barrier even for fast vegetation dynamics and is the focus of the present study. These two studies thus explore different aspects of a more general parameter space involving biophysical interactions. However, by characterizing the conditions leading to a low-elevation mode during fast vegetation growth, we provide a sufficient condition for the existence of a low-elevation barrier. Indeed, a slower vegetation recovery will only increase the dominance of the low-elevation mode.

Furthermore, the stochastic dynamics of dune growth given here, and the probabilistic characterizations thence derived, relate to a single point, as stated at the outset. The observational quantity corresponding to this model would be a time series of heights for a dune that is (piecewise) homogeneous in the y direction. In particular, the predicted steady-state distribution should be compared with the empirical distribution function of such a time series with a suitably large number of sample points. This comparison would allow for parameter estimation or model validation. However, as was previously cautioned, a dune in the mixed regime presents a nontrivial distribution structure and potentially takes a very long time to explore its state space if the modes are well separated, so that the observational series needed to resolve it would be very long indeed. Somehow, the longitudinal distribution of dune heights must be made to serve instead. This will be the topic of a forthcoming paper.

In summary, here we developed and solved a master equation describing the temporal evolution of the PDF of barrier elevation, represented by the height of its coastal dune. We find, consistent with earlier work (2), that barriers can be low elevation, high elevation, or mixed (bimodal PDF), depending on two control parameters that characterize the competition between vertical accretional and erosional processes in terms of the ratio of the temporal and spatial scales of HWEs and coastal dunes. The transient PDF and derived quantities such as intermodal transition times or after-storm recovery times, average overwash frequency, and overwash probability before dune recovery provide a quantitative description of barrier state and its resilience/vulnerability. These quantities can be used to evaluate important and varied aspects of barrier response such as the rate of barrier migration due to overwash-related sand transport, the effectiveness of dune protection in reducing back-barrier flooding, and the effects of potential interventions to accelerate after-storm recovery. Furthermore, this model allows the study of coastal remediation strategies using the initial condition to parameterize single interventions or modifying the effects of erosional events to account for multiple interventions. Our analytical results open the door to simplified mean-field models of barrier and barrier systems including dune–beach interactions and the coupled barrier–marsh–lagoon system.

## Materials and Methods

### Derivation of the Master Equation.

The probability density

Similarly to the stochastic soil moisture dynamics at a point (4), Eq. **2** can be written as a Chapman–Kolmogorov forward equation for the evolution of the PDF of barrier elevation **14**) and integrating leads to Eq. **3**.

### Definition of the Auxiliary Function ϕ ( ξ ) .

The function

### Average Transport Rate due to Overwashes.

Following energy considerations (2), the transport rate

## Data Availability.

All study data are included in this article.

## Acknowledgments

O.D.V. and I.R.-I. acknowledge the support of the Texas A&M Engineering Experiment Station.

## Footnotes

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^{1}To whom correspondence may be addressed. Email: oduranvinent{at}tamu.edu or irodriguez{at}ocen.tamu.edu.

Author contributions: O.D.V., B.E.S., and I.R.-I. 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.

Published under the PNAS license.

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- Earth, Atmospheric, and Planetary Sciences