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C.C., Reviewers: paper. and y the wrote I.R.-I., and data, K.A.R., analyzed research, T.R., contributions: Author an form follows the where recovery of dune curve undisturbed saturation dune-growth the exponential that and suggest (10) (9) simulations data numerical curve Recent dune-recovery 1C). after-storm (Fig. undisturbed the Vir- of see the predictions in islands; data barrier from ginia (obtained poten- HWEs of dune-overtopping periods of tially return calculated impacts superimposing when little evident of coast with the of locations backshore in high-elevation (9). the fast Oregon on initial as be and such been (8), can HWEs, have development growth frequency embryo-dune and dune effects hinder intensity erosional storm to the all, reported of After evidence HWEs? or of process dune-building the uvv o bu 0 eoea xrm vn ed oanother to leads event then extreme 1C). can an (Fig. dunes before overwash y Mature 100 eroded). dune about partially for the be survive (although safer can low a front is dune enter and the probability base to reach overtopping don’t seem the HWEs they most and m; where crest decades 1.5 few to a of 1 potentially recovery. period about dune overwash, reach down an dunes slowing after and Once over- erosion y could widespread 4 1A) to about leading Fig. for (e.g., embryo-dunes HWEs islands 1B), top barrier typical Fig. Virginia m) (e.g., (2 the Virginia height islands, flooding dune Metompkin maximum and a Hog coastal and for of (9), characteristic Oregon m/y), in (0.5 dunes rate rate dune-growth dune-growth large undisturbed tively the of terms in and height, dune owo orsodnemyb drse.Eal drniettm.d or [email protected] Email: addressed. be may [email protected]. correspondence whom To work.y this to equally contributed K.A.R and T.R. rsnl iteadd o eedo ia ag n wave and range tidal on sur- depend change not events that do regime. high-water find and we little of world, prisingly Using properties the dunes. statistical around embryo locations the small several of from enough development large data but the dune, mature prevent a high-water to overtop random to recovery of small frequency this too and that events size show the we by storm controlled Here, to is rise. sea-level sandy and coastal of impacts resiliency protect thus short-term and dunes the habitats coastal underpins of unique recovery after-storm build accretion. 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EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES ABC 4 4 Erosion of mature High-Water Events (HWEs) Erosion of 3.5 dunes Maximum Dune Height 3 No erosion embryo-dunes due to HWEs 3 Mature dune 3 Erosion of Erosion of embryo- 2.5 2 mature dunes dunes Embryo-dunes due to extreme Inter-arrival HWE size (S) l 2 ( t) 2 events Beach berm Beach Elevation 1.5 1 Beach- 1 forming 1 Washover fan events 95% confidence interva

Elevation relative to MSL (m) Elevation relative to MSL HWEs return period

0.5 0 Elevation relative to the beach (m) Theoretical return period Total water level Saturated dune growth 0 0 0 50 100 150 200 250 300 350 0 20406080100 Time (days) Distance from shoreline (m) Time after overwash (yr)

Fig. 1. (A) Example of predicted daily maximum total water levels ηd(t), representative of conditions in the Virginia barrier islands, showing HWEs above a typical beach elevation (B). Interarrival times ∆t and size or mark S of HWEs are also shown. (B) Typical cross- profiles of coastal dunes and a washover fan in Metompkin , Virginia, illustrating potential morphological effects of HWEs. (C) Illustration of the dune erosional regimes that emerge after superimposing return periods of HWEs potentially overtopping dunes in the Virginia barrier islands (symbols) to the undisturbed after-storm dune-recovery −t/T curve (black solid line): h(t) = H(1 − e d ) (10), with maximum dune height H = 2 m relative to the beach (as in B), dune-formation time Td = H/G, and dune-growth rate G = 0.5 m/y (9).

The competition between dune formation and water-driven arbitrary threshold elevation Z . For a given Z , we character- erosion has important implications for the stability of bar- ize a particular HWE i by the time it starts ti , the time to the rier islands, with low-intensity HWEs preventing dune recov- ery 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 A 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 B as clusters of the daily maximum of total water levels (ηd ) above a threshold Z representing a coastal feature potentially overtopped by water (Fig. 1 A and B). The total water level η at the shoreline is defined as the sum of the still-water level ηS , which is influenced by astronomical tides and meteorological effects such as storm surges and is character- ized by an effective tidal amplitude At , and the 2% exceedance wave runup ηW containing the effects of wave setup and excursion (see Materials and Methods for details). Wave√ runup was found to scale as ηW = a(β)HR (13–16), where HR = Hs L0 is a runup parameter defined in terms of the deep-water signifi- cant wave height Hs and wavelength L0, and a(β) is an empirical function of the beach slope β calibrated by using field data (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. 2A; see SI Appendix, Table S1 and Fig. S1 for further details). These Fig. 2. (A) Global map of the sites analyzed: Elfin , Alaska (AK); locations represent mostly wave-dominated sandy with Monterey , California (CA); South Beach, Oregon (OR); St. Petersburg, a relatively wide range of tidal and wave conditions (Fig. 2B). In Florida (FL-SP); Canaveral, Florida (FL-CC); Boston/Plum Island, Mas- what follows, we use local values of beach slopes (SI Appendix, sachusetts (MA); Padre Island, Texas (TX-PI); Galveston Island, Texas (TX-GV); Metompkin Island, Virginia (VA); Kailua Beach, Hawaii (HI); Tweed Heads, Table S2 and Fig. S2). Australia; and Grado, Italy (see SI Appendix, Fig. S1 for further details). (B) At each location, HWEs are identified as the discrete set of Mean significant wave height Hs (±SD) versus tidal amplitude At at each clusters of daily maximum total water levels ηd (t) exceeding an site (see Materials and Methods for calculation of At ).

2 of 6 | PNAS Rinaldo et al. https://doi.org/10.1073/pnas.2013254118 Probabilistic structure of events controlling the after-storm recovery of coastal dunes Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 h frequency distributions the probability the times by interarrival described of are site each HWEs. at HWEs of Properties Statistical Results 1A). event: (Fig. the during attained threshold the to “mark,” or event next distributed exponentially legend. with 2 Fig. process the Poisson in as marked are Abbreviations a distributions, sizes. as exponential for modeled elevation prediction be reference the (i.e., with consecutive site agreement λe between The each time above 4). at HWEs minimum Fig. for elevation only the are representative shown for Data a account average. denotes to overbar The d events. 2 by reduced size normalized rbblsi tutr feet otoln h fe-tr eoeyo osa dunes coastal of recovery after-storm the controlling events of structure Probabilistic shape al. null et Rinaldo with model Poisson-GP and a independent defined by statistically (2). as parameter represented considered HWEs be be Consecutive can thus tests). may statistical analyzed before of sites result are all the functions in see distribution 3; distributions probability (Fig. exponential both with that consistent find we tion), above HWEs series. time of number the is that elevation Note threshold sites. the of function HWEs consecutive between function time distribution rival cumulative the minus (one 3. Fig. A B −λ∆t o iheog lvtos(e eo o rprdefini- proper a for below (see elevations enough high For xmlso h opeetr uuaiedsrbto function distribution cumulative complementary the of Examples and ¯ i S S −1 1 + i IAppendix, SI endb h aiu oa ae ee relative level water total maximum the by defined , S e λ(Z / −S λ S itrria time) (interarrival / a ipyb siae as estimated be simply can .Teatrs en nearvltmshv been have times interarrival means asterisk The (B). ¯ S 1 = ) epciey(ahdlns,sget htHE can HWEs that suggests lines), (dashed respectively , ∆t / n sizes and ∆ IMethods SI t n ensize mean and h rqec n nest of intensity and frequency The Z λ*∆t S and h envle,gvnby given values, mean The . ∆t Z * i n is 3adS4 and S3 Figs. and n a ayaogthe among vary can and T ≡ = ∆t F stedrto fthe of duration the is t ftersae interar- rescaled the of ) i S +1 */ t i (Z

EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES 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 HR (Eq. 3) leading to the extreme runups behind HWEs (Fig. 7). Indeed, the distribution of the daily maximum −HR/`R of HR seems to have an exponential tail (∝ e ) with a rela- tively constant characteristic length `R independent of the mean HR (Fig. 7 A and B). This exponential tail becomes even more pronounced when considering only values of the runup param- eter HR associated with total water levels (or HWEs) above the reference beach elevation Zr (Fig. 7 C and D). Therefore, by filtering out the bulk of the distribution (which drives the mean HR), a natural beach (interpreted as being characterized by Zr ) essentially amplifies the tail of the distribution of the runup parameter, which then determines the properties of the size distribution of HWEs. The scaling√ and distribution properties of the extreme values of HR = Hs L0 is not obviously related to the distributions of either the significant deep-water wave height Hs or wavelength L0. In that regard, we can consider the runup parameter HR an additional quantity characterizing deep-water waves.

Conclusions Here, we defined and analyzed the statistical properties of global HWEs flooding beaches around the world. These events poten- Fig. 5. Reference elevation Zr , defined from a constant HWE frequency λr tially control after-storm dune recovery, wave-driven nuisance as λ(Zr ) = λr , versus a characteristic range of beach elevation, with mean Zb, measured from cross-shore DEM profiles at some of the sites analyzed (SI Appendix, Fig. S5). All elevations are relative to MSL. The uncertainty in Z (error bars) is given by ∆Z = S ∆λ and represents the statistical r r λr uncertainty of λ (Fig. 4A). A

Interestingly, the reference elevation Zr can be interpreted as a characteristic beach elevation (Fig. 5 and SI Appendix, Fig. S5), which suggests a causal relation between HWEs and beach mor- phodynamics. In this interpretation, λr can be understood as a critical frequency of HWEs, above which accretional events dominate, increasing beach elevation, and below which erosional events dominate, decreasing beach elevation, in agreement with recent measurements of poststorm beach recovery (18). For simplicity, and based on this interpretation of Zr as a ref- erence beach elevation, in what follows, we will refer to S r as the “mean intensity” of HWEs above (or flooding) the beach, where by beach, we are referring to the particular elevation Zr . B Effects of External Drivers on the Distribution Parameters. Tides and waves are the main drivers of HWEs, and their effect on the distribution parameters (Zr and S r ) can be parametrized by the tidal amplitude At and the average wave runup ηW at each site (Materials and Methods and Eq. 2). We find that the reference beach elevation Zr exceeds the tidal amplitude, and the excess beach elevation Zr − At increases with the average runup ηW (Fig. 6A). In contrast, the mean intensity of HWEs flooding the beach (S r ) does not depend on the tidal range and lacks a clear trend with the average runup ηW (Fig. 6B). In fact, S r changes surprisingly little between sites, despite a sixfold variation on mean significant wave height Hs and runup ηW (Fig. 2B).

Homogeneity of Global HWEs. Assuming Zr captures the natural beach elevation, the frequency λ of HWEs overtopping an eleva- tion z = Z − Zr relative to the beach only depends on the mean Fig. 6. Reference beach elevation Zr above the tidal amplitude At increases intensity S r of HWEs flooding the beach (Eq. 1). The relatively with the average wave runup ηW (A), whereas the mean intensity of HWEs ¯ constancy of S r , and indeed the mean size of HWEs in general above the beach, Sr , does not seem to depend on ηW (B). Sites names are (Fig. 4B), for the locations analyzed thus suggests an interesting placed close to the respective symbols. Abbreviations are as in the Fig. 2 homogeneity of global HWEs in natural beaches. legend.

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EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES amplitude At is then half the difference between the average maximum ACKNOWLEDGMENTS. T.R., I.R.-I., and O.D.V. were supported by and the average minimum. the Texas A&M Engineering Experiment Station. K.A.R. was sup- Data Availability. Data have been deposited in Texas Data Repository ported by a fellowship from the Hagler Institute at Texas A&M Dataverse at https://doi.org/10.18738/T8/HVKLND. University.

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