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(Fig. barrier low the when is recovery flooding dune elevation after-storm events disrupt group can low-intensity which second intraannual first beach, The of The dunes. consists mature (3). storms) group eroding large dunes potentially (e.g., and mature events overtopping high-intensity of ele- interannual beach height of level, consists the water total and maximum vation, between relation based groups the broad on two into divided be elevation—can threshold associated be can 1A). respectively cases (Fig. states, (2) barrier extreme low-elevation and and two migration high-elevation with landward These to rapid prone biodiversity. and in low low resulting elevation and/or overwash, island ecosystem frequent keeps rich which fre- a be eroded, impacts, can support dunes quently storm slow-growing and contrast, resist beach In all), development. islands at dunes, human dry case (if well-developed which or a slowly have in migrate storm to of island, next tend the presence the hits will before the (HWE) recover event and the can high-water ecosystem, dunes depends sand, plant Fast-growing thus fine (1). dune-building growth a of their of availability and establishment when water- sand the form random on dunes wind-blown and Vegetated trap barrier) 1A). plants (Fig. a events on erosional feature driven natural highest (the B islands Barrier the characterizing rate finally barriers. transport We of migration and drivers. landward frequency aver- external and overwash probability to overwash age tip- the response for a expressions barrier explicit suggesting derive in parameters, control point impacts— the storm ping relax- with to time—a rapidly resiliency recovery barrier’s changes bimodal, after-storm characterizing mixed, average high- time the a ation a find or toward barrier, We low-elevation state. converges a solution barrier, elevation control transient plant the the the on of Depending stress parameters, of formation. and function availability, dune sand a anchoring rise, turn ecosystem level dune in sea are maximum of which rate the time, the to formation and dune by (HWEs) and intensity controlled events height average are high-water the dynamics of relating The frequency point. numbers a dimensionless (PDF) at two function elevation describing density barrier probability equation the of master of a dynamics con- develop stochastic traditional the we with Here understood models. properly tinuous be in stochastic cannot erosional is and between dynamics nature barrier balance behind The ele- processes impact. accretional barrier to and storm thus tied the and condition during overwashes a vation storm-driven barrier, of stable frequency vertically the of migrating, existence slowly the and on a surges depend storm functions These from erosion. communities wave-driven coastal systems Barrier interior stresses. protect external also low- man- survive and and create reefs, develop Murray) can that oyster Brad groves marshes, A. features and salt Camporeale coastal where Carlo environments by ubiquitous energy reviewed are 2020; 29, islands June review Barrier for (sent 2020 11, November Rodriguez-Iturbe, Ignacio by Contributed 08540 NJ Princeton, University, Princeton a elevation Dur island Orencio barrier of dynamics Stochastic eateto ca niern,TxsAMUiest,CleeSain X78333;and 77843-3136; TX Station, College University, A&M Texas Engineering, Ocean of Department eetmaueet ftesohsi rpriso intraan- of properties stochastic the of measurements Recent given a above levels water total of clusters by HWEs—defined ewe h omto fvgttddnso foredunes or dunes competition vegetated the of by formation determined the is between elevation island arrier 01Vl 1 o e2013349118 1 No. 118 Vol. 2021 | nVinent an ´ tcatcdynamics stochastic a,1 ejmnE Schaffer E. Benjamin , | osa dunes coastal | atrequation master b n gai Rodriguez-Iturbe Ignacio and , 1 ino h cross-shore the elevation of barrier tion the general, In Elevation Barrier for Equation landward Master the driving rate barriers. transport of overwash migration average frequency, of the and calculation dis- probability and the overwash probability allows times, transient elevation recovery after-storm barrier the of of a function knowledge for moisture tribution soil The way of model (4). the stochastic the dynamics opens of lines barrier/dune charac- result level the the along and of This water elevation, evolution event temporal intensity. maximum the the of and the model of probabilistic size as duration defined its the during is terizes beach HWE the a above marks. of distributed mark exponentially with The process Poisson marked a ulse eebr2,2020. 28, December Published be thus can (ρ) rate char- growth a dune The by has approximated shoreline). overtopping steady wave a (for of absence time the acteristic in growth dune overwash. an low-elevation of outcome bare the a thus and (i.e., condition dune the fan that washover such a area), to relative elevation given a at island barrier h a of of position position point the highest also along-shore the is (1), location crest this dune Since the dune. a form to enough location description. (zero-dimensional) point two-dimensional complex a complex to propose this with problem we reduce calculated follows, to what be approximations In several to 2). has (1, models evolution eco-morphodynamic temporal its and ulse ne the under Published interest.y competing no University. declare Duke authors A.B.M., The and Turin; of University Polytechnic research, C.C., Reviewers: performed paper. research, y the designed wrote and I.R.-I. data, and analyzed B.E.S., O.D.V., contributions: Author owo orsodnemyb drse.Eal drniettm.d or [email protected] Email: addressed. be may [email protected]. correspondence whom To (y fet fptnilitretost ceeaeafter-storm accelerate the to and interventions recovery. flooding, potential of back-barrier of effectiveness reducing effects the in migration, recovery, protection landward equation after-storm dune barrier this of of of rate estimation barrier the solution quantitative in The point the drivers. equa- tipping allows a external stochastic of to a presence response by the suggests described that be tion barrier can that processes find behavior we the Here stochastic bar- of nature. complexity stochastic in their the and and uncertainty by involved supply amplified The sand is of response deterioration. rier lack rapid they and However, undergo rise impacts. level could storm sea from by defense threatened are are of and line ecosystems first coastal our important sustain islands Barrier Significance ueia iuain 1 n edmaueet 5 show (5) measurements field and (1) simulations Numerical cross-shore given a at form dunes assume we 1, ref. Following , t = ) x z c (x (y c ittdb hr eeaincnsrielong survive can vegetation where by dictated ), , b eateto ii n niomna Engineering, Environmental and Civil of Department y NSlicense.y PNAS T , d t ) n auae tamxmmdn height dune maximum a at saturates and a,1 .Frsmlct,w osdrdune consider we simplicity, For 1B). (Fig. y y https://doi.org/10.1073/pnas.2013349118 a erpeetdb h ueheight dune the by represented be can x n along-shore and h 0 = z tapoint a at ersnsteasneo a of absence the represents y oiin(i.1B) (Fig. position (x , y ) safunc- a is y | f6 of 1 H
EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES AB
Fig. 1. (A) Simplified dynamics of barrier elevation. Extreme HWEs, i.e., large storms, erode the mature dunes that define a high-elevation barrier, along with the dune-building vegetation; whereas frequent low-intensity HWEs disrupt after-storm dune recovery and keep the island in a low-elevation bare state. Sand supply increases dune growth and promotes a high-elevation state. Sea level rise promotes a low barrier by increasing plant stress and reducing the availability of dry sand. Examples of high and low barriers are from Virginia. (B) Geometrical description of coastal dunes where xc is the cross-shore position of dune crest h. Image is from Cedar Lakes, TX.
∂f ∂ + + dh/dt ≡ ρ(h) = (H − h)/Td . [1] = − [ρ (ξ)f (ξ, t)] − λ (ξ)f (ξ, t), [3] ∂t ∂ξ
+ The undisturbed dune growth after an overwash (h(0) = 0) thus where ξ = h/H is the rescaled dune height, ρ (ξ) = 1 − ξ is the rescaled dune growth rate (Eq. 1), and the term λ+(ξ) = follows the curve h(t) = H 1 − e−t/Td . In a first approxima- + −ξ/ξc ¯ + λ0 e , with ξc = S/(rH ) and λ0 = λ0Td , represents the tion we neglect the complexities of vegetation dynamics during rescaled frequency of events eroding a dune of rescaled height the recolonization of the washover fan (2, 5) and consider the ξ. For simplicity, we keep the rescaled time as t. initial plant establishment is much faster than the ensuing dune Integrating Eq. 3 over ξ and using the normalization condition growth. The implications of this assumption are explored in 1 R f (ξ, t)dξ = 1, we get the boundary condition at ξ = 0, Discussion and Conclusions. 0 The along-shore profile of barrier elevation is assumed to Z 1 consist of different random realizations of the stochastic dune ρ+(0)f (0, t) = λ+(ξ)f (ξ, t)dξ ≡ λ+(t), [4] elevation h(t) at a point. These dynamics are the result of 0 dune-forming processes, characterized by the continuous wind- driven deposition rate ρ(h) that increases dune size by dh in a which states that the probability density of having no dunes time interval dt (Eq. 1), and random erosional HWEs decreas- f (0, t) equals the ratio of the expected overwash frequency λ+ + ing dune size by an amount ∆h > 0 when an event i occurs at and the maximum dune growth rate ρ (0) = 1. time t = ti , Control Parameters. The evolution of the probability density func- tion f (ξ, t) is completely characterized by two dimensionless dh = ρ(h)dt − ∆h(h, t). [2] parameters: the rescaled mean size (intensity) of HWEs, ξc = S¯/(rH ), also representing the rescaled height of a dune eroded ¯ + We model HWEs as a marked Poisson process with frequency λ0 by the average HWE (S), and the rescaled frequency λ0 = λ0Td and exponentially distributed sizes (“marks”) S above the beach representing the ratio between the characteristic dune formation (3). Consistent with our point description, we neglect horizontal time Td and the average time 1/λ0 between HWEs. Global mea- dune erosion—which could be relevant in determining the out- surements of HWEs (3) suggest both S¯ and λ0 (in the range come during long or closely spaced storms—and assume dune 0.2 to 0.4 m and 1 to 2 mo−1, respectively) change little with height remains unchanged if no significant overtopping occurs wave and tidal conditions. Therefore, the control parameters −1 + (S < rh), whereas the dune is completely eroded otherwise (S > ξc ∝ H and λ0 ∝ Td can be interpreted as proxies for dune rh), with overwash parameter r between 1 and 2 (6). This is size (H ) and formation time (Td ). Note that dune size depends consistent with semiempirical models of dune erosion (2) and on plant zonation and thus on the stress of the plant ecosys- represents a sensible idealization of both field data (6) and con- tem anchoring dune formation (1), whereas dune formation time ceptual storm impact scales for barrier islands (7). For simplicity depends on the wind regime and the availability of dry sand in the nomenclature, in what follows we refer to any significant (1). Both quantities can thus be affected by sea level rise as the overtopping event (S > rh) as an overwash event, even for negli- water table becomes shallower and the stress on salt-intolerant gible dune height (h ∼ 0) where the term flooding could be more vegetation increases. appropriate. The stochastic erosional dynamics can be included into a Solution, Interpretation, and Derived Quantities Chapman–Kolmogorov forward equation for the evolution of the Transient Solution. Eq. 3 is solved using the method of charac- probability density function (PDF) of barrier elevation f (h, t) teristics, where the curve ξ∗(t) = 1 − e−t divides the (ξ, t) plane (Materials and Methods). After rescaling dune height by H and into two regions mapping either to the initial condition f (ξ, 0) for ∗ ∗ time by Td , the master equation for h > 0 reads ξ ≥ ξ (t) or to the boundary condition f (0, t) for ξ < ξ (t). Here
2 of 6 | PNAS Vinent et al. https://doi.org/10.1073/pnas.2013349118 Stochastic dynamics of barrier island elevation Downloaded by guest on September 28, 2021 Downloaded by guest on September 28, 2021 tcatcdnmc fbririln elevation island barrier of dynamics Stochastic al. et Vinent elevation initial the form the of conditions initial on focus we (H size dune maximum changing of effects the show right and values expected frequencies eemnsi rwhcurve growth deterministic (ξ overwash (ξ an project restoration of outcome the resenting 2. Fig. ihteaxlayfunction auxiliary the with time before ρ overwash an having where tteboundary the at + h rnin D for PDF transient The For (ξ )dt ξ < ξ p oo lt ftetmoa vlto ftePF(f PDF the of evolution temporal the of plots Color (t hspoaiiycnb rte as written be can probability this , , λ ξ ∗ 0 + 0 (t = ) = ξ ¯ f h rnin D eed ntevalues the on depends PDF transient the ), (t λ (ξ .Tergoso h aaee pc orsodn oahg,lw n ie bmdlPF are r ihihe.Arw nbottom on Arrows highlighted. are barrier PDF) (bimodal mixed and low, high, a to corresponding space parameter the of regions The ). ξ 0 e T , f 0 = − d t (ξ p = ) R clms n ecldma W intensities HWE mean rescaled and (columns) ξ 0 (t , t 0 = n a h form the has and t λ > , = ) f + ξ ξ (0, (ξ 0). 0 0 = ) ξ φ(ξ ( p o en rddwiefloigthe following while eroded being not ξ t ≥ t ))dt (t (t − ) φ(ξ ξ , 1 = ) endin defined ∗ ξ t ersnstepoaiiyo not of probability the represents 0 (t 0 (ξ (t ) ) t δ ))/φ(ξ )) − sn h definition the Using . (ξ stepoaiiydniyof density probability the is φ(ξ (1 − − ξ f )/ρ aeil n Methods. and Materials 0 (t (ξ 0 ξ ), 0 = 0 )), )= 0) , + )e (ξ i.2 rdune or 2) Fig. ; −t ), : δ (ξ (ξ , t ,dn rwhtm (T time growth dune ), )o ecldbrirelevation barrier rescaled of )) − ξ 0 f rep- ), (0, dξ ξ c [5] [7] [6] = t = ) ¯ S /(rH (at ξ ssrcl ioa for f bimodal at strictly minimum is a has solution This where taysaesolution steady-state Solution. Steady-State equation condition integral normalization the ∞ rw)(aussono o n et epciey.Sldlnsso the show lines Solid respectively). left, and top on shown (values (rows) ) min ξ a igehg-lvto oe(at mode high-elevation single a has d ≥ 0 = ,adfeuny(λ frequency and ), t 1 0 ). (ξ (i.e., = ) Z ξ 0 fe noews (Eqs. overwash an after t − λ f φ(ξ ∞ 0 + n(1 ln (ξ ≥ ∗ = ) (t e f 1/ξ − ∞ 0 0 Taking ))f n nest ie,ma size mean (i.e., intensity and ) 0 ξ : c h values The ). ξ < Z hr sasnl o-lvto mode low-elevation single a is there ) (0, 0 1 min t ρ ∂ φ(ξ − + f R < (ξ 0 /∂ https://doi.org/10.1073/pnas.2013349118 1 t ) 0 7 f ) ξ For 1. )dt t min (ξ dξ and 0 = , 0 t 1 = f = o ifrn ecldHWE rescaled different for 8) )dξ −1 (0, nEq. in ξ ξ c ρ − min t φ(ξ 1 = ln + ξ ) p (ξ 1 = r bandfrom obtained are λ (t ≤ earv tthe at arrive we 3, edn othe to leading , ) ) 0 + , ¯ S 0 . ,weesfor whereas ), ξ fHWEs. of ) n therefore and 0 (i.e., ). PNAS λ | 0 + f6 of 3 ≤ [9] [8] 1)
EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES Intermodal Transition Times and Barrier Elevation Regimes. In the After-Storm Recovery Time and Tipping Point in Barrier Response. parameter subset where the steady-state PDF exhibits bimodality The mean excursion time T l within the low-elevation mode can (0 < ξmin < 1), the mean excursion time T l below the mini- also be interpreted as the mean after-storm dune (or barrier) mum of the steady state (ξmin) provides an indication of the recovery time. Therefore, the intermodal transition time pro- timescale associated with bimodal switching, from the low- to the vides a characterization of the resiliency of the barrier under high-elevation mode. a combination of external and intrinsic conditions described Following ref. 8, T l is calculated as the ratio of the frac- by the control parameters (Fig. 3). For T l . 1, barrier eleva- tion of time the system spends below ξmin, given by the tion recovers quickly after an overwash and can be classified R ξmin 0 0 as resilient, whereas for T l 10 recovery takes much longer cumulative distribution P∞(ξmin) = f∞(ξ )dξ , and the & 0 as a now vulnerable barrier experiences frequent overwashes. mean rate ν(ξmin) of up-crossings of the level ξmin, ν(ξmin) = The faster-than-exponential increase of the recovery time in this ρ+(ξ )f (ξ ) 9 T min ∞ min . Substituting Eq. into the definition of l region (Fig. 3) can be interpreted as the crossing of a tipping gives point in barrier response. Z ξmin 0 The characteristics of the tipping point can be derived analyti- 1 φ(ξ ) 0 T l = + 0 dξ . [10] cally for ξc 1. In that case, the exponential integral Ei (1/ξc ) φ(ξmin) ρ (ξ ) 0 (1/ξc ) in Eq. 17 can be approximated by ξc e , leading to T l ∝ + λ ξc β(ξc ) Outside the bimodal region, T l ≡ 0 when f∞ has a single high- e 0 , where β is a weak function of ξc . In a first approxi- + elevation mode (ξmin ≤ 0 for λ0 ≤ 1), and T l → ∞ when f∞ has mation, the tipping point in barrier response, roughly defined by + 1/ξc a single low-elevation mode (ξmin ≥ 1 for λ0 ≥ e ). the boundary T l ∼ 10 limiting resilient barriers, is thus given by a constant value of the product of the two control parameters: The behavior of T l suggests three distinct regimes for the + dynamics of barrier elevation, what we call a “high,” a “low,” and ξc λ0 (Fig. 3). a “mixed” barrier (Figs. 2 and 3). In the cases where T l 1, The emergence of strongly time-separated modes (T l 1) the process of modal switching is long relative to the explicit above the tipping point has important implications for the sys- timescales of the dynamics, namely TD = 1 (by definition of the tem. It may inform approaches to remediation and management, rescaling) and 1/λ+ < 1, and the barrier remains low elevation in which circumstance it would be useful to know how high a 0 dune needs to be to be likely to persist, but it also presents a and prone to overwashes (“low barrier”). For T 1, the oppo- l quandary: The observation window in time necessary to resolve site is true; modal switching is fast and the barrier tends to be the predicted distribution may be impractically long. Rather, spa- high elevation most of the time (“high barrier”). Finally, for tial statistics must serve as a proxy, as highlighted in Discussion intermediate timescales (T l ∼ 1 to 10) both modes are relevant and Conclusions. and the barrier remains in a mixed (bimodal) state alternating between high and low elevations (“mixed barrier”; Fig. 2). Note Average Overwash Frequency and Transport Rate. In addition to that these regimes include the transient dynamics and do not nec- describing the natural variability of barrier elevation, the tran- essarily correspond to the modal properties of the steady-state sient PDF f (ξ, t) also defines the expected overwash frequency solution. at time t: λ+(t) = f (0, t) (Eq. 4). This allows the calculation of + the average overwash frequency hλ iT during a time interval T + 1 R T + 1 R T after an overwash as hλ iT = T 0 λ (t)dt = T 0 f (0, t)dt, which provides information about the role the intermittent pres- ence or absence of dunes can play in promoting long-term barrier habitat resilience. 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 Q S (t) per + + HWE can be approximated as Q S (t) = Q0λ (t)/λ0 with scaling term Q0. Thus, the average transport rate hQiT during a time interval T after an overwash is
+ + hQiT = Q0hλ iT /λ0 [11]
+ and the cumulative transport rate resulting from all HWEs λ0 T + + is simply hQiT λ0 T = Q0hλ iT T .
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 pe (t, ξ0) of an overwash taking place before a time t for a dune of initial elevation ξ0:
Fig. 3. Contour lines of the mean excursion time Tl within the low- pe (t, ξ0) = 1 − p(t, ξ0) = 1 − φ(ξ(t))/φ(ξ0), [12] elevation mode, also identified as the mean after-storm recovery time, rescaled by the dune formation time T , as a function of the two control d where p is the probability of no erosion (Eq. 6) and ξ(t) = parameters. Regions corresponding to resilient high barriers (Tl . 1), vul- −t 1 − (1 − ξ0)e is the deterministic growth of the initial con- nerable low barriers (Tl & 10), and mixed (bimodal) barriers are highlighted. The white region corresponds to λ+ ≥ e1/ξc where T → ∞ and thus dunes dition ξ(0) = ξ0. Defining the dune recovery height as ξr = 0 l −1 never recover. Arrows show the effects of dune and HWEs characteristics on 1 − e , the relevant probability to evaluate the vulnerabil-
the control parameters (see Fig. 2 for more details). ity of a dune of size ξ0 is the probability per = pe (tr , ξ0) of
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Fig. formation dune for required recov- slow is the which first, (2), 1A): vegetation (Fig. growth of dynam- dune ery vegetation in for mechanisms after-storm slowdown complementing of the two propose controls they Together, the ics. on model focused process-based previous (2) a of those complement results Our Conclusions and to Discussion approximation overwash. probable good highly a or negligible provides of and regions the 4) constrain Fig. in constant lines of (contour lines the describes thus imation and where recovers, dune condition the (1 ln before + 1 occurring overwash an tcatcdnmc fbririln elevation island barrier of dynamics Stochastic al. et Vinent parameters control the of function ξ a as recovery, dune before place c B A For sexpected, As o niiildn size dune initial an for ξ c nrsos omr nes n rqetHE and/or HWEs frequent and intense more to response in , γ (A ξ E c sawa ucinof function weak a is ξ − and 0 i ξ (x ξ (t for p 0 1 otu ie fteprobability the of lines Contour B) r e ) = ) r ξ stedtriitcrcvr iestsyn the satisfying time recovery deterministic the is c and ≈ p 1) = e ξ 1 r .Arw hwteefcso ueadHWEs and dune of effects the show Arrows (B). 0.15 r ≈ − . nrae ihbt oto parameters, control both with increases ξ e 0 exp x ξ /x 0 0 = .Interpreting 4B). (Fig. dunes initial of ) 1, nEq. in n ucinof function a and (A) 0 −λ p ξ e 0 + r c ξ and c h relation The 17. a eapoiae yan by approximated be can e −ξ p ξ 0 e 0 r /ξ n eueteapprox- the use we and , nteprmtrspace parameter the in c γ p (ξ e r c fa vrahtaking overwash an of , ξ λ 0 ) 0 + λ , 0 + n h initial the and ξ c ∝ λ e 0 + ξ [13] t 0 r and p p λ /ξ e e = 0 + r r c (∆h Equation. Master dune the of Derivation Methods and Materials barrier–marsh– includ- coupled systems the barrier and simpli- system. and lagoon interactions multiple to barrier dune–beach door for of the account ing models open to mean-field results events fied analytical erosional Our of interventions. effects using mod- strategies the or interventions remediation ifying single inter- parameterize coastal to of condition potential this initial study the Furthermore, of the recovery. effects allows after-storm reduc- model the accelerate in and to overwash-related protection ventions to flooding, dune due of back-barrier response migration effectiveness ing barrier barrier the of of transport, used rate aspects sand be the varied as can and such quantities important evaluate These to resilience/vulnerability. state barrier its of before description times, and probability quantitative a recovery overwash provide and recovery after-storm and dune HWEs such frequency, or quantities of overwash times derived scales average transition and spatial intermodal PDF and transient as temporal terms The the in dunes. of competition processes coastal ratio the erosional the and characterize of accretional that low vertical depending parameters be PDF), between We control can (bimodal dune. barriers two mixed that coastal or on (2), elevation, its barrier work high of of earlier elevation, height PDF with a the consistent the of find, of by topic evolution represented the temporal elevation, the be describing will tion This instead. serve to paper. forthcoming heights made dune long that be of very so distribution be must separated, would longitudinal it the well resolve Somehow, are to indeed. to modes nontrivial needed time the series a long if observational presents the very space a state regime its takes mixed explore potentially the parame- and number previously structure in for large was distribution dune as allow suitably However, a would a validation. cautioned, model comparison with or This series estimation ter distribution points. time empirical sample a the of such with homogeneous of compared (piecewise) function be is time should that a observa- tribution dune be The a outset. would the for the model in heights at this of stated to derived, series as corresponding point, thence quantity single characterizations tional a probabilistic to the relate and low- here, the of dominance veg- the slower for increase a mode. condition only Indeed, elevation barrier. will sufficient low-elevation recovery a a etation provide of during we existence mode the low-elevation parameter growth, studies a charac- vegetation general two to by fast leading These more However, conditions study. a the interactions. present terizing of biophysical the aspects involving of space different dynam- focus explore vegetation the fast thus can is for find even and we barrier ics low-elevation which ero- a (proto-dune), water-driven to dunes direct in lead vegetated second, accretion small and sand of (2), wind-driven sion plants on of depend absence to the assumed was and itiuino h size the of distribution where nsmay eew eeoe n ovdamse equa- master a solved and developed we here summary, In given growth dune of dynamics stochastic the Furthermore, = h )atraHEi ie by given is HWE a after 0) δ y steDrcdladsrbto n eue h exponential the used we and distribution delta Dirac the is (.) en ihrcmltl rdd(∆h eroded completely either being f E ieto.I atclr h rdce taysaedis- steady-state predicted the particular, In direction. (∆h, h) = = δ δ (∆h) (∆h) S fHE,wt mean with HWEs, of Z 0 1 rh − e e − ¯ S − ¯ S S h rbblt density probability The rh ¯ S dS https://doi.org/10.1073/pnas.2013349118 + + δ δ (h (h ¯ S − − . = ∆h) ∆h)e rnteoe tall at eroded not or h) Z − rh ∞ rh ¯ S , e − ¯ S PNAS f ¯ S S E (∆h, dS | fa of h) f6 of 5 [14]
EARTH, ATMOSPHERIC, AND PLANETARY SCIENCES 2 Similarly to the stochastic soil moisture dynamics at a point (4), Eq. 2 can QS = (S − h) /T, where T is a scale parameter with dimension of time. The be written as a Chapman–Kolmogorov forward equation for the evolution expected value of QS per HWE is obtained by integrating over all possible of the PDF of barrier elevation f(h, t): overwashes:
∂f ∂ − S Z H / Z ∞ ¯S = − [ρ(h)f(h, t)] − λ0f(h, t) f(h H, t) e ∂t ∂h QS(t) = QS(S, h) dSdh. [18] H ¯S Z ∞ 0 rh 0 0 0 0 + λ0 f(h , t)fE(h − h, h )dh . [15] h ¯ Substituting QS, introducing new variables η = (S − h)/S and ξ = h/H, and
Substituting fE (Eq. 14) and integrating leads to Eq. 3. using r = 1 give
¯2 Z ∞ Z 1 Definition of the Auxiliary Function φ(ξ). The function φ(ξ) is defined as S 2 −η −ξ/ξc QS(t) = η e dη e f(ξ, t)dξ [19] T 0 0 ξ λ+(ξ0) − R dξ0 0 + 0 φ(ξ) = e ρ (ξ ) [16] Z 1 + Q0 + λ (t) = + λ (ξ)f(ξ, t)dξ = Q0 + , [20] λ0 0 λ0 and can be written in terms of the exponential integral Ei(x) = − R ∞ −1 −x + + −ξ/ξc −x x e dx as where we used the definition λ (ξ) = λ0 e and define Q0 = ¯S2 R ∞ 2 −η T η e dη . 1 0 + − 1 1 − ξ φ(ξ) = exp −λ e ξc E − E . [17] 0 i ξ i ξ c c Data Availability. All study data are included in this article.
Average Transport Rate due to Overwashes. Following energy considera- ACKNOWLEDGMENTS. O.D.V. and I.R.-I. acknowledge the support of the tions (2), the transport rate QS during a single overwash (S > rh) scales as Texas A&M Engineering Experiment Station.
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