Vegetation controls on the maximum size ofcoastal dunesOrencio Durán1,2 and Laura J. Moore

Department of Geological Sciences, University of North Carolina at Chapel Hill, Chapel Hill, NC 27599

Edited* by Andrea Rinaldo, Laboratory of Ecohydrology (ECHO/IIE/ENAC), Ecole Polytechnique Federale Lausanne, Lausanne, Switzerland, and approvedAugust 22, 2013 (received for review May 14, 2013)

Coastal dunes, in particular foredunes, support a resilient ecosys-tem and reduce coastal vulnerability to storms. In contrast to drydesert dunes, coastal dunes arise from interactions betweenbiological and physical processes. Ecologists have traditionallyaddressed coastal ecosystems by assuming that they adapt topreexisting dune topography, whereas geomorphologists havestudied the properties of foredunes primarily in connection tophysical, not biological, factors. Here, we study foredune de-velopment using an ecomorphodynamic model that resolves thecoevolution of topography and vegetation in response to bothphysical and ecological factors. We find that foredune growth iseventually limited by a negative feedback between wind flow andtopography. As a consequence, steady-state foredunes are scaleinvariant, which allows us to derive scaling relations for maximumforedune height and formation time. These relations suggest thatplant zonation (in particular for strand “dune-building” species) isthe primary factor controlling the maximum size of foredunes andtherefore the amount of sand stored in a coastal dune system. Wealso find that aeolian sand supply to the dunes determines thetimescale of foredune formation. These results offer a potentialexplanation for the empirical relation between beach type andforedune size, in which large (small) foredunes are found on dis-sipative (reflective) beaches. Higher waves associated with dissi-pative beaches increase the disturbance of strand species, whichshifts foredune formation landward and thus leads to larger fore-dunes. In this scenario, plants play a much more active role inmodifying their habitat and altering coastal vulnerability thanpreviously thought.

ecomorphodynamic modeling | dune stabilization | sediment budget

Dune height and dune recovery following storms are criticalin determining coastal vulnerability to climate-change–

induced shifts in forcing (e.g., sea-level rise and changing storms).Coastal dunes arise from interactions between ecological andphysical processes yet the mechanisms involved in dune formationand the impact of climate change on these mechanisms has beenpoorly understood. Here we argue that plant zonation, rather thansediment supply, controls coastal vulnerability to storms by de-termining maximum dune size. In addition, plant zonation mayalso control the resiliency of coastal environments to climatefluctuations by altering the dune mobility threshold, potentiallyleading to dune destabilization.Foredunes, the first shore-parallel dune ridge encountered

landward of the shoreline, are a crucial part of coastal landscapes.As natural barriers, they increase biodiversity by sheltering moresensitive inland ecosystems from impacts of the sea while alsoproviding protection from storm-induced overwash, which isparticularly important on barrier islands. Coastal dune formationand evolution result from complex interactions among coastalplant communities, aeolian and subaqueous sediment transport,fluid dynamics, coastal and beach topography, and storms (1–4).Because so many different factors may contribute to dune mor-phology, identifying primary controls of foredune developmentand recovery following storms is particularly challenging.Foredunes are formed by the continuous accumulation of

wind-blown beach sand, which is trapped by burial-tolerant

vegetation. Ecologists have studied the response of plant com-munities (e.g., spatial sorting, zonation, and diversity) to physicaland chemical gradients (including sand burial, wind exposure,salt spray, soil moisture, underground water salinity, soil pH, andnutrients) many of which are affected by the topography (5–10).From an ecological perspective, once pioneer salt-tolerant strandspecies colonize a shore in the presence of active aeoliantransport, trapped sand acts as a positive selection mechanismfor burial-tolerant “dune-building” grasses (5, 6, 8). The resultingforedune further reduces salinity and landward sand transport,thus creating favorable conditions for new species, ecologicalcompetition, and plant succession (5, 6).Although the relevance of dune-building grasses has been

recognized (3, 11), most geomorphological research has focusedon the role of sand supply (a combination of sand availability andwind-transport potential) as a function of beach morphology andwind regime in the dune-forming process (1, 2, 4, 12, 13). Theempirical finding that foredune size correlates with beach type,with foredunes up to 10 times higher on dissipative beaches thanon reflective ones (1, 2), combined with conceptual models thatpredict enhanced aeolian transport on dissipative beaches (1, 14,15), has motivated the common assumption that maximumforedune size is primarily controlled by sand supply (1, 4, 15, 16).However, this explanation is problematic based on reports oflarge foredunes on dissipative beaches with a negligible sandinput (17, 18) and findings of very small foredunes formed ona reflective beach under high sand supply (19). In addition, thereseems to be no general empirical relation between foredune sizeand beach morphology, i.e., beach width and slope (15, 20, 21),characteristics traditionally used, along with grain size, as a proxyfor potential sand transport (which is notoriously difficult tomeasure at the beach).In comparison with plant zonation and sand supply, the role of

plant communities in foredune development has yet to be wellstudied. For example, there is evidence that plant physiology canalter foredune aspect ratio (22) or connectivity, i.e., either forminga continuous shore-parallel ridge or discontinuous “hummocky”dunes (3). However, there is no clear understanding of the in-dividual role of the different physical and biological factors onforedune size. Fundamental questions remain unanswered: whatlimits foredune size, do coastal dunes reach a steady state, andwhat controls foredune formation time?The inherent complexity of the experimental/field study of

aeolian bedforms with or without vegetation (ripples, dunes,megadunes, foredunes, etc.), has led to the use of modeling asa viable alternative to empirical approaches (23–31). Followingvan Dijk et al. (24), the process-based model developed by Kroyet al. (25) was the first to quantitatively reproduce the steadystate of crescent dunes (32). Further iterations of the model

Author contributions: O.D. and L.J.M. designed research, performed research, analyzeddata, and wrote the paper.

The authors declare no conflict of interest.

*This Direct Submission article had a prearranged editor.

Freely available online through the PNAS open access option.1Present address: Center for Marine Environmental Sciences, University of Bremen,D-28359 Bremen, Germany.

2To whom correspondence should be addressed. E-mail: oduran@marum.de.

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successfully addressed dune collisions (33), Martian dunes (34),and dune formation under bimodal winds (35) (see ref. 36 fora recent review). Durán and Herrmann (27) added another layerof complexity to this model by incorporating vegetation dynamicsand were able to successfully predict the stabilization of mobiledunes as a result of vegetation growth (37).Here, we extend the model of Durán and Herrmann (27, 38)

to include the ecological and physical effects of a coastal envi-

ronment, and we quantify the influence on foredune develop-ment of key interactions between wind flow, sand transport, to-pography, the shoreline, and vegetation. We then demonstrate thatthe inherent complexity of coastal ecomorphodynamic processescan be greatly reduced by focusing on a few fundamental quantities.

Formation and Stabilization of ForedunesThe coastal dune model consists of differential equations for thephysical and biological processes describing aeolian sand transporton a vegetated surface at the shore during low tide. It resolves thecoevolution of the sand-surface elevation h relative to the watertable at low tide, and the vegetation-cover fraction ρveg undera constant onshore wind, characterized by the surface shearstress τ0 on a flat bed (see Appendix for the model description).For a typical simulation, aeolian transport begins at the fore-

shore (during low tide), slightly above the water table, at the firstlocation where the wind shear stress is above the transportthreshold τt. Sand flux then steadily increases up to the maxi-mum, saturated value that the wind can sustain (Fig. 1A). Undera constant onshore wind, sand blows continuously across thebeach into the backshore where it is trapped by dune-buildingvegetation and a foredune begins to form (Fig. 1 A and B). Weassume that the spatial distribution (zonation) of the vegetationis characterized by the minimum distance from the shoreline(called vegetation limit Lveg) needed by the plants to survive longenough to build a mature foredune. (We are not aware of anyfield measurements to quantify this distance.) Closer to theshoreline, plant growth is hampered by wave runup, soil salinity,storms, etc., and any incipient foredune will be short lived.Initial foredune growth is then driven by the abrupt sand-flux

decrease induced by the vegetation at the vegetation limit (Fig. 1C–E). Plants act as roughness elements that absorb part of themomentum transferred to the sand surface by the wind, ef-fectively reducing the surface shear stress and thus the sand-transport rate. Once a proto-dune emerges, its evolution isdetermined by its interaction with the wind flow, with the vege-tation playing a secondary role as a passive roughness elementanchoring the dune crest and thus preventing dune motion.Growth of the foredune produces a deceleration of the flow up-wind thereby reducing the surface shear stress—which is roughlyproportional to the wind velocity—(Fig. 1D) and the sand flux

C D E

GF

B

Awind

Fig. 1. Mechanisms behind foredune formation and stabilization. (A) Sim-plified coast used as initial condition for the model. For visual reference,the foreshore is shown in blue. (B) Stationary foredune for Lveg =25 m andundisturbed wind shear stress τ0 =3τdt , where τdt is the threshold shearstress for dry sand (vegetation cover fraction is green). (C ) Evolution of thecentral cross-shore slice of the foredune elevation, (D) wind shear stress,and (E ) sand flux. (F) Evolution of the foredune height H rescaled by themaximum height Hmaxð△Þ, the computed wind shear stress at the shore-line τin rescaled by the undisturbed shear stress over a flat bed τ0ð○Þ andthe computed sand flux into the foredune qin rescaled by the saturated(maximum) flux q0

sat over a flat bed ð□Þ. Solid lines correspond to expo-nential fits with constant relaxation time. (G) Proportionality between therescaled wind shear stress at the shoreline τin=τ0 and the rescaled foreduneheight H=Hmax. The dashed line is the rescaled transport threshold for drysand τdt =τ0. For a realistic timescale it is assumed transport only takes place20% of the time.

A

B

Fig. 2. (A) Foredune profiles from the numerical model for different winds,characterized by the ratio of the undisturbed to threshold shear stress τ0=τdt ,and vegetation limits Lveg. (Inset) Rescaled profiles. (B) Evolution of foreduneheight for different vegetation limits and τ0=τdt = 4. (Inset) Rescaled evolu-tion curves; Hmax and Tf are given by Eqs. 2 and 3.

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(Fig. 1E), until the shear stress at the shoreline is below thethreshold for transport and sand flux ceases. At this point, there isno net sand supply to the backshore feeding dune growth and theforedune reaches a steady state of maximum height (Fig. 1F).Foredune development can be captured by a simplified analytical

morphodynamic model based on the linear relation for the topo-graphic forcing on the wind flow, as shown in Fig. 1G, where thewind shear stress at the shoreline τin decreases with foredune heightH. The topographic forcing is a function of the foredune windwardslope, which can be approximated as 2H=Lveg, where the vegetationlimit Lveg approximates the position of the crest relative to theshoreline. The topographically induced reduction of the un-disturbed wind shear stress τ0 at the shoreline can then be ap-proximated as τin ≈ τ0ð1− βðH −HcÞ=LvegÞ, where β and Hc areconstants obtained from the numerical simulations (for simplicitywe will consider Hc = 0 in what follows).The wind shear stress at the shoreline τin determines the

ratio of the actual sand flux qin crossing the shoreline to reachthe foredune and the potential sand supply q0sat, defined asthe maximum sand flux on a beach without foredunes:qin = q0satðτin − τtÞ=ðτ0 − τtÞ.The foredune height H for a given cross-shore profile results

from mass conservation at the dune crest: ∂H=∂t=−∂q=∂xjx=Lveg,

which can be rewritten as ∂H=∂t≈ α−1qin=Lveg after assumingthe flux gradient is essentially given by the total influx to thebackshore qin decaying to zero over a length αLveg with aproportionality constant α. After substituting τinðHÞ into qin,the equation for the foredune height can be rearranged intothe form

dH=dt≈ ðHmax −HÞ=Tf ; [1]

with a maximum size Hmax and formation time Tf defined as

Hmax ≡ β−1Lveg�1− τdt =τ0

�[2]

Tf ≡ αHmaxLveg=q0sat: [3]

Here β= 4:2 and α= 0:3 are obtained from the numerical simu-lations and τtd is the transport threshold for dry sand. We assumethat during most transport events the threshold is close to that ofdry sand τt ≈ τdt .Eq. 1 integrates as the exponential relaxation HðtÞ=

Hmax½1− expð−t=Tf Þ� in agreement with simulation results (Fig.1F). Therefore, as Fig. 2 shows, foredune evolution can besolely characterized by the maximum size Hmax and the for-mation time Tf .From the sand-transport model (Appendix) the potential sand

supply can be derived from the wind regime as:

q0sat = rtQ�τ0=τ

dt − 1

�; [4]

where Q is a dimensional constant and rt is the fraction of thetime the undisturbed wind is above the transport threshold. Inthe numerical simulations we use a constant wind regime andthus rt = 1 by definition (for the figures we use rt = 0:2). However,under real conditions, this is not the case and we define τ0 as theshear stress corresponding to the “formative wind,” i.e., the windthat contributes most to sand transport and dune formation, andrt the fraction of the time this wind is present.

Discussion and ImplicationsThe negative feedback between the topography and the windflow that eventually limits foredune growth has two importantconsequences. The first one is the scale invariance of the steady-state foredune profile (as evidenced by Fig. 2A). Analogous tomobile crescent dunes (25), scale-invariant fordunes result fromthe scale invariance of the flow field within the turbulentboundary layer, by which both small and large dunes deflect thewind in a similar way. The scale invariance, as confirmed bymeasured foredunes (Fig. 3A), suggests a characterization offoredune development based on the windward slope. For typicalwinds ðτ0 ≥ 3τdt Þ, simulated foredunes reach the steady state forwindward slopes in the range 18–25°, in agreement with meas-urements (Fig. 3C). Indeed, Arens et al. (17) and Bauer et al. (18)reported no transport activity during favorable wind conditionson the windward side of steep dunes (∼27° and ∼22° respectively;

A

B

C

Fig. 3. Scale invariance. (A) Mature foredunes: simulation (solid line) andmeasurements from Australia ð△Þ (39) ð■○▲Þ (1), Canada (×) (40), Floridað•Þ(41), and Netherlands ð□⋆Þ (17). (B) Incipient foredunes: simulation (dashedline) and measurements from Australia ð□○△Þ (2) and Netherlands ð⋆Þ (17,42). A simulated mature foredune is shown for comparison (solid line). No-tice that the model underestimates the downwind dune profile as it simu-lates sand transport via saltation and neglects suspended sand transport,which can be significant (and occur above the vegetation canopy) duringstorms (42). (C) Evolution of the characteristic slope of a simulated dune,defined as 2H=Lveg (solid line). Superimposed symbols: slopes of measuredmature (steady) and incipient (unsteady) foredunes, from A and B, re-spectively. Sand repose (avalanche) angle is shown for comparison (dottedline). All simulations were performed for τ0 = 4τdt . For simplicity the origin ofthe x coordinate is set at the foredune crest.

Fig. 4. Measurements on a cross-shore slice of a growing foreduneðH= 3:8 mÞ (17): rescaled profile (dashed line), vegetation cover (gray areas),and sand flux q rescaled by its maximum value at the beach qbeachð○Þ (thescattering is due to different wind velocities). Solid lines: simulation forτ0=τdt =3 and Lveg =25 m. Flux reduction in nonvegetated areas is due toflow–form interaction; closer to the crest is due to vegetation.

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Fig. 3C). In contrast, flatter foredunes have room to grow, asthose in Fig. 3B where sand accretion has been reported (2, 42).The second consequence is that some of the details of how

vegetation grows are irrelevant to the resulting foredune mor-phology as long as the typical vegetation growth rate, defined bythe ratio of the maximum plant height and typical growth timeHveg=tveg, is much higher than the maximum erosion/depositionrate, which is a necessary condition for dune stabilization (27).Measurements of wind deflection and sand accretion upwind

of foredunes provide additional support for foredune stabiliza-tion driven by the flow–form interaction. Arens et al. (17) andHesp et al. (40) reported wind deflections at a distance about5− 8H (H is dune height) upwind of the foredune crest, a valuecomparable to 8H found on crescent dunes and reproduced bysimulations (32). Furthermore, Arens (42) and Nordstrom (43)have reported sand deposition as far as 10H upwind of the crestof low foredunes attributed to topographic effects, also inagreement with simulations (Fig. 4).

Plant Zonation and Foredune Size. The scalings in Eqs. 2 and 3explicitly relate the morphological and dynamical properties offoredunes to wind and transport regimes, characterized by theundisturbed shear stress τ0 and the potential sand supply q0sat,respectively, and the interaction of the vegetation with theshoreline, which defines the plant zonation described by Lveg.From Eq. 2, the maximum foredune height ðHmaxÞ is primarilycontrolled not by the sand supply, as traditionally assumed (1, 2,4), but by the zonation of dune-building plants, i.e., the vegeta-tion limit Lveg, with a secondary contribution from the averagewind intensity during transport events. Because the steady stateof a foredune is determined by its slope, the farther from theshoreline a foredune forms, the higher it can be. Therefore, bydetermining where an incipient foredune can develop relative tothe shoreline, pioneer strand species play a critical role in con-trolling dune size. This coupling has important consequencesbeyond morphology because foredunes strongly modify the abi-otic gradients that ultimately determine how the associatedcoastal ecosystem develops (9, 10).The causal relationship between plant zonation and foredune

size provides a simple explanation for the empirical correlation

between beach type and foredune size (1). We know that dissi-pative beaches have larger average wave heights than reflectivebeaches (1), and that several limiting factors for vegetation growthare related to average wave height, e.g., wave runup during ex-treme tides and storms, soil salinity, and salt spray (2). As a con-sequence, plant zonation is wider on dissipative beaches than it ison reflective beaches (2, 15, 44). Therefore, foredunes tend to belarger on dissipative beaches than on reflective beaches. Thisconclusion is quantitatively supported by field data, which suggestsa linear correlation between average wave height and vegetationlimit, and hence between wave height and foredune height (Fig. 5),and it is consistent with measurements of foredune size and beachcharacteristics by Saye et al. (20).In contrast to foredune height, foredune formation time is

limited by potential sand supply and decreases with the transportfrequency rt (Eqs. 4 and 3). Therefore, in places with wider

Fig. 5. Foredune height versus characteristic wave height ðHbÞ for beachesin Australia (□, modified from ref. 2), Oregon (○, ref. 21), and Brazil (•, ref.45). The solid line corresponds to the model prediction for a wind τ0 =2:6τdtand assuming plant zonation increases linearly with wave height asLveg =bHb, with a fit parameter b= 35.

A

B

C

D

E

Fig. 6. Plant zonation and coastal dune fields. Different model outcomesstarting from a flat bed with an undisturbed wind shear stress above themobility threshold for vegetated dunes such that foredunes can becomeunstable (τ0 = 5τdt for a vegetation growth rate Hveg=tveg = 0:15 m=d, downfrom 0.3 m/d in previous simulations): (A) spontaneous nucleation ofa transgressive dune field in the absence of vegetation (46); (B) stableforedune for a narrow plant zonation; (C) unstable foredune, followed byblowouts, parabolic and crescent dunes, for an intermediate plant zonation;and (D) unstable foredunes and formation of a transgressive dune field fora wide zonation. (E) Evolution of the total volume per unit area depositedby aeolian transport for different vegetation limits. (Inset) Rescaled volumeper unit area. For a realistic timescale it is assumed transport takes place 20%of the time.

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zonation for strand species but low sand-transport frequency(e.g., as a result of wave inundation) dunes have the potential tobe large, but they will grow slowly. In this case, recovery followingdune erosion/destruction will also be slow and as a consequence,in some cases, even large dunes may be highly vulnerable tochanges in the frequency of intense storms.

Plant Zonation and Coastal Dune Fields. In addition to influencingcoastal vulnerability to storms, the relationship between plantzonation and foredune size has important implications for theaeolian sediment budget at the coast (i.e., total amount of sanddeposited landward by aeolian transport) and thus the potentialdevelopment of transgressive dune fields. Following Durán andHerrmann (27), permanent vegetated dunes, such as foredunes,can be destabilized and become mobile once the potential ero-sion/deposition rate is sufficiently high relative to the vegetationgrowth rate. Under the condition of a constant sand supply fromthe shore, destabilization of foredunes can lead to the formationof transgressive dune fields (29).Simulation results suggest that pioneer species zonation in-

directly affects this mobility threshold (Fig. 6). All other factorsbeing equal, an incipient foredune that forms where plant zo-nation is narrow would result in a reduced sand flux relative to anincipient fordune that forms where zonation is wider. A lowersand flux implies a lower potential erosion/deposition rate, whichincreases the mobility threshold for vegetation and leads toa more stable foredune (Fig. 6B). In contrast, foredunes formingwhere zonation is wider can become unstable for strong windsand generate a wide range of mobile and quasimobile dunemorphologies (e.g., blowouts, parabolic dunes, and crescentdunes; Fig. 6 C and D). This transition can be enhanced by thephysical factors, such as water availability and wind exposure, thatfurther stress dune plants on large foredunes, thereby reducingplant growth rate and further decreasing the mobility threshold.In addition to affecting the onset of coastal dune-field for-

mation, plant zonation also determines the volume of sandtransported inland by controlling the storage capacity (size) offoredunes (Fig. 6E). As a consequence, physical factors, such asbreaker height, although acting at a local scale, could potentiallylead to ecological and morphological changes at a much largerscale. This picture is consistent with field observations and con-ceptual models (2, 3, 15) in which the degree of mobility andextension of coastal dune fields, from stable foredunes to para-bolic to crescent dunes fields, is correlated with beach type. Al-though most transgressive fields are found on dissipative beaches(high waves, wide zonation), reflective beaches (low waves,narrow zonation) may have just a single foredune ridge (2).Therefore, plant zonation may not only control coastal vulner-ability to storms by determining maximum foredune height, itmay also control the resiliency of coastal environments to climatefluctuations by altering the mobility threshold, potentially lead-ing to dune destabilization.

Appendix: Coastal Dune ModelInitial Condition. The sand-surface elevation hðx; yÞ relative to themean water-table level Hwater during low tide is initially definedby an inclined foreshore, hðx; yÞ= x tan  θ for x< xshore, and bya flat backshore, hðx; yÞ=MHWL−Hwater for x> xshore (MHWL:mean high water level). Here, x is the cross-shore distance andxshore the position of the shoreline, defined by the MHWL asxshore = tan  θ−1ðMHWL−HwaterÞ. Notice that the origin of the xcoordinate ðx= 0Þ is by definition at h= 0. In the simulations,θ= 28 is the beach slope and MHWL−Hwater = 0:3 m is therelative water-table depth.

Upwind Boundary Condition. We assume, as a first approximation,that the foreshore is stable ð∂h=∂t= 0Þ, i.e., aeolian erosion isbalanced by accretion in the swash zone. As a result, the simu-lated foreshore acts as a sand reservoir supplying an unlimitedamount of sediment to the backshore, effectively feeding duneformation.

Fluid dynamics. The model uses a linear solution of the Reynolds-averaged Navier–Stokes equations for the turbulent boundarylayer over smooth terrain (45) to calculate the perturbation δτ ofthe wind shear stress induced by the topography h. The surfaceshear stress τ is τðhÞ= τ0 + τ0δτðhÞ (see refs. 35, 46 for details).For lee slopes steeper than the separation angle ∼20°, nonlinear

hydrodynamic effects are simply modeled by a separation stream-line below which wind and flux are set to zero. Each streamlineis defined by a third-order polynomial connecting the brink withthe ground at the reattachment point (46).

Shear Stress Partition. For randomly distributed plants, and as-suming the effective shelter area for one plant is proportional to itsbasal area, the fraction τs of the surface shear stress acting on thesand decreases with the local vegetation cover fraction ρveg as (47)

τs = τ=�1+Γρveg

�; [5]

where Γ is a dimensionless “roughness factor,” which describesthe effectiveness of the vegetation in slowing down the flow andthus in trapping sand. In the model, Γ= 16 is calculated fromvalues of plant form drag and geometry reported for creosotecommunities (see ref. 27 and references therein; it is reasonableto expect a similar value for coastal grasses and desert bushesdue to a roughly similar plant geometry).

Effect of Wetting on the Transport Threshold. We further considerthat at the shore, transport is naturally limited by the elevation hrelative to the water table, as the transport threshold τt is muchhigher for wet grains than for dry ones. This relation is capturedby the simple phenomenological expression

τtðhÞ= τdt +�τwt − τdt

�expð−h=hwÞ; [6]

where τtd and τwt = 10τtd are the threshold for dry and wet sand,respectively, and hw = 0:05 m characterizes the decrease in watercontent of the sand as a function of elevation.

Sand Transport. The sand flux is determined from the shear stressat the sand surface τs (Eq. 5), the surface gradient ∇h, and thetransport threshold τt (Eq. 6). It is well known that the sand fluxq over an erodible surface increases with the distance downwindas the saltation process spatially adjusts to the wind forcing (25).This effect is modeled as

∇ · q= qð1− q=qsatÞ=lsat; [7]

which describes the spatial relaxation of the sand flux toward anequilibrium “saturated” value qsat over a saturation length lsat(48). The saturated flux and saturation length are defined as:qsat =Qðτs − τtÞ=τdt and lsat =Lτdt =ðτs − τtÞ, where Qð∇hÞ andLð∇hÞ are slope-dependent dimensional functions (46).For slopes steeper than the angle of repose 34°, an addi-

tional dissipative flux models the surface relaxation due to ava-lanches (46).

Surface Dynamics. The sand-surface elevation h is then updatedusing mass conservation

∂h=∂t= −∇ · qΘðx− xshoreÞ; [8]

where the Heaviside function ΘðxÞ (1 for x> 0 ; 0 otherwise) dis-tinguishes the dynamics of the nonerodible foreshore ðx< xshoreÞfrom the backshore ðx> xshoreÞ. For simplicity in the formulationq is defined as a volume, not mass, flux.

Vegetation Dynamics. To represent vegetation dynamics in a sim-plified way, we assume a single generic grass species with a coverfraction that is sensitive to erosion and accretion and that can

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increase up to the maximum cover ρveg = 1 during a characteristictime tveg. We further assume that growth is also sensitive to thedistance from the shoreline such that plants can only growth land-ward of the vegetation limit Lveg :

dρvegdt

=

�1− ρveg

�

tvegΘ�x− xshore −Lveg

�− γH−1

veg

����∂h∂t

����; [9]

where γ is plant sensitivity to sand erosion/accretion and Hveg isthe maximum plant height. Both the growth time tveg and thesensitivity γ are a function of the erosion/accretion rate ∂h=∂t,which can be varied for different plant species. However, we findthat the results presented here are independent of these consid-erations. Therefore, for simplicity and in agreement with (27),we use a constant growth time tveg = 3 d, and sensitivity γ = 1,with Hveg = 1 m.

Integration. The model is integrated within a 2D domain largeenough to include the resulting morphology. The grid spacing

and time step are typically ∼ 1/4 m and ∼ 1/2 h, respectively,and are selected to resolve the smallest length and temporalscales involved in the problem, the saturation length lsat andtime l2sat=qsat.

Parameters. Model outcomes are investigated as function of thevegetation limit Lveg and the imposed onshore wind, character-ized by the undisturbed shear stress τ0, which is assumed con-stant throughout each simulation. Time in the model is thusshorter than in more realistic wind conditions, where the windintensity fluctuates daily and seasonally, with the conversionfactor being loosely given by the fraction of the time rt the wind isabove the transport threshold. For the figures we use rt = 0:2.

ACKNOWLEDGMENTS. We thank Don Young and John Bruno for providinginsights on dune grasses that assisted us in this work. Funding was providedby the Department of Energy’s Office of Science through the Coastal Centerof the National Institute for Climatic Change Research at Tulane University,the Virginia Coast Reserve Long-Term Ecological Research Program (NationalScience Foundation Grant DEB–1237733) via a subaward from the Universityof Virginia and the University of North Carolina at Chapel Hill.

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