Model for the genesis of coastal dune fields with vegetation

Model for the genesis of coastal dune fields with vegetation

Geomorphology 129 (2011) 215–224 Contents lists available at ScienceDirect Geomorphology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o ...

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Geomorphology 129 (2011) 215–224

Contents lists available at ScienceDirect

Geomorphology j o u r n a l h o m e p a g e : w w w. e l s ev i e r. c o m / l o c a t e / g e o m o r p h

Model for the genesis of coastal dune fields with vegetation Marco C.M. de M. Luna a,⁎, Eric J.R. Parteli b,c, Orencio Durán d, Hans J. Herrmann a,c,e a

Departamento de Física, Universidade Federal do Ceará, 60455-760, Fortaleza, CE, Brazil Programa de Pós-Graduação em Engenharia Química, Universidade Federal do Ceará, 60455-900, Fortaleza, CE, Brazil c National Institute of Science and Technology for Complex Systems, Rua Dr. Xavier Sigaud 150, 22290-180 Rio de Janeiro, RJ, Brazil d Laboratoire de Physique et Mécanique des Milieux Hétérogènes, UMR CNRS 7636, ESPCI, 10 rue Vauquelin, 75231 Paris Cedex 05, France e Institut für Baustoffe IfB, ETH Hönggerberg, HIF E 12, CH-8093, Zürich, Switzerland b

a r t i c l e

i n f o

Article history: Received 3 February 2010 Received in revised form 28 January 2011 Accepted 31 January 2011 Available online 4 February 2011 Keywords: Coastal dunes Parabolic dune Wind erosion Sand transport Dune model

a b s t r a c t Vegetation greatly affects the formation and dynamics of dune fields in coastal areas. In the present work, we use dune modeling in order to investigate the genesis and early development stages of coastal dune fields in the presence of vegetation. The model, which consists of a set of coupled equations for the turbulent wind field over the landscape, the saltation flux and the growth of vegetation cover on the surface, is applied to calculate the evolution of a sand patch placed upwind of a vegetated terrain and submitted to unidirectional wind and constant sand influx. Different dune morphologies are obtained, depending on the characteristic rate of vegetation growth relative to wind strength: barchans, transverse dunes with trailing ridges, parabolic dunes and vegetated, alongshore sand barriers or foredunes. The existence of a vegetation-free backshore is found to be important for the nucleation timescale of coastal dune generations. The role of the sand influx and of the maximum vegetation cover density for the dune shape is also discussed. © 2011 Elsevier B.V. All rights reserved.

1. Introduction Sand dunes appear very commonly along coasts (Pye, 1983; Hesp et al., 1989; Nordstrom et al., 1990; Arens, 1996; Hesp, 2002; Bailey and Bristow, 2004; Levin and Ben-Dor, 2004; Hesp and Martinez, 2008; Provoost et al., 2009). They are formed by the aeolian transport of sand from the beach, and may assume different forms depending on the wind directionality, on the amount of sediment available for transport and on the growth of vegetation. In non-vegetated fields where the wind regime is unidirectional, barchans and transverse dunes are the characteristic dune shapes (Bagnold, 1941). However, in coastal areas where humidity is typically high, aeolian transport competes against the growth of vegetation. The transformation of a barchan into a parabolic dune due to the growth of plants on the dune arms is a well-known consequence of this competition and has been studied by many authors (Tsoar and Blumberg, 2002; Durán and Herrmann, 2006a; Baas and Nield, 2007). Indeed, a quantitative understanding of the dynamics of coastal dune fields as function of the local environmental conditions is of crucial importance for coastal management. In this work we present insights gained from a physically based model for aeolian transport and dune formation, into the genesis and dynamics of dunes evolving with growing vegetation.

⁎ Corresponding author. E-mail address: [email protected] (M.C.M. de M. Luna). 0169-555X/$ – see front matter © 2011 Elsevier B.V. All rights reserved. doi:10.1016/j.geomorph.2011.01.024

Dunes cover extensive areas along the northeastern Brazilian coast, which is characterized by highly unidirectional wind regime – governed by easterly strong trade winds – and by the presence of vegetation (Jimenez et al., 1999; Barbosa and Dominguez, 2004; Durán et al., 2008; Levin et al., 2009; Tsoar et al., 2009). Currently active dunes, which comprise barchans, barchanoids and sand sheets, migrate on top of older generations of stabilized bedforms (Levin et al., 2009), and may advance more than several kilometers inland. In most areas, active dunes are separated from the coast by a deflation plain of 500 m to 2 km wide. Rates of dune migration vary significantly over the year (Jimenez et al., 1999; Levin et al., 2009). Indeed, average wind power displays a strongly seasonal behavior: most aeolian activity is restricted to the dry season between August and December, whereas the other half of the year wind velocities are mostly below threshold (Jimenez et al., 1999). Rainfall, inversely correlated with wind power, vanishes in the dry period and reaches maximum values of 400 − 600 mm month− 1 (with an average of about 250 mm month− 1) in the wet season. Time-series of wind speed and direction, rainfall and evaporation rates for the dune fields in north-eastern Brazil have been presented in previous works (Jimenez et al., 1999; Parteli et al., 2006; Levin et al., 2009). The dune areas along the Brazilian coast are of particular economical interest since they are a tourist attraction and constitute potential aeolian parks (Floriani et al., 2004). The competition between landward dune migration and vegetation growth results in different types of dunes as illustrated in Figs. 1–3. Near the city of Natal, at 5°49′ S, 35°11′ W, shore-parallel dune ridges covered with vegetation (“foredunes” (Hesp, 2002)) constitute one characteristic

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Fig. 1. Coastal dunes competing against vegetation in the city of Natal, in the State of Rio Grande do Norte, Brazil. The image shows dunes migrating from the beach into the continent, near 5°49′ S, 35°11′ W. These dunes are classified as foredunes (Hesp, 2002).

Fig. 3. Coastal dunes in Pirambu in the State of Sergipe, Brazil, near 10°36′ S, 36°41′ W. Parabolic dunes are the typical dune shape. The arms of the dunes, stabilized by the vegetation, point into the direction from which the effective sand-moving winds blow.

dune morphology (c.f. Fig. 1). Due to the dense vegetation cover, the sand transported onto the continent accumulates in the form of large barriers that can, in some cases, reach heights of several tens of meters (Hesp, 2002). More to the south, at Pirambu (Fig. 2), and along the Ceará coast, where winds are typically stronger (Levin et al., 2007) barchans and parabolic dunes are observed. Far to the northwest we find the largest dune field of Brazil. Covering approximately 270 km2, the region known as “Lençóis Maranhenses” has an annual rainfall of about 2000 mm and is the coastal dune field with the strongest sandmoving winds in northeastern Brazil (Levin et al., 2007). Transverse dunes and chains of barchans extending over several kilometers alternating with freshwater lagoons that form in the rainy seasons (c.f. Fig. 3) are the dominant dune morphologies at Lençóis (Gonçalves et al., 2003; Parteli et al., 2006; Levin et al., 2007). The competition between vegetation growth and aeolian transport of sand in coastal areas can, thus, lead to different dune morphologies. Recent models based on cellular automata could shed some light into the morphology and dynamics of aeolian landscapes with vegetation growth (Nishimori and Tanaka, 2001; Baas, 2002; Baas and Nield, 2007; Nield and Baas, 2008a,b). In the present work, we use a model for dune formation recently developed (Sauermann et al., 2001; Kroy

et al., 2002; Schwämmle and Herrmann, 2005; Durán and Herrmann, 2006b) and applied with success in the investigation of different dune morphologies (Sauermann et al., 2003; Parteli et al., 2006, 2009), in order to study the genesis and early stages of evolution of coastal dune fields with vegetation growth. In a recent work, the dune model, which incorporates a mathematical description of saltation sand transport and of the turbulent wind flow at the scale of dunes, has been extended to calculate the transition of a barchan into a parabolic dune due to vegetation growth (Durán and Herrmann, 2006a). The aim of the present study is to investigate the formation of a dune field starting from a flat hill of sand on the beach, as well as the dependence of the field morphology on the main parameters of the vegetation cover, the wind strength and the amount of sand available for transport. In fact, little attention has been given by modelers to the early developmental stages of coastal dune fields. Field studies on formative processes of coastal dune fields have also been few, in part because most dune fields consist of “mature” bedforms that provide little insight as to dune morphodynamics during the genesis of the field (Kocurek et al., 1992). The dune fields of the northeastern Brazilian coast are some of the areas where one can observe the inception of coastal dunes competing with vegetation growth. Therefore, we start our research through using numerical modeling to reproduce the morphology of the dunes in these fields. The calculations aim to address specific questions regarding coastal dune formation, as for example the role of the initial sand volume and vegetation growth rate for the emergence and morphology of along-shore dune barriers or foredunes (Hesp, 2002), as well as the factors controlling dune height and interdune scaping in a dune field emerging with vegetation growth. This paper is organized as follows. In the next section, we present a brief description of the dune model. The simulations are explained in detail, then, in Section 3. In Section 4 we present and discuss our results. Conclusions are given in Section 5. 2. The model

Fig. 2. Coastal dune field “Lençóis Maranhenses” in the State of Maranhão, Brazil. Barchanoids and transverse dunes intercalated by freshwater lagoons extending over several kilometers form the characteristic pattern of this dune field located at 2°33′ S, 42°59′ W.

The dune model combines a quantitative description of the turbulent wind field over the terrain with a continuum saltation model, which encompasses the evolution of the sand surface due to erosion and deposition and also accounts for avalanches and flow separation at the dune's lee. In the model, the vegetation cover acts as a rough patch that modifies the wind field and is allowed to grow at a rate that depends on the local erosion and deposition. The model

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consists of solving the following steps computationally in an iterative manner. 2.1. Wind shear stress The average wind shear stress over the topography is calculated by solving the set of analytical equations developed by Weng et al. (1991) for the turbulent wind over dunes or smooth hills. In the absence of dunes, the wind velocity increases logarithmically with the height above the ground. However, a dune introduces a perturbation into the wind field. The analytical model of Weng et al. (1991) is employed in order to calculate the Fourier-transformed components of the shear stress perturbation, namely: ( ˜ 2 ˜τˆ = hs kx 2 −1 + x 2 j→ k j U ðlÞ

! ) j→ j2 l K1 ð2σ Þ k σ ; + 2ln z0 K0 ð2σ Þ k2x

 pffiffiffi  ˜ k k 2 pffiffiffi h s x y 2 2σK1 2 2σ ; τ˜ˆ y = 2 j→ j U ð l Þ k

ð1Þ

ð2Þ

ð3Þ

→ where τ0 is the undisturbed shear stress over the flat ground. A fraction of the total shear stress τtot, however, is absorbed by the vegetation covering the sand surface. In the presence of vegetation, the actual shear → stress, τ, on the sand grains reduces to (Raupach et al., 1993):   −1 → → τ = τtot = 1 + ρυ βmσ υ ;

ð4Þ

where ρυ is the vegetation cover density, συ is the ratio of plant basal to frontal area, β is the ratio of plant to surface drag coefficients and the parameter m accounts for the non-uniformity of the shear stress (Raupach et al., 1993; Wyatt and Nickling, 1997). 2.2. Sand flux → The flux of saltating grains, q ðx; yÞ, is calculated using the shear → stress field τ obtained in Section 2.1. The saltation cloud is considered as a thin fluid-like moving layer that can exchange sand with the immobile sand bed. Once saltation starts, the density of saltating grains increases exponentially as a result of the multiplicative process inherent to the splash events at the grain-bed collisions (Bagnold, 1941). However, the increase in the number of grains in saltation yields a decrease in the wind strength within the saltation layer, since more momentum is transferred from the wind to accelerate the particles. After a saturation distance, the wind is just strong enough to sustain saltation, whereas the sand flux achieves a saturation value, qs. By using mass and momentum conservation for the saltation cloud and the sand bed, and by explicitly accounting for the particle–fluid interactions and the saturation transients of the flux, the following equation for the sand flux is derived (Sauermann et al., 2001):  →→  → → ∇ ⋅ q = 1− j q j = qs j q j = ls ;

where the saturated flux qs =(2υsα/g)ρfluidu2*t[(u*/u*t)2 −1] and the chari   h acteristic length of flux saturation ls = 2υ2s α = gγ = ðuT = uTt Þ2 −1 are pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi both written in terms of the wind shear velocity uT = τ = ρfluid , where ρfluid =1.225 kg/m3 is the air density. The parameters α=0.43 and γ=0.2 are determined empirically (Sauermann et al., 2001; Durán and Herrmann, 2006b), whereas g = 9.81 m/s2 is gravity and u*t = 0.22 m/s is the minimal threshold shear velocity for saltation. The average velocity of the saltating grains, vs, is computed by taking the steady-state wind velocity within the saltation layer (Sauermann et al., 2001; Durán and Herrmann, 2006b), whereas the transient of flux saturation is characterized by the length scale ls . 2.3. Surface evolution

where x and y are the components parallel and perpendicular, → respectively, to the wind direction, k is the wave vector, and kx and ky ˜ is the are its coordinates in the Fourier space. In Eqs. (1) and (2), h s Fourier transform of the height profile, l is the inner layer depth of the flow, U is the normalized vertical velocity profile and L is a characteristic length scale given by 1/4 the mean wavelength of the pffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi Fourier representation of the height profile. Further, σ = iLkx z0 = l, K0 and K1 are modified Bessel functions, and z0 is the aerodynamic → roughness of the surface. The local shear stress τtot ðx; yÞ is obtained, then, from the equation:  → → τ0 j → τ0 = j → τ0 j + τˆ ; τtot = j →

217

ð5Þ

The evolution of the sand surface is calculated from the sand flux → q ðx; yÞ by using mass conservation: →→ ∂h = ∂t = − ∇⋅ q = ρsand ;

ð6Þ

where ρsand = 1650 kg/m3 is the bulk density of the sand. Eq. (6) implies that erosion (deposition) takes place wherever the flux increases (decreases) in wind direction. 2.4. Avalanches and flow separation Wherever the local slope exceeds the angle of repose of the sand, i.e. 34°, the unstable surface relaxes in the form of avalanches in the direction of the steepest descent. Avalanches are regarded as instantaneous since their timescale is much smaller than that of dune migration. The streamlines of flow separation are, then, introduced at the dune's lee. Each streamline, which connects the slip face at the brink of the corresponding longitudinal slice of the dune with the ground at the reattachment point, is fitted by a third order polynomial, the parameters of which are chosen in accordance with numerical and wind-tunnel simulation results, as described in detail by Kroy et al. (2002). The separation streamlines define the separation bubble at the dune's lee, inside of which the flow and the sand flux are set to zero. 2.5. Vegetation growth The evolution of the vegetation cover is affected by the changes in the local surface profile. The vegetation height, hυ, is allowed to grow up to a maximum height Hυ within a characteristic growth time Tυ (Richards, 1959), and is related to the vegetation cover density ρυ introduced in Eq. (4) through the equation: 2

ρυ = ðhυ =Hυ Þ :

ð7Þ

The vegetation growth follows the equation (Durán and Herrmann, 2006a): dhυ = dt = Vυ ð1−hυ = Hυ Þ− j∂h = ∂tj;

ð8Þ

where the characteristic growth velocity Vυ of the vegetation height hυ encodes information about the environmental conditions that affect the growth process (Bowers, 1982; Danin, 1991; Hesp, 1991). Further, the growth velocity Vυ is set to zero at those places where sand erosion occurs (∂ h/∂ t b 0). 3. Simulations Calculations are performed using open boundaries, a constant sand influx qin at the inlet of the dune field and a wind of constant upwind shear velocity u*. The dune field has a length of 512 m transverse to

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the wind and 1024 m in the wind direction. We start with a transverse dune which has a Gaussian profile in the wind direction, a height of 1.5 m and a length of 80 m, and is placed at the beginning of the field. The dune surface is modulated with random fluctuations, the amplitude of which is equal to one grain diameter, 250 × 10− 6 m. The evolution of the dune field is studied for different values of influx qin ranging from 0 to the saturated flux qs(u*), and for different wind shear velocities u* between 0.30 and 0.70 m/s. The parameters of the vegetation are the maximum height of the vegetation (Hυ) and the growth velocity Vυ = Hυ/Tυ, where Tυ is the characteristic time needed for the vegetation to reach the height Hυ (Durán and Herrmann, 2006a). The calculations are performed for different values of Hυ: 20 cm, 50 cm, 100 cm and 800 cm. Further, different values of vegetation growth rate are investigated: Vυ = 2 m/ year, 12 m/year, 24 m/year and 36 m/year. We also study the behavior of the dune field as a function of the distance ΔL, measured from the inlet, from which the vegetation is allowed to grow. The distance ΔL may be affected by different factors, such as tides, water salinity, invading species and soil moisture (Hesp, 2002). We perform simulations for the different values of ΔL between 20 m and 100 m.

The growth of vegetation acts as a stabilizing agent for coastal dune sand. Vegetation cover decreases the air shear stress locally, thus trapping the sand and leading to local deposition. Plants first grow at those locations where erosion or deposition is small, e.g. on the terrain surrounding the dunes and on the barchan horns. The growth of vegetation inhibits sand transport downwind from the beach, thus promoting the accumulation of sand at the beach and the formation of an upwind sand barrier. This sand barrier or “foredune” (Hesp, 2002) increases in volume as sand is incising from the inlet and is trapped by the vegetation. Once the dunes reach a large enough size, they leave the foredune and migrate downwind, as shown in Fig. 5. This figure shows calculations for different values of relative sand influx qin/qs and for a constant vegetation growth rate Vυ = 12 m/year and shear velocity u* = 0.38 m/s. It is interesting to note that, when there is vegetation growth, dunes develop even with a small sand influx qin/ qs = 0.05 (Fig. 5a). As the value of influx increases, dunes emerge at a higher rate due to the faster accumulation of sand at the foredune (c.f. Fig. 5b,c). The dunes display, on average, the same characteristic height (of about 6–8 m), independent of the influx value, as shown in Fig. 6. This is in contrast to the situation of non-vegetated fields, where dunes emerge only if there is a high influx, and the dune height

4. Results and discussion 4.1. Genesis of a sand dune field: the role of sand influx and vegetation growth We study, firstly, the evolution of the field without vegetation growth. An upwind shear velocity of u* = 0.38 m/s, which is a typical value for sand-moving winds in coastal dune areas (Tsoar et al., 2009), is taken, while the relative sand influx qin/qs is varied. It is well known that a flat sand hill submitted to a unidirectional sand-moving wind and undersaturated flux is unstable and evolves into a dune that migrates downwind (Sauermann et al., 2001; Kroy et al., 2002; Andreotti et al., 2002). Indeed, for a low influx qin ≈ 0, the sand sheet at the beach develops into small, fast migrating dunes a few tens of centimeters high. Since the input of sand into the field is negligible compared to the sand loss, the small dunes decrease further in volume until they disappear. The sand is completely blown out and no dune field forms. However, a different scenario is obtained when qin/qs = 1, as can be seen in Fig. 4. When the flux is saturated, sand deposition at the upwind side of the beach prevents erosion of the dune's foot, in such a way that the sand hill cannot develop into a migrating dune. The small dunes developing on the surface detach from the sand sheet and migrate away on the bedrock. Due to the high influx, the dune size is increasing with the distance downwind, as can be seen in Fig. 4. Indeed, since the flux is saturated, dunes will steadily emerge from the sand sheet and advance with increasing size downwind. Similar results are obtained for different u* N u*t, provided there is a high upwind sand flux, i.e. qin/qs → 1.

Fig. 4. Simulation of a dune field with a sand influx qin/qs = 1.00, u* = 0.38 m/s and no vegetation growth. Arrows indicate the wind direction. Dunes emerge from the instabilities of the upwind sand sheet and migrate downwind. The dune height is about 5 m at a distance of 1 km from the inlet. Both axes are in units of meters.

Fig. 5. Dune formation with vegetation growth for different values of sand influx: qin/ qs = 0.05 (a), 0.20 (b) and 1.00 (c). Wind direction is from left to right and both axes are in units of meters. In contrast to the non-vegetated field of Fig. 4, here dunes leave the beach with a height of around 6–8 m. Further, no significant dependence of the average dune height with distance is found within the first 1 km downwind. Calculations were performed using a wind shear velocity of u* = 0.38 m/s, vegetation growth rate of Vυ = 12 m/year and maximum height of Hυ = 1.0 m. The length of the sand sheet at the entrance without vegetation growth is ΔL = 80 m.

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219

Heigth of Dunes (m)

10

8

6 qin/qsat = 0.05

4

qin/qsat = 0.20 qin/qsat = 0.60

2

qin/qsat = 1.00 0

10

20

30

40

50

Time (years) Fig. 6. The plot shows the largest dune height in the vegetated coastal dune fields of Fig. 5 as a function of time, for different values of sand influx qin/qs: 0.05, 0.20, 0.60 and 1.00. We see that the maximum height is roughly independent of the influx value.

increases with distance downwind (c.f. Fig. 4). In agreement with field observations (Kocurek et al., 1992), the calculations show that vegetation may contribute to the formation of coastal dunes by promoting sand deposition and accumulation in coastal areas with low sand influx. 4.2. Influence of beach width on the morphology of vegetated coastal dune fields

4.3. Influence of the maximum height of the vegetation In the model, the local vegetation cover density ρυ is related to the local height hυ of the plants through Eq. (7), where Hυ is the maximum height the vegetation can reach. Fig. 9 shows calculations using different values of Hυ, with u* = 0.38 m/s, Vυ = 12 m/year, ΔL = 80 m and saturated influx.

Fig. 7. Dependence of the field morphology on the width of the vegetation-free backshore, namely ΔL = (a) 20 m, (b) 40 m and (c) 100 m. Wind is blowing to the right. Calculations were performed with saturated influx qin = qms, Vυ = 12 m/s, Hυ = 1.0 m and u* = 0.38 m/s.

The morphology of the coastal dune field changes as Hυ is varied. When Hυ – which is in fact related to the maximum density of the vegetation cover – is sufficiently small (about 20 cm), the amount of sand trapped by the vegetation at the beach is small, and the behavior

12

Height of Dune (m)

The width ΔL of the vegetation-free backshore plays an important role for the sediment supply and dune formation (Davidson-Arnott and Pyskir, 1988; Nordstrom et al., 1990). The larger the nonvegetated backshore, the higher the alongshore accumulation of sand and the growth of the foredune, and the more sand can be transported from the beach inland. Fig. 7 shows calculations performed with different values of ΔL under saturated flux and values of shear velocity of u* = 0.38 m/s, vegetation growth rate of Vυ = 12 m/year and maximum vegetation height of Hυ = 1.0 m. Dunes emerging from the beach with non-vegetated width ΔL = 20 m (c.f. Fig. 7a) display an average size of 1–5 m. Dunes migrate several hundreds of meters downwind forming incipient trailing ridges of stabilized, vegetated sand at their horns. When ΔL increases, dunes emerge at a higher rate and the distance between dunes in the field diminishes, as can be seen from the simulations with ΔL = 40 m and 100 m, respectively, in Fig. 7a–c. Indeed, the average size of dunes forming at the beach increases with ΔL, as can be seen in Fig. 8, because the larger the total volume of sand incoming from the non-vegetated backshore, the larger the dunes escaping the upwind sand patch at the inlet and entering the vegetated area. However, the maximum size of dunes is obviously limited by the total volume of sand available at the beach — which has a length of Δx = 80 m (c.f. Section 3). When the threshold distance (ΔL) for vegetation growth, which is measured from the inlet of the simulation area, becomes of the order of (or larger than) the total width of the beach (Δx), no further increase in dune size with ΔL is observed: for ΔL = 60, 80 or 100 m, barchans with an average height of ∼ 8 m are the dominant dune type (Fig. 8).

10 8 6 4 0

20

40

60

80

100

ΔL (m) Fig. 8. Largest dune height in the coastal dune fields of Fig. 7 as a function of ΔL. The dune height increases with ΔL because the larger the area free of vegetation growth at the beach, the more sand is accumulated.

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Fig. 9. Calculations for different values of maximum vegetation height Hυ: (a) 20 cm, (b) 50 cm and (c) 800 cm, with qin = qs, Vυ = 12 m/s, ΔL = 80 m and u* = 0.38 m/s. Wind direction is from left to right and both axes are in units of meters.

of the field is close to the one in Fig. 4: small crescent dunes with height of 2 m or less emerge from the sand sheet and migrate downwind with increasing size with distance from the beach (c.f. Fig. 9a). When Hυ is increased, the efficiency of the vegetation in trapping sand also increases. This can be understood by noting that, due to Eq. (8), the vegetation growth rate dhυ/dt is higher the larger the maximum Hυ compared to hυ. For Hυ = 50 cm, dunes emerge with larger average height (of about 6 m) and trailing ridges with little vegetation cover as the dunes migrate inland (c.f. Fig. 9b). For Hυ = 8 m (Fig. 9c), the average height of the emerging dunes amounts to 7–10 m. 4.4. The role of vegetation growth rate and wind speed As shown in a recent work (Durán and Herrmann, 2006a), the transformation of a barchan of volume V into a parabolic dune is controlled by the value of the fixation index, Θ = Q sV

−1 = 3

−1

Vυ ;

ð9Þ

where Q s(u*) = qs/ρsand is the saturated (bulk) sand flux. The barchan– parabolic transition takes place if Θ b Θc, where the threshold value Θc is about 0.5 (Durán and Herrmann, 2006a). In this way, the stabilization dynamics of a barchan of given volume V depends fundamentally on the growth velocity Vυ of the vegetation cover and the wind speed u*.

Fig. 10. Coastal dune formation calculated for different rates of vegetation growth, Vυ (m/year): (a) 2.0, (b) 12.0, (c) 24 and (d) 36, and for constant u* = 0.38 m/s. Wind direction is to the right and both axes are in units of meters. The different dune morphologies, i.e. barchans, crescent dunes with trailing ridges, parabolic dunes and foredunes correspond to fixation index values Θ = 3.5, 0.47, 0.16 and 0.07 (c.f. Eq. (9)), respectively from(a) to (d). Calculations are performed with ΔL = 80 m, Hυ = 1.0 m and saturated influx.

Fig. 10 shows the behavior of the coastal dune field for the different rates of vegetation growth, Vv, for a constant value of wind speed u* = 0.38 m/s and incoming saturated flux (qin = qs). We see that, when Vυ is sufficiently small (e.g. Vυ = 2 m/year, c.f. Fig. 10a), barchans of increasing size with distance are obtained, a scenario that resembles the one of the non-vegetated field in Fig. 4. Indeed, the height of dunes emerging from the beach is around 2 m. For barchans with heights of ∼ 6 m far inland from the beach, the corresponding value of the fixation index Θ is around 3.5, so parabolic dunes are not

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observed. As Vυ increases to 12 m/year, the vegetation barrier at the downwind front of the sand backshore becomes more efficient in trapping sand, in such a way that the height of the dunes on the beach is larger (around 8 m). In this case, a fixation index Θ ≈ 0.47 is obtained for the emerging dunes. When Θ is close to the critical value Θc = 0.5, the timescale of dune stabilization diverges (Durán and Herrmann, 2006a). Dunes in Fig. 10b migrate downwind with approximately constant height, although some incipient trailing ridges may be seen. When Vυ increases further to 24 m/year, the height of the emerging dunes also increases (to around 12 m), which yields Θ ≈ 0.16. Parabolic dunes are obtained (c.f. Fig. 10c), the height of which diminishes with distance — at a distance of 1 km downwind from the beach, dune height is about 4 m. The field dynamics result, thus, from the interplay of two different processes related to the characteristic growth rate Vυ of the vegetation cover. On one hand, the fixation index Θ, which determines the transition of a barchan into a parabolic dune (Durán and Herrmann, 2006a), scales with 1/Vυ (c.f. Eq. (9)). On the other hand, the higher Vυ the larger the size of the dunes emerging from the beach. However, due to Eq. (9), an increase in the average dune volume leads, again, to a decrease in Θ, thus anticipating the barchan–parabolic transition. One dramatic consequence of the reduction of the stabilization timescale with Vυ is the formation of large vegetated foredunes (Hesp, 2002), as exemplified in Fig. 10d. For Vυ = 36 m/year, dunes appear along the beach with heights between 15 m and 20 m, which gives Θ ≈ 0.07. In this case, the height of the upwind vegetated sand barrier increases in time, and the downwind front of the parabolic dunes advances inland with decreasing velocity. It is interesting that the formation process of the foredune is in good agreement with the conceptual model developed from field observations by Hesp (2002). In Fig. 10d, the longitudinal dune profile increases sharply downwind from the front of the backshore, reaching a maximum height of 25 m after about 50 m from the beach and decreasing further inland, such as in real examples of coastal foredunes (Hesp, 2002). Similar field morphologies are observed by varying the wind shear velocity u* for constant Vυ = 12 m/year (Fig. 11). For u* = 0.70 m/s, aeolian drag overcomes the action of vegetation growth and the characteristic dune type is the barchan (c.f. Fig. 11a), with the dune size increasing with distance downwind. For u* = 0.50 m/s (c.f. Fig. 11b), crescent dunes are still the dominant morphology, however, incipient trailing ridges also form. By decreasing u* to 0.38 m/s, barchans transform into parabolic dunes (c.f. Fig. 11c), and for low enough values of u*, alongshore vegetated foredunes are obtained (c.f. Fig. 11d). In fact, field morphodynamics are, again, controlled by the fixation index Θ (Eq. (9)), which scales with the saturated flux, which in turn is a function of the wind shear velocity. It is important to emphasize that the variable of the vegetation which enters the calculation of the shear stress on the sand grains is ρυ, the vegetation cover density (c.f. Eq. (4)), which is related to the vegetation height (hυ) through Eq. (7). Thus, in the model equations, the characteristic growth velocity of the vegetation (Vυ), defined as a vertical growth rate, in fact represents an effective horizontal (propagation) growth rate of the vegetation cover (Durán et al., 2008). Measured values of Vυ may achieve several decimeters per year (Reitz et al., 2010), although higher growth rates have been also reported (Hesp, personal homepage). Indeed, real values of Vυ should be, in general, smaller than the values used in the simulations. The calculations consider upwind sand-moving winds of constant strength (u*), while in reality wind speed is a varying function of time, and u* is only a fraction of the time above threshold (Liu et al., 2005; Levin et al., 2009). Because the value of u* used in the simulations means an effective wind speed value, which represents an average over sandmoving wind velocities (Parteli et al., 2009), Vυ cannot be directly compared to measured growth rates of vegetation — the choice for the value of Vυ in the simulations is also based on the assumption of constant u* above threshold. Again, the important quantity for dune

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Fig. 11. Calculations with different wind shear velocities u* (m/s): (a) 0.70, (b) 0.50, (c) 0.38 and (d) 0.30. Wind is blowing to the right and both axes are in units of meters. Parabolic dunes and foredunes appear as u* decreases since the lower u* the smaller the fixation index Θ (Eq. (9)). Calculations are performed with Vυ = 12 m/year, ΔL = 80 m, Hυ = 1.0 m and saturated influx.

morphology is the model fixation index (∝ Q s/Vυ, c.f. Eq. (9)), i.e. the ratio of net sand transport and vegetation growth rates, which encompasses the critical rate of erosion/deposition for dune stabilization due to vegetation growth (Reitz et al., 2010).

5. Conclusions The genesis of coastal dune fields was studied through using a model for aeolian saltation and dune formation that accounts for

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Fig. 12. On the left, simulations of coastal dune fields obtained with (a) u* = 0.38 m/s, Hυ = 100 cm and Vυ = 1 m/year; (b) u* = 0.38 m/s, Hυ = 100 cm and Vυ = 12 m/year; (c) u* = 0.38 m/s,Hυ = 50 cm and Vυ = 12 m/year, and (d) u* = 0.38 m/s, Hυ = 100 cm and Vυ = 36 m/year. Wind direction is from left to right and both axes are in units of meters. All simulations were performed with incoming saturated influx, and with ΔL = 80 m. On the right we see images of real dune fields, namely (a) and (b) the dune field in Lençois Maranhenses, Northeast of Brazil; (c) White Sands in New Mexico, USA and (d) coastal dunes near Broughton Islands, Eastern Australia.

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vegetation growth. The morphology and dynamics of dunes emerging from a sand patch under unidirectional wind was investigated. The results of the calculations may be summarized as follows: 1. In the absence of vegetation growth, coastal dune fields are formed when there is a high influx of sand. The calculations shed light on the formative stages of coastal dunes emerging from a beach under saturated influx: small instabilities with heights of a few centimeters develop on the sand patch; these instabilities evolve into small dunes a few tens of centimeters high, which migrate downwind increasing size with distance; the average size of the dunes increases downwind due to the large influx. 2. In areas of low influx, vegetation growth may enhance deposition, thus promoting the nucleation of dunes even if sand supply is low. The formation dynamics of the dune field starts with an increasing alongshore accumulation of sand and the appearance of a sand barrier or foredune. Dunes grow in size and escape the barrier migrating downwind into the vegetated area. As the influx increases, individual barchans tend to link laterally forming crescentic ridges. The width (ΔL) of the non-vegetated backshore determines the timescale of dune growth at the beach. The larger ΔL, the higher the rate of sand accumulation and dune growth, and the smaller the average distance between dunes. 3. The wind shear velocity (u*) and the characteristic vegetation growth velocity (Vυ) fundamentally control the morphology of the coastal dune field. Both quantities are encoded in the fixation index Θ (Eq. (9)), which controls the transformation of barchans (Θ N 0.5) into parabolic dunes (Θ b 0.5) (Durán and Herrmann, 2006a). Furthermore, the calculations show that, when there is a sand source, the average volume V of emerging dunes depends both on the vegetation growth rate relative to the wind strength and on the maximum vegetation cover density, which is related to the maximum vegetation height Hv through Eq. (7). In fact, the greater the ability of the plants to trap sand, the larger the dunes. Indeed, an increase in dune volume implies a decrease in Θ (c.f. Eq. (9)), thus favoring dune stabilization. The model reproduces different coastal dune field morphologies observed in nature: (i) non-vegetated barchans and transverse dunes; (ii) barchans with incipient trailing ridges; (iii) parabolic dunes and (iv) alongshore vegetated sand barriers or foredunes, respectively, for increasing growth rate of the vegetation cover relative to the wind strength. Further, the dune shape is a function of the density of vegetation cover, sand supply and influx. In Fig. 12 typical simulation outcomes are compared to images of real coastal dune fields. In conclusion, the model has been applied to the study of coastal dune field morphology and dynamics. The model could be employed as a helpful tool in the investigation of the past development stages of coastal landscapes, as well as to predict the evolution and migration of dunes in coastal areas. In fact, important features need to be added to the model in the future, as for example time variations of wind direction and intensity, as well as the raising and sinking of the water table level, which strongly affects local transport rates. The evolution of coastal dune fields can be greatly influenced by the formation of interdune freshwater ponds (Levin et al., 2007, 2009) and interdune flats of wet or damp surface (Kocurek et al., 1992) as a result of seasonal variations of rainfall, evaporation rates and groundwater level. Furthermore, large seasonal variations in wind strength u* are typical in many coastal areas. Whereas the value of u* used in the model means an effective (average) velocity of winds above the threshold u*t, wind shear velocities u* are often below threshold during the rainy seasons (Jimenez et al., 1999). Such variations in wind strength and groundwater level have important implications for the timescale of dune formation and dune field dynamics and must be included in order to improve the quantitative assessment of coastal dune fields.

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