# Steady-State Dune Morphodynamics Through the Barchan-Parabolic Transition

Jeanne Alkalla, Clément Narteau, Olivier Rozier, Jean Vérité, Sylvain

Courrech Du Pont

► To cite this version:

Jeanne Alkalla, Clément Narteau, Olivier Rozier, Jean Vérité, Sylvain Courrech Du Pont. Steady-State Dune Morphodynamics Through the Barchan-Parabolic Transition. Journal of Geophysical Research: Earth Surface, 2025, 130, 10.1029/2024JF008220. insu-05080902

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Submitted on 23 May 2025

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Distributed under a Creative Commons Attribution 4.0 International License
# JGR Earth Surface

## RESEARCH ARTICLE

10.1029/2024JF008220

**Key Points:**

* The unstable nature of parabolic dune size is demonstrated using a cellular automaton dune model that incorporates vegetation

* Numerical simulations are performed to obtain steady-state dunes through the barchan-parabolic transition

* These steady states are analyzed to link dune shape, vegetation growth and sediment transport

**Correspondence to:**

J. Alkalla,
alkalla@ipgp.fr

**Citation:**

Alkalla, J., Narteau, C., Rozier, O., Vérité, J., & Courrech du Pont, S. (2025). Steady-state dune morphodynamics through the barchan-parabolic transition. *Journal of Geophysical Research: Earth Surface, 130*, e2024JF008220. https://doi.org/10.1029/2024JF008220

Received 9 DEC 2024
Accepted 25 MAR 2025

**Author Contributions:**

**Conceptualization:** Clément Narteau

**Data curation:** Jeanne Alkalla,
Olivier Rozier

**Formal analysis:** Jeanne Alkalla,
Clément Narteau, Sylvain Courrech du
Pont

**Funding acquisition:** Clément Narteau

**Investigation:** Jeanne Alkalla

**Methodology:** Jeanne Alkalla,
Clément Narteau, Olivier Rozier

**Project administration:** Clément Narteau

**Resources:** Clément Narteau

**Software:** Jeanne Alkalla, Olivier Rozier

**Supervision:** Clément Narteau,
Olivier Rozier, Sylvain Courrech du Pont

**Validation:** Jeanne Alkalla,
Clément Narteau, Jean Vérité,
Sylvain Courrech du Pont

**Visualization:** Jeanne Alkalla,
Olivier Rozier, Jean Vérité

**Writing – original draft:** Jeanne Alkalla,
Clément Narteau

**Writing – review & editing:**
Jeanne Alkalla, Clément Narteau,

© 2025. The Author(s).
This is an open access article under the
terms of the Creative Commons
Attribution License, which permits use,
distribution and reproduction in any
medium, provided the original work is
properly cited.

<figure><img src="image_2.png" /></figure>

# Steady-State Dune Morphodynamics Through the Barchan-Parabolic Transition

Jeanne Alkalla¹, Clément Narteau¹, Olivier Rozier¹, Jean Vérité¹, and
Sylvain Courrech du Pont²

¹Institut de Physique du Globe de Paris, Université Paris Cité, CNRS, Paris, France, ²Laboratoire Matière et Systèmes Complexes, Université Paris Cité, CNRS, Paris, France

**Abstract** Dunes in the presence of vegetation exhibit a variety of shapes, from barchan to parabolic forms. Given the rapid fluctuations in environmental conditions across space and time, it is challenging to ascertain whether these dune shapes are merely transient or indicative of a dynamic equilibrium between sediment transport and vegetation growth. In this study, plant cover is introduced into a 3D cellular automaton dune model to numerically investigate the influence of vegetation on dune morphodynamics. Numerical simulations show that isolated parabolic dunes are unstable, increasing or decreasing in size according to the volume of vegetated sediment they remobilize downstream or deposit upstream in their horns. When specific boundary conditions are applied, dunes converge on steady states that accurately capture the barchan-parabolic transition. Most of the isolated dune shapes observed in nature are reproduced at steady state in the model by increasing the intensity of vegetation processes, from the typical migrating barchans to the fully stabilized parabolic dunes. As the impact of vegetation increases, the steepness of the steady-state dune slopes changes, and the crest curvature reversal occurs. This ensures that all the longitudinal sections of the dune migrate at the same rate by reorienting the transverse sediment fluxes. In the model parameter space, sudden jumps in steady-state properties are associated with the stabilization of upstream horns or crest curvature reversals. These results illustrate why transitional dune shapes between barchans and parabolic dunes are less common in natural environments where environmental conditions are heterogeneous and variable in both space and time.

**Plain Language Summary** Dune stabilization with plants represents an effective means of combating desertification and coastal erosion. Similar to barchans without vegetation, parabolic dunes are isolated aeolian bedforms that facilitate the analysis of dune forms in the presence of vegetation. The present study develops a numerical dune model to analyze the permanent feedback between plant cover and sand transport. The model is validated by examining the properties of isolated dunes with respect to the intensity of vegetation processes, in particular when their morphology changes from the crescent shape of barchans (arms point downwind) to the parabolic forms (horns are stabilized by vegetation upwind). These dunes are commonly reported on Earth and can be observed in the model at steady state. Various parameters are quantified to characterize the barchan-parabolic transition (e.g., dune slopes, crest curvature). The results demonstrate that the changes in shape are linked to the reorganization of transverse sand fluxes on the back of the dunes. This compensates for the loss of sand along the barchan arms or vegetation growth on the parabolic dunes edges. This theoretical understanding of steady-state parabolic dunes could prove to be an important step in the management of coastal and desert areas subject to high climatic variability.

## 1. Introduction

Parabolic dunes are found in different environments on Earth, in coastal or desert areas (Figure 1), as soon as vegetation can stabilize aeolian sands (Courrech du Pont et al., 2024; Goudie, 2011; Hesp, 2013; Hesp et al., 2022; Lancaster et al., 2016; Yan & Baas, 2015). Their morphodynamics are closely related to those of barchan dunes, which are fundamental to the understanding of dune physics. Although barchan and parabolic dunes are both crescent-shaped, their horns, on either side of a central body, point in opposite directions. In the absence of vegetation, a barchan migrates at a speed inversely proportional to its height by recycling and transporting up its stoss slope the sediment stored in its slip face (Bagnold, 1941). On both sides, the longitudinal sections of the barchan decrease in height and aspect ratio, until the slip face disappears (Elbelrhiti et al., 2005; Zhang et al., 2010). These sections no longer act as sediment traps. Instead, they emit sand and form arms oriented in the wind direction. In the absence of coupling between the longitudinal sections, the observed decrease in height
Olivier Rozier, Jean Vérité,
Sylvain Courrech du Pont

<figure><img src="image_3.png" /><figcaption>**Figure 1.** Parabolic dunes in vegetated dune fields around the world: (a) White Sands, USA (32.90°N, 106.23°W). (b) Wyoming, USA (41.95°N, 108.62°W). (c) Brazilian coast (10.60°S, 36.64°W). (d) Inner Mongolia, China (43.01°N, 119.54°E). (e) South Australia (34.91°S, 135.89°E). (f) Alberta, Canada (54.94°N, 118.35°W). Flux roses indicate the direction toward which the wind blows. They are computed from the surface wind data of the ERA5-land climate reanalysis using the formalism of Gao et al. (2021). Black arrows show the resultant drift directions.</figcaption></figure>

should induce an increase in migration rate. The barchan shape in the direction perpendicular to the wind is such that the sand losses and the decrease in height are compensated for by transverse flux, resulting in the migration of the dune as a whole with a steady shape (Lima et al., 2024; Zhang et al., 2014). This eventually results in the characteristic crescent shape with arms pointing downwind.

In the presence of vegetation, plants colonize the lateral edges of dunes, thereby fixing these areas of lower sediment transport via their roots and/or their impact on the wind shear velocity. While the central body continues to migrate, horns are stabilized by surrounding vegetation and point against the wind, eventually leading to the parabolic shape (Durán et al., 2005; Lancaster & Baas, 1998; Reitz et al., 2010; Wiggs et al., 1995). Transitions between barchan and parabolic dunes are frequently observed in terrestrial dune fields, where aeolian bedforms display a wide range of shapes and aspect ratios depending on climatic conditions and sediment availability.
Field measurements and satellite imagery have provided a better understanding of the barchan-parabolic transition by identifying key parameters such as vegetation cover and wind speed (Anthonsen et al., 1996; Barchyn & Hugenholtz, 2015; Han et al., 2022; Hugenholtz & Wolfe, 2005; Lancaster & Baas, 1998; Lee et al., 2019; Marín et al., 2005; Reitz et al., 2010; Tsoar & Blumberg, 2002; Xu et al., 2015; Zhang et al., 2020). All these observations have been complemented by numerical modeling, which is an effective approach for investigating changes in environmental conditions. As in the study of non-vegetated dunes, continuous and discrete models have been developed. Continuous approaches encode the impact of vegetation by reducing the bed shear stress or the sand flux according to the local vegetation cover, which itself evolves in response to erosion and deposition rates (de M. Luna et al., 2011; Durán & Herrmann, 2006; Nishimori & Tanaka, 2001). In discrete approaches, topography results from an assemblage of elementary cells, but vegetation cover is always modeled by scalar variables that controls erosion, transport and deposition probabilities (Barchyn & Hugenholtz, 2012a, 2012b; Nield & Baas, 2008; Pelletier et al., 2009; Yan & Baas, 2018). All of these methods produce consistent models of vegetated dunes with respect to wind strength and vegetation cover. They provide a comprehensive overview of the transition from barchan to parabolic dunes, and convincingly explain the coexistence of different types of dunes when they stabilize under the influence of vegetation or are reactivated by aeolian transport (Hanoch et al., 2018; Yizhaq et al., 2007, 2009). However, the analysis of vegetated dune patterns in steady-state regimes has received relatively little attention. This is likely due to the fact that, like for barchans (Zhang et al., 2010), numerical models require to be run under specific boundary conditions.

Under a unidirectional wind regime and a homogeneous sediment supply, an isolated barchan dune is subjected to a continuous input of sediment, which is proportional to its width (Hersen, 2004; Zhang et al., 2014). This input is permanently compensated by sediment losses that occur at the tip of the arms. Nevertheless, the width of these arms varies little with respect to dune size, such that the rate of sediment loss increases more slowly than the input rate as the width of the dune increases. This results in the unstable nature of the size of barchans. There is a threshold width below which a barchan decreases in size, and eventually disappears (Courrech du Pont et al., 2024; Hersen, 2004). Above this threshold, the barchan will continue to grow until other processes potentially limit its size (Andreotti et al., 2009; Courrech du Pont et al., 2024; Gao, Narteau, & Rozier, 2015; Gunn et al., 2022). In a steady state, when the size, shape, and migration rate of the dunes are constant, the incoming flux is balanced by the losses. This equilibrium is unstable and, as a result, is not observable in natural systems. Nevertheless, this equilibrium can be attained in a three-dimensional cellular automaton dune model, in which all sedimentary cells ejected downstream can be randomly and uniformly reinjected upstream (Zhang et al., 2010, 2014). Once the steady state is reached, it is therefore possible to investigate how the shape and dynamics of the barchans vary with flow strength and dune size.

In what follows, plant cover is incorporated into this dune model (Section 2) and the numerical approach is validated by examining the morphodynamics of vegetated dune fields (Section 3). The unstable nature of the size of parabolic dunes is observed, and the boundary conditions of the model are modified to obtain them at steady state (Section 4.1). These steady states are analyzed through the barchan-parabolic transition to explain the dynamic interactions between the intensity of vegetation processes, sediment transport and dune shape (Section 4.2).

## 2. A Real-Space Cellular Automaton Dune Model With Vegetation

The Real-Space Cellular Automaton Laboratory (ReSCAL) is a generator of 3D stochastic models (Rozier & Narteau, 2014). Each model is constructed from a finite number of states distributed on a cellular grid (Figures 2a and 2b). At the elementary length scale $l_0$ of this cubic lattice of size $H_s \times W_s \times L_s$ (height × width × length), a cell is a real-space representation of the physical environment.

The dynamical system is based on the evolution of pairs of nearest-neighbor cells called doublets. Each individual physical process is associated with a set of doublet transitions and a characteristic transition rate expressed in units of $t_0^{-1}$, where $t_0$ is the characteristic time scale of the model. The iteration scheme behaves like a Markov process which is fully defined at each time step by the cellular space (i.e., all the doublets capable of transition) and the corresponding transition rates (Narteau et al., 2001, 2009; Rozier & Narteau, 2014). This approach retains the advantages of self-organizing processes within conventional cellular automata, while also proposing a continuous time setting. Here, we take advantage of the modular structure of ReSCAL to introduce plant cover into the model used to analyze bedforms, which has proven effective in investigating aeolian dune morphodynamics over a wide
<figure><img src="image_4.png" /><figcaption>**Figure 2.** A real-space cellular automaton model for sediment transport in presence of vegetation. (a) A parabolic dune in the 3D cubic lattice of the model of sediment transport. (b) A vertical slice of cells aligned with the wind. (c) Different possible states of an elementary cell. Doublet transitions and the associated transition rates for panel (d) the model of sediment transport and (e) vegetation dynamics. Transition rates that depend on the bed shear stress are denoted by \* (see also Table 1).</figcaption></figure>

range of wind regimes and sand supply conditions (Gao, Narteau, Rozier, & Courrech du Pont, 2015; Lü et al., 2017; Lucas et al., 2015; Rozier et al., 2019; Rubin et al., 2024; Vérité et al., 2024; Zhang et al., 2010; Zhang et al., 2012).

The ReSCAL dune model combines two cellular automata simulating sediment transport and turbulent flow (Narteau et al., 2009). Although these two cellular automata can operate independently, the strength of the complete model lies in their couplings, which introduce a permanent feedback of an evolving topography on the flow, and the dependence of the transport rate on flow strength. These feedback mechanisms will be modulated by the presence of vegetation.

## 2.1. A Model of Sediment Transport With Vegetation

In the model of sediment transport, cells can be in one of the three states: fluid, mobile, and immobile sediment (Figures 2c and 2d). For the design of the various landscapes over which dunes can migrate, an additional state, ground, represents a consolidated bed surface. Individual physical processes and the corresponding transition rates include erosion, $\Lambda_e$ (immobile → mobile sediment), deposition, $\Lambda_c$ (mobile → immobile sediment), longitudinal transport, $\Lambda_t$ (motion of mobile sediment), and transverse transport through lateral diffusion, $\Lambda_d$. In order to account for gravity-driven granular avalanches, these physical processes are combined with a horizontal diffusion mechanism with a threshold slope corresponding to an angle of repose of 35° (Gao et al., 2016).

Two new states are introduced in the model of sediment transport to study the effect of plant cover on dune morphodynamics: vegetated sediment and vegetated ground (Figure 2c). We only consider the feedback of plant cover on transport, without investigating the structure and the impact of below-ground (rooting) or above-ground (canopy, roughness) biomass. In practice, when a sediment cell is vegetated, it is not capable of movement.
Consequently, simulations are conducted to investigate the trapping of sediment by vegetation, rather than the plant system itself. As for the model of sediment transport, vegetation cover is entirely governed by the dynamics of short-range interactions. However, in this case, the interactions are limited to those between vegetated and non-vegetated cells.

As illustrated in Figure 2, two additional pairs of doublet transitions have been incorporated to accurately represent the competition between vegetation growth and sediment transport in both horizontal and vertical directions. They are associated with four new transitions and the corresponding transition rates. Two for vegetation growth (immobile → vegetated sediment) in the horizontal and vertical directions ($\Lambda_{hg}$ and $\Lambda_{vg}$, respectively). Two others for vegetation death (vegetated → immobile sediment), due to sandblasting in the horizontal direction ($\Lambda_{hd}$), and burial in the vertical direction ($\Lambda_{vd}$). Immobile sedimentary cells can also spontaneously and randomly vegetate, $\Lambda_{rg}$, modeling wind sowing. Then, depending on sediment flux and flow strength, vegetation can appear, spread or decline on a flat bed and along dune slopes.

## 2.2. The Fluid Flow and the Bed Shear Stress

To simulate fluid flow, a lattice gas cellular automaton based on discrete motions of a finite number of fluid particles is implemented (Frisch et al., 1986). At each iteration, particles can move from cell to cell along the direction specified by their velocity vector. Depending on the configuration of fluid particles at each node of the cubic lattice, the collision dynamics can modify the velocity vectors of the involved particles. The repetition of propagation and collision phases produces a flux of particles that is converted into a fluid velocity field $\vec{u}$ by averaging velocity vectors over space and time at the elementary length scale $l_0$ of the sediment transport model.

At the surface of the sediment layer (vegetated or not), all fluid particles that collide with sedimentary cells bounce back along their incident direction. This results in no-slip boundary condition. The velocity field is null at the sediment-fluid interface and there is a permanent feedback of the topography on the fluid flow. It spontaneously produces flow recirculation on the lee side of dunes and an increase in flow velocity on their stoss side (Zhang et al., 2010, 2012). It also generates a logarithmic vertical velocity profile, which is characteristic of turbulent flows. The bed shear stress is derived from the velocity gradient,

$$ \tau_s = \tau_0 \frac{\partial \vec{u}}{\partial \vec{n}}, \qquad (1) $$

where $\vec{n}$ is the normal to the topography and $\tau_0$ is the stress scale of the model. In the absence of transport for vegetated cells, plant cover is not incorporated into the calculation of bed shear stress. Consequently, vegetated cells exert the same influence on fluid flow as other sedimentary or ground cells within the model.

## 2.3. The Control of Fluid Flow on Sediment Transport and Vegetation Growth

The flow continuously modifies the topography through sediment transport, where the bed shear stress is high enough to mobilize sediment cells. The erosion rate $\Lambda_e$ has a continuous and piece-wise linear relationship with the bed shear stress $\tau_s$ according to

$$ \Lambda_e(\tau_s) = \Lambda_0 \begin{cases} 0 & \text{for } \tau_s < \tau_1, \\ \frac{\tau_s - \tau_1}{\tau_2 - \tau_1} & \text{for } \tau_1 \le \tau_s \le \tau_2, \\ 1 & \text{for } \tau_2 < \tau_s, \end{cases} \qquad (2) $$

where $\Lambda_0$ is a constant rate, $\tau_1$ the threshold shear stress for motion inception, and $\tau_2$ a parameter to adjust the slope of the linear relationship. For consistency, it is assumed that $\tau_2 \gg \tau_s$ and $(\tau_2 - \tau_1)/\tau_0 = 100$. By definition, $(\tau_s - \tau_1)$ is the excess shear stress. In practice, the threshold shear stress, $\tau_1$ is used to modify the ability of the flow to generate transport in the model: the higher the $\tau_1$-value, the smaller the strength of the flow (Narteau et al., 2009). The diffusion rate also depends on flow strength assuming the same linear dependence as for the erosion rate (Equation 2).
To account for wind erosion and plant abrasion, there is a negative dependence of flow strength on vegetation growth:

$$ \Lambda_{hg}^*(\tau_s) = \Lambda_{hg} \left\{ \begin{array}{ll} \frac{1}{\tau_2 - \tau_s} & \text{for } \tau_s \le \tau_1, \\ \frac{\tau_2 - \tau_s}{\tau_2 - \tau_1} & \text{for } \tau_1 \le \tau_s \le \tau_2, \\ 0 & \text{for } \tau_2 < \tau_s. \end{array} \right. \quad (3) $$

Other transitions do not explicitly depend on the basal shear stress, but are still coupled to the sediment transport because they require the presence of a neighboring cell in a particular state. For example, burial transition leading to vegetation death necessitates the deposition of a sediment cell on top, while vertical growth requires first a horizontal transition of vegetation growth.

## 2.4. Setting the Length and Time Scales of the Model

As shown in Narteau et al. (2009), a size-selection mechanism occurs during the development of incipient dunes in the model, and it was found that there is a most unstable wavelength for which the dune growth rate is the highest. Dune instability theory enables to link this characteristic wavelength for dune formation in the model to physical quantities that encompass contributions from both transport and flow dynamics (Courrech du Pont et al., 2024; Elbelrhiti et al., 2005; Lü et al., 2021). Accordingly, the model can be used to reproduce dune patterns in all types of natural environments as long as the elementary length and time scales, {$l_0, t_0$}, as well as the threshold shear stress value, $\tau_1$, are defined from field measurements of the most unstable wavelength, together with the mean flow strength and the associated sediment transport rate. The same length and time scales are used in the model with vegetation. Nevertheless, in this case, when different flow strengths are considered (i.e., different $\tau_1$-values), the time scale changes and the values of the transition rates for the vegetation processes must be modified according to the $t_0$-value. Here, for the sake of simplicity, the flow strength is kept constant by considering the same value of the threshold shear stress in all numerical simulations.

In what follows, all results are presented in units of {$l_0, t_0$}, but it is useful to give in physical units the orders of magnitude of the transport and vegetation processes that can be accessed by the model using the parameter values given in Table 1. A transport threshold of $\tau_1/\tau_0 = 1$ corresponds to a dimensionless shear velocity, $u_*/u_c$, of 5.6. For a transport threshold wind speed of 3.52 m s⁻¹ and an aerodynamic roughness of 10⁻³ m (Courrech du Pont et al., 2024), it yields a wind blowing permanently at a speed of 19.6 m s⁻¹ at a height of 10 m. Under these strong wind conditions, $l_0 = 0.5$ m, $t_0 = 5.3 \times 10^{-5}$ yr, and a steady-state barchan dune of height 29 $l_0$ (14.5 m) migrates at a rate of $4.3 \times 10^{-2} l_0 t_0^{-1}$ (406 m yr⁻¹). Considering a flat surface composed of ground cells under the same flow strength and no sediment transport (i.e., no vegetation death by burial or sandblasting), vegetation spreads at a rate of $1.7 \times 10^{-4} l_0 t_0^{-1}$ (1.6 m yr⁻¹) for $\Lambda_{hg}t_0 = 10^{-4}$. For $\Lambda_{hd}t_0 = 0.1$, a vegetated sediment surface retreats at a rate of $3.1 \times 10^{-3} l_0 t_0^{-1}$ (28 m year⁻¹) when subjected to a constant sand flux of 0.1 $l_0^2 t_0^{-1}$ (1.29 m² day⁻¹). In 2D, when a steady-state dune migrates over a vegetated surface, the height at which vegetation is preserved on the slip face increases with $\Lambda_{vg}$ without affecting significantly the migration rate as long as it does not reach the upper part of the dune. Vegetation reaches heights of 4 and 8 $l_0$ (2 and 4 m) on the dune slopes for $\Lambda_{vg}t_0$-value of 0.06 and 0.20, respectively.

All these values are obtained under ideal conditions using specific model configurations. In comparison, the full dynamics of the dune model in the presence of vegetation involves multiple feedback mechanisms, which can be challenging to anticipate. For this reason, they are explored below via a thorough numerical investigation, keeping all the parameters shown in Table 1 constant, except for $\Lambda_{vg}$ and $\Lambda_{hd}$. The outputs of the numerical simulations can be used to study dune morphodynamics under constant wind conditions when vegetation growth and sandblasting vary in intensity.

## 3. Vegetated Dune Fields and Stability of Parabolic Dune Size

Figure 3 shows vegetated dune fields forming from a flat sediment bed for three different values of $\Lambda_{vg}$, the parameter that controls the vegetation growth on dune slope. For the low value $\Lambda_{vg}t_0 = 0.02$, a barchan dune field
<table><caption>**Table 1**

Units and Values of the Parameters of the Vegetated Dune Model</caption><thead><tr><th>Variable</th><th>Description</th><th>Value</th><th>Units</th><th>Dependence</th></tr></thead><tbody><tr><td colspan="5"><strong>Elementary units</strong></td></tr><tr><td>l<sub>0</sub></td><td>Length</td><td></td><td>[L]</td><td></td></tr><tr><td>t<sub>0</sub></td><td>Time</td><td></td><td>[T]</td><td></td></tr><tr><td>τ<sub>0</sub></td><td>Stress</td><td></td><td>[M][L]<sup>-1</sup>[T]<sup>-2</sup></td><td></td></tr><tr><td colspan="5"><strong>Transport processes</strong></td></tr><tr><td>Λ<sub>i</sub></td><td>Transition rate for transport</td><td>6</td><td>t<sub>0</sub><sup>-1</sup></td><td></td></tr><tr><td>Λ<sub>0</sub></td><td>Transition rate for erosion</td><td>4</td><td>t<sub>0</sub><sup>-1</sup></td><td>Shear stress</td></tr><tr><td>Λ<sub>c</sub></td><td>Transition rate for deposition</td><td>2</td><td>t<sub>0</sub><sup>-1</sup></td><td></td></tr><tr><td>Λ<sub>g</sub></td><td>Transition rate for gravity</td><td>10<sup>3</sup></td><td>t<sub>0</sub><sup>-1</sup></td><td></td></tr><tr><td>Λ<sub>d</sub></td><td>Transition rate for diffusion</td><td>0.02</td><td>t<sub>0</sub><sup>-1</sup></td><td>Shear stress</td></tr><tr><td>Λ<sub>a</sub></td><td>Transition rate for avalanches</td><td>1</td><td>t<sub>0</sub><sup>-1</sup></td><td>Slope</td></tr><tr><td>a</td><td>Transport coefficient</td><td>0</td><td></td><td></td></tr><tr><td>ε</td><td>Erosion coefficient</td><td>10<sup>-4</sup></td><td></td><td></td></tr><tr><td>τ<sub>1</sub></td><td>Transport threshold</td><td>1</td><td>τ<sub>0</sub></td><td></td></tr><tr><td>τ<sub>2</sub> - τ<sub>1</sub></td><td>Linear slope of erosion rate</td><td>100</td><td>τ<sub>0</sub></td><td></td></tr><tr><td>θ<sub>c</sub></td><td>Slip angle</td><td>35</td><td>Deg</td><td></td></tr><tr><td colspan="5"><strong>Vegetation processes</strong></td></tr><tr><td>Λ<sub>vg</sub></td><td>Transition rate for vertical growth</td><td>0.5<sup>a</sup></td><td>t<sub>0</sub><sup>-1</sup></td><td></td></tr><tr><td>Λ<sub>vd</sub></td><td>Transition rate for vertical death</td><td>0.1</td><td>t<sub>0</sub><sup>-1</sup></td><td></td></tr><tr><td>Λ<sub>hg</sub></td><td>Transition rate for horizontal growth</td><td>10<sup>-4</sup></td><td>t<sub>0</sub><sup>-1</sup></td><td>Shear stress</td></tr><tr><td>Λ<sub>hd</sub></td><td>Transition rate for horizontal death</td><td>0.1<sup>a</sup></td><td>t<sub>0</sub><sup>-1</sup></td><td></td></tr><tr><td>Λ<sub>rg</sub></td><td>Transition rate for random seeding</td><td>10<sup>-4</sup></td><td>t<sub>0</sub><sup>-1</sup></td><td></td></tr><tr><td>b</td><td>Random seeding coefficient</td><td>10<sup>-2</sup></td><td></td><td></td></tr></tbody></table>

Note. The model of sediment transport is the same as in Narteau et al. (2009). <sup>a</sup>Unless specified otherwise in the text.

develops (Figure 3a). Barchan dunes of different sizes constantly interact and evolve through collisions and sediment fluxes emitted from their arms. These output fluxes and the migrating barchans reduce vegetation cover, which is only present in areas with low sediment transport. For the intermediate value $Λ_{vg}t_0 = 0.70$, a field of parabolic dunes emerges (Figure 3b). They deposit sediment upstream along their horns, while their central body kills the plants they bury, and feed on the sediment stabilized by the dunes that preceded them (Hesp et al., 2022). For the high value $Λ_{vg}t_0 = 2.00$, vegetation processes dominate and the entire dune pattern is stabilized (Figure 3c). All the sediment is distributed along parabolic dune horns that have formed pinhead structures of equal height as their central body becomes fully vegetated. Taken together, these simulations show that the model can reproduce various barchan and parabolic dune fields observed on Earth, as well as the different mass exchange mechanisms between individual dunes in each case.

In order to estimate the mass balance of a parabolic dune as it loses sediment upstream along its horns and feeds on the vegetated sediment on which it migrates, we examine an isolated bedform approaching a vegetated sedimentary bed for $Λ_{vg}t_0 = 0.6$. The initial condition is a conical heap of sediment cells located on a non-vegetated surface composed of ground cells (Figure 4). When the newly formed dune migrates onto the vegetated surface, it rapidly acquires a parabolic shape, as vegetation stabilizes the horns from the sides, and the central dune body kills the vegetation by burying it. If the vegetated surface covers a thin layer of sediment ($≤0.25 l_0$), the dune body decreases in size and the horns converge towards each other to eventually form a fully stabilized pinhead deposit structure (Figure 4a). If the vegetation covers a sedimentary bed with a thickness of 3 $l_0$, the newly formed dune grows in size as it remobilizes entirely the sediment bed cleared of vegetation (Figure 4b). In both cases, the amount of sediment captured in the horns per length unit is the same. The increase or decrease in dune size is
<figure><img src="image_5.png" /><figcaption>**Figure 3.** Vegetated dune fields according to the intensity of vegetation growth: (a) Population of barchan dunes for $\Lambda_{vg}t_0 = 0.02$. (b) Population of parabolic dunes for $\Lambda_{vg}t_0 = 0.70$. (c) Fully stabilized dune field for $\Lambda_{vg}t_0 = 2.00$. In all cases, the initial condition is a sedimentary bed with an average thickness of 0.5 $l_0$ using the parameter values shown in Table 1.</figcaption></figure>

<figure><img src="image_6.png" /><figcaption>**Figure 4.** The unstable nature of parabolic dune size. Evolution of a sediment pile migrating on panel (a) a vegetated ground (no sediment available) and (b) a vegetated sediment surface with a thickness of 3 $l_0$. (c) Number of mobile sediment cells as a function of time for the shrinking (orange) and growing (purple) parabolic dunes shown in panels (a) and (b), respectively. As parabolic dunes migrate over a vegetated surface and deposit sediment upstream in their horns, their size can only decrease or increase according to the amount of sediment they remobilize.</figcaption></figure>
<figure><img src="image_7.png" /><figcaption>**Figure 5.** A parabolic dune at steady state. (a) Formation of a steady-state parabolic dune from a conical sediment heap for $\Lambda_{vg}t_0 = 0.6$. The conical heap has a radius of 30 $l_0$ and a height of 40 $l_0$, for a volume of $3.8 \times 10^4 l_0^3$. White rectangles at $t=0$ show the sediment reinjection zones upstream and downstream of the initial sediment heap. (b) Volume, $V_d$, (c) length, $L_d$, and (d) migration rate, $c$, of the dune with respect to time. Gray dashed lines in panels (b), (c) and (d) correspond to the time of the dunes shown in panel (a).</figcaption></figure>

therefore entirely controlled by the thickness of the sediment bed that the dune is likely to remobilize during its migration (Hesp, 2013; Hesp et al., 2022). Consequently, the morphodynamics of parabolic dunes, like barchans, are unstable and controlled by sediment supply. Nevertheless, in presence of vegetation, this sediment supply does not depend only on the input sediment flux from upstream, but also on the downstream amount of sediment stabilized by vegetation that can be remobilized by the dune.

**4. Steady-State Dune Morphodynamics Through the Barchan-Parabolic Transition**

To produce isolated steady-state dunes in presence of vegetation, we complement the numerical approach developed by Zhang et al. (2014) and Rozier et al. (2019) for barchans and elongating dunes, respectively. The initial condition is a conical heap of sediment located on a non-vegetated surface (Figure 5).

This dune loses sediment through its arms or horns, downstream or upstream, depending on whether it takes on a barchan or parabolic shape. In both cases, sediment cells leaving the system are randomly and uniformly reinjected to compensate for these losses. This reinjection takes place both upstream and downstream of the dune, so that it can be fed regardless of its shape (barchan or parabolic) and the density of vegetation in the system. The rectangular reinjection remains fixed in the reference frame of the center of mass of the dune (Figure 5a at $t=0$). Their cross sections perpendicular to the flow are identical to those of the initial conical sediment heap, which may be thinner or wider than the evolving dune. However, all dunes reach a steady state with a constant shape, volume, $V_d$, length, $L_d$, and migration rate, $c$ (Figures 5a–5d). These steady states may differ quantitatively depending on the details of the sand supply conditions, for example, the size of the reinjection zone.
<figure><img src="image_8.png" /><figcaption>**Figure 6.** Different steady-state dune shapes through the barchan-parabolic transition: (a) Steady-state dune shapes for an increasing Λ<sub>vg</sub>-value. (b) Mean dune shapes calculated from 500 dune profiles separated by 500 t<sub>0</sub> after reaching steady state. (c) Comparison with similar dune shapes on Earth (Credits Google Earth). From top to bottom, examples are taken from Western Sahara (27.23°N, 12.48°W), the Kubuqi (40.49°N, 107.37°E) and the Gurbantünggüt (46.03°N, 88.86°E) deserts in China, the Saskatchewan province in Canada (59.04°N, 108.54°W), Nevada in the USA (41.09°N, 117.67°W), and from the K'gari island in Australia (25.66°S, 153.03°E).</figcaption></figure>

**4.1. Identification of Steady-State Dune Shapes**

When increasing the Λ<sub>vg</sub>t<sub>0</sub>-value from 0 to 2.5, Figure 6a shows that the model reproduces a wide range of steady-state dune shapes illustrating the barchan-parabolic transition. Once the steady states have been reached, long-term statistical analysis of the dune topography allows for the estimation of the mean value and dispersion of different morphological parameters that characterize this transition in the model (Figures 6b and 7). These time-averaged properties, and any instantaneous shapes produced by numerical simulations, can also be compared with
<figure><img src="image_9.png" /><figcaption>**Figure 7.** Properties of active dunes at steady state as a function of vegetation growth. (a) Absolute value of crest curvature |γ|, (b) density of vegetation on the surface ρ, (c) dune height H_d and the (d) transverse 2H_d/W_d and (e) longitudinal dune slopes H_d/L_d with respect to Λ_vg t_0. Initial conditions are specified in Figure 5. The circles and error bars show the mean values and standard deviations calculated from 500 elevation maps of the dune, separated by intervals of 500 t_0 after reaching steady state. The color of the circles is assigned according to the curvature γ, considering a logarithmic color scale in [-10⁻¹, 10⁻¹]. The insets show how the dune dimensions are measured. The dashed lines show the Λ_vg and γ-values at which abrupt transitions in dune shape are observed.</figcaption></figure>

aerial images to ensure that the model captures much of the diversity of isolated dunes observed on Earth in the presence of vegetation (Figure 6c).

For near zero Λ_vg t₀-values, the typical crescent shape of barchans is systematically observed in the simulations. The sediment fluxes emitted along the arms are high enough to kill vegetation downstream. For Λ_vg t₀ = 0.12, vegetation stabilizes the outer face of the barchan arms, there is no further loss of sediment downstream, and vegetated horns are deposited upstream as the dune migrates. Its central body retains its crescent shape, giving rise to a barchan-parabolic dune. For Λ_vg t₀ = 0.14, vegetation growth is more intense at the edge of the dunes and along the slip face. The vegetated horns trap more sediment upstream, while the central body has a crestline almost perpendicular to the flow, forming transverse-parabolic dunes. For Λ_vg t₀ = 0.18, the dune at steady state has completely reversed its crestline curvature and widens significantly to form a wide parabolic dune. For Λ_vg t₀ > 0.22, all steady-state dunes exhibit a parabolic shape. Then, there is a threshold Λ_vg-value for which vegetation processes take over, and the parabolic dune pattern becomes fully stabilized. Altogether, these different examples demonstrate that the majority of dune shapes characterizing the barchan-parabolic transition
<table><caption>**Table 2**
Classification of Steady-State Dune Shapes in the Model</caption><thead><tr><th>Dune shape</th><th>Name</th><th>Curvature γ [l<sub>0</sub><sup>-1</sup>]</th></tr></thead><tbody><tr><td></td><td>Barchan</td><td>0.02 ≤ γ</td></tr><tr><td></td><td>Barchan-parabolic</td><td>0.003 ≤ γ &lt; 0.02</td></tr><tr><td></td><td>Transverse-parabolic</td><td>0 ≤ γ &lt; 0.003</td></tr><tr><td></td><td>Wide parabolic</td><td>-0.003 ≤ γ &lt; 0</td></tr><tr><td></td><td>Parabolic</td><td>-0.06 ≤ γ &lt; -0.003</td></tr><tr><td></td><td>Stabilized parabolic</td><td>No active crestline</td></tr></tbody></table>

Note. Crest curvature ranges are determined according to the shape at steady state.

can exist at steady state in the model. Nevertheless, within the range of $Λ_{vg}$-values examined, some shapes are observed more frequently than others.

The crest curvature, $γ$, is used to quantify dune shape through the barchan-parabolic transition. It is defined as the quadratic coefficient of the second order polynomial that best fits the crestline of the main dune body, 50 $l_0$ on either side of the central section of the dune. As illustrated in Table 2, each dune type can be associated with a range of curvature values (Figure 6).

## 4.2. Morphological Properties of Steady-State Dunes

Figure 7 shows the variation in steady-state properties of active dunes when the $Λ_{vg}t_0$-value increases from 0 to 1.10 in steps of 0.01. As a result, the density of vegetation on the dune increases continuously (Figure 7b). Interestingly, the transition between a migrating parabolic and a fully stabilized dune does not correspond to complete vegetation cover, but rather to an average coverage of 0.55 (i.e., $Λ_{vg}t_0 = 1.11$). When migration is observed, the distribution of vegetation is, of course, not homogeneous over time and on the different parts of the dune (see examples in Figure 4). However, this

suggests that at steady state, approximately half of the surface area of the dune has to transport sediment to ensure migration and offset plant growth. The existence of a threshold in vegetation cover, as well as its variations with respect to the $Λ_{vg}$-value, indicates that transitions between dune shapes can be either continuous (smooth) or discrete (abrupt) in the presence of vegetation.

While vegetation density increases continuously, two discontinuous changes in dune shape are observed, one for horn stabilization, the other for curvature reversal. These abrupt variations are particularly evident when examining the height of the central section and the mean dune slopes (Figures 7c–7e). In contrast to barchans, which display a decrease in height and a reduction in steepness in the direction perpendicular to the wind as vegetation growth increases, parabolic dunes show an increase in height and steepness (Figures 7c and 7d). This is associated with the change in the sign of the crest curvature. As vegetation becomes more established on barchan and parabolic dunes, the longitudinal slope becomes steeper (Figure 7e).

In order to account for positive and negative feedbacks between transport and vegetation growth, steady-state dune properties are computed within the {$Λ_{hd}$, $Λ_{vg}$} parameter space by varying $Λ_{vg}t_0$ from 0 to 1.4 in steps of 0.01 and $Λ_{hd}t_0$ from 0 to 0.6 in steps of 0.02 (Figure 8). Consistent with the behaviors described in Figure 7, Figure 8 shows that there is always a transition from barchan to fully stabilized parabolic dunes as the intensity of vegetation growth (i.e., $Λ_{vg}$-value) increases, whatever the intensity of sandblasting (i.e., $Λ_{hd}$-value). However, the abrupt transitions associated with the horn stabilization, curvature reversal, and complete dune stabilization occur at increasing $Λ_{vg}$-values when $Λ_{hd}$-value increases, thereby demonstrating the antagonistic effect of these two parameters on vegetation cover and transport. Furthermore, it is uncommon to observe curvature values below 0.003 $l_0^{-1}$ in steady-state conditions, and the number of instances where transverse and wide parabolic dunes can be computed is limited. Thus, the numerical simulations reveal a sharp transition from barchan-parabolic to parabolic dune types when either the values of $Λ_{vg}$ and $Λ_{hd}$ increase.

In response to the influence of vegetation, all the changes in dune shape result in modifications of the transverse sedimentary fluxes. In all cases, these enable the dune to migrate as a whole at a constant speed and in a steady shape.

## 4.3. Dune Morphodynamics Through the Barchan-Parabolic Transition

### 4.3.1. Sediment Fluxes on Barchan and Parabolic Dunes

The sediment flux may be decomposed into flow-parallel and flow-normal components to estimate how they contribute to the observed dune morphodynamics. Figures 9a and 9b illustrate the flux vectors measured at steady state in the numerical simulations over a barchan and a parabolic dunes, respectively. The magnitude of the transverse fluxes on the lee slope of dunes is always greater than that of the stoss slope.
<figure><img src="image_10.png" /><figcaption>**Figure 8.** Curvature of active dunes at steady state in the {$\Lambda_{hd}$, $\Lambda_{vg}$} parameter space. The curvature is computed from the crestline of the main dune body, 50 $l_0$ on either side of the central section of the dune. As in Figure 7, dashed lines at low, intermediate, and high $\Lambda_{vg}$-values show the contour lines for $\gamma l_0$ = {0.02, 0, -0.06}. These correspond to abrupt transitions in dune shape, occurring when horns begin to stabilize, crestlines undergo a curvature reversal, and dunes fully stabilize.</figcaption></figure>

On the stoss side, there is a transverse flux of sediment from the central body towards the arms for the barchan, and from the horns towards the central body for the parabolic dune. This is the opposite on the lee face for both barchan and parabolic dunes. Thus, the flow-normal component of the sediment flux invariably undergoes a change of sign at the crest, whatever the sign of the crest curvature. The integrated mass balance of the central section of the dunes, calculated over the distance in the direction of flow, consistently indicates a net loss of sediment (Figure 9c). At steady state, it is compensated by the input sediment flux, $Q_{in}$, which may originate from either the free flux from upstream or from the vegetated sediment layer on which the dune is migrating.

### 4.3.2. Migration Rates

According to the principle of mass conservation in 2D, the migration rate, $c$, of an isolated dune at steady state can be expressed as $Q_{crest}/H_d$, where $Q_{crest}$ is the flux at the crest and $H_d$ the height of the dune. Consequently, the longitudinal dune aspect ratio is the determining factor in dune migration rate, given the wind speed-up (i.e., the increase in wind speed above a positive slope) and the subsequent increase in sand flux at the crest. As illustrated in Figure 9d, this dependence is modulated in 3D through the barchan-parabolic transition as a consequence of the transverse sediment fluxes.

For barchans, the central section loses sediment to the outer sections on the stoss side, and gains sediment on the lee side. In this case, the migration rate of barchans in numerical simulations is slightly larger than that predicted by $Q_{crest}/H_d$. This is the inverse situation for parabolic dunes (Figure 9d), for which the transverse fluxes feed the central section on the stoss slope and deplete it on the lee side. The discrepancy between $Q_{crest}/H_d$ and the actual migration rates is even more pronounced when vegetation cover increases. In addition, it is noteworthy that this discrepancy tends to zero at crest reversal ($\Lambda_{vg}t_0 \approx 0.16$).

A detailed examination of the shapes and dynamics of dunes in a steady state, along with the associated sediment fluxes, offers now a more comprehensive understanding of dune morphodynamic equilibria in the model with vegetation.
<figure><img src="image_11.png" /><figcaption>**Figure 9.** Sediment fluxes on dunes at steady state. (a) Barchan at $\Lambda_{vg}t_0$ = 0. (b) Parabolic dune at $\Lambda_{vg}t_0$ = 1. Contour lines show an elevation of 2 $l_0$. (c) Cumulative transverse flux of the central longitudinal section of the dunes (shaded areas in panels (a) and (b)) as a function of distance in the wind direction. At steady state, the overall sediment loss is compensated by the constant injection flux $Q_{in}$ (see panel (e)). (d) Dune migration rate $c$ predicted from the sand flux at the crest and the dune height as a function of the measured migration rate. The difference reflects the cumulative mass balance on the dune stoss slopes (see panel (c)). Arrows show the trend as the $\Lambda_{vg}$-value increases and the curvature decreases. (e) Input sand flux as a function of $\Lambda_{vg}t_0$. As in Figure 7, dashed lines show the discontinuous transitions in dune shape associated with the horn stabilization, curvature reversal and the complete stabilization of the dune.</figcaption></figure>

### 4.3.3. Morphodynamic Equilibria

In the absence of vegetation, dunes invariably evolve into the characteristic crescentic shape of barchans. For a small increasing $\Lambda_{vg}$-value, the sand flux emitted along the arms reduces. This is associated with a rapid decrease in the input sediment flux for a small increasing $\Lambda_{vg}$-value (Figure 9d).

Above a threshold $\Lambda_{vg}$-value, vegetation traps sediment on the edge of the dune, resulting in the stabilization of horns as the dune is migrating. This sudden reversal in direction of the sediment loss from downwind to upwind is associated with a minimal near-zero magnitude for this loss. In this case, dune migrates as a whole at a constant speed, almost without loss. Such a balance is highly tenuous, explaining why the transition from barchan to barchan-parabolic shape is so abrupt.

As the intensity of vegetation growth continues to increase, it stabilizes higher and higher parts of the lateral edges of the dunes, slowing them down considerably. Subsequently, the crest curvature undergoes a reversal, which results in a deceleration of the central body and an acceleration of the lateral sections by redirecting the transverse sedimentary flux. According to the increase in $\Lambda_{vg}$-value, it corresponds to the second abrupt transition in dune shape after which the parabolic shape is established. Increasing further the intensity of vegetation growth results in steeper transverse dune slopes (Figure 7e), which amplifies the lateral redistribution of sediment (i.e.,
deceleration of the central body and acceleration of the lateral sections). Meanwhile, the sand loss does not significantly change with the $Λ_{vg}$-value, mainly because the migration rate decreases as the horns are getting thicker.

The parabolic shape is, like the barchan shape, resilient to small perturbations. If the crest curvature of parabolic dune is increased above the steady-state value, the transverse sediment fluxes from the horns to the central zone on the stoss side would increase. As a result, the central zone would slow down and the horns would speed up, reducing the dune curvature back to equilibrium. Hence, like barchans in the absence of vegetation, parabolic dunes are unstable in size (Section 3), but they are stable in shape (attractor) explaining why they are the most common type of dunes observed in vegetated areas.

In contrast to barchans, parabolic dunes have to migrate to lose sediment along their horns. Ultimately, once a threshold $Λ_{vg}$-value is exceeded, the dune is fully stabilized by vegetation. It corresponds to the third abrupt transition in dune shape. It occurs for a finite dune size and a finite, non-zero migration rate. This can be explained as a positive feedback loop between vegetation growth and dune migration. In order to avoid encroachment from surrounding vegetation, the dune must migrate and remobilize sediment. However, as vegetation cover increases, the migration rate decreases, and it cannot be sustained above a critical $Λ_{vg}$-value.

# 5. Concluding Remarks

## 5.1. A New and Complementary Numerical Model

This study presents an extended version of a cellular automaton dedicated to dune morphodynamics, which now accounts for the couplings between sediment transport and vegetation. While the initial model has previously demonstrated its capacity to quantitatively predict numerous characteristics of dune patterns in the absence of vegetation (e.g., superimposed bedforms, elongation, multiple crest orientations), our results illustrate that it is now also capable of reproducing a range of dune patterns observed in presence of vegetation under unidirectional wind regimes. Based on the self-organization mechanisms inherent to this discrete approach (Baas, 2002; Baas & Nield, 2007), there is the potential for the modeling of vegetated dune patterns under a variety of wind regimes and conditions of sediment availability. In contrast to continuous models (Durán & Herrmann, 2006), feedbacks between the sediment environment and the vegetation are not explicit components of the model. Instead, these feedbacks occur spontaneously through collective behaviors that arise from the dynamics of interactions between populations of cells.

The results of numerical simulations indicate that, despite the stable and attractive shape of barchan and parabolic dunes, these bedforms are inherently unstable in size. It can thus be concluded that, under natural conditions, isolated barchan and parabolic dunes can only undergo a decrease or an increase in size, until they reach a minimum or a maximum size, which depends on their environment. Their unstable nature exhibits a mirror symmetry in the exchange of mass with the surrounding environment (Figure 10). The barchans feed on the upstream sediment flux, and loss sediment downstream along their arms. In contrast, parabolic dunes rework the vegetated sedimentary surface on which they migrate downstream (Hesp et al., 2022), and lose material upstream along their horns, where the sediments are captured by vegetation. These erosion and deposition dynamics specific to parabolic dunes represent an essential ingredient that must be quantitatively taken into account in models and field studies. Indeed, vegetated dune morphodynamics depend not only on the scale-invariant flow, but also on the intensity of vegetation processes and the amount of sediment that can be reworked downstream.

In the numerical simulations, the horns are depositional structures that can serve as markers of the intensity of vegetation processes at the time of their formation. Their spacing can be analyzed to track variations in parabolic dune size as they migrate, potentially giving access to the amount of sediment remobilized by the upstream dune. While the principle of conservation of mass can still be employed to estimate the migration rate of parabolic dunes as $Q_{crest}/H_d$ (Figure 9d), there is also a systematic negative deviation from this theoretical prediction, which increases with increasing vegetation cover.

The model developed is, in many ways, a simplified representation of the inherent complexity of vegetated dunes in their natural environment. This is particularly evident with regard to the new transitions that deal with vegetation, which do not account for the impact of groundwater levels, seasonal fluctuations or the presence of diverse plant species (Nield & Baas, 2008). These sophistications can be incorporated at a later stage to assess
<figure><img src="image_12.png" /><figcaption>**Figure 10.** Symmetry in the unstable nature of barchan and parabolic dune size. In this schematic diagram, *Q*<sub>in/dune</sub> and *Q*<sub>out/dune</sub> are the input and output fluxes in the dune reference frame. In this mobile reference frame, which takes migration into account, the shape and input/output fluxes of the barchan and parabolic dunes exhibit mirror symmetry. In the case of barchan dunes, the input/output fluxes are controlled by the strength of the flow. In the case of parabolic dunes, the input/output fluxes are controlled by migration, which determines the amount of sediment captured by vegetation upstream along the horns or remobilized downstream.</figcaption></figure>

their influence on dune morphodynamics. Thus, it will be possible to evaluate out-of-equilibrium dynamics, which will be characterized by transient behaviors, hysteresis loops, and new time scales that should depend on steady-state properties analyzed in this study. At this stage, the model shows that taking account of vegetation using global parameters is sufficient to reproduce most of the dune types usually described in nature through the barchan-parabolic transition.

## 5.2. Vegetated Dune Morphodynamics

Parabolic dunes are typically classified according to their shape and total length after complete stabilization (Pye, 1982; Yan & Baas, 2015). Following, Courrech du Pont et al. (2024), we present here a complementary classification based on their dynamical behavior at steady state. Accordingly, parabolic dunes are classified as migrating dunes, which exhibit distinctive shapes and migration rates as the result of a morphodynamic equilibrium with the intensity of the vegetated processes.

The full range of steady-state dune patterns obtained by the numerical model through the barchan-parabolic transitions indicates that there are three main morphological transitions according to an increase in vegetation cover: (a) the stabilization of the horns, (b) the curvature reversal of the crestline and (c) the complete stabilization of the dunes. As shown in Figures 7 and 9, they are associated with abrupt changes in dune shape, which are accompanied by sudden variations in migration rate, transverse sediment fluxes, and the exchange of mass with the surrounding environment (i.e., input sediment flux). As a consequence, the vegetated dune shapes computed at steady state in the vicinity of these abrupt transitions are only obtained over narrow ranges of parameter values. In natural settings, where environmental conditions are likely to vary in both space and time, this may explain why transverse-parabolic or wide parabolic dunes (see Table 2) are rarely observed in comparison with barchan and parabolic dunes, which exhibit in the model smooth and continuous changes in morphology as the vegetation cover varies.

Altogether, the steady-state dune shapes allow to capture the dynamic interaction between topography, shear stress and transport through the barchan-parabolic transition. As vegetation processes intensify on barchans, the output sediment flux along the arms decreases significantly. Once vegetation has begun to stabilize the horns, the barchan-parabolic shape emerges spontaneously when the output sediment flux reverses its direction. At some point, vegetation slows down the lateral edges of the dunes, *Q*<sub>crest</sub>/*H*<sub>d</sub>, and there is an abrupt shift to parabolic shapes as the curvature of the crestline reverses. In this new configuration, as in the case of barchans, transverse fluxes perpendicular to the direction of the wind transfer sediment from, what would be in the absence of lateral
fluxes, the slowest to the fastest sections. Transverse sediment fluxes thus control the steady-state shape of isolated dunes through the barchan-parabolic transition.

This study also demonstrates that there is a minimum size for parabolic dunes, which is associated with a non-zero migration rate and a threshold in vegetation cover. The dependence of this minimum dune size to the intensity of the vegetation processes now needs to be quantitatively investigated in natural environments in which climatic variations and heterogeneous conditions give rise to a wide range of dune migration rates. Consequently, it could be employed as a proxy for the competition between transport and vegetation, allowing the models to be resized using a similar approach to that used for the characteristic wavelength $\lambda_{max}$, in the context of dune formation.

## Data Availability Statement

All study data are included in the article or can be reproduced from the source code, available at https://www.ipgp.fr/rescal/.

## Acknowledgments

We acknowledge financial support from the French National Research Agency (ANR-17-CE01-0014/SONO and ANR-23-CE56-0008/EOLE), the UnivEarthS LabEx program (ANR-10-LABX-0023), and the IdEx Université de Paris (ANR-18-IDEX-0001). Numerical computations were partly performed on the S-CAPAD/DANTE platform, IPGP, France. Satellite images in Figures 1 and 6 are courtesy of Google Earth.

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