Dune formation under bimodal winds

Abstract

The study of dune morphology represents a valuable tool in the investigation of planetary wind systems—the primary factor controlling the dune shape is the wind directionality. However, our understanding of dune formation is still limited to the simplest situation of unidirectional winds: There is no model that solves the equations of sand transport under the most common situation of seasonally varying wind directions. Here we present the calculation of sand transport under bimodal winds using a dune model that is extended to account for more than one wind direction. Our calculations show that dunes align longitudinally to the resultant wind trend if the angle θ_w between the wind directions is larger than 90°. Under high sand availability, linear seif dunes are obtained, the intriguing meandering shape of which is found to be controlled by the dune height and by the time the wind lasts at each one of the two wind directions. Unusual dune shapes including the “wedge dunes” observed on Mars appear within a wide spectrum of bimodal dune morphologies under low sand availability.

The majority of the world's sand dunes are formed by winds blowing from more than one wind direction: bimodal winds, which oscillate seasonally between two main directions, are typical of the largest sand deserts of our planet (1–7). Whereas the well-known barchans and transverse dunes appear under unimodal winds for low and high sand availability on the ground, respectively, seif dunes (Fig. 1) are the prevailing dune type under bimodal wind regimes. These dunes form in the absence of vegetation and align parallel to the resultant wind trend, developing a sharp crest which explains the term “seif” (“sword” in Arabic) and an intriguing meandering shape which has challenged geomorphologists for decades. In recent years, understanding the dynamics of seif dunes has become an issue of major interest also for planetary scientists: fields of seif dunes occur seldom on Mars (8, 9), which led to the hypothesis that Mars wind systems are essentially unidirectional. Is the appearance of seif dunes, indeed, related to the wind regime only? Moreover, are seif dunes the only dune type shaped by bimodal winds? The results of the present work provide insights into these challenging questions.

Fig. 1.

Bimodal winds lead to formation of seif dunes at Bir Lahfan, Sinai, Egypt (image courtesy of Google Earth), which elongate in the resultant transport direction. The sand rose (right corner at top) shows that the wind blows from two main directions. Each arm of the sand rose indicates the direction from which the wind blows, and the length of each arm is proportional to the potential rate of sand transport from the direction indicated by the arm (1). The arrow indicates the resultant sand transport trend.

It is still unknown whether other factors rather than wind directionality could also be relevant for dune alignment (10, 11) and which factors determine the meandering shape of seif dunes. Besides, very little is known about the dune morphology resulting from bimodal winds under conditions of low sand availability. In spite of many theoretical advances in the past, a model that solves both the three dimensional equations of aeolian sand transport and computes the evolution of dunes under bimodal wind regimes was lacking (12–17).

In the present work, we extend a well-developed and tested dune model (18, 19) to investigate the formation of dunes under bimodal wind regimes. This model, which combines a set of well-established analytical equations for the turbulent wind field over the topography (20) with a continuum model for saltation (18)—the motion of grains in ballistic trajectories producing a splash of new ejected particles after grain-bed collisions (21–23)—reproduces quantitatively the shape of barchans and transverse dunes (24–26) and has become a powerful tool in the investigation of the large time scale processes involved in the formation of aeolian landscapes. Here the dune model is extended in order to account for more than one wind direction. In order to achieve a directionally varying flow, the field is rotated, whereas the wind direction is kept constant. Our aim is to find, using dune simulations, the shape of seif dunes, the conditions for their development, and the dune types that appear in their place when sand availability is low.

This paper is organized as follows. In the next section we present the model equations and describe in detail how the simulations are performed. Next, the results of our calculations are presented and discussed. We investigate the formation of desert seif dunes by performing calculations on a dense sand bed exposed to a wind of oscillating direction. Finally we study the formation of dunes under bimodal wind regimes when the sand available for transport is scarce.

The Model

The dune model consists of a system of continuum equations in two space dimensions that combines a description of the average turbulent wind shear force above the dune with a continuum saltation model which allows for saturation transients in the sand flux. The model can be sketched as follows.

1. First, the wind over the surface is calculated with the model of Weng et al. (20), as detailed elsewhere (19). In the turbulent atmospheric boundary layer, where sand transport takes place, the wind speed increases logarithmically with the height z above the flat ground. The presence of a dune or hill, however, introduces a perturbation into the wind field. The Fourier-transformed components of this perturbation are (20)

   

   

   where x and y are parallel, respectively, perpendicular to the wind direction; $\sigma =\sqrt{\text{i}L{k}_x{z}_0/l}$, K ₀, and K ₁ are modified Bessel functions; and k _x and k _y are the components of the wave vector $\overrightarrow{k}$, i.e., the coordinates in Fourier space. ${\tilde{h}}_{\text{s}}$ is the Fourier transform of the height profile, U is the vertical velocity profile which is suitably non-dimensionalized, l is the depth of the inner layer of the flow, and L is a typical length scale of the hill or dune and is given by 1/4 the mean wavelength of the Fourier representation of the height profile. The aerodynamic roughness, z ₀, i.e., the apparent roughness of the surface in the presence of the saltating grains, is normally larger than the surface roughness of the immobile bed, which is of the order of 10–30 times the average grain diameter. The value z ₀ = 1.0 mm, which is consistent with measurements (21, 26, 27), is used in the calculations. The local shear stress at each position (x,y) of the field is obtained from the equation $\overrightarrow{\tau }=\left|{\overrightarrow{\tau }}_0\right|({\overrightarrow{\tau }}_0∕\left|{\overrightarrow{\tau }}_0\right|+\overrightarrow{\widehat{\tau }})$, where ${\overrightarrow{\tau }}_0$ is the undisturbed shear stress over a flat ground.
2. Next, the local sand flux is calculated for each position (x,y), using the shear velocity $u_{*}=\sqrt{\tau /{\rho }_{\text{fluid}}}$, where τ is the wind shear stress obtained in the calculation of step (i) and ρ_fluid = 1.225 kg / m³ is the air density. The sand bed represents an open system which can exchange grains with the moving saltation layer, for which the erosion rate—which is the difference between the vertical flux of ejected grains and the vertical flux of grains impacting onto the bed—at any position (x,y) represents a source term. As the number of grains launched into saltation increases, the wind transfers more momentum to accelerate the grains, thus leading to a decrease of the wind strength within the saltation layer (22). After a distance called saturation length, the wind strength is just sufficient to sustain saltation and the flux achieves a maximum value. By taking the interactions between the particles and the fluid and the transient of flux saturation into account, the following differential equation is obtained for the sand flux $\overrightarrow{q}\left(x,y\right)$ (18):

   

   where q _s = (2v _sα/g)ρ_fluid u _*t ²[(u _*/u _*t)² − 1] is the saturated flux and ℓ_s = (2v _s ²α/gγ)/[(u _*/u _*t)² − 1] is called saturation length, whereas α = 0.43 and γ = 0.2 are empirically determined model parameters, g is gravity and u _*t = 0.22 m/s is the minimal threshold shear velocity for saltation. The mean grain velocity at saturation, v _s, is calculated numerically from the balance between the forces on the saltating grains using the reduced wind velocity within the saltation layer (18, 26). It should be emphasized that Eq. 3, which gives the local height-integrated flux $\overrightarrow{q}\left(x,y\right)$ calculated from the local shear stress at the position x,y, has a general structure (the well-known logistic equation) which is independent of the precise form of q _s or ℓ_s. The divergence of the flux vanishes if $\left|\overrightarrow{q}\left(x,y\right)\right|=q_{\text{s}}$ (the saturated flux), whereas the transient of flux saturation due to the cascade process of saltation is encoded in the saturation length ℓ_s (18).
3. The local change in surface height h(x,y) is computed from mass conservation, by using the local flux $\overrightarrow{q}\left(x,y\right)$, for each position (x,y) of the field:

   

   where ρ_sand = 1650 kg/m³ is the bulk density of the sand.

4. If sand deposition leads to slopes that locally exceed the angle of repose, 34°, the unstable surface relaxes through avalanches in the direction of the steepest descent, and the separation streamlines are introduced at the dune lee (19). Each streamline is fitted by a third-order polynomial connecting the brink with the ground at the reattachment point (19), and defining the “separation bubble,” in which the wind and the flux are set to zero. The model is evaluated by performing steps 1–4 computationally in an iterative manner.

Calculations are performed using open boundaries, a constant sand influx q _in between 20% and 40% of the maximum flux q _s, which correponds to typical condition on interdune areas (28), and a wind of constant upwind shear velocity u _* = 0.4 m/s, which is value characteristic for real dune fields (1). We also studied different u _* > u _*t but found no qualitative difference in the result of crest alignment. To simulate the bimodal wind, we rotate the field by an angle θ_w, keeping the wind direction constant. The wind lasts in each direction for a time T _w; then the field is rotated to the other direction. In this manner, the separation bubble adapts to the wind direction after rotation of the field.

Results and Discussion

We investigate, first, the evolution of a flat sand bed submitted to a bimodal wind regime. We find that the resulting dune morphology depends crucially on the angle θ_w between the wind directions: Whereas transverse dunes are obtained if θ_w is accute, the resulting bedforms align longitudinally to the resultant wind direction if θ_w > 90°. Typical simulation outcomes obtained with θ_w = 40° and θ_w = 120° are shown in Figs. 2 A and B, respectively. The results of our simulations are consistent with experimental and field observations: Longitudinal seif dunes (Fig. 1) occur, in fact, in areas of bimodal wind regimes with θ_w > 90° (1, 2, 5, 6). Longitudinal ripples and subaqueous dunes also appear to develop when θ_w > 90° (10, 11). However, it was not possible to conclude, neither from experiments nor from numerical calculations using celular automata models (12, 14, 15), whether other factors besides the divergence angle could also compete to form longitudinal dunes. For instance, it has been suggested that the wind strength could play an important role for dune alignment (10). By using our model, we have tested a wide range of wind speed values u _* > u _*t, characteristic time T _w and sand influx q _in. We have also considered the situation where those quantities are not equal for both wind directions. In all cases, the conclusion is the same regardless of the particular choice of the parameters. Our numerical outcomes support the hypothesis that dunes align normal to the maximum gross transport, i.e., the transport perpendicular to the bedform trend, measured without regard to direction of transport (10).

Fig. 2.

Dunes emerging on an initially flat sand bed under bimodal wind regime. In all images, arrows indicate the wind directions. (A) Snapshot of simulation with u _* = 0.4 m/s, T _w ≈ 12 days and θ_w = 40° at time T = 150T _w. (B) Seif dunes obtained from the simulation with same parameters as in A but with θ_w = 120°, at T = 300T _w. (C) seif dunes of height ≈20 m obtained using different values of T _w with θ_w = 120° and T = 300T _w. Results are expressed in terms of t _w = T _w/T _m where T _m ≈ 10⁸s is the migration (or reconstitution) time of a transverse dune with the height of the seif dunes shown in the images; dune fields in A–C have length of 1 km. (D) Top view of a seif dune with height of ≈25 m within a dune field obtained with θ_w = 120° and t _w ≈ 10⁻². Small transverse dunes, with height of a few meters, inciding obliquely onto the seif dune in the center can be seen. The image has dimensions 1700 m × 400 m. (E and F) Crest position y _crest (E) and crest height h _crest (F) of the seif dune shown in (d) as function of the position in the resultant wind direction, x (to the right in the image).

The dune shape depends strongly on the characteristic time T _w of the bimodal wind, as shown in Fig. 2 C. When T _w is low, straight longitudinal dunes, which practically do not meander, are obtained. As T _w increases, sinuous seif dunes, as those observed in real fields, develop. Furthermore, when T _w becomes too large, the longitudinal dunes disappear giving place to transverse dunes oriented perpendicularly to the wind components, as can be seen from Fig. 2 C. It is interesting to notice that Bagnold (21) already noted different seif shapes according to the prevailing long-period wind regime, still the author was not able to establish any dependence on the wind regime in a quantitative manner. We can understand the results of our calculations by recalling the concept of relaxation or reconstitution time (11, 29, 30)—the time required to reshape a dune after the change in wind direction. Under given wind strength u _*, a transverse dune of width L has migration velocity v _d and reconstitution time T _m ≡ L/v _d. We define the rescaled time of the bimodal wind,

If T _w is of the order of or larger than T _m (t _w ≥ 1.0), then crescent dunes oriented transversely to the wind components appear, whereas longitudinal alignment is obtained for t _w < 1.0. The value of T _m in fact means the reconstitution time L/v _d of a transverse dune which has the height H of the dunes in the field considered, whereas both L ∼ H and v _d ∼ 1/H are well-known functions of H (21, 31). Therefore, it is possible to understand why small barchans may be found in fields of large bimodal sand dunes. The time a dune needs to reach its equilibrium form after a change in wind direction in fact increases with the square of the dune height (31). Small bedforms (the so-called surface waves) oriented transversely to the bimodal wind components develop on the interdune areas inciding obliquely onto the flanks of the seif dunes (Fig. 2 D). Such small bedforms are ubiquitous in most real seif dune fields (2, 6). The small bedforms, which are practically non-existent if T _w is too small, become increasingly large the higher T _w.

The well-known meandering of seif dunes developing on sand beds arises due to the alternate action of obliquely inciding winds onto both flanks of the dune. Both wind directions compete to shape crescent dunes of oscillating orientation along the dune crest. When t _w is low (≈ 10⁻⁴) the crescent bedforms are, indeed, too incipient for the sinuosity to develop and the seif dunes look straight. As t _w increases, the crescent dunes develop further, and the surface along the seif dunes adapts alternately to both wind directions yielding the seif dunes the characteristic meandering. Seif dunes as those shown in Figs. 2 B, C Center, and D are found when t _w is within the range 10⁻³ < t _w < 10⁻¹. Indeed, we could not find a significant dependence of the average wavelength of the meandering on the bimodal wind timing for t _w within the aforementioned range. Taking into account the morphologic dimensions of crescent dunes (32, 33), we expect the typical wavelength of the meandering to be approximately one order of magnitude larger than the dune height. For instance, dunes of ≈20 m height in fact meander with a wavelength of 100 – 200 m typically (Fig. 2 E), which is in agreement with the situation in real dune fields (2). The best fit to our data using the empirical expression S = kt _w ^a H ^b leads to k ≈ 9, a ≈ −0.005, b ≈ 1.0.† In this manner, whereas the characteristic time of the bimodal wind is found to control the development of the seif meandering, the average wavelength of the dune sinuosity within the range 10⁻³ < t _w < 10⁻¹ appears to be determined by the dune height. As a result of the meandering, peaks and saddles modulate the dune height along the crest line (Fig. 2 F), as observed in the field (2, 6). Finally, as t _w approaches unity, dunes orient transversely to each wind component. Our conclusions are supported by simulation results obtained with different values of θ_w > 90°.

The calculations do not only give insight into the dune meandering but they also enable a quantitative estimate of the bimodal wind timing T _w in a field where seif dunes display a given shape, on the basis of Fig. 2. Once T _m is known, T _w can be estimated from Eq. 5, taking the value of t _w that corresponds to the dune shape according to Fig. 2 C. Indeed, it is important to notice that other factors, as the secondary flow along the crest line of seif dunes, also play an important role for the shape of seif dunes (2, 5). Thus, the model should be improved in order to account for secondary flow effects and interdune transport in order to improve the quantitative assessment of seif dunes. Nevertheless, the conclusion that the dune meandering is controlled by T _w should be robust with respect to secondary flow effects. Further, our results provide evidence that seif dunes may appear without regard of secondary flow and that the critical angle for dune alignment is independent of the nature of the flow at the dune lee.

Exotic Dunes When the Sand Is Scarce.

Whereas seif dunes form when the sand amount is high, which dune forms appear when the wind is bimodal but sand availability is low? We perform calculations starting with a sand hill of Gaussian shape, in the same manner as done to calculate barchans; however, now we use a bimodal wind. Again, the bimodal dune shape varies according to the dune volume, because the larger the dune the longer time is needed for a given change to be achieved under the same wind strength. We define T _m ≡ L/v _d the migration time of a barchan dune which has length L and migration velocity v _d and is obtained starting with the Gaussian hill using an unidirectional wind. The different dune shapes obtained in the calculations are shown in Fig. 3 Lower as function of θ_w and t _w (Eq. 5). We see that when t _w ≥ 1.0, a barchan is obtained. Furthermore, a variety of dune shapes appear in a bimodal wind regime in which t _w < 1.0. In a general fashion, the dunes obtained are oriented transversely to the resultant wind direction if θ_w is smaller than 90°, while longitudinal bedforms appear for higher angles.

Fig. 3.

Bimodal sand dunes on bedrock. (Upper) Images of dunes on Mars, near (A) 71.7°N, 51.3°W (“occluded” barchan dunes) (B) 48.6°S, 25.5°W (“wedge” dunes) (C) 49.6°S, 352.9°W, and (D) 48.7°S, 167.4°W (“drop” dunes). (Image courtesy of NASA/JPL/MSSS). (Lower) Bimodal sand dunes obtained with different values of θ_w and t _w = T _w/T _m, where T _m is the migration time of the barchan dune. Arrows indicate the wind directions. Dunes indicated by letters compare with the dunes in the images on top. Simulation snapshots taken at time T of the order of (1 – 10) T _m. The shapes in the diagram are independent of the dune scale, which is included in T _m.

Some of the bimodal dune shapes obtained in the calculations resemble exotic Mars dune shapes as those seen in the images of Fig. 3 Upper. Such dunes appear on bedrock where barchans should be the only dune type observed if the wind direction was constant. While the Martian north polar “occluded” barchans (Fig. 3 A) are obtained with θ_w between 40° and 80°, for θ_w ≈ 100° we find the intriguing “wedge” shaped dunes (Fig. 3 B). For t _w of the order of 10⁻⁴, dome dunes, i.e., dunes without slip faces or horns, are obtained when θ_w < 90°, while the “drop dunes” (Fig. 3 D) appear when θ_w = 120°. Our calculations reveal that these dune shapes, which have been classified as “transitional” (34), in reality do not change in time. In contrast, the seif dunes (Fig. 3 C′) keep elongating in time. At very high angles (i.e., θ_w close to 160° or 180°) reversing dunes are obtained, which resemble linear dunes that don't meander (35).

The occluded barchan morphology shown in the diagram of Fig. 3 appears as consequence of lateral diffusion at the windward foot of the barchan: The bimodal wind, inciding obliquely from alternating directions and forming an accute divergence angle, dislodges sand from the upwind foot of the barchan to the dune sides, yielding the rounded barchan shape. We see that the slip face of the occluded barchan gets smaller the larger θ_w, which is consequence of the transport of sand from both sides of the dune into the dune lee. For θ_w > 90°, an incipient tail develops at the lee side leading to the wedge dune morphology. In fact, when θ_w is within the range 90° < θ_w < 110°, a mixed state is found where both transverse and longitudinal alignments are visible. Such a mixed state has been reported in experiments with ripples on a sand bed (10, 11), whereas the corresponding dune morphologies on the bedrock are presented in Fig. 3. For increasing θ_w, the downwind tail of the wedge dune elongates, leading to a longitudinal dune with slip face at both flanks. However, since the dune is evolving on the bedrock and there is no mobile sand around, the surface along the longitudinal axis of the dune cannot meander. This result is consistent with the conclusions from our simulations of seif dunes on sand bed (Fig. 2), i.e., seif dunes meander as a result of the oblique incidence of sand onto both dune flanks. Furthermore, as can be seen in Fig. 3, no slip face appears if t _w is too small, i.e. of the order 10⁻⁴ or smaller, since, for such a short timing of the bimodal wind, the excess of sand deposited from one direction onto the dune crest is always removed by the wind blowing from the other direction before the surface slope reaches the critical value for avalanches.

In conclusion, by solving the three-dimensional equations of sand transport under bimodal wind regimes, we find that the condition for longitudinal alignment of aeolian dunes, θ_w > 90°, is independent of sand availability, sand properties, or atmospheric conditions. Indeed, we have verified the nondimensional phase diagram of Fig. 3 by performing simulations under atmospheric conditions of both Earth and Mars. Our calculations revealed that the wedge and drop dunes observed on Mars (36, 37) in fact belong to a broad class of dune morphologies that appear under bimodal wind when the sand availability on the ground is low. In fact, one of these dune shapes (the wedge) could be reproduced in flume experiments by Hersen (38). It would be interesting to perform systematic experiments with bimodal flow as the turntable or flume ones in order to confirm the findings of Fig. 3. Furthermore, the development of linear dunes from asymmetric barchans, which get one of their horns progressively elongated due to changes in the wind direction (4, 39), poses one challenging problem that should be addressed in the future with the help of the dune model. The model should be further improved in order to account for more complex wind regimes and for secondary flow patterns at the dune lee. Moreover, the model can be extended in order to study the formation of accumulating (star) dunes under multidirectional winds.

Application to Planetary Dunes and Wind Systems.

The phase-diagram of Fig. 3 contains useful informations with potential applications to the investigation of planetary wind systems. In fact, each bimodal dune morphology is unambiguously associated with one single set (θ_w, t _w), whereas t _w = T _w/T _m and T _m encodes the informations on the wind strength u _*, the volume V of the dune, grain size and density as well as atmospheric viscosity and density of the physical environment considered. Indeed, in order to obtain the characteristic time T _w of the bimodal wind, the migration time T _m = [L/v _d] of a barchan with volume V shaped under those conditions of wind and atmosphere must be known. However, since for a barchan of volume V, L ∝ V ^1/3 and v _d ∝ q _s/[ρ_sand V ^1/3] (where q _s is the saturated flux and ρ_sand is the bulk density of the sand) (21, 31), T _m scales in fact with [V ^2/3ρ_sand]/q _s. Thus, the migration time T _m of a barchan dune of volume V in an arbitrary physical environment can be obtained from the equation T _m = T _m ^E·[q _s ^Eρ_sand]/[q _sρ_sand ^E], where the upperscript E refers to the corresponding quantity for a terrestrial barchan dune of same volume, in such a manner that T _w can be obtained from the equation

The timing of the planetary bimodal wind, T _w, may be obtained from the diagram of Fig. 3 and from Eq. 6 as follows. (i) First, calculate the migration time, T _m ^E, of a terrestrial barchan which has the volume, V, of the planetary dune considered and occurs in a terrestrial field where the saturated flux is q _s ^E. (ii) Next, the saturated flux q _s at the physical environment considered must be calculated, and thus we need to know the average shear velocity u _* of the sand-moving winds at the planetary dune field—for instance, Martian sand-moving winds have u _* values between 2.5 m / s and 3.5 m/s (40–43). (iii) Finally, the value of t _w, obtained from the diagram in fig. 3, is substituted into eq. (6).

The value of T _w considered in the calculations of the present work must be interpreted as an effective time during which the wind strength has the constant value u _* above the threshold for saltation, u _*t. Thus, T _w is normally smaller than the real time of the wind changes, since u _* is commonly most of the time below u _*t. In fact, both T _m and T _m ^E refer to the situation in which the wind blows steadily with the average shear velocity, u _*, of the sand-moving winds. Indeed, the actual migration time of a barchan accounts for the frequency with which sand-moving winds occur, i.e., ${\tilde{T}}_{\text{m}}^E=T_{\text{m}}^E/f^E$ and ${\tilde{T}}_{\text{m}}=T_{\text{m}}/f$, where f ^E and f are the fraction of time the wind is above the threshold u _*t on Earth and in the planetary environment considered, respectively. Therefore, if, for example, ${\tilde{T}}_{\text{m}}^E$ is substituted into Eq. 6 rather than T _m ^E, then the right-hand-side of Eq. 6 must be multiplied by f ^E. Whereas the terrestrial f ^E ≈ 10% typically (1, 44), the value of f at a given location on Mars, for example, can be estimated from missions' data on the average frequency and duration of the strongest wind gusts, during which u _* > u _*t at that location (40, 42). Furthermore, if the right-hand-side of Eq. 6 is multiplied by f ⁻¹, then the value of T _w obtained from Eq. 6 means the actual timing of the local bimodal wind system.

On Mars, sand induration by ice or mineral salts could also compete to form straight linear dunes, rounded barchans and domes (45), and thus both mechanisms (sand induration and bimodal wind regimes) probably act simultaneously on the red planet. Indeed, here we have shown that many exotic Martian dune forms might result from bimodal wind regimes alone without necessity of sand induration. The low sand availability in many Martian sand systems yields an explanation for the common appearance of dune shapes as those in Fig. 3 on Mars: if more sand were available, then seif dunes—rather much more typical on Earth—would form in their place.