Unravelling raked linear dunes to explain the coexistence of bedforms in complex dunefields

Abstract

Raked linear dunes keep a constant orientation for considerable distances with a marked asymmetry between a periodic pattern of semi-crescentic structures on one side and a continuous slope on the other. Here we show that this shape is associated with a steady-state dune type arising from the coexistence of two dune growth mechanisms. Primary ridges elongate in the direction of the resultant sand flux. Semi-crescentic structures result from the development of superimposed dunes growing perpendicularly to the maximum gross bedform-normal transport. In the particular case of raked linear dunes, these two mechanisms produces primary and secondary ridges with similar height but with different orientations, which are oblique to each other. The raked pattern develops preferentially on the leeward side of the primary ridges according to the direction of propagation of the superimposed bedforms. As shown by numerical modelling, raked linear dunes occur where both these oblique orientations and dynamics are met.

ARTICLE

Received 16 May 2016 | Accepted 12 Dec 2016 | Published 27 Jan 2017

DOI: 10.1038/ncomms14239 OPEN

Unravelling raked linear dunes to explain the coexistence of bedforms in complex duneelds

Ping L1,2, Clment Narteau3, Zhibao Dong1,2, Olivier Rozier3 & Sylvain Courrech du Pont4

Raked linear dunes keep a constant orientation for considerable distances with a marked asymmetry between a periodic pattern of semi-crescentic structures on one side and a continuous slope on the other. Here we show that this shape is associated with a steady-state dune type arising from the coexistence of two dune growth mechanisms. Primary ridges elongate in the direction of the resultant sand ux. Semi-crescentic structures result from the development of superimposed dunes growing perpendicularly to the maximum gross bed-form-normal transport. In the particular case of raked linear dunes, these two mechanisms produces primary and secondary ridges with similar height but with different orientations, which are oblique to each other. The raked pattern develops preferentially on the leeward side of the primary ridges according to the direction of propagation of the superimposed bed-forms. As shown by numerical modelling, raked linear dunes occur where both these oblique orientations and dynamics are met.

1 Department of Geography, Shaanxi Normal University, 620 Changan West Avenue, Xian, Shaanxsi 710119, China. 2 Key Laboratory of Desert and Desertication, Northwest Institute of Eco-Environment and Resources, Chinese Academy of Sciences, 320 West Donggang Road, Lanzhou, Gansu 730000, China. 3 Institut de Physique du Globe de Paris, Sorbonne Paris Cit, Universit Paris Diderot, UMR 7154 CNRS, 1 rue Jussieu, 75238 Paris Cedex 05, France.

4 Laboratoire Matire et Systmes Complexes, Sorbonne Paris Cit, Universit Paris Diderot, UMR 7057 CNRS, Btiment Condorcet, 10 rue Alice Domon et Lonie Duquet, 75205 Paris Cedex 13, France. Correspondence and requests for materials should be addressed to P.L. (email: mailto:[email protected]

Web End [email protected] ) or to C.N. (email: mailto:[email protected]

Web End [email protected] ).

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14239

Raked linear dunes display asymmetric morphologies with long primary ridges breaking down only on one side into periodic secondary ridges with a perpendicular alignment

to the main crest and a semi-crescentic shape. This unique pattern provides an intriguing case study to begin to examine how primary and secondary bedforms with similar heights but different trends can coexist in the same wind regime.

A major question in arid zone geomorphology is to determine to what level active duneelds are in dynamic equilibrium with the current wind regime or the result of longer-term processes like climate change1. This is a challenging task given the broad spectrum of observed dune shapes, which are primarily controlled by sand availability and wind directional variability2,3. If crescentic barchans (low sand availability, unidirectional wind) and star dunes (high sand availability, multidirectional winds) occur under two extreme and opposite sets of conditions, linear dunes are often sub-divided into different categories according to their shape4,5 (simple, compound and complex) or orientation with respect to the resultant sand ux direction6,7 (transverse, oblique and longitudinal). Unfortunately, all these classications fail to take account of dynamics and, without independent chronological data810, it is difcult to interpret or reconstruct the duneeld history.

Rather than a single growth mechanism, it was recently shown3,11 that dunes can either extend in the direction of the resultant sand ux in zones of low-sediment supply12 (that is, the ngering mode) or grow perpendicularly to the maximum gross bedform-normal transport13 in conditions of high-sediment supply (that is, the bed instability mode). These two distinct dune growth mechanisms determine both dune shape and orientation. Taking into account the feedback of dune aspect-ratio on the magnitude of the ow14, these orientations can be analytically derived from the wind regime to be compared with observations in the eld. Most importantly for our present purpose, the coexistence of the two dune growth mechanisms can theoretically lead to the development of different types of steady-state bedforms through spatial changes in sand availability or the emergence of secondary structures15. Given the apparent complexity of most duneelds on Earth, none of these interacting steady-state dune patterns have been identied and quantied so far in modern sand seas. It is an important issue because the coexistence of different bedform alignments has also been associated with changes in wind regime16.

Raked linear dunes have been recognized for the rst time in the Kumtagh desert17 (Xinjiang province, China), where they cover an area of more than 2.4 104 km2. This remote and

arid region of China has been the subject of intensive investigation in recent years1822. These studies provide unique sets of data, but none of them have identied the mechanism for dune formation and the subsequent dune morphodynamics. Instead, most of them have concentrated on zibars and other grain-size segregation patterns23, which are obvious in interdune areas.

Considering the traditional classication, raked linear dunes seem to be a specic case of compound/complex linear dunes. Their most distinguishable feature is a lateral asymmetry in the distribution of secondary ridges. At the duneeld scale, primary ridges display all the characteristics of linear dunes. They are relatively straight, extending over distances of kilometres parallel to one another with a low ratio of dune to interdune areas24. On the top of them, secondary ridges almost perpendicular to the main dune alignment develop only on one side of the dune. These secondary ridges have approximately the same height as the primary ridges and they resemble half crescent, so that the entire pattern is similar to a train of barchans connected to each other along the same arm.

Here we study the morphodynamics of raked linear dunes in the Kumtagh desert from two satellite images taken at a time interval of 8 years. Dune shapes, orientations and dynamics (elongation and migration) are compared with theoretical predictions derived from local wind data to show that raked linear dunes are a steady-state dune type in dynamic equilibrium with the modern wind regime. Then, we numerically and analytically investigate the conditions for the emergence of this dune type to demonstrate that the raked pattern result from the coexistence of two dune growth mechanisms.

ResultsMorphodynamics of raked linear dunes. Figure 1a,b shows aerial views of raked linear dunes in the Kumtagh desert (see also Supplementary Figs 13). At different sites in interdune areas of the Kumtagh desert, two wind towers of 2 m height have recorded the mean wind speed and direction every 15 min during 2 years from 2007 to 2009. Figure 1c shows the wind roses and the distributions of sand ux direction. This eastern end of the Tarim basin is exposed to a trimodal wind regime with a widely spread dominant wind direction from the north and two secondary peaks associated with easterly and westerly winds. Dune orientations predicted from these wind data are a northsouth alignment for bedforms growing in the bed instability mode and an extension in the southwesterly direction for dunes growing in the ngering mode (Fig. 1d and Table 1 in Methods section). These orientations are in general agreement with the alignments of the primary and secondary ridges observed in satellite images (Fig. 1e). These ndings suggest that raked linear dunes can form by elongation and that secondary ridges can result from the development of superimposed bedforms growing perpendicularly to the maximum gross bedform-normal transport. In the framework of the two dune growth mechanisms, the two dune orientations form an oblique angle Da 703, which leads to an angle DaQ 52

between the resultant sand ux at their crests because of the speed-up effect in multidirectional wind regimes15 (see Methods section). Then, as the superimposed bedforms nucleate and grow on the primary ridges, they migrate from one side of the linear dunes to the other. According to this migration, the regular pattern of secondary ridges is observed on the leeward slope of the primary ridges (that is, the northwestern side of the linear dunes in Fig. 1). Hence, the complex asymmetric pattern of raked linear dunes may naturally arise from the coexistence of two dune growth mechanisms with both oblique orientations and dynamics.

Independently of the winds, such a scenario can be quantitatively investigated from dune morphodynamics using eld data and satellite images from the Kumtagh desert (Fig. 2a,b). Figure 2c shows that the wavelength of the semi-crescentic secondary bedforms is linearly related to the height of the linear dunes. This relationship is similar to that observed for longitudinal dunes breaking in barchans under unidirectional wind regime2527, suggesting that the same mechanism is responsible for the singular asymmetry of raked linear dunes. In addition, the widths WB of the semi-crescentic structures exhibit a normal distribution with a mean value around 37 m and a s.d. close to 3 m across the entire duneeld (Fig. 2d). The linear dunes reach a homogeneous width WF with a mean value of 26 m (Fig. 2e) away from the growing tips, which may change shape over time (see Methods section). Using a pair of Google Earth images acquired in 2008 and 2014, we estimate the elongation rate e 13.23 m per year at the tip of linear dunes and the

migration rate c 61 m per year of the secondary ridges along

the same direction (Fig. 2f). Despite more variation in the elongation rate distribution, which can be explained by the loss of sediment at the dune tips during periods of constant wind

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a

b

1 km

North

Windvelocity (i.e., m s1)

c d e

2 4 6 8 10 12 >12

200 m

0.025

Probability dist. funct.

North

0.02

0.015

0.01

0.005

2006

0

Wind tower

Wind tower

W1

Probability dist. funct.

0.025

200 m

0.02

0.015

North

0.01

0.005

South West

0 North East 2014

W2

2014

Sand ux direction

Figure 1 | Raked linear dunes in the Kumtagh desert. (a,b) Satellite and aerial views of raked linear dunes (401300 N, 921300 E). (c) Distributions of sand ux direction derived from the two wind towers W1 (9234037.6800 E 4018032.2100 N) and W2 (92333.8600 E 4015016.4300 N) located 6 km apart in interdune areas from 2007 to 2009 (blue dots). Red curves are the best ts using a three-component Gaussian mixture model (see Methods section).

(d) Predicted orientations for dunes growing by extension (ngering mode, black arrows) and perpendicularly to the maximum gross bedform-normal transport (bed instability mode, red arrows). Green arrows show the predicted resultant sand ux at the crest of dunes growing in the bed instability mode. By denition, this is also the propagation direction of these dunes. (e) Comparison between predicted and observed dune orientation at two different times.

QB at the crests of primary and secondary ridges can be directly derived from the equation of conservation of mass over the entire cycle of wind reorientation. Considering that the dunes elongate or propagate without changing shape, the relations are e QF/HF

and c/cos(DaQ) QB/HB, where cos(DaQ) is a correction to

account for the migration direction of the superimposed bed-forms in the bed instability mode and where HF and HB are the heights at the crests of primary and secondary bedforms, respectively. These heights HF and HB are derived from the measurements of the width WF of linear dunes (Fig. 2b,f) and from the width WB of half-crescentic structures (Fig. 2b,e) using the relations HF (tan(yF) WF)/2 and HB tan(yB) WB. Substituting

in the ux equations above gives QF (e tan(yF) WF)/2

and QI (c tan(Yb) WB)/cos(DaQ). Here we take yF 15

and yB 11, which are the usual reported values for the trans-

verse aspect-ratio of linear12 and barchan28 dunes. Figure 3 shows the comparison between the sand uxes computed from wind data and dune morphodynamics. The close agreement between these two independent estimates indicate that a hierarchy of bedforms can be used to provide independent quantitative constraints on sediment transport.

Conditions for steady-state raked linear dunes. Together, sand ux estimates (intensity and direction) and comparisons between predicted and observed dune orientations suggest that the raked linear dunes in the Kumtagh desert have reached a steady state,

Table 1 | Sand uxes and dune properties derived from wind data in the Kumtagh desert.

Variable Units Wind tower W1 Wind tower W2 DP m2 per year 47.08 49.02

RDP m2 per year 25.08 23.25

QI

!

m2 per year 57.95 53.98

!

QF

m2 per year 42.80 39.53

RDD 3 230.8 220.1 aI 3 122.1 108.7 aF 3 230.0 220.9 Da 3 72.1 67.8 DaQ 3 3.9 6.5

RDP/DP | 0.53 0.47 sF/sI | 0.55 0.53

Figure 1c shows the wind and sand ux roses for the wind towers W and W . See Methods section and equations (210) for the description of all the variables. All angles are measured counterclockwise from East. Following Gao et al.11, the wind speed-up have been computed with g 1.6. Dune orientations show little variation (o1) with respect to the g-value.

orientation, Fig. 2g shows that linear dunes elongate more than two times faster than the secondary ridges migrate.

Sand uxes on raked linear dunes. Using the morphometric parameters of raked linear dunes, the resultant sand uxes QF and

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a b

[afii9838]

H

WB WF

c d e

0.3

0.3

100

0.2

0.2

[afii9838] (m)

80

PDF

PDF

0.1

60

0.1

40 3 4 5 6 7

0 30 35 40 45

0 15 25 35

H (m)

WB (m)

WF (m)

f

g

11-Nov-2006

08-Jan-2014

0.4

PDF

0.2

0 0 5 10 15 20 25

Elongation and migration rates (m per year)

Figure 2 | Morphodynamics of raked linear dunes. (a,b) Characteristic length scales in raked linear duneelds. (c) Relationship between the mean wavelength l and the mean height H of secondary ridges17. The line is the best linear t with a y-intercept of 485 m, a value larger but on the order of the most unstable wavelength for the formation of dunes. Errorbars are one s.d.s in wavelength (blue) and height (magenta) measured from 50 secondary ridges. (d,e) Distributions of the widths WF and WB of primary and secondary ridges, respectively. (f) Evolution of dune shape from 2006 to 2014.

(g) Distribution of elongation and migration rates of primary and secondary ridges over the same time period. Dashed lines in d,e,g are the best ts to the data using normal distributions.

which is fully consistent with the recorded wind regime. Nevertheless, such a regular pattern has never been reported before in numerical and laboratory experiments investigating dune shape11,25,2932. The main reason is that all these studies have only concentrated on unidirectional and bidirectional wind regimes that seem to never satisfy the conditions for the development and stability of raked linear dunes (see Methods section). From our eld-based results and systematic exploration of dune shape in bidirectional wind regimes11, we infer that raked linear dunes can only be observed in zones of low sand

availability if three conditions are met (see Methods section and equations (210) for the derivation of all variables from wind data). First, the ratio sF/sI between the dune-height growth rates of the ngering and the bed instability modes has to be large enough to promote elongation. Second, the angle Da between the two dune orientations should be oblique, wide enough for the two modes to be distinguished and yet narrow enough to avoid segmentation of the primary ridges (especially for decreasing sF/sI-value). Finally, the angle DaQ between the migration direction of the superimposed bedforms and the elongation

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a

90

2 per year)

Q F(m

60

Q I(m

b

2 per year)

30

0

90

60

30

0 0 10 20 30

Dune number

40 50

Figure 3 | Predicted sand uxes using two independent dune growth mechanisms. (a) Resultant sand ux QI at the crest of dunes growing in the bed instability mode. (b) Resultant sand ux QF along the crest of elongating dunes. Solid (wind tower W1) and dashed (wind tower W2) lines are the sand ux predicted from the wind data shown in Fig. 1c. Dots are the sand ux derived from the morphodynamics of individual dunes between 2006 and 2014. The coloured areas show the 90% condence intervals centred on the mean sand ux values hQIi 48.3 m2s 1 and hQFi 34.0 m2s 1.

direction should be wide enough to preserve the continuity of primary ridges and yet narrow enough for the secondary ridges and the asymmetric pattern to fully develop.

To test these hypotheses, we use a cellular automaton dune model that accounts for feedback mechanisms between the ow and the evolving bed topography11,12,15,3335. Using a tridirectional wind regime similar to that observed in the eld, simulations are run for sF/sI 0.75, Da 42 and DaQ 9.3.

After a few cycles of wind reorientation a linear dune rapidly extends on the non-erodible bed away from the xed source of sediment (Fig. 4). As it continues to elongate, superimposed bedforms develop and grow preferentially on the leeward side of the linear dunes according to their direction of propagation. Then, a systematic asymmetry starts to develop despite the constant loss of sediment at the tips of the linear dune and semi-crescentic structures. Over longer times, the raked pattern extends across the entire domain, keeping a constant shape characterized by regularly spaced secondary ridges of constant height and width propagating only on one side of a well-established linear dune. The tip region left appart, the superimposed dunes do not escape the nger dunes, but migrate along it. It is worth noting that the raked linear duneeld in the Kumtagh desert does not exhibit isolated barchan either. As in the numerical simulations, this suggest that the mass balance between the primary and the secondary ridges has reached an equilibrium.

DiscussionNumerous observations of differing alignments between main and superimposed bedforms have previously been observed in both modern aeolian and subaqueous bedforms as well as in their ancient deposits. For example, Rubin and Hunter36 and Rubin37,38 present examples of deposit records where the two sets of bedforms coexist under (inferred) steady-state conditions. These previous studies attributed differences in orientations not to changes in ow direction through time but rather to differences between the ow generating the main bedforms (external to the dune eld) and the ow producing the superimposed dunes (ow within the internal boundary layer). Here we nd that, in multidirectional wind regimes, there is no

need for secondary ow or deection of lee-side ow to give rise to the coexistence of bedforms with similar heights and different trends. There is no such ingredient in our numerical model. Instead, considering specic boundary conditions and sediment properties23, all the independent variables required to produce the different types of dunes are incorporated into the function of ow directionality and intensity. This approach is especially justied when it comes to study primary and secondary ridges with the same height.

Our results show that raked linear dunes can be considered as a new class of composite bedforms, just as star dunes15. According to the usual classications, raked linear dunes can be considered as linear dunes asymmetrically indented by a train of superimposed oblique dunes. Such a hierarchy of bedforms is possible because primary and secondary ridges are associated with different dune growth mechanisms with specic orientation and dynamics. The main dunes are in supply-limited conditions and, as a result they elongate in the direction of the resultant sand ux (ngering mode). The superimposed dunes arise from the instability of a sand bed39 because the supply of sand from the main longitudinal dunes is essentially unlimited. Nevertheless, the simultaneous manifestation of these two modes of dune orientation occur only in limited angular ranges between crest alignments and their propagation or extension directions. As shown by numerical and analytical modelling, a tridirectional wind regime similar to that observed in the eld is needed to full these conditions. Unidirectional or bidirectional wind regimes cannot11 (see Methods section), which is consistent with the rareness of this dune type.

From the ratio between the elongation rate (E13 m per year) and the length of the raked linear duneeld (E100 km), the minimum time for the formation of the Kumtagh desert dunes under the present wind conditions is at least 5,500 years. Such a time scale and the principal wind directions are in good agreement with the presence and the alignment of yardangs at the northeastern end of the duneeld (Supplementary Fig. 4), indicating that the current wind regime may have prevailed since the Holocene thermal maximum.

More generally, this study illustrates how the combination of two dune growth mechanisms can generate new types of steady-state bedforms in dynamic equilibrium with a given multi-directional wind regime. Complex dune shapes with multiple crest orientations may develop from the coexistence of bedforms with the same15,40 or different3 growth mechanisms. Considering the possible number of combinations, it is impossible at this stage to generalize the dynamical behaviour observed in the Kumtagh to other environments where raked patterns have been observed. For example, there are similar dune shapes in Mali (Fig. 5a), the Namib Sand Sea41 (Fig. 5b) and Saudi Arabia42 (Fig. 5c). Nevertheless, these dunes are one order of magnitude larger than those in the Kumtagh desert and they develop in the middle of sand ow paths from other dune types over long time scales (4100 years) for which there is no wind data. As a consequence, their origin and stability need to be investigated further before any conclusion can be drawn. Indeed, they may also reect transitions in dune type across space or over time25,26. Another interesting systematic feature to be explored is the raked dune patterns on Titan43, the largest moon of Saturn, where trimodal wind regimes have been proposed44,45 to explain the eastward elongation of linear dune at a global scale.

Interacting bedforms have been a challenge for decades to address major issues on the dynamics of duneelds, their origin and evolution in presence of various environmental variables4649. The methodology based on the two dune growth mechanisms applies generally to dune landscapes in which different trends coexist. Here we show how it can be used to

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a

t/t0 = 1.6 104

100 l0

Flux rose

b

Predicted orientations

t/t0 = 3.1 104

c

t/t0 = 4.0 104

d

t/t0 = 4.5 104

e

t/t0 = 7.1 104

Figure 4 | Formation of steady-state raked linear dunes from a local sand source. (ad) Elongating raked linear dune and development of secondary bedforms. (e) A snapshot of the steady-state raked linear dune. Insets show the sand ux roses and the predicted orientations of dunes growing by extension (ngering mode, black arrow) and perpendicularly to the maximum gross bedform-normal transport (bed instability mode, red arrows). The green arrow shows the resultant sand ux at the crest of dune growing in the bed instability mode. By denition, this is also the migration direction of these dunes. The asymmetric dune pattern results from the obliquity between the two dune orientations and the oblique propagation of secondary bedforms.

analyse sediment transport and local climatic conditions. Although raked linear dunes remain small-scale features (E10 m height) compared with giant dunes50 (E100 m height), the current state of any duneeld is likely to have been reached through the coexistence of different types of bedforms, which should rst be considered with respect to the two growth mechanisms. Independently, many studies have concentrated on the impact of climate change on the observed dune morphologies16. It is now time to combine these knowledge to deepen the understanding of modern sand seas on Earth and other planetary bodies.

Methods

Saturated ux on a at sand bed. Using the wind data collected in the Kumtagh desert (Fig. 1c), Table 1 shows the predicted sand ux on a at sand bed and a set of variables relevant for dune morphodynamics: orientation, sand ux at the crest, migration direction and dune-height growth rate. Continuous wind measurements in the eld provide the wind speed ui and direction ~xi at different times ti, iA[1; N].

For each time step i, we calculate the shear velocity

ui

uik

log z=z0

r : 2

Using the gravitational acceleration g 9.81 m s 2, the grain to uid density ratio

rs=rf 2:05 103 and the grain diameter d 180 mm, we nd uc 0.19 m s 1,

which corresponds to a threshold wind speed of 3.6 m s 1 two metres above the ground. For each time step i, the saturated sand ux Qi

! on a at sand bed can be calculated from the relationship proposed by Ungar and Haff52 and calibrated by Durn53

Qsat u

25 rr

d

g

q

u2 u2c

for u uc;0 else:

3

Here the sand ux takes into account a dune compactness of 0.6.From the individual saturated sand ux vectors Qi

!, we estimate the norm of the mean sand ux on a at erodible bed, also called the resultant drift potential:

RDP P

N i2

Qi !dti

; 1

where z 2 m is the height at which the wind velocity ui has been measured,

z0 10 3 m the characteristic surface roughness and k 0.4 the von-Krmn

constant. The threshold shear velocity value for motion inception can be determined using the formula calibrated by Iversen and Rasmussen51

uc0:1

rsrf gd

PNi2 dti

; 4

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b

N

3 km

a

c

Figure 5 | Comparison of giant linear dune features in major terrestrial sand seas. (a) Erg Chech desert, Mali (24430 N, 3390 W). (b) Rubal-Khali desert, Saudi Arabia (20510 N, 54090 E). (c) Namib desert, Namibia (24150 S, 14590 E). Despite similar raked patterns, we cannot conclude at this stage that these dunes are in a steady state or in equilibrium with the current wind regime. Given their size and location along major sand ow paths, they can also result from transition in dune shape across space and over time.

where

dtiti ti 1:This quantity is strongly dependent on the function of wind directionality. We also calculate the drift potential, which is the average of individual norms and represent the mean volume of sand per unit width and time that have been transported through a vertical plane:

DP P

Dune orientation aI in the bed instability mode. In the bed instability mode, dunes grow in height from the erosion of the sand bed in the interdune areas. All winds contribute to the growth of the dune height. Thus, linear bedforms develop perpendicularly to the maximum gross bedform-normal transport13.

Considering the orientation angles yi of uxes Qci

!, we calculate Q>(a), the total sand ux perpendicular to the crest for all possible crest orientations aA[0; p]. Then, we identify the maximum value of Q>(a) that corresponds to the most probable crest orientation aI of dunes in the bed instability mode. Note that this procedure is the same as in Rubin and Hunter13 (that is, the gross bedform-normal transport rule), except that we take into account the increase of the sand uxes at the crest of dunes (that is, ga0). As detailed in Courrech du Pont et al.3 and Gao et al.11, it may signicantly change the predictions of dune orientations.

Dune orientation aF in the ngering mode. In the ngering mode, dunes extend in the direction of the mean sand ux. Hence, the orientation aF of nger dunes is the one for which the direction of the mean sand ux at the crest (that is, the time average of equation (6)) aligns with dune orientation.

In practice, we calculate Q>(a) and Q||(a), the total sand ux perpendicular and parallel to the crest for all possible crest orientations aA[0; 2p]. Then, we select the orientation aF for which the sediment ux perpendicular to the crest vanishes (that is, Q>(a) 0) and for which the ux parallel to the dune is positive (that is,

Q||(a)40). If more than one solution exists, as it is the case for star dunes15, we look for the angle at which the Q||-value is maximum. By denition, when there is no feedback of topography on the ow (that is, g 0 in equation (6)), the

orientation of the linear ngering mode aF is given by the resultant sand transport direction (also called the RDD). When the wind speed-up is taken into account, the dune orientation depends on the g-value; g 1.6 gives reasonable estimates of

dune orientation3,11.

From the predicted crest orientations aI and aF in the bed instability and the ngering modes, we calculate

DaaI aF; 8

the angle between the two bedform alignments (DaA[0; p/2]).

Sand ux at the crest of dunes in the bed instability and ngering modes.

Because the apparent dune aspect-ratio (that is, the aspect ratio seen by the wind) determines the increase in wind speed at the top of the dune, the magnitude and

N i2

Qi !

dti

: 5

Averaged over the entire time period, this quantity does not take into account the orientation of the sand uxes54.

The ratio RDP/DP is a non-dimensional parameter, which is often used to characterize the directional variability of the wind regimes55,56: RDP/DP-1 indicates that sediment transport tends to be unidirectional; RDP/DP-0 indicates that most of the transport components cancel each other. Finally, RDD is the resultant drift direction, that is, the direction of

PNi2 Qi

PNi2 dti

!dti.

Sand ux at the crest of dunes. A positive topography accelerates the wind, so that the sand ux over a dune depends on the dune shape. For 2D turbulent ows over low hills, Jackson and Hunt57 show analytically that the increase of wind velocity at the top of the hill, the so-called speed-up factor, is approximately proportional to the hump aspect ratio. Hence, at the rst order of the dune aspect ratio and neglecting the transport threshold, the sand ux Qci

! at the crest of the

dune takes the following form

Qci

! Qi

! 1 g sin yi a j j ; 6

where a is the orientation angle of the linear dune, Qi

! the sand ux on a at sand bed, yi the orientation angle of this ux vector and g the ux-up ratio:

gb

HW 7 where W is the width of the dune, H its height and b a dimensionless coefcient that accounts for all the other physical ingredients (for example, roughness) that affect the speed-up.

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ARTICLE NATURE COMMUNICATIONS | DOI: 10.1038/ncomms14239

the orientation of the time averaged sand ux at the dune crest Q I;F

f g

! is a function

of the dune orientation a{I,F}:

Q I;F

f g

! P

N i2

! 1 g sin yi a I;F f g

dt

Qi

i

; 9

PNi2 dti

where yi is the orientation angle of the ux Qi

! on a at sand bed. Then, DaQ is the angle between the direction of the resultant sand ux at the crest of dunes in the bed instability and the ngering modes (DaQA[0; p]).

Dune-height growth rate. All sand uxes perpendicular to the crest can contribute to dune growth. Considering the dune orientation a{I,F}, we calculate the

characteristic (growth) rate s{I,F} to build up a linear dune of height H and width W

in the bed instability or the ngering mode3,11:

s I;F

f g

1HW P

N i2

Qi !

1 g sin yi a I;F

f g

dt

i

sin yi a I;F

f g

: 10

Characterization of the wind regime. We use an Expectation-Maximization algorithm to t the ux orientation distribution by a Gaussian mixture model. Thus, we replace the real ux data by a limited number nY of normal distributions characterized by a mean orientation Yi, a standard variation si and a weight wi with i {1, y, nY}. Considering only time periods during which the wind velocity is

above a critical value for sediment transport (that is, u*4uc, see equation (3)), we assume that the probability distribution function of sand ux orientation Y may be described by a sum of normal distributions:

P Y

X

n

i1

PNi2 dti

: 11

The Expectation-Maximization algorithm is a natural generalization of maximum likelihood estimation to the incomplete data case. Basically, this is an iterative scheme that includes two different steps. Starting from initial guesses for the parameters wi, si and Yi, the (rst) Expectation-step is to compute a probability distribution over possible completions. In the (second) Maximization-step, new parameters are determined using the current completions. These steps are repeated until convergence.

Figure 1d shows how the sand ux orientation can be tted with a three component Gaussian mixture model (nY 3). For the wind tower W1, we nd

mean orientations Y{1,2,3} {250, 189, 13}, s.ds s

{1,2,3}

wisi

2p

p exp

Y Yi

2

2s2i

{25, 9, 21} and weights w{1,2,3} {0.48, 0.35, 0.17}. For the wind tower W2, we nd mean

orientations Y{1,2,3} {232, 190, 6}, s.d.s s

{1,2,3}

{32, 7, 17} and weights w{1,2,3} {0.50, 0.30, 0.20}. All angles are measured anticlockwise from east. The

northern wind is the strongest one. The secondary wind comes from the east. The weakest wind comes from the west.

Elongation rate and mean width of linear dunes. To estimate the elongation and the mean width of linear dunes from Google Earth images, we isolate the contours of the growing tip at two different times (Supplementary Fig. 5a). From the surface area covered by individual dunes, we compute dune orientation and locate the dune tip at each time. Then, we measure dune width perpendicularly to this orientation at different positions along the linear dune. Supplementary Fig. 5be shows the variation of dune width with respect to the distance from the latest dune tip. We measure the elongation using the distance between the dune tips at two different times. We estimate the mean dune width as the homogeneous width reached along the dune body away from the growing tip. From these measurements, we observe that the shape of the growing tip changes. This is not surprising given the function of wind directionality. For this reason, we could have measured elongation using the positions at which dunes reach their homogeneous width instead of the dune tips. Nevertheless, this procedure gives similar results but more uncertainty. One should also note that the sand uxes derived from these elongation rates are likely to be underestimated because of sand loss at the dune tips.

No raked linear dunes under bidirectional wind regime. Raked linear dunes have never been reported before in laboratory or numerical experiments investigating dune shape25,2932, even when the two dune growth mechanisms have been under consideration3,11. All these experiments have concentrated only on unidirectional or bidirectional wind regimes. There is clearly three dominant wind directions in the Kumtagh desert. Such a trimodal wind regime generates specic conditions, which are never met together in a bidirectional wind regime.

In the parameter space {y, N} of bidirectional wind regimes (y and N are the angle and the transport ratio between the two winds, respectively), {Da, sF/sI,

DaQ}-values can be computed analytically11 (Supplementary Fig. 6ac). For each variable, we report the range of values derived from the wind data in the Kumtagh desert to the parameter space {y, N} of bidirectional wind regimes. Thus, we

delineate different regions of the parameter space {y, N}. When we compare these regions, we observe that there is no overlap indicating that the specic conditions predicted from the recorded wind data cannot be reproduced by bidirectional wind regimes (Supplementary Fig. 6df). Therefore, there is a general incompatibility between the three variables and, even when they are considered two by two, regions of overlaps are small in size. Hence, bidirectional wind regimes cannot provide the {Da, sF/sI, DaQ}-values met in the raked linear duneeld of the Kumtagh desert.

Data availability. The data that support the ndings of this study are available from the corresponding author upon reasonable request. The numerical dune model has been built using the Real-Space Cellular Automaton Laboratory35 (ReSCAL), a free software under the GNU general public licence. The source codes can be downloaded from http://www.ipgp.fr/rescal

Web End =http://www.ipgp.fr/rescal .

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Acknowledgements

We acknowledge nancial support from the National Natural Science Foundation of China (n 41571008 and 1JY31JA51), the UnivEarthS LabEx programme of Sorbonne Paris Cit (ANR-10-LABX-0023 and ANR-11-IDEX-0005-02), the French National Research Agency (ANR-12-BS05-001-03/EXO-DUNES) and the French Chinese International laboratory SALADYN. Images of Figs 1a,e,2f and 5 are courtesy of Google Earth. Pictures of Figs 1b,2a,b are courtesy of George Steinmetz.

Author contributions

P.L. and C.N. designed the research. Z.D. and P.L. carried out the eld measurements. O.R. and C.N. developed the numerical code. P.L., C.N. and S.C.P. wrote the manuscript. All authors analysed the data and discussed the results.

Additional information

Supplementary Information accompanies this paper at http://www.nature.com/naturecommunications

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How to cite this article: L, P. et al. Unravelling raked linear dunes to explain the coexistence of bedforms in complex duneelds. Nat. Commun. 8, 14239 doi: 10.1038/ncomms14239 (2017).

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