Local Wind Regime Induced by Giant Linear Dunes: Comparison of ERA5-Land Reanalysis with Surface Measurements

Abstract

Emergence and growth of sand dunes results from the dynamic interaction between topography, wind flow and sediment transport. While feedbacks between these variables are well studied at the scale of a single and relatively small dune, the average effect of a periodic large-scale dune pattern on atmospheric flows remains poorly constrained, due to a pressing lack of data in major sand seas. Here, we compare local measurements of surface winds to the predictions of the ERA5-Land climate reanalysis at four locations in Namibia, both within and outside the giant linear dune field of the Namib Sand Sea. In the desert plains to the north of the sand sea, observations and predictions agree well. This is also the case in the interdune areas of the sand sea during the day. During the night, however, an additional wind component aligned with the giant dune orientation is measured, in contrast to the easterly wind predicted by the ERA5-Land reanalysis. For the given dune orientation and measured wind regime, we link the observed wind deviation (over $50^{\circ }$) to the daily cycle of the turbulent atmospheric boundary layer. During the night, a shallow boundary layer induces a flow confinement above the giant dunes, resulting in large flow deviations, especially for the slower easterly winds. During the day, the feedback of the giant dunes on the atmospheric flow is much weaker due to the thicker boundary layer and higher wind speeds. Finally, we propose that the confinement mechanism and the associated wind deflections induced by giant dunes could explain the development of smaller-scale secondary dunes, which elongate obliquely in the interdune areas of the primary dune pattern.

Appendices

Appendix 1: Linear Theory of Wind Response to Topographic Perturbation

Following the work of Fourrière et al. (2010), Andreotti et al. (2012) and Andreotti et al. (2009), we briefly describe in this appendix the framework for the linear response of a turbulent flow to a topographic perturbation of small aspect ratio. As a general bed elevation can be decomposed into Fourier modes, we focus here on a sinusoidal topography:

$$\begin{aligned} \xi = \xi _{0}\cos \left[ k\left( \cos (\alpha )y - \sin (\alpha )x\right) \right] , \end{aligned}$$

(3)

which is also a good approximation for the giant dunes observed in the North Sand Sea and South Sand Sea Station (Fig. 2 and Online Resource Fig. S14). Here, x and y are the streamwise and spanwise coordinates, $k=2\pi /\lambda$ the wavenumber of the sinusoidal perturbation, $\alpha$ its crest orientation with respect to the x-direction (anticlockwise) and $\xi _{0}$ its amplitude. The two components of the basal shear stress $\varvec{\tau } = \rho _{0} u_{*}{\varvec{u}}_{*}$, constant in the flat bottom reference case, can then be generically written as:

$$\begin{aligned} \tau _{x}&= \tau _{0}\left( 1 + k\xi _{0}\sqrt{{\mathcal {A}}_{x}^{2} + {\mathcal {B}}_{x}^{2}}\cos \left[ k\left( \cos (\alpha )y - \sin (\alpha )x\right) + \phi _{x}\right] \right) , \end{aligned}$$

(4)

$$\begin{aligned} \tau _{y}&= \tau _{0} \, k\xi _{0}\sqrt{{\mathcal {A}}_{y}^{2} + {\mathcal {B}}_{y}^{2}}\cos \left[ k\left( \cos (\alpha )y - \sin (\alpha )x\right) + \phi _{y}\right] , \end{aligned}$$

(5)

where $\tau _{0}$ is the reference basal shear stress on a flat bed. We have defined the phase $\phi _{x, y} = \tan ^{-1}\left( {\mathcal {B}}_{x, y}/{\mathcal {A}}_{x, y}\right)$ from the in-phase and in-quadrature hydrodynamical coefficients ${\mathcal {A}}_{x, y}$ and ${\mathcal {B}}_{x, y}$. They are functions of k and of the flow conditions, i.e the bottom roughness, the vertical flow structure and the incident flow direction, and the theoretical framework developed in the above cited papers proposes methods to compute them in the linear regime.

Following Andreotti et al. (2012), the effect of the incident wind direction can be approximated by the following expressions:

$$\begin{aligned} {\mathcal {A}}_{x}&= {\mathcal {A}}_{0}\sin ^{2}\alpha , \end{aligned}$$

(6)

$$\begin{aligned} {\mathcal {B}}_{x}&= {\mathcal {B}}_{0}\sin ^{2}\alpha , \end{aligned}$$

(7)

$$\begin{aligned} {\mathcal {A}}_{y}&= -\displaystyle \frac{1}{2}{\mathcal {A}}_{0}\cos \alpha \sin \alpha , \end{aligned}$$

(8)

$$\begin{aligned} {\mathcal {B}}_{y}&= -\displaystyle \frac{1}{2}{\mathcal {B}}_{0}\cos \alpha \sin \alpha , \end{aligned}$$

(9)

where ${\mathcal {A}}_{0}$ and ${\mathcal {B}}_{0}$ are now two coefficients independent of the dune orientation $\alpha$, corresponding to the transverse case ($\alpha =90$). In the case of a fully turbulent boundary layer capped by a stratified atmosphere, these coefficients depend on kH, $k z_{0}$, ${\mathcal {F}}$ and ${\mathcal {F}}_{{\text {I}}}$ (Andreotti et al. 2009). For their computation, we assume here a constant hydrodynamic roughness $z_{0} \simeq 1$ mm (Online Resource section 1). For the considered giant dunes, this leads to $k z_{0} \simeq 2 \cdot 10^{-6}$, as their wavelength is $\lambda \simeq 2.4$ km (or $k \simeq 2 \cdot 10^{-3}$ $\hbox {m}^{-1}$). Values of $z_{0}$ extracted from field data indeed typically fall between 0.1 mm and 10 mm (Sherman and Farrell 2008; Field and Pelletier 2018). Importantly, ${\mathcal {A}}_{0}$ and ${\mathcal {B}}_{0}$ do not vary much in the corresponding range of $k z_{0}$ (Fourrière et al. 2010), and the results presented here are robust with respect to this choice.

With capping layer height and Froude numbers computed from the ERA5-Land time series, the corresponding ${\mathcal {A}}_{0}$ and ${\mathcal {B}}_{0}$ can be deduced, as displayed in Online Resource Fig. S13. Interestingly, it shows similar regimes as in the diagrams of Fig. 8 and Online Resource Fig. S11a,b, supporting the underlying physics. However, the agreement is qualitative only. Further, the linearity assumption of the theoretical framework requires $\left( \vert \tau \vert - \tau _{0}\right) /\tau _{0} \ll 1$, which translates into $k\xi \sqrt{{\mathcal {A}}_{0}^{2} + {\mathcal {B}}_{0}^{2}} \ll 1$. In our case, the giant dune morphology gives $k\xi _0 \simeq 0.1$, which means that one quits the regime of validity of the linear theory when the coefficient modulus $\sqrt{{\mathcal {A}}_{0}^{2} + {\mathcal {B}}_{0}^{2}}$ becomes larger than a few units. In accordance with the theoretical expectations, these coefficients present values on the order of unity (${\mathcal {A}}_{0} \simeq 3$ and ${\mathcal {B}}_{0} \simeq 1$) in unconfined situations (Claudin et al. 2013; Lü et al. 2021). In contrast and as illustrated in Online Resource Fig. S13a,b, larger values are predicted in case of strong confinement, which does not allow us to proceed to further quantitative comparison with the data.

Finally, the linear model is also able to reproduce the enhancement of the flow deflection over the sinusoidal ridges when $\sqrt{{\mathcal {A}}_{0}^{2} + {\mathcal {B}}_{0}^{2}}$ is increased (Online Resource Fig. S13). Here, using $k\xi _0 \simeq 0.1$ to be representative of the amplitude of the giant dunes at the North Sand Sea station, the coefficient modulus is bounded to 10.

Appendix 2: Sediment Transport and Dune Morphodynamics

We summarise in this appendix the sediment transport and dune morphodynamics theoretical framework leading to the prediction of sand fluxes and dune orientations from wind data.

Sediment Transport The prediction of sand fluxes from wind data has been a long standing issue in aeolian geomorphological studies (Fryberger and Dean 1979; Pearce and Walker 2005; Sherman and Li 2012; Shen et al. 2019). Based on laboratory studies in wind tunnels (Rasmussen et al. 1996; Iversen and Rasmussen 1999; Creyssels et al. 2009; Ho et al. 2011), as well as physical considerations (Ungar and Haff 1987; Andreotti 2004; Durán et al. 2011; Pähtz and Durán 2020), it has been shown that the steady saturated saltation flux over a flat sand bed depends linearly on the shear stress:

$$\begin{aligned} \frac{q_{{\text {sat}}}}{Q} = \varOmega \sqrt{\varTheta _{{\text {th}}}}\left( \varTheta - \varTheta _{{\text {th}}}\right) , \end{aligned}$$

(10)

where $\varOmega$ is a proportionality constant, $Q = d\sqrt{(\rho _{{\text {s}}} - \rho _{0})gd/\rho _{0}}$ is a characteristic flux, $\varTheta = \rho _{0} u_{*}^{2}/(\rho _{{\text {s}}} - \rho _{0})gd$ the Shields number, and $\varTheta _{{\text {th}}}$ its threshold value below which saltation vanishes. $\rho _{{\text {s}}} = 2.6~{\text {g}}~{\text {cm}}^{-3}$ and $d=180~\mu {\text {m}}$ are the grain density and diameter, and g is the gravitational acceleration. The shear velocity, and consequently the Shields number as well as the sediment flux, are time dependent.

Recently, Pähtz and Durán (2020) suggested an additional quadratic term in Shields to account for grain-grain interactions within the transport layer at strong wind velocities:

$$\begin{aligned} \frac{q_{{\text {sat, t}}}}{Q} = \frac{2\sqrt{\varTheta _{{\text {th}}}}}{\kappa \mu }\left( \varTheta - \varTheta _{{\text {th}}}\right) \left( 1 +\frac{C_{{\text {M}}}}{\mu }\left[ \varTheta - \varTheta _{{\text {th}}}\right] \right) , \end{aligned}$$

(11)

where $\kappa = 0.4$ is the von Kármán constant, $C_{\mathrm{M}} \simeq 1.7$ a constant and $\mu \simeq 0.6$ is a friction coefficient, taken to be the avalanche slope of the granular material. The fit of this law to the experimental data of Creyssels et al. (2009) and Ho et al. (2011) gives $\varTheta _{{\text {th}}} = 0.0035$. The fit of Eq. 10 on these same data similarly gives $\varOmega \simeq 8$ and $\varTheta _{{\text {th}}} = 0.005$. The sand flux angular distributions and the dune orientations in Fig. 9 are calculated using this law (11). We have checked that using the ordinary linear relationship (10) instead does not change the predicted dune orientations by more than a few degrees.

Dune Orientations Dune orientations are predicted with the dimensional model of Courrech du Pont et al. (2014), from the sand flux time series computed with the above transport law. Two orientations are possible depending on the mechanism dominating the dune growth: elongation or bed instability. The orientation $\alpha$ corresponding to the bed instability is then the one that maximises the following growth rate (Rubin and Hunter 1987):

$$\begin{aligned} \sigma \propto \frac{1}{H_{d} W_{d} T}\int _0^T q_{{\text {crest}}}\vert \sin \left( \theta - \alpha \right) \vert \, {\text {d}}t, \end{aligned}$$

(12)

where $\theta$ is the wind orientation measured with respect to the same reference as $\alpha$, and $H_{d}$ and $W_{d}$ are dimensional constants respectively representing the dune height and width. The integral runs over a time T, which must be representative of the characteristic period of the wind regime. The flux at the crest is expressed as:

$$\begin{aligned} q_{{\text {crest}}} = q_{{\text {sat}}}\left[ 1 + \gamma \vert \sin \left( \theta - \alpha \right) \vert \right] , \end{aligned}$$

(13)

where the flux-up ratio $\gamma$ has been calibrated to 1.6 using field studies, underwater laboratory experiments and numerical simulations. Predictions of the linear analysis of Gadal et al. (2019) and Delorme et al. (2020) give similar results.

Similarly, the dune orientation corresponding to the elongation mechanism is the one that verifies:

$$\begin{aligned} \tan (\alpha ) = \frac{\langle q_{{\text {crest}}}(\alpha ) {\varvec{e}}_{\theta }\rangle \cdot {\varvec{e}}_{WE} }{ \langle q_{{\text {crest}}}(\alpha ) {\varvec{e}}_{\theta } \rangle \cdot {\varvec{e}}_{SN}}, \end{aligned}$$

(14)

where $\langle .\rangle$ denotes a vectorial time average. The unitary vectors ${\varvec{e}}_{WE}$, ${\varvec{e}}_{SN}$ and ${\varvec{e}}_{\theta }$ are in the West-East, South-North and wind directions, respectively.

The resulting computed dune orientations, blue and red arrows in Fig. 9, then depend on a certain number of parameters (grain properties, flux-up ratio, etc.), for which we take typical values for aeolian sandy deserts. Due to the lack of measurements in the studied places, some uncertainties can be expected. We therefore run a sensitivity test by calculating the dune orientations for grain diameters ranging from $100~\mu {\text {m}}$ to $400~\mu {\text {m}}$ and for a speed-up ratio between 0.1 and 10 (wedges in Fig. 9).