Analytical mesoscale modeling of aeolian sand transport

The mesoscale structure of aeolian sand transport determines a variety of natural phenomena studied in planetary and Earth science. We analyze it theoretically beyond the mean-field level, based on the grain-scale transport kinetics and splash statistics. A coarse-grained analytical model is proposed and verified by numerical simulations resolving individual grain trajectories. The predicted height-resolved sand flux and other important characteristics of the aeolian transport layer agree remarkably well with a comprehensive compilation of field and wind-tunnel data, suggesting that the model robustly captures the essential mesoscale physics. By comparing the predicted saturation length with field data for the minimum sand-dune size, we elucidate the importance of intermittent turbulent wind fluctuations for field measurements and reconcile conflicting previous models for this most enigmatic emergent aeolian scale.

Figure 1

(a) Coarse-grained numerical simulations (symbols) of the hop-height distribution $P_H\left(h\right)$ compared to the analytical model, Eq. (3) (solid lines), for various wind shear stresses $\tau$ $[{10}^2\tau /\left({\varrho }_{\text{g}}ga\right)=1,1.5,2,2.5$ from bottom to top]. The simulation computes the statistics of the ejected bed grains from the analytical model of Ref. [36], which yields a power-law distribution for low hop heights with exponent $\nu =1.4$. The left inset is a log-log representation of the same data, with logarithmic binning widths (data shifted vertically for better readability). (b) Trajectories reaching beyond the characteristic height $H$ of the transport layer die out exponentially. While $H$ was, for each data point, used as a free fit parameter to match the simulation data, its $\tau$ dependence is well reproduced by the more refined calculation in Sec. 3c (dashed line).

Figure 2

Simulation data (symbols) and analytical predictions obtained from Eqs. (3)–(10) (lines) for the (a), (b) height-resolved mass concentration ${\varrho }_H\left(z\right)$, (c), (d) flux $j_H\left(z\right)$, and (e), (f) grain-borne shear stress ${\tau }_{\text{g},H}\left(z\right)$. Normalizing the data by the height-integrated concentration ${\rho }_H$ and flux $q_H$, the grain stress at the ground ${\tau }_{\text{g},H}\left(0\right)$, and the characteristic transport-layer height $H$, a good collapse is achieved for various wind shear stresses $\tau$ (legend), as theoretically predicted. The solid lines correspond to the model approximation of shape-invariant trajectories with $\epsilon \left(h\right)=\epsilon \left(H\right)$ independent of $h$; the dashed lines represent Eq. (18), which invokes $\epsilon \left(h\right)\propto h^{-1/2}$, as appropriate for the majority of (low) trajectories, which collectively carry most of the momentum and therefore dominate the stress (see the inset of Fig. 9).

Figure 3

Simulation data (symbols) and analytical predictions obtained from Eqs. (3)–(10) (lines) for the height-resolved grain velocity $V_H\left(z\right)$ for various wind shear stresses $\tau$ (legend). Solid lines represent Eq. (21), which corresponds to the model approximation of shape-invariant trajectories with $\epsilon \left(h\right)=\epsilon \left(H\right)$ independent of $h$. Dashed lines show the long-trajectory estimate in Eq. (30) predicted from the wind-speed profile given in Eq. (26).

Figure 4

(a) Height-resolved wind speed for various wind shear stresses $\tau$ $[{10}^2\tau /\left({\varrho }_{\text{g}}ga\right)=1,1.5,2,2.5$ from bottom to top] from numerical simulations (symbols) and our model, Eq. (26), with (dashed lines) and without [solid lines, with first-order corrections shown in (b)] invoking the simplifying Owen hypothesis that the airborne shear stress at the bed is screened precisely to the threshold value ${\tau }_{\text{t}}$ for grain entrainment [42].

Figure 5

“Bare” and “dressed” wind-strength-dependent saturation lengths (linear and logarithmic axes scaling). Simulation data for the bare saturation length ${\ell }_{\text{sat}}$ (symbols) were obtained by fitting a linear relaxation process with decay length ${\ell }_{\text{sat}}$ to the response of the height-integrated flux $q\left(x\right)$ along the wind direction to a small step increase of the shear stress $\tau \left(x\right)$ at $x=0$ (notice the improving agreement with the theoretical expectation upon approaching the ideal limit of an infinitesimal step height, given in the legend). Its power-law decay in $\tau$ (b) close to the transport threshold ${\tau }_{\text{t}}=0.01{\varrho }_{\text{g}}ga$ gives way to a weak growth at larger $\tau$, in good accord with Eqs. (31), (32), with a numerical factor ${\ell }_{\text{sat}}/L=2$ (solid lines). Under realistic field conditions, the sharp singularity at $\tau ={\tau }_{\text{t}}$ is smeared out by intermittent wind-speed fluctuations, giving rise to the apparent “dressed” saturation length given by the right-hand side of Eq. (33), with a Weibull-distributed wind speed of variance $0.1{\langle u\rangle }^2$ [60] and the aerodynamic-impact threshold ratios ${\tau }_{\mathrm{ta}}/{\tau }_{\mathrm{t}}=1.27$ [obtained from our simulations; dashed line in panel (a)] and ${\tau }_{\mathrm{ta}}/{\tau }_{\mathrm{t}}=10$ (expected for wind-blown sand transport on Mars [53, 62]; dotted line).

Figure 6

(a) Theory and literature data for the height-resolved horizontal grain flux $j_H\left(z\right)$ (linear/logarithmic height axis; data extracted from independent sources vertically shifted, for better visibility). The simple analytical model with shape-invariant grain trajectories, Eqs. (3)–(9a) (solid lines), is fitted to wind-tunnel measurements by Rasmussen and Mikkelsen [64] (dots), Rasmussen and Sørensen [66] (squares), and field data by Namikas [49] (triangles) for various wind strengths $\tau$, using $\nu$ and $H$ as free global fit parameters. This yields $\nu =0.94$ and the data for $H$ shown in the inset (b) with the improved model prediction from Eq. (31) (dashed line). While laboratory data and theory agree well, field measurements (triangles) consistently find higher trajectories.

Figure 7

(a) Theory and literature data for the vertical sand flux ${\varphi }_H(z=0,\ell )$ through the sand bed at a downwind distance $\ell$ from the end of the sand bed (linear/logarithmic height axis; data extracted from independent sources vertically shifted, for better visibility). The simple analytical model with shape-invariant grain trajectories, Eq. (19), is fitted to wind-tunnel measurements by Ho et al. [41] (dots) and Rasmussen et al. [1] (squares), and to field data by Namikas [49] (triangles) for various wind strengths $\tau$, using the value $\nu =0.94$ obtained from Fig. 6. (b) The trajectory length $L$ (the only free fit parameter) compared to the theoretical expectation $H/\epsilon \left(H\right)\approx 10H$ [45] (dashed line). Again, laboratory data and theory agree well, while field measurements consistently find longer (and higher) trajectories.

Figure 8

Field data for the minimum dune length ${\mathcal{L}}_{\text{min}}$ reported by Andreotti et al. [54] (symbols) compared to the dressed saturation length $\langle \varphi {\ell }_{\text{sat}}\rangle /\langle \varphi \rangle$ from Eqs. (32), (31) (line). The expected linear relation is recovered for Weibull-distributed intermittent wind-speed fluctuations of variance $0.05{\langle u\rangle }^2$ [60] and an effective threshold shear stress ${\tau }_{\text{t}}^{\text{eff}}=0.12{\tau }_{\text{t}}$, suggesting a factor of proportionality of about 35 in Eq. (33), reasonably close to previous theoretical estimates [55, 56].

Figure 9

Trajectory aspect ratio $\epsilon \equiv h/l$ for various wind strengths, as obtained from our simulations. For most mesoscale measures, its dependence on the hop height $h$ is negligible (a), as is the weak dependence on the wind shear stress $\tau$, as illustrated for the aspect ratio $\epsilon \left(H\right)$ of the characteristic trajectory (b). However, the height-resolved grain stress, Eq. (18), crucially depends on the power-law decay for $h\ll H$, which is better resolved on the logarithmic scale in the left inset (same data logarithmically binned), solid and dashed lines representing Eq. (E7) (scaled by an empirical prefactor 0.65) and the power law $\epsilon \propto h^{-1/2}$, respectively.