Particle velocity distribution in saltation transport

We report on wind-tunnel measurements of particle velocity distribution in aeolian transport. By performing extended statistics, we show that for saltation occurring over an erodible bed the vertical lift-off velocity distributions deviate significantly from a Gaussian law and exhibit a long tail accurately described by a lognormal law. In contrast, saltation over a rigid bed produces Gaussian velocity distributions. These results strongly suggest that the deviation from Gaussian distributions is a consequence of the splash process which is exclusively present in saltation transport over an erodible bed. We further suggest that the non-Gaussian statistics is intimately related to the statistical properties of a single splash event which produces ejection of particles with lift-off velocities distributed according to a lognormal law. This lognormal behavior can be simply inferred from the propagation process of the impact energy through the granular bed which can be viewed as the analog of a fragmentation process. These findings emphasize the crucial role of the splash process in saltation transport.

Figure 1

Sketch of the wind-tunnel.

Figure 2

Vertical velocity distribution $p\left(v\right)$ of saltating particles located close to the ground (at $z=10d$ within a layer $dz=5d$) for various wind strengths measured in terms of the Shields parameter $S$. Fitted distributions: dashed line—Gaussian law with variance $s^2=53gd$ (${\chi }^2$ test: ${\chi }^2=0.35$), solid line—lognormal law with $\mu =1.6$ and $\sigma =1.0$ (${\chi }^2=0.17$), dot-dashed line—Weibull distribution, $p_W\left(v\right)\propto v^{k-1}{e}^{-{(v/\lambda )}^k}$, with $k=1.0$ and $\lambda =7.8\sqrt{gd}$ (${\chi }^2=0.20$), dot-dashed line—$\Gamma$ distribution, $p_G\left(v\right)\propto v^{k-1}{e}^{-v/\lambda }$, $k=1.1$ and $\theta =7.9\sqrt{gd}$ (${\chi }^2=0.20$), dotted line—linear combination of two Gaussian laws with variances $s_1^2=25gd$ and $s_2^2=625gd$ (${\chi }^2=0.17$). Inset: Variation of the variance of the distribution (i.e., $\langle v^2/gd\rangle$) with the Shields parameter $S$.

Figure 3

Distribution of the vertical particle velocity close to the ground (i.e., $z\approx 10d$) obtained over a rigid bed for various Shields numbers with a given mass flow rate $Q/{\rho }_{p}d\sqrt{gd}\approx 1.7$. Solid line represents a Gaussian distribution that best fits the upward velocities. Inset: Variation of $\langle v_{up}^2/gd\rangle$ with the Shields parameter.

Figure 4

Left panel: illustration of successive binary collisions produced by an impacting particle and leading to the ejection of a particle. Right panel: construction of the resulting collision tree; from a node (i.e., a collision) results two branches: the right one corresponds to the reflected particle and the left one to the target particle.

Figure 5

Distribution of path length $N\left(k\right)/N_{\mathrm{tot}}$ obtained from the discrete collision model for two different impact velocities $V_0/\sqrt{gd}=30,100$. Solid lines represent Gaussian fits.

Figure 6

Comparison of the vertical lift-off velocity distribution obtained in steady state saltation over erodible bed and that resulting from a splash event derived from the splash model [cf. Eq. (5)]. Lift-off velocities are evaluated at the height $z=10d$ within a layer $dz=5d$.