Booming Dune Instability

Sand avalanches flowing down the leeward face of some desert dunes spontaneously produce a loud sound with a characteristic vibrato around a well-defined frequency, a phenomenon called the “song of dunes.” Here, we show through theory that a homogenous granular surface flow is linearly unstable towards growing elastic waves when a localized shear band forms at the interface between the avalanche and the static part of the dune. We unravel the nature of the acoustic amplifying mechanism at the origin of this booming instability. The dispersion relation and the shape of the most unstable modes are computed and compared to field measurements.

Figure 1

Schematic of a booming avalanche. Elastic waves are guided along the surface and amplified by reflections on the shear band when they propagate up the slope.

Figure 2

Dispersion relation for $\mu =\tan \theta =0.6$, $\mathcal{A}=\mathcal{B}$, and without any dissipation. The dispersion relation of the first mode in the absence of avalanche [14] is shown in dotted lines. The presence of the shear band gives rise to two branches, and no more, shown in solid lines $\ominus$ and dashed lines $\oplus$. For each mode propagating upslope $\uparrow$ with an angular frequency $\omega >0$ and a growth rate $\sigma$, there exists a mode propagating downslope $\downarrow$ with an angular frequency $-\omega$ and a growth rate $-\sigma$. (a) Relation between $\omega$ and $k$, both rescaled using the avalanche thickness $H$. (b) Relation between $\sigma$ and $k$. (c) Relation between $\sigma$ and $\omega$.

Figure 3

(a),(b) Profiles of the displacement ($U$ and $W$) for the upward propagating mode associated to the first maximum of the growth rate ($\ominus$, $H^{5/6}\omega /\Gamma \simeq 0.68$ and $H^{5/6}\sigma /\Gamma \simeq 0.15$). (c) Profile $U(\zeta )$ for the second maximum of $\sigma$ ($\ominus$, $H^{5/6}\omega /\Gamma \simeq 0.95$ and $H^{5/6}\sigma /\Gamma \simeq 0.36$). (d),(e) Field measured profiles of the vibration amplitude along the $x$ and $z$ axis ($H\sim 35\pm 10\mathrm{mm}$), at 3 m from the front. The first mode (a),(b) is shown in solid lines for the surface vibration amplitude $|Z|=g\cos \theta /{\omega }^2$ predicted in 4.

Figure 4

(a) Growth rate $\sigma$ as a function of $kH$ taking dissipation into account ($\mathcal{N}=1$). The branches correspond either to upward ($\uparrow$, $\oplus$ in solid line and $\ominus$ in dotted line) or downward ($\downarrow$, $\oplus$ in dot-dashed line and $\ominus$ in dashed line) propagating modes. (b) Bifurcation diagram showing the local maxima of the growth rate (bold lines) and the marginal stability curves (thin lines) as functions of ${\mathcal{N}}^{-6/5}$. The two most unstable modes are the third (dot-dashed line) and fourth (solid line) maxima of the branch $\ominus$ and propagate upward ($\uparrow$).

Figure 5

(a) Typical displacement signal $U_x(t)$ measured with an accelerometer placed below a booming avalanche. Note the characteristic vibrato around a central frequency (here, $f=107\mathrm{Hz}$). (b) Profile of the displacement amplitude along the avalanche axis. The dotted line shows the best fit by the amplitude equation expected for a convective instability: $\ell {\partial }_x{U}_x=U_x(1-U_x^2/U_{\infty }^2)$, with $\ell =2.1\mathrm{m}$ and $U_{\infty }=130\mu \mathrm{m}$.