Synchronization due to Phase Locking

Andreotti proposed that a sand avalanche excites elastic waves at the surface of the dunes. These elastic waves exert feedback on the particle motion providing partial synchronization by a wave-particle mode locking. Andreotti (2004) measured the booming frequency (f = 100±5Hz) on a barchan dune in the Sahara desert in Morocco. He argued that the frequency is only dependent on the properties of the grain dynamics inside the avalanche, such as particle diameter (D= 0.18 mm) and shear velocity. The elastic waves are located close to the surface as the vibration quickly attenuates at a depth of 10 cm. The measured phase speed (40 m/s) is two orders of magnitude lower than the body wave velocity in quartz. Andreotti denotes the elastic waves as Rayleigh-Hertz waves and establishes a dispersion relation:

f =c2/3g1/6λ−5/6, (2.8)

with wavelengthλand speed of soundc= 230m/s obtained by fitting the dispersion relation.

Andreotti introduces the velocity gradient Γ as “the typical rate at which grains jump

over their neighbors and make collisions” and notes that it ”is independent of the flowing depth h.” The balance between potential and kinetic energy results in a limiting velocity between two adjacent grain layers equal to√gD. Laboratory experiments showed that the velocity gradient has exactly the same value as the booming frequency on sand dunes and therefore Andreotti proposed that the booming frequencyf is equal to the velocity gradient

Γ such that:

Γ'0.4

r

g

D =f. (2.9)

No rational for the factor 0.4 is given other than that it fits the data of frequency and diameter measured in the current study and the data point provided byCriswellet al.(1975)

and Lindsay et al. (1976) for shoveling sand (D = 0.380 mm andfbp = 66 Hz). Andreotti

argues that the collisions of grains excite larger-scale elastic waves that are synchronized by wave-particle mode locking. Andreotti also notes that for inhomogeneous flow, inertial effects have to be added to the gravity and hence the velocity gradient should be modified. In a follow-up paper (Bonneau et al., 2007), Andreotti and co-workers extended the Rayleigh-Hertz wave theory. They argued that the sand grains are stratified in layers due to gravity, resulting in refraction of waves to the surface. The sound wave speed, varying as

c∝P1/4 for large confining pressures and c∝P1/6 for small confining pressures, increases with depth and creates bending of rays toward the surface. The influence of the vanishing confining pressure near the surface on sound wave propagation remains unaddressed. The authors do not provide exact velocity profiles with depth or specify an exact length scale or velocity increase at which this process of bending of waves occurs.

Bonneau et al. argue that coherent elastic modes are excited as a result of nonlinear

Hertzian contact and dispersion. The elastic waves synchronize the collisions which excite the elastic wave resulting in an amplitude increase. A waveguide cutoff exists due to the finite depth of the layers, resulting in the inability to propagate sound. A coupling between the avalanching grains and surface elastic waves results in the song of the dunes.

The authors state that the non-propagative resonant mode is around 73 Hz, which cor- responds to a waveguide depth of 47 cm, “which is indeed the typical depth at which the first wet layer may be found on dry days.” This claim is unfounded as neither the velocity in the dune, nor the depth of the wet layer is measured in the current study and no references are made toward other studies in literature.

The authors identify two different length scales that are important in booming: the shearing of the particles on the grain-scale and a waveguide effect within the scale of the dune. The frequency of booming is determined solely from particle characteristics f = 0.4pg/D while the internal waveguide cutoff frequency influences the ability of the dune to boom due to the finite width of subsurface layers. In private communication,Andreotti acknowledged that field experiments showed that the booming frequency changes from day to day based on environmental parameters. Bonneauet al.(2007) did not conduct measurements of wave propagation of booming in situ and no rationale is provided as to whether the waveguide influences the booming frequency directly.

Andreotti and Bonneau (2009) shared recently a currently unpublished manuscript in

personal communication. The authors classify three contradictory dynamical mechanisms for the booming emission: (i) quasi-periodic stick-slip motion (Bagnold, 1966;Patitsas, 2008), (ii) incoherent source selected by resonance over the thickness of the avalanche (Douady

et al., 2006) or the thickness of a dry layer (Vriend et al.,2007) and (iii) selective acoustic amplifier emitting coherent elastic modes that synchronize grain motion (Andreotti, 2004;

Bonneau et al., 2007). The authors propose yet another mechanism based on a “linear

exponential growth of elastic waves within the flowing layer that is bordered by a thin shear band on the bottom of the avalanche. The authors assume in the theoretical derivation of the dispersion relation that the system homogeneous in time and lateral direction and changes properties with depth. The local maximum of the growth rate creates a mode that propagates and grows in one direction (either downslope or upslope). The manuscripts states that “guided modes are selected by the condition of constructive interference between plane waves as they bounce back and forth.” The shear band that separates the flowing and the static part of the avalanche induces a “coherent amplification of guided elastic waves.”