Continuum saltation model for sand dunes

Abstract

We derive a phenomenological continuum saltation model for aeolian sand transport that can serve as an efficient tool for geomorphological applications. The coupled differential equations for the average density and velocity of sand in the saltation layer reproduce both the known equilibrium relations for the sand flux and the time evolution of the sand flux as predicted by microscopic saltation models. The three phenomenological parameters of the model are a reference height for the grain-air interaction, an effective restitution coefficient for the grain-bed interaction, and a multiplication factor characterizing the chain reaction caused by the impacts leading to a typical time or length scale of the saturation transients. We determine the values of these parameters by comparing our model with wind tunnel measurements. Our main interest are out of equilibrium situations where saturation transients are important, for instance at phase boundaries (ground/sand) or under unsteady wind conditions. We point out that saturation transients are indispensable for a proper description of sand flux over structured terrain, by applying the model to the windward side of an isolated dune, thereby resolving recently reported discrepancies between field measurements and theoretical predictions.

The shape of barchan dunes

“…In particular, shape transitions from dunes (with slipface) to smooth heaps as a function of sand mass were predicted [1,15,16]. The model adapts, combines, and extends earlier theoretical developments in turbulence [12,26] and aeolian sand transport [19,22] into a consistent mathematical description, which allows to identify a weak spontaneous symmetry breaking of the turbulent air flow as the origin of the growth instability giving rise to structure formation. It moreover pinpoints the mutual competition of this symmetry breaking with saturation transients in the sand flux as the basic mechanism responsible for shape selection as a function of size and environmental parameters such as density of the air, wind speed or sand supply.…”

“…The new phenomenological length scale ℓ ⊥ can be expected to be a complicated function of the splash process, but will be proportional to (though supposedly numerically substantially smaller than) the average saltation length, i.e. the effective characteristic hopping length of the grains, which is predicted to be relatively weakly sensitive to changes in wind speed [15,22] for the same reason as the grain speed. Therefore, ℓ ⊥ will be treated as a constant phenomenological coefficient for the following considerations.…”

“…For any fixed given ratio τ /τ t of the wind shear stress τ to the threshold shear stress τ t for the mobilization of grains by the wind, the saturation length is predicted [15,22] to scale linearly in τ t divided by the gravitational acceleration g and the atmospheric density ρ a . According to conventional estimates from simple arguments for the aerodynamics on the grain scale, τ t = θρ ′ gd [2], i.e.…”

“…It moreover pinpoints the mutual competition of this symmetry breaking with saturation transients in the sand flux as the basic mechanism responsible for shape selection as a function of size and environmental parameters such as density of the air, wind speed or sand supply. The characteristic length scale provided by the saturation transients is called the saturation length ℓ s [15,16,22]. Due to the ballistic grain motion, it can be much larger than the grain diameter d (≈ 10 −4 m), which is the only elementary scale in the problem, and can thus compete with the turbulent symmetry breaking to cause scale dependence and shape transitions of dunes that are of the order of 10 2 m long.…”

Periodic saltation over hydrodynamically rough beds: aeolian to aquatic

“…This regime, called saltation, is the primary mode of transport in Aeolian sand transport, and has been investigated theoretically (Bagnold 1941, Bagnold 1966, Owen 1964, Ungar & Haff 1987, Sauermann et al 2001, Andreotti 2004, Jenkins et al 2010, experimentally (Bagnold 1941, Nalpanis et al 1993, Foucaut & Stanislas 1997, Iversen & Rasmussen 1999, Ho et al 2014) and numerically (Anderson & Haff 1988, Kok & Renno 2009). More references can be found in recent reviews on Aeolian transport (Durán et al 2011, Kok et al 2012, Valance et al 2015.…”