Abstract
Abstract. We describe the probabilistic physics of rarefied particle motions and deposition on rough hillslope surfaces. The particle energy balance involves gravitational heating with conversion of potential to kinetic energy, frictional cooling associated with particle–surface collisions, and an apparent heating associated with preferential deposition of low-energy particles. Deposition probabilistically occurs with frictional cooling in relation to the distribution of particle energy states whose spatial evolution is described by a Fokker–Planck equation. The Kirkby number Ki – defined as the ratio of gravitational heating to frictional cooling – sets the basic deposition behavior and the form of the probability distribution fr(r) of particle travel distances r, a generalized Pareto distribution. The shape and scale parameters of the distribution are well-defined mechanically. For isothermal conditions where frictional cooling matches gravitational heating plus the apparent heating due to deposition, the distribution fr(r) is exponential. With non-isothermal conditions and small Ki this distribution is bounded and represents rapid thermal collapse. With increasing Ki the distribution fr(r) becomes heavy-tailed and represents net particle heating. It may possess a finite mean and finite variance, or the mean and variance may be undefined with sufficiently large Ki. The formulation provides key elements of the entrainment forms of the particle flux and the Exner equation, and it clarifies the mechanisms of particle-size sorting on large talus and scree slopes. Namely, with conversion of translational to rotational kinetic energy, large spinning particles are less likely to be stopped by collisional friction than are small or angular particles for the same surface roughness.
Highlights
Sediment transport on steepland hillslopes involves a great range of scales of particle motions
As a step in this effort we show in the second companion paper (Furbish et al, 2021a) that the theory in this first paper is entirely consistent with data from laboratory and field-based experiments involving measurements of particle travel distances on rough surfaces
Our formulation of rarefied particle motions is based on a description of the energy balance of a cohort of particles treated as a rarefied granular gas, as well as a description of particle deposition that depends on the energy state of the particles
Summary
Sediment transport on steepland hillslopes involves a great range of scales of particle motions. (For an ordinary gas, the onset of rarefied conditions occurs when Kn 0.01.) We note that laboratory experiments (Kirkby and Statham, 1975; Gabet and Mendoza, 2012; Furbish et al, 2021a) and field-based experiments (DiBiase et al, 2017; Roth et al, 2020) designed to mimic particle motions and travel distances on hillslopes effectively focus on rarefied conditions These conditions represent one of the most fundamental of Earth surface processes imaginable – how individual sediment particles that are not transported by a fluid move down a rough inclined surface. The implication is that landform configurations and associated particle fluxes reflect an inherent variability (“weather”) that is just as important as the expected (“climate”) conditions in characterizing system behavior
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