Rarefied particle motions on hillslopes – Part 1: Theory

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.

Rarefied particle motions on hillslopes – Part 2: Analysis

“…Nonetheless, the analysis points to the need for further clarity concerning how particle size and shape in concert with surface roughness influence the extraction of particle energy and the likelihood of deposition. In the third companion paper (Furbish et al, 2021b) we show that the generalized Pareto distribution in this problem is a maximum entropy distribution (Jaynes, 1957a, b) constrained by a fixed energetic "cost" -the total cumulative energy extracted by collisional friction per unit kinetic energy available during particle motions. That is, among all possible accessible microstatesthe many different ways to arrange a great number of particles into distance states where each arrangement satisfies the same fixed total energetic cost -the generalized Pareto distribution represents the most probable arrangement.…”

“…As described in our first companion paper (Furbish et al, 2021a), we are focused on rarefied motions of particles which, once entrained, travel downslope over the land surface. This notably includes the dry ravel of particles down rough hillslopes following disturbances (Roering and Gerber, 2005;Doane, 2018;Doane et al, 2019;Roth et al, 2020) or upon their release from obstacles (e.g., vegetation) following failure of the obstacles (Lamb et al, 2011(Lamb et al, , 2013DiBiase and Lamb, 2013;DiBiase et al, 2017;Doane et al, 2018Doane et al, , 2019, and the motions of rockfall material over the rough surfaces of talus and scree slopes (Gerber and Scheidegger, 1974;Kirkby and Statham, 1975;Statham, 1976;Tesson et al, 2020).…”

“…The formulation of rarefied particle motions presented in the first companion paper (Furbish et al, 2021a) is based on a description of the kinetic energy balance of a cohort of particles treated as a rarefied granular gas and a description of particle deposition that depends on the energy state of the particles. 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.…”

“…The purpose of this second companion paper is to present an analysis of several data sets concerning particle motions on rough surfaces, as viewed through the lens of the theory presented in the first companion paper (Furbish et al, 2021a). In Sect.…”

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We examine a theoretical formulation of the probabilistic physics of rarefied particle motions and deposition on rough hillslope surfaces using measurements of particle travel distances obtained from laboratory and field-based experiments, supplemented with high-speed imaging and audio recordings that highlight effects of particle-surface collisions. The formulation, presented in a companion paper (Furbish et al, 2021a), is based on a description of the kinetic 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. Both laboratory and field-based measurements are consistent with a generalized Pareto distribution of travel distances and predicted variations in behavior associated with the balance between gravitational heating due to conversion of potential to kinetic energy and frictional cooling due to particle-surface collisions.

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