Rarefied Particle Motions On Hillslopes - Part 1: Theory

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 f(r)(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 f(r)(r) is exponential. With non-isothermal conditions and small Ki this distribution is bounded and represents rapid thermal collapse. With increasing Ki the distribution f(r)(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.