Climate change is causing increasingly widespread, frequent, and intense wildfires across the western United States. Many geomorphic effects of wildfire are relatively well studied, yet sediment transport models remain unable to account for the rapid transport of sediment released from behind incinerated vegetation, which can fuel catastrophic debris flows. This oversight reflects the fundamental inability of local, continuum-based models to capture the long-distance particle motions characteristic of steeplands. Probabilistic, particle-based nonlocal models may address this deficiency, but empirical data are needed to constrain their representation of particle motion in real landscapes. Here we present data from field experiments validating a generalized Lomax model for particle travel distance distributions. The model parameters provide a physically intuitive mathematical framework for describing the transition from light- to heavy-tailed distributions along a continuum of behavior as particle size increases and slopes get steeper and/or smoother. We show that burned slopes are measurably smoother than vegetated slopes, leading to 1) lower rates of experimental particle disentrainment and 2) runaway motion that produces the heavy-tailed travel distances often associated with nonlocal transport. Our results reveal that surface roughness is a key control on steepland sediment transport, particularly after wildfire when smoother surfaces may result in the preferential delivery of coarse material to channel networks that initiate debris flows. By providing a first-order framework relating the statistics of particle motion to measurable surface characteristics, the Lomax model both advances the development of nonlocal sediment transport theory and reveals insights on hillslope transport mechanics.

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.

Recent work has highlighted the significance of long‐distance particle motions in hillslope sediment transport. Such motions imply that the flux at a given hillslope position is appropriately described as a weighted function of surrounding conditions that influence motions reaching the given position. Although the idea of nonlocal sediment transport is well grounded in theory, limited field evidence has been provided. We test local and nonlocal formulations of the flux and compare their ability to reproduce land surface profiles of steep moraines in California. We show that nonlocal and nonlinear models better reproduce evolved land surface profiles, notably the amount of lowering and concavity near the moraine crest and the lengthening and straightening of the depositional apron. The analysis provides the first estimates of key parameters that set sediment entrainment rates and travel distances in nonlocal formulations and highlights the importance of correctly specifying the entrainment rate when modeling land surface evolution. Moraine evolution associated with nonlocal and nonlinear transport formulations, when described in terms of the evolution of the Fourier transform of the moraine surface, displays a distinct behavior involving growth of certain wave numbers, in contrast to the decay of all wave numbers associated with linear transport. Nonlinear and nonlocal formulations share key mathematical elements yielding a nonlinear relation between the flux and the land surface slope.

We examine probabilistic elements of soil particle ages and residence times measured from their entry into the mechanically active soil column, focusing on steady conditions near a hillslope crest, where particles steadily move into and through a soil mantle with fixed thickness in the presence of erosion. Our objective is to clarify consequences of the geometry of the particle trajectories and disturbance-driven mixing in relation to the arrangement of the surfaces through which the particles enter and leave the soil element. We introduce an Eulerian-Lagrangian description of particle motions to characterize effects of a nonuniform (two-dimensional) mean motion with superimposed depth-dependent mixing. With weak mixing, the forms and moments of the probability distributions of particle residence times and ages are distinct and are largely controlled by the geometry of the mean particle motion within the soil element. The average particle age is systematically less than the average residence time. These distributions converge to exponential forms with equal means only in the idealized limit of a "well-mixed" system. The analysis provides the foundation for considering particle weathering and chemical losses in relation to residence times and ages and for describing the spatiotemporal structure of tracers, for example, cosmogenic nuclide concentrations and optically stimulated luminescence particle ages, in relation to what these reveal about particle mixing.Mudd and Yoo (2010) summarize this topic in relation to the use of reservoir theory (e.g., Bolin & Rodhe, 1973;Eriksson, 1971) to calculate expected variations in the probability distributions of particle ages and residence Key Points:• We present an Eulerian-Lagrangian description of disturbance-driven transport with depth-dependent mixing • The mean particle age generally is less than the mean residence time; these converge with strong mixing • The analysis provides the foundation for examining the structure of cosmogenic nuclide and OSL tracers

We examine probabilistic elements of how cosmogenic nuclide (e.g., 10 Be) concentrations and optically stimulated luminescence (OSL) particle ages are distributed within a soil mantle near a hillslope crest as a consequence of disturbance-driven transport and particle mixing. We use an Eulerian-Lagrangian algorithm in which fluctuating particle motions, representing depth-dependent mixing, are superimposed on a two-dimensional mean motion. The intensity of mixing is characterized by a Péclet number involving the vertical speed of particles entering the soil mantle at the soil-bedrock interface, the mechanically active soil thickness, and a particle diffusivity at the soil surface. With weak mixing, the vertical profile of 10 Be concentration reflects the strong influence of the mean motion in which particles spend much of their lives in the higher part of the soil column with higher 10 Be production rates. With increasing mixing intensity, the profile becomes linear, then uniform, and the vertically averaged concentration is larger than that expected with one-dimensional motion. With weak mixing, particles possessing a finite OSL age tend to remain near the soil surface; with increasing mixing they become more uniformly distributed with depth. Depth-interval-averaged OSL ages increase linearly with depth and then become uniform with strong mixing. With moderate to strong mixing, the probability distribution of OSL ages is approximately exponential with an average much less than the mean residence time of particles. The formulation is consistent with profiles of 10 Be concentrations and interval-averaged OSL particle ages compiled from published data, suggesting moderate to strong mixing in the cases examined.