## 1. Introduction

[2] In absence of overland flow-driven or wind-driven transport, the movement of soil on landscapes requires some kind of disturbance (Figure 1). This disturbance arises in many ways leading to a wide range of length scales of displacement. In clay-rich soils mantling sloping landscapes, periodic wetting of the ground may cause swelling and downslope flow, but even as the soils remain wet, progressively increasing grain resistance may halt motion. Drying and cracking then resets the contacts and allows another period of flow in the next wet season [*Fleming and Johnson*, 1975]. This cycle operates over some length scale of displacement. Simple wetting expansion and drying collapse through a season can incrementally shift near surface soils short distances downslope [e.g., *Kirkby*, 1967]. Seasonal cycles of movement by ice-driven processes shift soils and during spring melt can give way as continuously moving solifluction lobes which may carry soil a considerable distance even on gentle slopes [e.g., *Washburn*, 1973]. Biota work the soil at a wide range of scales, leading to dilation and displacement downslope. Insects and worms may cause minor local displacement but through their persistent and pervasive activity cause significant movement [e.g., *Darwin*, 1881]. Burrowing animals can make an extensive network of tunnels and push piles of dirt meters downslope. The collapse of large trees may rotate and expose their root system and displace clumps of soil meters downslope [e.g., *Norman et al.*, 1995; *Gabet et al.*, 2003]. The exposed, locally steep, tree throw mound and the smaller annual burrow mounds are sites of accelerated rain splash, raveling and fine scale biotic disturbance. In effect, the biotic roughening of the ground surface by the local mound formation leads to accelerated soil movement. On sufficiently steep granular soils, fire may suddenly remove particles stored behind fallen woody debris and unleash particles to ravel downslope [e.g., *Roering and Gerber*, 2005], sometimes tens of meters. Shallow landslides may also initiate, mobilize, and redeposit on hillslopes. Soil movement, then, arises through the sum of stochastic processes, influenced by seasonal and biotic cycles, the integral of which is a net flux of soil which tends to increase with increasing hillslope gradient. The individual particle step lengths resulting from disturbances will vary greatly.

[3] On gentle hillslopes there is field evidence [e.g., *McKean et al.*, 1993] that the mean soil transport varies linearly with local gradient. On steeper slopes, however, theory and limited observations suggest that transport increases nonlinearly with slope [e.g., *Roering et al.*, 1999]. Increasing field and theoretical evidence indicates that flux also depends on active transport depth [*Heimsath et al.*, 1999; *Roering*, 2008, *Furbish et al.*, 2009]. In particular, *Furbish et al.* [2009] show that a diffusivity-like coefficient which takes into account the local slope depth product produces a sediment flux which varies linearly with local gradient. Both linear and nonlinear flux laws assume that transport depends on some “local” slope, although we lack theory for what sets the length scale over which that slope should be determined. The disturbance by biota creates an irregular ground surface, with locally steep piles of loose soil that diffuse downslope across the mean slope (Figure 1). Hence, the slope at any point may not represent the actively contributing slope-driving processes, and cannot account for travel distances resulting from disturbances. If we could monitor every particle on a hillslope where these disturbance-driven processes (often placed together under the term “creep”) occur, it is possible that long transport events occur with a finite, nonvanishing, nonexponentially decaying probability such that the pdf of transport distances is heavy tailed [e.g., *Tucker and Bradley*, 2010]. This conception of soil transport may not be well represented by a transport expression that relates flux to a “local” slope. Moreover, the possibility of heavy-tailed particle travel histories makes selecting a meaningful mean slope for the application of such local laws problematic. To date, empirical fitting procedures (reducing variance by increasing the length scale of averaging while trying to maintain local profile curvature) have been used for the estimation of the local slope; common methods include polynomial fitting and Gaussian filtering [e.g., *Roering et al.*, 1999; *Lashermes et al.*, 2007].

[4] Here we propose an alternative formulation of sediment transport on hillslopes which relies on the notion of nonlocal computation of sediment flux, reflecting the fact that mass flux at a point on the hillslope is being influenced by disturbances well upslope and not simply linked to local slope (and soil depth). Our analysis may also explain the variance in flux rate for a given local slope observed in some studies. Our theory, although not derived from physical considerations (e.g., involving balances of forces and resistances), presents a general mathematical framework within which the upslope influences to the sediment flux at a given point can be cast into a continuum constitutive law for sediment transport. Specifically, we propose a nonlocal formulation of transport laws which relies on an integral (non-Fickian) flux computation which explicitly takes into account the upslope topography from any point of interest. The proposed nonlocal transport model includes linear diffusive transport as a special case.

[5] The paper is structured as follows. In section 2, we formulate the nonlocal constitutive law for sediment transport on hillslopes and in section 3 we derive its steady state equilibrium profile under appropriate boundary conditions. In section 4 we interpret observed hillslope profiles in the Oregon Coast Range, in the Appalachians of Maryland and Virginia, and east of San Francisco (California) within the nonlocal transport formulation. In section 5 we compare the linear, nonlinear and nonlocal transport models in several ways. The most important result is that the linear nonlocal model gives rise to a nonlinear relationship between sediment flux and local slope, akin to that observed on steep slopes. In section 6 we demonstrate that applying the nonlocal flux model to an ensemble of hillslope profiles produces significant variability of sediment flux for a given value of local slope as a result of variations in upslope topography. In section 7, we discuss the relationship between the shape of the probability density function of the sediment displacement lengths (which dictate the microscopic behavior of the transport process but which are typically not measured) and the parameter *α* of the nonlocal transport model (which describes the macroscopic properties of the transport). In section 8 we present some preliminary thoughts as to the ability of the nonlocal transport formulations to circumvent the scale dependence of sediment flux computed using local, nonlinear models. We conclude that our model shows the possibility that nonlocal sediment transport processes may be important on hillslopes and warrant more consideration both in field studies and theoretically. Our model anticipates more process-based considerations that would account mechanistically for biotic disturbance and it suggests that models for transport and weathering of colluvial soils and geochronological analysis of particles on steep hillslopes should consider the possible effects of nonlocal transport.