## 1 Introduction

[2] Recent work on the topic of sediment transport on hillslopes suggests the need to distinguish between “local” and “nonlocal” transport processes and associated formulations of the sediment flux [*Foufoula-Georgiou et al.*, 2010; *Furbish and Haff*, 2010; *Tucker and Bradley*, 2010; *Gabet and Mendoza*, 2012]. The mathematical distinction between local and nonlocal transport is clear. Namely, the definition of local versus nonlocal transport centers on whether the sediment flux at a contour position *x* can be expressed as a unique function of local hillslope conditions at *x* (for example, the local land-surface slope), or whether the flux at *x* also depends on hillslope conditions a significant distance upslope or downslope of this position, in which case the flux at *x* must be expressed in a way that takes into account these nonlocal (upslope or downslope) conditions. A physical distinction between local and nonlocal transport is less clear, owing to the variety of processes that contribute to sediment transport on hillslopes and to the widely varying length scales of sediment motions during transport. Local transport generally involves sediment motions that are sufficiently small wherein their collective contribution to transport at a position *x* can be characterized in terms of local hillslope conditions at *x*. In contrast, nonlocal transport involves sediment motions that are sufficiently large that it becomes necessary to functionally relate these motions to upslope (or downslope) conditions, which may be different from those at *x*.

[3] For example, as summarized in *Furbish et al.* [2009a] and *Furbish and Haff* [2010], there is a long record of work suggesting that the rate of downslope transport of soil material by creep on soil-mantled hillslopes is approximately proportional to the local land surface slope. Inasmuch as soil creep involves the collective, quasi-random motions of soil particles associated with small-scale bioturbation, effects of wetting-drying [e.g., *Kirkby*, 1967] or freeze-thaw cycles [*Anderson*, 2002], where particle motions span pore to many-pore length scales during creation and collapse of porosity within the soil column, then a (local) linear relation between transport and land surface slope is well founded [*Furbish et al.*, 2009b]. Similarly, the downslope drift of soil particles associated with rain splash can be viewed as local transport, inasmuch as the sediment flux arising from this downslope drift is related to the local land-surface slope [*Furbish et al.*, 2007, 2009a; *Dunne et al.*, 2010]. Both of these examples by definition represent local formulations of transport with respect to slope.

[4] Particularly in steeplands, local formulations of transport may not adequately characterize transport behavior. With increasing steepness, soil motions associated with ravel [e.g., *Gabet and Mendoza*, 2012], soil slips, transport by fossorial animals [*Gabet*, 2000], or tree throw, [*Norman et al*., 1995] and possibly transport by patchy, intermittent surface flows following fire [*Roering and Gerber*, 2005], can involve downslope travel distances that are much larger than those associated with small-scale bioturbation or freeze-thaw acting within the soil column, or those associated with rain splash [*Roering et al.*, 1999; *Foufoula-Georgiou et al.*, 2010]. In these cases, the downslope flux of soil material past a given contour position *x* can involve motions that originate from near to far upslope and traverse significant distances downslope of *x* before coming to rest. Key qualities of transport (e.g., the amount of soil material mobilized or the travel distances of soil particles) may depend on hillslope conditions at the site where the motions begin, which are different from those at *x*. Equally important, these motions are patchy and intermittent, and mostly involve dispersal of soil material over the land surface, where material moves rapidly in comparison with the slower bulk soil motion arising from creation and collapse of porosity. In this situation, an appealing approach is to describe the flux in terms of a convolution integral which in principle weights the effects of the conditions at all positions upslope and downslope of *x* that contribute to the flux at *x* [*Foufoula-Georgiou et al.*, 2010; *Furbish and Haff*, 2010]. This represents a nonlocal formulation of transport.

[5] The differences between local versus nonlocal transport have far-reaching implications for the evolution of hillslope topography. If, for example, the downslope soil flux at a position *x* is proportional to the local land-surface slope, then that part of the local rate of change in the surface elevation determined by the divergence of the flux at *x* is proportional to the local derivative of the slope, independent of upslope and downslope conditions. (Hereafter, we refer to the local derivative of the slope as the land-surface “concavity,” carrying magnitude and sign). Moreover, this divergence is the same at any two sites with the same concavity. In contrast, if the soil flux at a particular hillslope position depends on the local and upslope conditions, then the rate of change in the surface elevation at two sites with identical local conditions (e.g., slope and concavity) but different upslope configurations might be entirely different.

[6] Because the topic of nonlocal transport, and its relation to local transport, is relatively new, there is uncertainty in the literature regarding the definitions and distinguishing features of local versus nonlocal formulations of transport. Our first objective therefore is to step through the essential elements that characterize local versus nonlocal formulations. We then focus on the idea of sediment disentrainment following mobilization—how sediment particles come to rest after traveling some distance downslope (or upslope) in relation to land-surface conditions experienced by the particles during their motions. Indeed, the disentrainment process has a central role in determining the distribution of particle travel distances [*Furbish and Haff*, 2010; *Furbish et al.*, 2012; *Roseberry et al.*, 2012], which, in turn, forms the basis of the (nonlocal) convolution form of the sediment flux. We specifically demonstrate how a probabilistic description of sediment disentrainment represents a unifying physical basis for explaining how different distributions of particle travel distances naturally arise from assumptions regarding their travel and disentrainment—from the scale of particle trajectories resulting from raindrop impacts to the scale of particle motions that approach the full length of a hillslope. Our description of nonlocal transport therefore goes far beyond recent treatments of this topic, which are limited to specific cases of particle behavior [*Foufoula-Georgiou et al.*, 2010; *Furbish and Haff*, 2010; *Tucker and Bradley*, 2010; *Gabet and Mendoza*, 2012].

[7] In section 2, we qualitatively illustrate the ideas of local and nonlocal transport, focusing on the functional relation of the flux to land-surface slope. This section provides a concise description of the essence of what distinguishes nonlocal from local formulations of transport. In section 3, we provide a definition of the sediment flux on hillslopes, formulating it as a quantity that is averaged over space and time in a way that takes into account the patchy, intermittent sediment motions that characterize many transport processes in steeplands. We then show how the flux, expressed as a convolution integral, formally derives from the distribution of particle travel distances, and how the quantity being convolved consists of the sediment mobilization rate. In section 4, we formulate the disentrainment rate function and its relation to the distribution of particle travel distances, and we present an example involving the effects of land-surface slope on disentrainment. Here we illustrate how the distribution of travel distances may be determined by conditions at the origin of motion or, alternatively, how this distribution may be altered due to the fact that particles experience changing conditions during their downslope motions. In section 5, we return to the definition of the sediment flux expressed in the form of a convolution integral and show how the flux varies depending on the form of the distribution of travel distances and the underlying description of disentrainment rates. We then show how the convolution can be recast in the form of an advection-diffusion equation in the case where the distribution of travel distances possesses finite mean and variance. In section 6, we compare the convolution-integral form of the flux with the advection-diffusion approximation of this convolution and illustrate how both yield a nonunique relationship between the sediment flux and the local land-surface slope. This suggests that the evolution of hillslope profiles involving nonlocal versus local transport are likely distinct. We then show that the advection-diffusion approximation involving the land-surface slope may be reinterpreted as a local formulation if the idea of a local formulation is generalized to include more than the land-surface slope. In section 7, we return to the physical distinction between local and nonlocal transport. Because the (nonlocal) convolution form of the flux is quite general, this distinction hinges on the scale of particle motions relative to the scale of resolution at which the factors controlling transport are defined or measured.