## 1. Introduction

[2] A central goal in geomorphology is to develop mathematical formulas that describe the rate of sediment flow across the Earth's surface as a function of factors such as topography, lithology, ecology, and climate. One of the most common methods involves combining an equation for continuity of mass at a point on the Earth's surface with a transport rate function that depends on properties such as local surface gradient and water flux [e.g., *Kirkby*, 1971]. For reasons of practicality, many of these flux expressions (known as Geomorphic Transport Laws, or GTLs) [*Dietrich et al.*, 2003] rely on two key assumptions. The first assumption is that the average sediment mass flow rate can be accurately approximated from a combination of terrain attributes, such as gradient, and hydraulic attributes, such as mean boundary shear stress, without needing to account directly for the momentum or kinetic energy associated with individual moving sediment particles. The second and closely related assumption is that these topographic and/or hydraulic attributes need only be measured at a single point in order to generate an accurate prediction of mass flow at that point. For example, the widely used diffusion equation analogy for erosion and deposition by soil creep:

where *z* is land surface height, *t* is time, and *K*_{c} is a rate constant, is based on the assumption that the sediment mass flow rate at a particular point (*x*, *y*) is determined by the land surface gradient at that point, ∇*z* = ∂*z*/∂*x* + ∂*z*/∂*y*, irrespective of the surrounding topography. Similar concepts have been applied to river and marine erosion and sedimentation, among other environments [e.g., *Slingerland et al.*, 1994].

[3] The purpose of this paper is to examine the extent to which these assumptions are justified in geomorphic transport systems, and where they are not, to suggest an alternative approach. The problem is examined with a particular eye toward sediment transport on hillslopes, but the essence of our results can be generalized to other geomorphic settings. The paper is organized around three main ideas. Section 2 reexamines the basis for traditional local flux GTLs. We argue that there are circumstances in which the local flux assumption underlying many GTLs is not justified, and for which a different approach is needed. Section 3 introduces a particle-based numerical modeling approach that eliminates (and also illuminates) restrictions imposed by the local flux assumption. Particle-based simulations of hillslope profile evolution under soil creep highlight relationships between grain motion dynamics, bulk transport rates, and hillslope morphology, and illustrate conditions under which standard local gradient theory is, and is not, appropriate. The simplicity and efficiency of the particle approach also makes it an attractive alternative to numerical solutions of partial differential equations for modeling hilslope evolution. We also briefly compare the model's dynamics with the predictions of fractional diffusion theory [*Foufoula-Georgiou et al.*, 2010], noting some similarities and as well as some important differences. Finally, section 4 addresses the need for a more general analytical (continuum) approach to geomorphic transport modeling that can handle situations in which a local flux assumption would be inappropriate. We briefly outline one possible approach that uses a probabilistic framework based on a type of master equation [*Furbish et al.*, 2009; *Ganti et al.*, 2010].