Trouble with diffusion: Reassessing hillslope erosion laws with a particle-based model

Authors

  • Gregory E. Tucker,

    1. Cooperative Institute for Research in Environmental Sciences, University of Colorado at Boulder, Boulder, Colorado, USA
    2. Department of Geological Sciences, University of Colorado at Boulder, Boulder, Colorado, USA
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  • D. Nathan Bradley

    1. Cooperative Institute for Research in Environmental Sciences, University of Colorado at Boulder, Boulder, Colorado, USA
    2. Department of Geological Sciences, University of Colorado at Boulder, Boulder, Colorado, USA
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Abstract

[1] Many geomorphic systems involve a broad distribution of grain motion length scales, ranging from a few particle diameters to the length of an entire hillslope or stream. Studies of analogous physical systems have revealed that such broad motion distributions can have a significant impact on macroscale dynamics and can violate the assumptions behind standard, local gradient flux laws. Here, a simple particle-based model of sediment transport on a hillslope is used to study the relationship between grain motion statistics and macroscopic landform evolution. Surface grains are dislodged by random disturbance events with probabilities and distances that depend on local microtopography. Despite its simplicity, the particle model reproduces a surprisingly broad range of slope forms, including asymmetric degrading scarps and cinder cone profiles. At low slope angles the dynamics are diffusion like, with a short-range, thin-tailed hop length distribution, a parabolic, convex upward equilibrium slope form, and a linear relationship between transport rate and gradient. As slope angle steepens, the characteristic grain motion length scale begins to approach the length of the slope, leading to planar equilibrium forms that show a strongly nonlinear correlation between transport rate and gradient. These high-probability, long-distance motions violate the locality assumption embedded in many common gradient-based geomorphic transport laws. The example of a degrading scarp illustrates the potential for grain motion dynamics to vary in space and time as topography evolves. This characteristic renders models based on independent, stationary statistics inapplicable. An accompanying analytical framework based on treating grain motion as a survival process is briefly outlined.

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:

equation image

where z is land surface height, t is time, and Kc 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].

2. Geomorphic Transport Laws Revisited

2.1. Meaning of the Exner Equation

[4] Continuity of sediment mass for a point on the Earth's surface is often written in the form of the Exner equation:

equation image

where z is land surface height relative to a datum in the underlying rock column and qs is a two-dimensional vector that expresses the magnitude and direction of bulk volumetric sediment transport per unit width [L2/T]. We can gain insight into the physical meaning of qs with help from the following thought experiment. Imagine standing at the finish line of a race with a stopwatch in hand, and an assistant with a scale just beyond the finish line. As each runner crosses the finish line, you mark the time and your assistant weighs the runner. After a certain time interval, Δt, has passed on your stopwatch, you add up the masses of the runners who crossed the line during the monitoring period. The result is an average mass flux, Qm = Mrunnerst. The magnitude of the flux of runners will depend on their speeds and on their concentration in the vicinity of the finish line. “Vicinity” in this case has a very specific meaning. Suppose that a runner is moving at a steady speed u, and that he crosses the finish line just at the end of our measurement period. At the moment you started the stopwatch, he would have been a distance uΔt from the finish line. This distance is a natural length scale associated with our averaging procedure. The time scale Δt must be large enough to ensure a smooth average but small enough that the average will not drift substantially during a single Δt.

[5] If we replace runners with sediment grains, and normalize by sediment bulk density, we have a working definition of qs. Seen in this light, qs can be understood as a statistical quantity: the average bulk volume of grains per unit width that passes a given point during a small time interval Δt. The distance over which our measurement records grain motion is λ = uΔt (where u of course is now the velocity of a sediment particle). Having established a working definition (or at least a word picture) for qs, the next step is to consider mathematical expressions for qs.

2.2. Locality Assumption

[6] To derive a mathematical expression for the time rate of change of surface height z, the most common approach is to combine the Exner equation (2) with a GTL that describes a functional form for qs. One of the most common GTLs is an expression for hillslope sediment transport that relates sediment transport rate to the product of local slope and specific overland flow discharge, q [L2/T]:

equation image

where K is an efficiency factor, and m and n reflect assumptions in the derivation of the expression and usually fall in the range 1–2 [e.g., Carson and Kirkby, 1972; Prosser and Rustomji, 2000]. The case m = 0, n = 1, K = Kc yields the well-known slope linear soil creep law that is the basis for equation (1).

[7] A critical ingredient of transport laws like (3) is the locality assumption: the formula implies that the sediment transport rate at a point can be predicted from the slope and discharge measured at that same point. This amounts to assuming that both the momentum and the concentration of particles depend only on the slope and discharge (and perhaps other properties) within the immediate vicinity of the point in question, irrespective of the transport history those particles may have experienced when they were farther away. To be slightly more precise, the locality assumption holds that (1) the characteristic distance λ is small relative to the scale of the system of interest (such as the length of a hillslope) and (2) particle momentum and concentration, and therefore qs, at a point x are dictated by properties (slope, discharge, etc.) within a small strip of ground between xλ and x + λ.

[8] The locality assumption is widely used in transport theory across the physical sciences. Some classic examples include mass transport by molecular diffusion (Fick's law), momentum transport by molecular interaction in viscous fluids (Newton's law of viscosity), heat conduction (Fourier's law), flow of fluid in porous media (Darcy's law), and contaminant transport by porous media flow (standard advection-dispersion theory). In each of these examples, a net flux of mass, momentum, or energy is proportional to the local spatial gradient of that quantity. Furthermore, each example describes a bulk flux that arises from quasi-random motions in a many particle system. An underlying assumption in these transport laws, as in statistical mechanics generally, is that one does not need to know the position and momentum of every particle in order to predict with high accuracy the state of the system at a point. Thus, the widespread use of local, gradient-based geomorphic transport laws like (3) falls comfortably within a well-established tradition. Before examining the conditions under which the locality assumption is justified for hillslope transport processes, it is useful to take a closer look at the general statistical basis for continuum transport laws.

2.3. Sediment Motion as a Random Walk

[9] If one knew the position and momentum of every sediment grain, and the effective rules for interaction between grains, one could potentially compute qs from first principles. In fact, this is the basis for the discrete element method, which has been applied to a number of problems in geomorphology [e.g., Haff and Anderson, 1993; Campbell et al., 1995; Schmeeckle and Nelson, 2003; McEwan et al., 2004]. On the other hand, there are obvious practical problems in solving the full equations of motion for all of the sediment grains in a landscape. One alternative is to treat grain motions statistically, by analogy to the statistical mechanics of gases. Einstein [1937] pioneered the treatment of bed load sediment motion as a random walk process, and the approach has been adopted for problems including hillslope transport [Jyotsna and Haff, 1997; Foufoula-Georgiou et al., 2010], sediment dispersion [e.g., Sayre and Hubbell, 1965; Bradley et al., 2010; Ganti et al., 2010], and bioturbation [e.g., Meysman et al., 2003, 2008]. More generally, it has been recognized for over a century that there is a close connection between random walk processes and diffusion phenomena [see, e.g., Hughes, 1995]. For example, consider a group of particles that undergo random displacements (“hops”) in one dimension every time interval δt, with a mean displacement length μ and variance σ2. In the limit of many particles and many hops, the bulk behavior at scales much larger than μ and δt is very accurately described (in one dimension) by an advection-dispersion equation:

equation image

where C(x, t) is the concentration of particles [M/L3] at point x and time t, v is drift velocity and D is an effective diffusion coefficient. The drift and diffusion coefficients are directly related to the random walk statistics by

equation image
equation image

The corresponding mass flux [M L−2 T−1] is

equation image

which, for a symmetric random walk (μ = 0), bears an obvious similarity to Fick's law and Darcy's law, for example. Notice the locality assumption implied by (7): the mass flux at a particular point depends on the concentration gradient in an infinitesimally small region around that point.

[10] This illustration of the connection between random walks and local diffusion is just one simple example from a very large literature on random walks [see, e.g., Hughes, 1995; Metzler and Klafter, 2000, 2004]. For our purposes, there are two key points. First, continuum, locally based geomorphic transport laws like (3) can have a solid statistical physics foundation, as long as the characteristic time and space scales of grain motion are small relative to the scales of the geomorphic system of interest. Second, it follows that GTLs for many sediment transport processes could in principle be derived from the statistical physics of sediment grain motion, just as Einstein [1937] derived his famous bed load transport formula from random walk theory. Such an approach does not eliminate the need for empiricism, but it does effectively shift the level of empiricism a step closer to the process level. Instead of measuring bulk transport coefficients (such as K in 3) one measures the frequency, mean, and variance (or their heavy-tailed equivalents) [e.g., Ganti et al., 2010] of particle displacement events, and ultimately tries to establish the dependence of those factors on various physical controls [e.g., Furbish et al., 2007]. However, such an approach would require demonstrating that the motion statistics do in fact satisfy the locality constraints. As it turns out, a growing number of physical, chemical, and biological systems have been discovered that do not satisfy these constraints.

2.4. Locality and Nonlocality in Transport Systems

[11] The locality assumption embedded in gradient-based transport models like Fick's law is appropriate when there is a clear gap between the microscales associated with particle motion and the macroscales associated with the system of interest. In Fickian diffusion, the microscopic length scale is the mean free path of a molecule. In homogeneous porous media flow, it is the characteristic scale of a representative elementary volume: larger than a grain but much smaller than an aquifer [e.g., Middleton and Wilcock, 1994]. This gap in scale between micro and macro allows one to express averages of mass flux in terms of properties defined at a single point: properties such as gradient and conductivity are measured over an area so small relative to the size of the system of interest that it can for all practical purposes be treated mathematically as if it were a point.

[12] However, not all transport systems posses a clear gap in space and time scales between particle motions and the system as a whole [West et al., 2003]. During the past several decades, a number of physical and biological transport systems have been identified that exhibit nonlocal behavior in space and/or time. Spatial nonlocality arises when particle velocities are correlated in time or when particle travel distances are likely to be large relative to the scale of the system [Schumer et al., 2001, 2009]. More formally, spatial nonlocality occurs when the mean square (or second moment) of particle travel distance diverges (i.e., tends to infinity), which is a defining characteristic of probability distributions that have a “fat” or “heavy” tail. Nonlocality in time is similar and can be caused by long-term correlation (“long memory”) or by long periods of immobility.

[13] In the geosciences, one of the best known examples of nonlocal transport is the motion of a tracer through a heterogeneous aquifer. The nonlocal behavior can be caused either by anomalously fast transport pathways or by traps that hold the fluid immobile for long periods. For example, Benson et al. [2001] modeled the nonlocal dispersion of a groundwater tracer through the aquifer at the MADE site using a heavy-tailed probability distribution of travel distance. Berkowitz et al. [2002] argued that nonlocal transport is better modeled by assuming a heavy-tailed distribution of trapping times. Sornette [2004] attributes nonlocal transport in fractured aquifers to a heavy-tailed distribution of fracture lengths, which gives rise to a broad distribution of trapping times. Kirchner et al. [2000] found evidence of nonlocal transport in the distribution of travel times of rainwater through a catchement. Examples of nonlocal motion from biology include the travels of spider monkeys through Mexican forests [Ramos-Fernández et al., 2004] and the flight of an albatross [Viswanathan et al., 1996]. These are just a few of many examples from diverse physical and biological systems (see, e.g., Metzler and Klafter [2000, 2004], West et al. [2003], and Sornette [2004] for reviews).

[14] It is important to recognize that nonlocality does not imply any failure of mass or momentum conservation; rather, it simply means that one cannot accurately estimate the concentration and momentum of particles at a particular point based solely on properties measured at that point. To return again to the foot race example, a nonlocal runner would be one whose speed and direction could not be predicted solely based on, say, the slope of the hill or the roughness of the ground immediately beneath her feet. With a large group of nonlocal runners, one could not even predict their average flux at a particular point without knowing something about, for example, the distance they have already covered, the roughness of the preceding segments, the number who started the race, the day's temperature history, and so on.

2.5. Do Geomorphic Systems Exhibit Nonlocality?

[15] The recognition that so many transport systems show nonlocal, heavy-tailed behavior raises the question of whether sediment transport may behave similarly, and under what circumstances. What sediment systems possess clear scale gaps that permit the locality assumption and what systems lack such gaps? There is mounting evidence that nonlocal transport can occur in fluvial sediment transport by running water, as evidenced by anomalous scaling of variance with time [Nikora et al., 2001, 2002] and heavy-tailed sediment velocity distributions [Bradley et al., 2010]. Kirkby [1991, 1992] noted the importance of long-distance particle transport by introducing GTLs that include a parameter for sediment transport distance, while Meysman et al. [2003, 2008] developed nonlocal models for sediment mixing by bioturbation. Stark et al. [2009] suggested several reasons why nonlocal transport may be the norm rather than the exception in fluvial systems, and developed a fractional derivative model for longitudinal profile evolution that reproduces typical observed characteristics such as slope area scaling. Foufoula-Georgiou et al. [2010] showed that a fractional (nonlocal) model of equilibrium hillslope sediment transport is consistent with observed hillslope profiles.

[16] As an example, consider the case of soil creep. Soil creep, the gradual (and sometimes not so gradual) downslope motion of soil, arises from quasi-random displacements of soil particles by processes such as animal burrowing, vegetation growth and decay, raindrop impact, dry ravel, expansion and contraction of soil due to wetting and drying, and growth and melting of interstitial ice lenses. (Note that this definition of soil creep is different from true rheologic creep, in that it involves net motion arising from particle mixing under gravitational pull, rather than plastic deformation of a coherent mass). On relatively gentle slopes, most of these processes have characteristic length scales that are small relative to the length of the hillslope. For example, soil excavated by burrowing animals is typically displaced on length scales of order centimeters to tens of centimeters, while hillslopes typically have characteristic length scales of tens to hundreds of meters. The net rate of downslope transport by these processes depends strongly on the local topographic gradient, as expressed by the linear soil creep law:

equation image

For relatively gentle slopes, the short length scale of grain displacement relative to hillslope length appears to justify the use of a local gradient operator.

[17] As slope angle increases, however, there is evidence for substantial increases in both the frequency and average displacement distance of transport events in creep-related processes, including dry ravel [Gabet, 2003] and raindrop impact [Furbish et al., 2007], as well as acoustically stimulated disturbance in the laboratory [Roering et al., 2001; Roering, 2004]. For example, Gabet's [2003] flume experiments on ravel showed a nonlinear increase in mean transport distance with gradient. Moreover, as slope gradient approaches the angle of repose, various forms of landsliding can come into play, with displacement length scales comparable to that of the hillslope as a whole. Thus, it is expected that as gradient increases, the locality assumption behind equation (8) becomes increasingly questionable. In section 3, the nature and implications of this transition from local to nonlocal transport are studied using a simple model of sediment particle motions on a hillslope.

3. Particle-Based Modeling of Hillslope Evolution

3.1. Model Description

[18] The model is designed as a simple analogy for a host of different processes that displace sediment grains on hillslopes. It is similar in spirit to the famous sand pile model of Bak et al. [1987] in that particle motion depends on local microtopography, but in this case the system is driven not by mass input from above but by random, gradient-dependent disturbance-induced motions of the particles themselves. It is also similar to the particle model used by Jyotsna and Haff [1997] to study smoothing and roughening processes on slopes.

[19] The hillslope is represented as a pile of two-dimensional particles (Figure 1). Each particle has a dimensional width ε and a dimensionless width of unity. These particles undergo quasi-random motions according to the following rules: (1) during each iteration, a particle and a direction (left or right) are selected at random; (2) depending on the outcome of a random number toss (described below), the selected particle may move one particle length in the given direction (Figure 1); (3) the particle will continue moving in the same direction and with the same probability dependence until coming to rest or exiting the base of the slope. Under rule 3, the probability of continued motion is recalculated from the same distribution as for initial motion, though if desired one could assign a higher probability to a moving particle as a way of accounting its momentum. These rules are intended to capture, in a simple way, two observed aspects of soil creep transport: first, motion is initiated at or near the surface by a quasi-random disturbance event, such as growth of an ice crystal, excavation of a burrow, or uprooting of a tree, and second, the point-to-point displacement of the disturbed sediment depends on the local microtopography. The probabilistic, slope-dependent treatment of grain motion events is consistent with (though simpler than) the rate process theory approach used by Roering [2004] to derive grain velocity profiles in a disturbance-driven granular shear flow. The random direction is intended to mimic the fact that many disturbance events involve uphill as well as downhill motion; for example, trees and the soil attached to their roots sometimes fall upslope, while burrowing animals sometimes place excavated soil on the upslope side of their burrows. Note that the model has a built-in bias toward downhill events, because uphill events are more likely to fail the first motion test and remain stationary.

Figure 1.

Schematic illustration of the particle-based hillslope model. Gray represents particle selected for motion, and dashed outline represents potential destination.

[20] The slope is N particles long plus an additional boundary particle on each side, so that the dimensionless length of the domain from the center of the left boundary particle to the center of the right boundary particle is L = N + 1. The model cycles through a series of global iterations. The dimensional duration of a global iteration is τ, while the dimensionless equivalent is unity. During each global iteration, N sites are selected at random as candidates for particle motion. For each site i selected, a direction (d, equal to 1 or −1) is also selected at random. The probability p that the top particle will move one space in direction d depends on the altitude of the particle, Zi, relative to that of the adjacent particle, Zi+d, as follows (Figure 1):

equation image

The choice of probability values is not critical to the model's behavior, as long as the likelihood of motion increases with local height difference; these particular values were chosen to give, on average, a motion probability that ranges from 0 to 1 as slope angle increases from zero to 45 degrees.

[21] If initial motion occurs, the particle is moved one unit and the probability of motion is tested again. The process repeats until either the particle comes to rest or it reaches one of the model edges. A single “hop” event consists of the total distance traveled from start to stop, and can range in size from one unit to the total length of the domain. Note that not all disturbance events will generate hops, because particles that fail their first motion test will not move at all.

[22] One can explore a variety of different boundary conditions. We begin with the case in which the system is driven by steady lowering at the two boundaries (Figure 1). The base level is lowered by one unit (or equivalently, the interior is raised) every T global iterations, so that the dimensional lowering rate is Ud = ε/Tτ, and the dimensionless equivalent is U = 1/T. Starting from a flat surface, the system eventually reaches an equilibrium in which the total flux of particles from the hillslope, qs (particles per global iteration), balances the input from below, or qs = N/T. The second boundary condition examined represents the case of scarp degradation, in which one half of a horizontal domain is raised to an initial elevation Z0, forming a scarp which then begins to degrade. The third boundary condition studied is the case of a hypothetical cinder cone, in which particles are ejected with random launch angle θ and initial velocity v0 from the point x = 0 at the left edge of the domain. Ejected particles follow ballistic trajectories until striking the ground surface. Upon landing, a direction of motion is chosen on the basis of the local topography: at landing point i, the direction of motion is to the right (increasing x) if Zi−1Zi+1 and to the left otherwise. The particle is then tested for motion using the rules described above, until it comes to rest or exits the domain. The height of the launch point at i = 1 is maintained at the height of the adjacent particle; in other words, Z1 = Z2.

3.2. Steady Hillslope Topography and Transport Statistics

[23] The topography and motion statistics that emerge from the base level–driven model show a range of behavior that depends on the dimensionless parameters L and U. These two parameters govern the dimensionless steady state relief Zmax. At small U, and when L is relatively short, hillslope shape is parabolic, reflecting the diffusive character of the system (Figure 2). The mean hop length (excluding zero-motion events) is on the order of 3.2 particle widths, and the probability distribution of displacement length has a thin (exponential) tail (Figure 3).

Figure 2.

Simulated hillslope with L = 1280 and T = 1000. Height and horizontal distance are normalized to total hillslope length.

Figure 3.

Normalized frequency distribution of hop length, λ, for L = 1280 and T = 1000. Note the exponential tail. In this example, about three quarters of the time the selected particle fails to move at all (shown on the plot as zero hop length). The mean hop length, excluding zero-motion events, is 3.2; the mean of all events, including zero motion, is 0.81.

[24] At relatively low L and U the model hillslope shows a strong scale separation between average hop length (order 100 particles) and system length (order 103 particles in this example). Transport rates at steady state are well approximated by the linear creep law (one-dimensional version of equation (8)) with a constant coefficient D. At steady state, with the boundary conditions used in Figure 2, the expected relationship between transport rate and gradient is

equation image

where x is distance from the left-hand boundary and qs is positive to the right. A good prediction of particle discharge at a point can be obtained by measuring the gradient around that point over a length scale much larger than a single particle but much smaller than the length of the hillside (Figure 4), indicating that a locality assumption would be reasonable under these conditions provided that one chose an appropriate scale. Using too small a length scale would generate a large degree of scatter, while using too large a length scale would amount to fitting a straight line to a curving hillside. The expression of the linear transport scaling shown in Figure 4 can be seen in the topography: the solution to equation (10) is a parabola, as expected.

Figure 4.

Steady state particle transport rate (qs) as a function of surface gradient for the low L and U case shown in Figure 2. Particle discharge is measured by tracking the cumulative number of particles passing through each location, while slope is measured over a distance of 24 particles centered at each point. The discrete jumps in slope represent increments of 1/24. Straight line is the relationship predicted by equation (10) with D = 2.6ε2/τ calculated by fitting a parabola to the crest height in Figure 2.

[25] In contrast to the diffusive behavior at lower values of L and U, when these parameters are large, model hillslopes become planar in form (Figure 5). In this situation, the system exhibits angle-of-repose behavior in which many particles travel the full distance between their point of origin and the system boundary in a single hop event. The hop length distribution (Figure 6) is broad, and truncated at the system half-length. The fact that the hop length distributions are sensitive to the overall system scale is indicative of nonlocality: there is a high probability that any particle passing through a given point on the slope will have begun its hop a significant distance upslope. Thus, the situation illustrated in Figures 5 and 6 lacks a scale gap between particle motions and system length, and the transport rate is not accurately described by a local linear gradient law (10). Interestingly, the model's high-gradient behavior is consistent, at least qualitatively, with fractional diffusion theory [Foufoula-Georgiou et al., 2010], while its low-gradient behavior is consistent with standard diffusion theory.

Figure 5.

Simulated hillslope with N = 1280 and T = 10.

Figure 6.

Normalized frequency distribution of hop length for T = 10 and L = 160, 320, 640, 1280, 2560, and 5120. Note the dependence of the distribution on the scale of the hillslope (L).

[26] The relationship between mean transport rate and gradient at high U (Figure 7) resembles nonlinear flux gradient curves that have been used to model hillslope evolution, with a linear relationship between local gradient and flux when gradients are relatively low and a nonlinear relationship when gradients approach a threshold (in this case on the order of unity) [Hanks and Andrews, 1989; Anderson and Humphrey, 1989; Howard, 1994; Gabet, 2000; Roering et al., 2001]. However, the particle model's behavior calls into question the use of a local (albeit nonlinear) transport law. In the model, as gradient increases, the characteristic motion range undergoes a transition from being largely independent of system length to being largely controlled by system length. Particles passing through a given point have an increasingly high probability of originating from a distant (≫ε) position. Thus, local gradient plays a decreasing role in controlling flux as overall slope angle increases. Under these conditions, a locality assumption would not be justified, because the flux at a point x depends not just on the gradient of immediately surrounding topography but on the entire portion of the slope between x and the ridge top. For example, if one were to remove the upper half of the hillslope, the transport rate along the lower slope would immediately drop without any change at all in the local gradient. This length-dependent flux behavior is also a feature of the fractional-order hillslope transport model explored by Foufoula-Georgiou et al. [2010].

Figure 7.

Steady state particle transport rate (absolute value of qs) as a function of surface gradient magnitude for the high L and U case shown in Figure 5. Particle discharge is measured by tracking the cumulative number of particles passing through each location, while slope is measured over a distance of 24 particles centered at each point. The discrete jumps in slope represent increments of 1/24.

3.3. Transport Properties Evolve in Time and Space

[27] An interesting property of the model, and probably of many natural hillslopes, is that the transport statistics depend on the topography, and can therefore vary in both time and space. This makes the geomorphic problem somewhat different from the class of problems for which the transport statistics are stationary and independent of the state of the system. For example, transport and dispersion in heterogeneous aquifers have been modeled using heavy-tailed velocity distributions in which the tail of the distribution depends on the aquifer's geology but not on variables that typically change substantially over time, such as flow rate, hydraulic head, and tracer concentration [e.g., Benson et al., 2001]. Thus, the analogy between an aquifer and a hillslope is imperfect because the transport dynamics on a hillslope can change significantly as the topography evolves.

[28] To illustrate the potential for coevolution of topography and transport statistics, Figure 8 shows an example in which the particle model is configured to represent a degrading fault scarp. Scarp degradation is a classic problem in hillslope geomorphology, and is widely used as a method for dating events such as earthquakes and sea and lake level high stands. Initially, the model scarp is so narrow (one particle wide) that horizontal particle motions are short range and the process has a local, diffusive character (Figure 8a). As the scarp widens, a region of long-distance motion develops in the area where slopes are relatively high. This region of long-range, nonlocal transport gives rise to a pronounced asymmetry in the scarp form, with a curvilinear upper face and a planar to slightly concave upward rampart (Figure 8b). Similar asymmetric forms have been documented on natural degrading scarp forms, such as marine terrace risers [e.g., Rosenbloom and Anderson, 1994]. Indeed, hillslope profiles with this general form are widespread in landscapes prone to scarp retreat, such as the Colorado Plateau in the western United States. The origins of such asymmetries have been attributed to various factors, such as lithology contrast or limited regolith production. These interpretations may well be correct, but the simulation in Figure 8 suggests that scarp profile asymmetry can also arise purely in response to long-distance grain motions on a steep, evolving scarp face bounded by low-gradient upper and lower surfaces.

Figure 8.

Particle model simulation of an evolving fault scarp in (a) its initial state and after (b) 300, (c) 700, and (d) 10,000 global iterations.

[29] As the scarp continues to degrade, the steep scarp face widens and the region of long-distance transport widens accordingly (Figure 8c). As the gradient continues to shrink, however, the particle hops shorten, ultimately returning to purely short-range, diffusive-style behavior (Figure 8d). Thus, we have a situation in which close feedback between topography and particle motion statistics creates a system that can exhibit both local and nonlocal behavior at different times and places. This raises an interesting challenge regarding how one models such a system. We discuss two possible solutions below. First, however, it is worth considering the particular nature and origin of nonlocal behavior in this type of geomorphic system, as it suggests one possible strategy for mathematical modeling.

3.4. Ballistic Versus Near-Ground Motion

[30] A system of particles may be said to exhibit nonlocality when relatively distant particles influence one another on time scales that are short relative to the macroscopic time scale. In the case of geomorphic systems, two working definitions of “relatively distant” might be (1) distances much longer than the characteristic wavelength of microtopography (if there is one) and (2) distances over which macroscopic properties such as the mean angle of a hillslope or the width of a river channel show a detectable trend: as, for example, the systematic downslope increase in gradient on a convex upward hillslope like the one in Figure 2.

[31] Interestingly, there are at least two end-member forms of long-distance, nonlocal transport that can occur in a geomorphic system: one that is truly ballistic, and one that involves near-ground motion and close interaction with microtopography. To appreciate the difference between these forms of long-distance transport, consider the example of a volcanic cinder cone (Figure 9). At a point on the flank of a cinder cone, particles can arrive by two pathways. One path involves ballistic trajectories: a particle is ejected from the cone with some initial velocity and ejection angle, and it soars through the air until striking the ground (Figure 9, path A). The particle's arrival is independent of the details of topography between the vent and the point of impact, and in that sense it might be said to be strongly nonlocal. The second pathway involves motion along the ground, as in the example of a particle that strikes the ground high up on the steep flank of a cone and rolls downward some distance before coming to rest (Figure 9, path B). Although the distance of rolling may be large relative to the width of the cone, and thereby satisfy our definition of nonlocal, its motion is strongly contingent on the microtopography along the pathway from the impact point to the resting point. In that sense, near-ground motion may be thought of as weakly nonlocal. The essence of the difference is this: if one were to excavate a trench between the vent and the base of the cone, the ballistic particles would not be affected in the least, but the rolling particles would be trapped because they are sensitive to the short-wavelength topography along the path of motion. Analytical models of near-ground sediment motion should ideally account for this trapping potential.

Figure 9.

Particle simulation of a cinder cone, illustrating the difference between strongly nonlocal, ballistic pathways (path A) and weakly nonlocal, along-ground pathways (path B).

[32] The examples of a fault scarp (Figure 8) and ballistic versus ground motion (Figure 9) also illustrate a potential challenge in the application of fractional derivatives to geomorphic systems [e.g., Bradley et al., 2010; Foufoula-Georgiou et al., 2010; Stark et al., 2009]. In fact, the question of whether one could derive a fractional transport model from the statistics of sediment motion was one of the original motivations for creating the particle model. As shown for example by Schumer et al. [2003], a fractional derivative can be seen as a weighted function of a variable (such as gradient) over a broad area, and in that sense represents a powerful tool for modeling nonlocal transport [Stark et al., 2009; Foufoula-Georgiou et al., 2010; Ganti et al., 2010]. Moreover, fractional derivatives need not be purely heuristic devices for fitting data; just as one can derive an integer-order advection-diffusion equation from Brownian motion, one can also derive a fractional equivalent from a heavy-tailed random walk process. Therefore, if one had reason to believe that sediment motions for a particular process were heavy tailed, one could obtain a corresponding fractional derivative to express their transport. However, a key underlying assumption in using a derivative of order α is that the jump statistics are stationary in space and time. This condition may not apply to near-ground motion on an evolving topography, because the hop length distribution is a function of topography itself.

[33] In some cases, fractional models may be able to account for spatially varying α indirectly, via the imposed boundary conditions. Foufoula-Georgiou et al. [2010] noted that when a fractional diffusion equation is applied to a hillslope profile, the boundary conditions lead to an effective value of α that declines from 2 at the ridge, reaching a constant <2 only far from the ridge. However, such behavior would not be sufficient to account for the transport dynamics in the scarp example (Figure 8), where the scarp face is separated from its upper and lower boundaries by low-gradient, Fickian domains, and where nonlocal transport occurs only in the time period during which the scarp is relatively steep. An interesting question for further research is whether it is possible to derive composite fractional/integral transport models in which the effective value of the tail exponent α responds dynamically to the topography itself, such that one could transition smoothly between Fickian and heavy-tailed behavior as the landform evolves.

3.5. A Useful Alternative to Differential Equations?

[34] Particle-based models like the one developed here provide an interesting alternative to traditional numerical solutions to continuum equations. Similar rule-oriented, particle-based, and cellular models have become increasingly popular in geomorphic analysis [e.g., Anderson, 1990; Chase, 1992; Werner and Hallet, 1993; Jyotsna and Haff, 1997; Crave and Davy, 2001; Favis-Mortlock et al., 2000; Kessler and Werner, 2003]. The model presented here is able to capture a surprisingly rich range of behavior, including local and nonlocal dynamics and a variety of hillslope forms (Figures 2, 5, 8, and 9) using a very simple framework. Implementation is straightforward, and computation times are reasonable (in fact, performance is generally better than comparable forward difference solutions to 1D linear or nonlinear diffusion equations). A common objection to rule-based models is that their parameters cannot be directly measured, but there are now a number of studies demonstrating the feasibility of measuring grain motions in the field and the laboratory [Schmidt and Ergenzinger, 1992; Hassan et al., 1999; Habersack, 2001; Ferguson et al., 2002; McNamara and Borden, 2004; Roering, 2004; Furbish et al., 2007]. Displacements of grain aggregates such as burrow mounds can also be estimated in the field [Gabet, 2000]. Discrete particle models, which use contact mechanics of particle-particle interactions in combination with classical equations of motion, have also shown promise as sources of grain motion statistics [Radjai et al., 1999; Silbert et al., 2001]. Thus, particle-based sediment transport models are not necessarily any less fundamental than differential equation models, and they offer some interesting advantages.

4. A Probabilistic Strategy for Continuum Modeling of Long-Distance Transport on Hillslopes

[35] Despite the advantages of particle-based models, there are good reasons to pair them with analytical, continuum-type GTLs. Among other considerations, analytical expressions have a compact form and can be solved exactly for certain cases. This section briefly examines one alternative method for formulating analytical GTLs in cases where long-distance (or long-memory) sediment motion renders diffusion-type equations inapplicable.

[36] Furbish et al. [2009] introduced the use of a master equation to describe geomorphic systems. The type of master equation relevant for our purposes is a probabilistic expression for continuity of mass at a point on the earth's surface, given random arrivals and departures of sediment grains. The framework involves three ingredients: (1) a form of mass continuity that separates local erosion and deposition rates; (2) a description of the potential pathways along which a sediment grain may move, given that it is eroded at a particular location; and (3) a probabilistic treatment of the distance that a grain will move along a particular path. A complete development and application of this approach lies beyond the scope of this paper; here, the aim is simply to provide a working outline that begins to address the problem of coevolving morphology and grain dynamics.

4.1. Mass Continuity

[37] Although mass continuity for geomorphic systems is often written in terms of the divergence of sediment discharge, ∇ · qs, some authors use a framework in which local erosion and sedimentation rates are treated independently [e.g., Kirkby, 1992, 2003; Crave and Davy, 2001; Veldkamp and van Dijke, 2000; Ganti et al., 2010]. Using this approach, the time rate of change of surface altitude z at a point is given as

equation image

where equation image is the rate of vertical motion relative to a fixed datum (such as the rate of tectonic uplift relative to mean sea level), equation image is the deposition rate, and equation image is the erosion rate. For the model in Figure 2, for example, equation image is equal to one particle per T iterations (ε/), equation image is equal, on average, to one particle per iteration (ε/τ) times the probability that a particle at that locale will move, and equation image depends on the erosion rates at all other locations and the probability that particles eroded at those locations will come to rest at the point in question. The deposition flux is the most complicated of the three, and it is the one we focus on next.

4.2. Potential Transport Paths

[38] Imagine that a sediment grain is set in motion by some event, such as a raindrop striking the soil [e.g., Furbish et al., 2007] or a turbulent sweep imparting momentum to the grain. A potential transport path is defined as a potential trajectory that such a sediment grain could follow after being mobilized at a particular starting point r = (x, y). Each transport path T is associated with a starting location, r0 = (x0, y0), and functions x(s, T) and y(s, T), which describe the x and y positions along path T as a function of total distance s from the starting point. The function s(r, T) represents the distance along path T from the starting point r0 to point r (assuming that path T includes r).

[39] In a quasi one-dimensional framework like that of the particle model illustrated above, there are only two transport paths associated with each location: one to the left and one to the right. On a topographic surface, an infinity of potential transport pathways radiate out from each (x, y) location. In practice, however, we can anticipate that many of these will be so unlikely as to be ignored. For instance, a particle undergoing motion in a river might reasonably be assumed to follow the course of the river downstream rather than skipping over the divide to the next drainage basin.

[40] With this framework in mind, one way to express the average depositional flux equation image (r) at a particular point r is as follows. One adds up (integrates) the contribution from every other relevant origin point, r0, on the landscape. For each contributing origin point, assuming that there is a countable number of paths that intersect point r, one adds up the contributions from each path. The average sediment flux from point r0 to point r along path T is the erosion rate at point r0 times the probability that particles will follow path T times the probability that they will land at point r. Written in symbolic form:

equation image

where PT (Tr0) is the probability a particle will follow path T given that it begins motion at point r0, s (r, T) is the pathwise distance to point r along path T, and PS (s (r, T), T) is the probability that a particle following that path will move exactly the distance from r0 to r. The integral is taken over the set of all contributing origin points, R. The next step is to define the probability density function PS describing how far a particle will move along a given path.

4.3. Sediment Motion as a Survival Process

[41] Stark and Guzzetti [2009] used the concept of a survival process, a probabilistic framework that expresses the likelihood that an entity will “survive” a given series of random events, in an analysis of landslide development. The same concept can be used to develop the probability density function PS(s, T) for sediment motion along a particular transport path T.

[42] Consider a sediment grain following a trajectory that crosses a series of potential trapping points. Each trapping point represents a stable position at which a particle could come to rest, such as a trio of clasts on the bed of a gravel river or a shallow depression on a hillslope. Let the probability of being trapped while crossing a particular position i be pi. In principle, pi should depend on the depth of the trap relative to the size of the grain [Roering, 2004] and possibly also on the grain's momentum. The probability that a grain will successfully traverse N locations without becoming trapped is

equation image

The probability of being trapped at exactly the Nth location is equal to the probability of successfully crossing the first N − 1 and then sticking at the Nth:

equation image

If the average distance between traps, δ, is known, then the number of traps crossed can easily be converted into a distance, s, using s = δN.

[43] In general, the shape of P(N) will depend on the nature of the microtopography (and also possibly the grain's momentum history) along the transport path, which makes specifying P(N) somewhat complicated. In some cases, however, it may be feasible to characterize the scale and effective magnitude of traps statistically using measured samples of microtopography. The simplest case would be one in which the scale, δ, and the trapping probability, p, were effectively constant in space and time. These conditions might apply, for example, to a planar surface with a characteristic microtopographic wavelength. In such a case, in the limit of many traps the movement of a grain is equivalent to a Poisson process, with an exponential probability density function for traversing a distance s before becoming trapped:

equation image

where sm = δ/p is the mean travel distance. Note that although in this case the distribution is thin tailed, this by itself does not guarantee locality in the motion statistics, as the mean hop length can still be large relative to the system length. If the trapping probability varies systematically in space, different travel distance probability functions can arise. For example, by analogy to Stark and Guzzetti's [2009] landslide propagation model, if the trapping probability decreases systematically along the path (because, for example, the slope angle increases), the travel distance probability density function can have a power law tail. This type of framework retains the simplicity and empirical basis of a statistical physics view of sediment transport while allowing for long-distance or correlated grain motions whose paths depend on topography and hydraulics.

5. Conclusions

[44] A probabilistic, particle-based model of hillslope evolution illustrates some potential connections between the statistics of grain motion and the macroscopic patterns of landform evolution. The model provides a simple analogy for the processes of grain displacement associated with soil creep and ravel. The dynamics that emerge can be mainly local or mainly nonlocal, depending on the slope angle and length. Furthermore, the degree of nonlocality in grain motions can vary in both time and space. Purely local dynamics collapse, as expected, to a diffusion-like process, which gives rise to convex upward parabolic hillslope forms under conditions of steady and uniform erosion rate. This provides additional support for the diffusion equation analogy for soil creep on gentle slopes.

[45] At high gradients, many grains move a large fraction of the slope length, introducing an element of nonlocality. In this regime, the diffusion analogy no longer applies. Instead, the model tends to produce planar slopes and an apparent strongly nonlinear relationship between grain flux and local gradient. However, this relationship is not purely causative: in the nonlocal regime, the magnitude of the particle flux at a point depends not only on gradient but also on the distance to the ridge top and on the topography between the ridge top and the point in question. To the extent that this model is a reasonable description of the dynamics of actual hillslopes, it implies that the success of nonlinear diffusion models in explaining slope forms may be partly a case of getting the right answer for the wrong reason. Nonlinear diffusion models are undoubtedly useful; that is, they do an excellent job of reproducing both natural and experimental slope forms and dynamics [e.g., Roering et al., 2001; Roering, 2008], but one should be cognizant of this potential limitation.

[46] Despite its simplicity, the particle model appears to reproduce some interesting aspects of observed slope forms, including an asymmetry in the early stages of degradation of a steep scarp. It thus suggests the hypothesis that such asymmetries can arise solely from the dynamics of grain displacement, even in the absence of factors such as limited regolith thickness. Moreover, the simplicity and computational efficiency of the particle model suggests a useful role for such models as a complement to continuum approaches in numerical modeling of landform evolution.

[47] The model we have presented also raises an interesting mathematical question that to our knowledge has not yet been explored: what type of continuum model best describes a dispersive system in which the motion statistics are themselves dependent on the system state, both in time and space? We suspect that there may be many analogous systems in which transport statistics vary according to system state variables. For example, the rheology of creeping flow in solids can change from Newtonian to non-Newtonian in response to increasing stress, as dislocation creep begins to play a role [e.g., Turcotte and Schubert, 2002]. Similar behavior can be expected from waterborne sediment transport: as fluid stress increases, grain suspension becomes increasingly important and grain motion lengths tend to increase.

[48] The nature of nonlocal transport dynamics in the particle model, and by analogy in some natural systems, is such that the transport length scale is contingent on microtopography along the potential transport pathway. This suggests one possible approach to analytical modeling of geomorphic transport systems that include elements of nonlocality. The approach uses the concept of potential transport paths, which are functions that describe the potential trajectory of a grain across the land surface together with a probability density function for deposition at each point along the path. Such a density function can, in principle, be derived by treating a grain's motion as a survival process. Potential transport paths can be combined with a mass continuity framework with separate erosion and deposition terms, and with an integral expression for deposition fluxes.

[49] Nonlocal particle motion dynamics are likely to be common to many geomorphic transport systems, and can potentially lead to patterns of landform evolution that depart from the predictions of purely local models. Thus, there is a need to identify transport systems that exhibit nonlocal behavior and develop mathematical models to represent them. By combining grain-scale experiments, field studies, and computer models, it may ultimately be feasible to develop a statistical physics basis for geomorphic transport laws, incorporating the full spectrum of local and nonlocal dynamics.

Acknowledgments

[50] David Furbish aided our thinking by generously sharing an unpublished manuscript on modeling sediment transport using a master equation. We are grateful to Colin Stark for fruitful discussions and an early copy of his paper on landslide propagation. Reviews from Efi Foufoula-Georgiou, Garry Willgoose, and two anonymous reviewers helped us tremendously in sharpening the paper. This research was supported in part by the National Science Foundation (EAR-0621199) and the U.S. Army Research Office (47033-EV). The authors would also like to thank NCED (National Center for Earth-surface Dynamics – an NSF Science and Technology Center at the University of Minnesota funded under agreement EAR-0120914) and the Water Cycle Dynamics in a Changing Environment hydrologic synthesis project (University of Illinois, funded under agreement EAR 0636043) for cosponsoring the STRESS (Stochastic Transport and Emerging Scaling on Earth's Surface) working group meeting (Lake Tahoe, November 2007) that fostered the research presented here.

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