### Abstract

- Top of page
- Abstract
- 1. Introduction
- 2. Nonlocal Transport
- 3. Behavior of the Convolution
- 4. Comparison With Related Formulations and Field Observations
- 5. Discussion and Conclusions
- Appendix A:: Averaging of Land Surface Elevation
- Acknowledgments
- References

[1] In soil-mantled steeplands, soil motions associated with creep, ravel, rain splash, soil slips, tree throw, and rodent activity are patchy and intermittent and involve widely varying travel distances. To describe the collective effect of these motions, we formulate a nonlocal expression for the soil flux. This probabilistic formulation involves upslope and downslope convolutions of land surface geometry to characterize motions in both directions, notably accommodating the bidirectional dispersal of material on gentle slopes as well as mostly downslope dispersal on steeper slopes, and it distinguishes between the mobilization of soil material and the effect of surface slope in giving a downslope bias to the dispersal of mobilized material. The formulation separates dispersal associated with intermittent surface motions from the slower bulk behavior associated with small-scale bioturbation and similar dilational processes operating mostly within the soil column. With a uniform rate of mobilization of soil material, the nearly parabolic form of a hillslope profile at steady state resembles a diffusive behavior. With a slope-dependent rate of mobilization, the steady state hillslope profile takes on a nonparabolic form where land surface elevation varies with downslope distance *x* as *x*^{a} with *a* ∼ 3/2, consistent with field observations and where the flux increases nonlinearly with increasing slope. The convolution description of the soil flux, when substituted into a suitable expression of conservation, yields a nonlinear Fokker-Planck equation and can be mapped to discrete particle models of hillslope behavior and descriptions of soil-grain transport by rain splash as a stochastic advection-dispersion process.

### 1. Introduction

- Top of page
- Abstract
- 1. Introduction
- 2. Nonlocal Transport
- 3. Behavior of the Convolution
- 4. Comparison With Related Formulations and Field Observations
- 5. Discussion and Conclusions
- Appendix A:: Averaging of Land Surface Elevation
- Acknowledgments
- References

[2] It is often assumed that the rate of downslope transport of soil material by creep on soil-mantled hillslopes is approximately proportional to the local land surface gradient. 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 linear relation between transport and land surface gradient is well founded [*Furbish et al.*, 2009b]. There is evidence, however, that in steeplands a simple linear dependence of transport on local gradient gives way to a nonlinear relation [*Andrews and Bucknam*, 1987; *Roering et al.*, 1999] wherein the time-averaged flux is approximately linearly proportional to gradient for small values of the gradient, but increases rapidly as the gradient approaches a critical value. With increasing gradient, soil motions associated with ravel [e.g., *Roering and Gerber*, 2005], soil slips, transport by fossorial animals (e.g., gophers) [*Gabet*, 2000] or tree throw [*Norman et al.*, 1995], and possibly including ravel or transport by patchy, intermittent surface flows following fire, can involve downslope travel distances that are much larger than those associated with small-scale bioturbation or freeze-thaw acting within the soil column [*Roering et al.*, 1999; J. Roering, personal communication, 2009]. In these cases, the downslope flux of soil material past a given contour position *x* [L] can involve motions that originate from near to far upslope and traverse significant distances downslope of *x* before coming to rest. Equally important, these motions for the most part involve dispersal of material over the surface, where the durations spent in crossing *x* are relatively brief as soil material “whizzes” downslope, that is, material is dispersed rapidly when viewed relative to the slower bulk soil motion arising from creation and collapse of porosity.

[3] Owing to the patchiness and intermittency of soil motions on soil-mantled hillslopes, describing the soil flux as a function of hillslope geometry and related environmental factors is fundamentally a stochastic problem. Here we formulate a probabilistic description of the soil flux in order to characterize the collective effect of patchy, intermittent motions. This is an elaboration of the formulation provided by *Foufoula-Georgiou et al.* [2010], wherein the flux of soil material at a given hillslope position is expressed as a nonlocal weighting (convolution) of the upslope serial configuration of land surface gradient. Key points of our analysis include (1) a description of the flux that includes both upslope and downslope convolutions of land surface geometry in order to characterize motions in both directions, notably accommodating the bidirectional dispersal of material on gentle slopes near hillslope crests as well as mostly downslope dispersal on steeper slopes; (2) a distinction between the mobilization of soil material (defined as the volume of sediment activated per unit area per unit time) which, in itself, does not lead to dispersal, and the effect of surface slope in giving a downslope bias to the dispersal of mobilized material; (3) a description of how this downslope bias is related to the idea of a critical slope as introduced by *Roering et al.* [1999]; (4) a separation of dispersal associated with intermittent surface motions from the slower bulk behavior associated with small-scale bioturbation and similar dilational processes operating mostly within the soil column [*Furbish et al.*, 2009b]; and (5) a simple formalism for thinking about how descriptions of the flux must involve both spatial and temporal averaging.

[4] We show that with a uniform rate of mobilization of soil material, despite downslope dispersal over significant distances, the form of a hillslope profile at steady state is nearly parabolic with an apparent diffusivity that depends on the system size. In contrast, with a slope-dependent rate of mobilization, the steady state hillslope profile takes on a nonparabolic form where land surface elevation varies with downslope distance *x* as *x*^{a} with *a* ∼ 3/2, a departure from the “classic” parabolic relation that is predicted when the flux of soil material is assumed to be proportional to the local land surface slope. We also illustrate that the description of the flux of soil material as a convolution, when substituted into a suitable expression of conservation, yields a nonlinear Fokker-Planck (advection-dispersion) equation. The form of this equation is significant because it emphasizes that the embedded coefficients (homologous to the drift velocity and the diffusivity in continuum-based advection-dispersion equations) are obtained as first and second moments, respectively, of a distribution of displacement distances, it illustrates how spatial variations in the rate of mobilization of soil material figure into net transport, and it shows that the flux of soil material in general possesses a nonunique relation to local land surface slope.

[5] The results of the analysis mimic key field observations, notably land surface profiles of hillslopes with rounded crests and nearly linear slope forms away from the crest under steady state conditions, where the downslope flux increases nonlinearly with increasing slope, consistent with the nonlocal theory of *Foufoula-Georgiou et al.* [2010] and as proposed by *Roering et al.* [1999]. The analysis also can be mapped to closely related formulations, including the discrete particle model of hillslope behavior introduced by *Tucker and Bradley* [2010] and the description of soil grain transport by rain splash as a stochastic advection-dispersion process introduced by *Furbish et al.* [2009a]. The basic probabilistic description of soil material motions is a straightforward statement of conservation of mass, so ingredients of the analysis also may be adaptable to stochastic formulations of sediment transport by surface flows on hillslopes [e.g., *Lisle et al.*, 1998].

### 5. Discussion and Conclusions

- Top of page
- Abstract
- 1. Introduction
- 2. Nonlocal Transport
- 3. Behavior of the Convolution
- 4. Comparison With Related Formulations and Field Observations
- 5. Discussion and Conclusions
- Appendix A:: Averaging of Land Surface Elevation
- Acknowledgments
- References

[52] Our formulation of the intermittent flux *q*_{i}(*x*, *t*) involves two essential parts. The first part, outlined in section 2.2, consists of a probabilistic description of soil material motions along the *x* coordinate (Figure 3). Although kinematic in nature, this description is a straightforward statement of conservation of mass. Once mobilized, soil material has to go somewhere, whether downslope or upslope, nearby or far away. As such, (13) and (14) represent flux forms of the Master Equation, the most general statement of conservation possible. In the case of (13), nothing is assumed *a priori* about the spatial, fractional rates of deposition *P* and *N* or the associated distributions of the displacement distances *r* and *s*, other than mobilized material must be partitioned between upslope and downslope motions in a way that conserves mass. Moreover, this formulation leaves open the possibility that *P* and *N* may vary with conditions (e.g., land surface slope or other factors) downslope and upslope of the position of origin. In the case of (14), this statement of conservation is only slightly restrictive in that it assumes that *P* and *N* and the associated distributions of the displacement distances depend on conditions at the position of origin.

[53] The rate of mobilization *E* likewise is kinematic, and is entirely analogous to the entrainment rate (per unit area) in entrainment forms of the Exner equation as applied to river sediment transport (e.g., see reference to *Tsujimoto* [1978] in the work of *Parker et al.* [2000] and *Ganti et al.* [2010]). In detail, this rate depends on the physics involved in each process. At the scale of rain splash transport, for example, the rate of mobilization of soil grains is a function of raindrop size and intensity as well as soil and vegetation properties. With bioturbation, the rate of mobilization reflects the details of biotic activity; and with soil slips this rate is a function of the frequency and spatial distribution of failures associated with precipitation events. As used here, therefore, *E* represents an amalgamation of these process, although the mobilization of soil material may involve a dominant process.

[54] A key assumption of the probabilistic formulation of *q*_{i}(*x*, *t*) is that, once mobilized, soil material is dispersed “instantaneously” over *x*. Indeed, this assumption is implied in related formulations that involve a convolution of sediment particle entrainment [*Parker et al.*, 2000; *Ganti et al.*, 2010] or soil material transport [*Foufoula-Georgiou et al.*, 2010]. It is justified if dispersal occurs far more rapidly than significant changes in morphology, that is, if the amount of material undergoing motion per unit time is a small proportion of the total mass of the evolving morphologic feature (e.g., bed forms or the hillslope). This assumption, however, leads to an important difference from traditional probabilistic descriptions of conservation and transport (i.e., the classic Master and Fokker-Planck equations). Namely, in the traditional case, the flux involves time derivatives of the first and second moments of the distribution of particle displacements, leading to the mean drift speed and the dispersion coefficient [*Risken*, 1984; *Ebeling and Sokolov*, 2005; *Furbish et al.*, 2009a, 2009b]. In contrast, herein the time dimension defining the rate of dispersal (and flux) is embedded in the rate of mobilization *E*, where the magnitude of dispersal involves the average *μ*_{λ} and variance *σ*_{λ}^{2} of displacements. Thus, in (32) and (33), *Eμ*_{λ} is like the product of an active layer thickness and a drift speed and *Eσ*_{λ}^{2} is like the product of an active layer thickness and a dispersion coefficient.

[55] The second essential part of our formulation of the intermittent flux *q*_{i}(*x*, *t*) consists of adding effects of land surface slope. This part is semiempirical, albeit involving a physical basis for certain transport processes. It appeals to the idea that, although in detail numerous factors may influence the probabilities of motion in the positive or negative *x* directions, *p* and *n*, and the fractional rates of deposition, *P* and *N*, effects of land surface slope must enter the formulation at lowest order. As described in sections 2.3 and 4.1, the basis for *p* and *n* as a function of slope is well founded, theoretically and empirically, for processes such as rain splash and tree throw [*Furbish et al.*, 2007, 2009a; *Dunne et al.*, 2010; *Norman et al.*, 1995]; and slope fundamentally controls the likelihood of downslope motion of soil slips and ravel [*Gabet*, 2003], as well as the likelihood of downslope dispersal of soil material by gophers [*Gabet*, 2000; *Gabet et al.*, 2003]. Similarly, the basis for *P* and *N* is well founded for rain splash, specifically that these spatial rates are set by the *e*-folding length scales *λ*_{r} = 1/*P* and *λ*_{s} = 1/*N* at the position of origin of grain motions. For larger-scale processes of dispersal, *P* and *N* may well vary with downslope land surface conditions at the scale of land surface roughness (e.g., involving the filling of divots) when viewed over short travel distances, as in the particle model of *Tucker and Bradley* [2010]. The assumption that *P* and *N* vary only with land surface slope at the position of origin is equivalent to assuming that this behavior emerges when the ensemble of motions (including filling of small divots) is averaged over the local roughness scale (section 2.1). In fact, assuming deposition is proportional to small-scale roughness, then for a given (averaged) slope the spatial, fractional rates *P* and *N* should be constant if roughness is homogenous. We also note that, as used herein, the critical slope *S*_{c} is a loosely constrained coefficient, although it does have a physical basis in the idea of a critical slope as described by *Roering et al.* [1999] and in the friction angle described by *Gabet* [2003] for dry ravel. The effect of *S*_{c} in the simulations (section 3.2) and in the comparisons with field observations (section 4.2) is to modulate the effect of slope *S* in setting *P* and *N* locally.

[56] Inasmuch as the fractional rates of deposition, *P* and *N*, vary with land surface slope as in (19) and (20), then aside from effects of local roughness, the possibility exists that for processes involving large travel distances, *P* and *N* may vary with conditions away from the position of origin in the presence of significant profile curvature. The essence of this idea is embodied in (21) and (22), where dispersal of sediment is explicitly influenced by the serial configuration of land surface slope downslope and upslope from its position of origin. For the steady state profiles considered above (sections 3.2 and 3.3), where the land surface slope varies “slowly” (except possibly near the hillslope crest), the assumption that *P* and *N* are determined locally is reasonable. But certainly there are profile configurations, notably involving unsteady conditions, where relatively sharp changes in surface slope occur over distances that are comparable to travel distances, e.g., local convexities with steepening by channel downcutting, or concavities near the bases of hillslopes in zones of soil thickening. Then, for example, material raveling from a position just upslope of a convexity would likely be dispersed downslope (on the steeper, lower part) over a greater distance than would otherwise be anticipated based on the land surface slope at the origin of the raveling material, and vice versa for a position upslope of a concavity. The formulation embodied in (21) and (22) therefore merits attention, but with the caveat that we need further clarification of the physics of deposition in relation to land surface conditions, including slope, to elaborate this higher-order formulation.

[57] Perhaps the least constrained part of the formulation is that the rate of mobilization *E* has a slope-dependent part, namely *E*_{1}∣*S*∣, although the influence of this on hillslope form is key. In effect this assumption suggests that the frequency and magnitude of processes such as soil slips and ravel, and perhaps tree throw, increase with hillslope steepness [*Norman et al.*, 1995; *Gabet*, 2003]. Or, once soil material is disturbed, say, by biotic activity, the likelihood of remobilization and dispersal of this material from disturbed sites (e.g., mounds, ravel accumulations) increases with steepness. With a uniform rate of mobilization (*E* = *E*_{0}, *E*_{1} = 0), the predicted steady state hillslope profile is nearly parabolic, consistent with the long-standing idea that hillslopes subject to “local” transport processes such as rain splash (with *E*_{1} = 0) appear “diffusive.” With *E*_{1} > *E*_{0}, then as described in sections 3.2 and 3.3, the intermittent flux *q*_{i}(*x*) varies with slope as ∣*S*∣*S* as in (47) so that the predicted steady state hillslope profile varies as *x*^{3/2}. Thus, the assumption that *E* has a slope-dependent part may be viewed as an hypothesis: Land surface profiles that vary with distance *x* between *x*^{2} and *x*^{3/2} reflect varying mixtures of transport processes. On the one hand, if these processes involve a relatively uniform rate of mobilization (e.g., rain splash) or mostly produce bulk motion associated with creation and collapse of porosity within the soil column, then the steady state profile would tend toward an *x*^{2} relation. On the other hand, if these processes mostly involve intermittent surface motions where the averaged rate of mobilization is slope dependent, the profile would likely have an *x*^{3/2} relation.

[58] Fitted profiles (section 4.2) reasonably match the measured hillslope profiles described by *Roering et al.* [1999] and *McKean et al.* [1993], where the land surface indeed declines as *x*^{3/2} over the lower parts of the hillslopes (Figures 9b and 10b). There is significant uncertainty in the selected parametric values (other combinations could possibly provide equally reasonable fits), and in the assumption that the rate of soil production at the two sites is uniform. Although we did not fully explore possible combinations of parametric values, using the reported values of *W* and Γ nonetheless requires in both cases that *E*_{1} ≫ *E*_{0} with relatively small *K*_{s}, consistent with the idea that a significant fraction of the transport at these sites involves intermittent surface motions whose frequency of occurrence increases with hillslope steepness. Inasmuch as an *x*^{3/2} relation is a robust result under these conditions, then the qualitative fit to the profiles described by *Roseberry* [2009] (Figure 12b) similarly suggests the presence of slope-dependent mobilization of soil material. We note, however, that with field evidence suggesting slope-dependent mobilization, the absence of an *x*^{3/2} relation would suggest unsteady conditions.

[59] *Foufoula-Georgiou et al.* [2010] note that if the flux is expressed as a (nonlocal) convolution of land surface slope *S*(*x*, *t*), then the flux does not possess a unique relation to the local slope. This result also can be inferred from the differential form of the flux *q*_{i}(*x*, *t*) given by (32). Namely, substituting (34) and (35) into (32) gives

For a uniform rate of mobilization (*E* = *E*_{0}) and nearly uniform slope such that ∂(*S*^{2})/∂*x* is negligibly small, *q*_{i}(*x*, *t*) possesses an approximate one-to-one relation with slope given by the advective terms on the right side of (51). If, however, ∂(*S*^{2})/∂*x* is not small or *E* varies with slope, then the flux at any position *x* depends on derivatives of the slope, that is, curvature. This means that, depending on the specific history of hillslope evolution, a given value of the slope *S*(*x*, *t*) may be associated with any physically realizable curvature, in which case derivative terms in (51), in addition to the advective term involving *S*^{2}, can become sufficiently important that the flux does not possess a unique relation to the slope.

[60] As noted in several sections above, key parametric quantities in the formulation are reasonably well constrained for certain transport processes, whereas others are not. For example (sections 2.3 and 4.1), the form of the probability distribution of grain displacements, the length scale *λ*_{0}, and the critical slope *S*_{p} have a solid theoretical and empirical underpinning for the process of rain splash transport [*Furbish et al.*, 2007, 2009a; *Dunne et al.*, 2010], and field observations suggest that *S*_{p} as well as characteristic dispersal distances can be estimated for transport associated with fossorial animal behavior [e.g., *Gabet*, 2000] and tree throw [*Norman et al.*, 1995]. The critical slope *S*_{c}, although defined above (section 2.3) within a probabilistic framework, must physically represent the slope coinciding with an effective dynamic friction angle, and thus it coincides with the critical slope introduced by *Roering et al.* [1999], and with the friction angle estimated by *Gabet* [2003] for the specific case of dry ravel. (The observation of *Gabet* [2000], that sediment excavated by gophers on steep slopes tends to ravel downslope, is also consistent with this idea of a critical slope, although we reiterate that *S*_{c} nominally represents a condition where the local slope is sufficiently steep that all sediment moving downslope remains in motion.) Depending on the environmental setting, individual processes may give rise to “characteristic” distributions of travel distances and probabilities of downslope versus upslope motion as a function of slope, which again points to the need for clarification of the physics of deposition in relation to land surface conditions. Whereas the formulation hinges on the idea that land surface slope must enter this problem at lowest order, elaboration of this point will likely involve conditions of surface roughness and characteristics of the transported sediment.

[61] We have focused here on variations in land surface geometry that might arise from a nonlocal formulation of the flux of soil material. Going beyond land surface, this type of formulation is a starting point for describing the transport and dispersal of “tracers” [e.g., *Michaelides et al.*, 2010], whether regarded as specific soil constituents (e.g., organic carbon and other nutrients, grain size fractions) or exotic substances (e.g., contaminants, radionuclides, rare Earth elements). This opens the possibility of examining the coupled behavior of land surface geometry and tracers to constrain model ingredients [*Furbish*, 2003; *Roering et al.*, 2004; *Mudd and Furbish*, 2006] as well as, for example, the role of soils, and soil transport, in carbon sequestration [e.g., *Yoo et al.*, 2006; *Roseberry*, 2009]. The probabilistic basis of the formulation might also be adapted to transport by surface flows, where sediment mobilization depends on fluid stresses, possibly involving detachment by rain splash.