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Keywords:

  • sediment transport;
  • master equation;
  • Fokker-Planck equation

Abstract

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

  1. Top of page
  2. Abstract
  3. 1. Introduction
  4. 2. Nonlocal Transport
  5. 3. Behavior of the Convolution
  6. 4. Comparison With Related Formulations and Field Observations
  7. 5. Discussion and Conclusions
  8. Appendix A:: Averaging of Land Surface Elevation
  9. Acknowledgments
  10. 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 xa 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].

2. Nonlocal Transport

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

2.1. Soil Flux as an Averaged Quantity

[6] Consider a convex planar hillslope (Figure 1), and let x, y and z [L] denote associated Cartesian coordinates. The horizontal x axis is positive in the downslope direction with origin (x = 0) at the hillslope crest. The foot of the hillslope is positioned at x = X. The horizontal y axis is positive toward the left when looking downslope, with origin (y = 0) along the hillslope “axis.” For later reference a convenient hillslope “width” b [L] is defined by y = ±b/2. The z axis is positive upward. Letting t [t] denote time, then z = ζ(x, y, t) [L] denotes the local elevation of the land surface.

image

Figure 1. Definition diagram of convex planar hillslope and associated xyz coordinate system.

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[7] For any position x, let V(x, t; t0) [L3] denote the total (solid) volume of sediment that has crossed x downslope at time t since some initial time t0, over the width b. The instantaneous volumetric flux Q(x, t) = dV/dt [L3 t−1] and the flux per unit width q(x, t) = (1/b)Q(x, t) [L2 t−1]. In subsequent sections we relate this one-dimensional flux q(x, t) to the configuration of the land surface along x, specifically the slope ∂ζ/∂x = S(x, t). This requires that we define the width b, the slope S(x, t) and the flux q(x, t) in a way that they may be considered as varying smoothly with x and with time t, taking into account disturbance-scale roughness of the land surface.

[8] The land surface elevation ζ(x, y, t) consists of a large-scale variation in elevation (the hillslope “form”) with superimposed small-scale roughness created by disturbances that mobilize and disperse sediment. The average of ζ(x, y, t) over y involving a width b is

  • equation image

Inasmuch as b is sufficiently large that (1) adequately averages over divot-scale roughness, and sufficiently small that (1) does not average over larger-scale topographic variations (e.g., swales), then ζ(x, t) varies smoothly with x so that S(x, t) = ∂ζ(x, t)/∂x also varies smoothly with x (Appendix). In effect the one-dimensional quantities ζ(x, t) and S(x, t) may be envisioned as pertaining to averaged conditions along a flow tube [e.g., Heimsath et al., 2005] of width b. We note that because surface roughness reflects the disturbances responsible for transport, the scale and character of roughness may vary with the extant process, and any continuum description of transport behavior (e.g., “diffusive” transport) by definition occurs (i.e., must be viewed as such) at a scale larger than the roughness [Jyotsna and Haff, 1997]. We also note that the choice of b has implications for field measurements. In general, for comparison with the predictions of the work presented here, the spatial resolution of surface processes in the field should not be smaller than b. For example, if the roughness is at a scale of mounds and pits on a surface where transport is dominated by tree throw, then measurement of transport rates using Young pits, with their typically much smaller dimensions, would not be warranted.

[9] The preceding suffices for a definition of the slope S(x, t) at any instant t. But at any position x, the land surface elevation ζ(x, y, t) varies temporally with the creation and smoothing of roughness, including the divots and mounds associated with intermittent surface motions. We therefore redefine ζ(x, t) as

  • equation image

assuming that b does not vary with time. Here, T is a suitable averaging period, which we elaborate further below. Meanwhile, the slope S(x, t) = ∂ζ(x, t)/∂x at x is now interpreted as a local value averaged over disturbance-scale roughness, including short-term (temporal) variations in this roughness.

[10] Turning to the sediment flux per unit width q(x, t), this flux involves disparate scales of motion, and, as fully described in later sections, it is useful to separate q(x, t) into two parts. Here we refer to the flux associated with dilation-driven motions (creation and collapse of porosity) as a bulk flux qs(x, t) to emphasize that particle motions occur mostly within the soil column, involving length scales smaller than the active soil thickness. We refer to the flux associated with faster whizzing motions as an intermittent surface flux qi(x, t) to emphasize that these motions mostly involve dispersal over the land surface, and are patchy in space and time. Over the period of time that particles reside on a hillslope, from initial entrainment into the active soil to delivery to the channel, these particles may participate in both types of motion and be exchanged between the bulk soil and the surface numerous times.

[11] The instantaneous flux, q(x, t) = qs(x, t) + qi(x, t), is small most of the time inasmuch as soil motion across x involves (over the width b) the slow bulk flux qs(x, t). But q(x, t) may be intermittently large due to the flux qi(x, t). To envision this, consider the sum of the two fluxes, bulk and intermittent (Figure 2). In this conceptual example, the intermittent flux involves short-lived events that are relatively infrequent (Figure 2a), but whose average magnitude is 20 times that of the bulk flux, so that the contribution of the intermittent flux to the total sediment volume V(x, t; t0) crossing x over the entire time period (tt0) is similar to the contribution of the bulk flux (Figure 2b). As a continuous expression in space and time, q(x, t) is equivalent to the average slope of (1/b)V(x, t), that is, q(x, t) = (1/b)[V(x, t) − V(x, t0)]/(tt0).

image

Figure 2. Schematic diagram showing (a) the sum of a low-magnitude bulk flux qs(x, t) and an intermittent flux qi(x, t) involving events that are relatively infrequent, but whose average magnitude (selected from an exponential distribution) is 20 times that of qs(x, t) so that (b) the contribution of qi(x, t) to the total sediment volume V(x, t) = Vs(x, t) + Vi(x, t) crossing x over the entire time period (tt0) is similar to that of qs(x, t).

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[12] Following this idea, and for consistency with the averaged slope S(x, t), we redefine q(x, t) as

  • equation image

The averaging period T in both (2) and (3) must be long compared to the duration of individual sediment-moving events, but shorter than the time required for a significant change in the hillslope form. For soil-mantled hillslopes where the active soil may be viewed as a thin skin relative to the total hillslope relief, a reasonable measure of T is the mean residence time (or “turnover” time) of soil particles, TH/Γ, [Furbish and Fagherazzi, 2001; Mudd and Furbish, 2006], where H [L] is a characteristic (average) soil thickness and, as described below, Γ [L t−1] is the rate of soil production. For hillslopes undergoing bulk diffusive transport, the residence time T typically varies over 103–104 years [Furbish and Fagherazzi, 2001], long enough to smooth divots but much shorter than time scales of hillslope relaxation [Mudd and Furbish, 2007].

[13] The definitions (2) and (3) represent a quasi-steady approximation, similar to that adopted in Reynolds averaging of turbulent flow quantities. The land surface elevation ζ(x, t), slope S(x, t) and flux q(x, t) represent “sliding” averages centered amidst conditions before and after time t. A model of unsteady hillslope evolution therefore pertains to how these (smooth) average quantities change rather than to changes in actual (instantaneous) conditions.

[14] To complete this section, consider steady state conditions as viewed over a time scale longer than the averaging time scale T. In a fixed reference frame the rate of downcutting by the channel at the foot of the hillslope is equal in magnitude to the uniform uplift rate W [L t−1] such that the channel position is fixed. Or, in a reference frame that moves with the channel, the rate of downcutting is steady such that W represents a relative, uniform upward motion. By definition the land surface elevation ζ(x, t) = ζ(x) in either reference frame is unchanging and the steady volumetric (solid) flux per unit width q(x) at position x is

  • equation image

Here, cη = 1 − ϕ is the volumetric particle concentration just beneath the active soil layer involved in transport (where ϕ is the porosity). If the base of this active layer, z = η(x), coincides with the soil-saprolite (or soil-bedrock) interface, then Γ denotes a uniform rate of conversion of saprolite (or bedrock) with concentration cη to soil, that is, the soil production rate [Carson and Kirkby, 1972; Heimsath et al., 1997]. If the base of the active layer η(x) is well above bedrock, then Γ is the rate of entrainment of material with concentration cη into active transport. In either case, cηΓ = cηW may be envisioned as the volume of solid material crossing the fixed position η(x) per unit area per unit time. Here we note that one can define q(x) as a solid volume flux, as above [see also Furbish et al., 2009b], or as a soil volume flux (solid plus pores, in which case (4) would appear as q(x) = (cη/cs)Wx = (cη/csx, where cs is the vertically averaged soil concentration above η(x)), and in either case maintain internal consistency within expressions of continuity. We prefer the solid volume flux in that it is less liable to be misleading because (1) worms, beetles, tree roots, etc., move particle mass around, not porosity (although one can write a statement of conservation of porosity that is coupled with conservation of particle mass) and (2) this form of the flux is how we write the Exner equation as applied to river sediment transport, and it is reasonable to adopt a common approach.

2.2. Probabilistic Formulation of Soil Flux

[15] With reference to Figure 3, consider transport associated with intermittent surface motions along a coordinate x. For material mobilized at x = u, let r [L] denote a distance of travel in the positive x direction, and let s [L] denote a (positive) distance of travel in the negative x direction. Further, let p(u) denote the probability that motion is in the positive x direction, and let n(u) denote the probability that motion is in the negative x direction. Thus, p(u) + n(u) = 1.

image

Figure 3. Definition diagram for transport of material along the coordinate x involving the probability densities fr(r, u) and fs(s, u) of travel distances r = vu and s = uw for motions originating at position x = u. The densities fr(r, u) and fs(s, u) represent probability distributions (per unit distance) of where material comes to rest.

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[16] Let Fu(v) denote the probability that a particle originating at u (r = 0) is deposited at a distance less than or equal to v = u + r, so that Ru(v) = 1 − Fu(v) is the probability that a particle moves beyond v. Note that Fu(v) is identical to the probability Fr(r, u) that a particle originating at u is deposited at a distance less than or equal to r, so by definition the probability density of r is fr(r, u) = dFr/dr = dFu/dv = −dRu/dv [L−1]. Let P(v) [L−1] denote the spatial, fractional rate of deposition, namely

  • equation image

from which it follows that

  • equation image

Integrating (6) from u (r = 0) to v (v = u + r), noting that Fu(u) = 0, then gives

  • equation image

In turn, let Fu(w) denote the probability that material originating at u (s = 0) is deposited at a distance greater than or equal to w = us, so that Ru(w) = 1 − Fu(w) is the probability that material moves beyond w (in the negative x direction). The probability density of s is fs(s, u) = dFs/ds = dFu/dw = −dRu/dw [L−1]. Let N(w) [L−1] denote the associated spatial, fractional rate of deposition. By a development similar to that above we then obtain

  • equation image

[17] We now assume that once soil is mobilized from position u (Figure 3), it is dispersed “instantaneously” over x, that is, on a whizzing time scale that is much shorter than the time required for a significant change in hillslope geometry. This key idea, elaborated below, bears on interpreting rates of dispersal and transport. If E(u) [L t−1] denotes the (solid) volume of soil at u that is mobilized per unit area per unit time, then bE(u)dudt is the material mobilized within the small interval du during dt. Conservation requires that with dispersal of this material during dt, the part passing position v = x in the positive x direction from u < x is bE(u)p(u)Ru(x)dudt, or

  • equation image

and the part passing position w = x in the negative x direction from u > x is bE(u)n(u)Ru(x)dudt, or

  • equation image

The total volume of material passing position x in the positive x direction during dt is

  • equation image

and the volume of material passing position x in the negative x direction during dt is

  • equation image

The volume flux past x is Q(x) = Q+(x) + Q(x). Summing (11) and (12) and dividing by bdt then gives the intermittent flux qi(x) at x, namely

  • equation image

This is a flux form of the Master Equation [Risken, 1984; Ebeling and Sokolov, 2005; Furbish et al., 2009a, 2009b], which illustrates that the flux at x is influenced by motions originating at positions both to the left and right of x.

[18] For later reference, if the spatial rates of deposition, P(v) and N(w), are determined locally by conditions at the position of origin u rather than varying with v or w, that is, P = P(u) and N = N(u), then (13) simplifies to

  • equation image

This has the form of the sum of two convolutions, namely

  • equation image

with the kernels

  • equation image

We turn now to the probabilities p(u) and n(u), the spatial, fractional rates of deposition P(v) and N(w), and the rate of mobilization E(u), as these are influenced by land surface slope.

2.3. Soil Flux as a Convolution of Slope

[19] We assume that the probabilities p(u) and n(u) are related in a simple way to the local land surface slope S(u) = ∂ζ(u)/∂x, namely

  • equation image

and

  • equation image

where Sp is the (positive) magnitude of a critical slope. In addition to satisfying the condition that p(u) + n(u) = 1, note that when S(u) = 0, p(u) = n(u) = equation image, so motion is partitioned equally between the positive and negative x directions. When S(u) [RIGHTWARDS ARROW]Sp, p(u) [RIGHTWARDS ARROW] 1 and n(u) [RIGHTWARDS ARROW] 0, so motion is entirely in the positive x direction; and when S(u) [RIGHTWARDS ARROW] Sp, p(u) [RIGHTWARDS ARROW] 0 and n(u) [RIGHTWARDS ARROW] 1, so motion is entirely in the negative x direction. These limits thus represent a condition where the slope is sufficiently steep that motion is entirely in the downslope direction. At the scale of rain splash transport, for example, mobilization of grains by raindrop impacts on dry sand leads to grain motions that are almost entirely downslope for ∣S∣ = Sp ∼ 0.58 (30°) [Furbish et al., 2007; Dunne et al., 2010, Figure 7; Furbish et al., 2009a, Figure 1]. Indeed, the linear model provided by Dunne et al. [2010] for the fraction F of sediment splashed downslope, their equation (4b), has the same form as (17) (or (18)), where the fit to available splash data for ∣S∣ < 0.47 (25°), namely F = 0.5 + 0.86∣S∣, is entirely consistent with F = p(u) = (1/2)(1 + ∣S∣/Sp) with 0.5/Sp = 0.5/0.58 = 0.86. At a larger scale, Norman et al. [1995] observe that soil mounds and pits associated with tree throw increase in size with increasing slope, and suggest that the dispersal of disturbed soil material mostly involves wasting back into the pits on gentle slopes, whereas with slopes of ∣S∣ = Sp ∼ 1.1 (47°), material is mostly dispersed downslope by slumps or wash. At an intermediate scale, Gabet [2000] observes that even with gentle slopes gophers distribute soil material mostly downslope (presumably to prevent sloughing of material back into the hole), that the average downslope displacement of material increases with steepness, and that at slopes steeper than ∼0.6, excavated sediment tends to ravel downslope. The functions in (17) and (18) may or may not have this linear form in detail [e.g., Dunne et al., 2010], but this form must represent the first-order effect of surface slope on the probabilities p(u) and n(u).

[20] Turning to the spatial, fractional rates of deposition P(v) and N(w), and letting λ0 denote a characteristic length scale of dispersal on a horizontal surface, we assume that these rates vary with land surface slope as

  • equation image

and

  • equation image

where Sc is the magnitude of a critical slope. The middle part of (19) is equivalent to writing P(v) = 1/λr with the dispersal length scale λr = λ0[ScS(v)]/[Sc + S(v)], so the bracketed part, giving the range λr = λ0 for S(v) = 0 to λr [RIGHTWARDS ARROW] equation image for S(v) [RIGHTWARDS ARROW]Sc, effectively modulates the length scale λ0 (Figure 4). Similar comments pertain to N(w) = 1/λs. The limits S(v) [RIGHTWARDS ARROW]Sc and S(w) [RIGHTWARDS ARROW] Sc represent a condition where the local slope is sufficiently steep that all sediment moving downslope remains in motion. The critical slope Sc in (19) and (20) is generally larger than Sp in (17) and (18). It nominally coincides with the critical slope introduced by Roering et al. [1999], and with the friction angle introduced by Gabet [2003] for the specific case of dry ravel. Note that this formulation for P(v) and N(w) does not explicitly involve sediment momentum, but implicitly involves a friction-like behavior [e.g., Gabet, 2003] inasmuch as the dispersal length scale decreases with decreasing steepness.

image

Figure 4. Plot of dimensionless dispersal length scales 1/0 = λr/λ0 and 1/0 = λs/λ0 versus normalized slope ∣S/Sc∣.

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[21] Substituting (17) through (20) into (13) then gives

  • equation image

where the kernels

  • equation image

so that the dispersal of sediment is explicitly influenced by the serial configuration of land surface slope downslope and upslope from its position of origin. Looking downslope, if the land surface becomes less steep with downslope distance, the fractional deposition increases with distance. If the land surface monotonically steepens with downslope distance, the fractional deposition decreases as “runout” distances increase. An important feature of this formulation is that, in relating the rates of deposition to the land surface configuration, as in (19) and (20), nothing is assumed a priori regarding the forms of the distributions of travel distance, fr(r, u) and fs(s, u).

[22] We now simplify the formulation in anticipation of comparing it with related work and field observations. Let LS = ∣S∣/∣dS/dx∣ [L] denote a length scale over which the land surface slope is relatively uniform. If the downslope length scale of dispersal λr = 1/P (or λs = 1/N) ≲ LS, then we may assume that the spatial rates of deposition, P and N, are effectively determined locally by the slope S(u) at the position of origin u rather than varying with v or w, that is, P = P(u) and N = N(u). With this simplification (21) becomes

  • equation image

and the kernels in (22) become

  • equation image

Now, in contrast to (21) and (22), by relating the rates of deposition to the local land surface slope at position u, the forms of the distributions of travel distance, fr(r, u) and fs(s, u), are explicitly exponential, albeit varying with hillslope position u as slope, and therefore the dispersal length scales λr and λs, vary with u.

[23] The rate of mobilization of material E(u) is uninfluenced by land surface slope on gently sloping areas where processes of mobilization like bioturbation simply bring transportable material to the surface (with local dispersal) without regard for slope. The ejection and radial “splash” of soil grains by raindrop impacts [Furbish et al., 2009a] is a good example of mobilization of material independently of slope. But it is also likely that, with increasing steepness, the rate of mobilization is influenced by land surface slope. This is certainly the case for mobilization involving soil slips, dry ravel [Gabet, 2003], and release of material temporarily stored in mounds or behind plant stems that are removed by fire or death [Roering and Gerber, 2005]. We thus suppose that

  • equation image

where E0 [L t−1] is the rate of mobilization on a flat surface and E1 [L t−1] modulates the first-order effect of land surface slope, a point that is elaborated below. We turn now to the addition of bulk transport.

2.4. Soil Flux as a Sum of Bulk and Intermittent Parts

[24] Although time is not explicitly included in the formulation above, we may now appeal to the quasi-steady approximation outlined in section 2.1 and express the flux in terms of both position and time. Here we also assume that the flux q(x, t) at x may be partitioned into two parts, a bulk part qs(x, t) and an intermittent surface part qi(x, t), as described in section 2.1. This partitioning is based on a natural separation of operative length and time scales, in that small-scale biological activity (e.g., worm behavior, root growth) “deep” within the active layer at a position x − Δx cannot influence the flux at x, whereas intermittent (surface) soil motions originating at x − Δx can contribute to the flux at x if these motions involve travel distances r, s ≥ Δx for Δx on the order of the thickness H [L] of the active layer of transport [Furbish et al., 2009b]. We further assume that qs(x, t) varies linearly with land surface slope S(x, t) inasmuch as H is uniform and steady, set by the depth of biological activity [Roering et al., 2002; Gabet et al., 2003; Roering, 2008; Furbish et al., 2009b]. Thus,

  • equation image

where Ks [L2 t−1] is a diffusivity for bulk transport. Substituting (26) into a suitable statement of conservation of mass assuming uniform Ks and active soil thickness H then gives

  • equation image

where it is immediately apparent that, as the surface mobilization rate E(u, t) goes to zero and qi(x, t) vanishes, properties of “local” diffusive behavior are recovered, that is, land surface elevation ζ(x, t) satisfies a diffusion-like equation. We show next that the divergence ∂qi(x, t)/∂x in (27) leads to a nonlinear advection-dispersion equation.

2.5. Convolution Recast as a Fokker-Planck Equation

[25] The formulation above can be recast into the form of an advection-dispersion (Fokker-Planck) equation. The value of this bears on illustrating the connection between the formulation and related work on sediment transport, and it provides advantage in numerical simulations of land surface evolution in that the divergence of the soil flux at position x can be calculated using “local” information near x rather than “nonlocal” information over all positions x as required by the convolutions (23) and (24). Starting with (15), and recalling that the displacement distances r = xu (u < x) and s = ux (u > x), a change of variables gives

  • equation image

Expanding the products hr(r, xr)E(xr)p(xr) and hs(s, x + s)E(x + s)n(x + s) as Taylor series about x to first order then leads to

  • equation image

Letting P(x) = 1/λr(x) and N(x) = 1/λs(x) in the kernels (24), and recalling that fr(r, x) and fs(s, x) are the exponential probability densities of r and s (which may vary with x), then λr(x)fr(r, x) = hr(r, x) and λs(x)fs(s, x) = hs(s, x). Substituting gives

  • equation image

Each of the first two integrals in (30) by definition equals unity, and the second two integrals define the first moments (means) of r and s, namely, λr(x) and λs(x). Thus,

  • equation image

The first term on the right side of (31) is advective and the second is dispersive. The bracketed part of the first term is merely the weighted average μλ [L] of the displacement distances r and s, that is, μλ(x) = λr(x)p(x) − λs(x)n(x). The parenthetical part of the second term is a weighted variance σλ2 [L2] of these distances, that is, σλ2(x) = [λr(x)]2p(x) + [λs(x)]2n(x). Because in this formulation material is considered to move “instantaneously” during dt, a downslope velocity U (or “drift” speed) is only implied, nominally defined by Udt = μλ. Similarly, a diffusivity Ki is nominally defined by Kidt = σλ2. We may thus rewrite (31) as

  • equation image

illustrating how a spatial variation in either the mobilization rate E(x) or the variance σλ2(x) can effect a dispersive flux in the absence of average motion (μλ = 0) [Furbish et al., 2009a] (section 3.2). In turn, appealing to the quasi-steady approximation and substituting (32) into (27) (and neglecting the bulk term),

  • equation image

which is a specialized Fokker-Planck equation. The form of (33) is significant because it emphasizes that the embedded coefficients (homologous, respectively, to the drift velocity and the diffusivity in continuum-based advection-dispersion equations), specifically μλ and σλ2, are obtained as moments of a distribution of displacement distances, and that these moments as well as the mobilization rate E are entirely inside the spatial derivatives, a point that is elaborated below.

[26] The length scales λr(x) and λs(x) are expressed in terms of the local slope S(x) using (19) and (20), and the probabilities p(x) and n(x) are obtained from (17) and (18). In the case of transport by rain splash, the rate of grain mobilization E(x) varies with raindrop intensity and grain properties, independently of slope [Furbish et al., 2009a], where in turn the raindrop intensity may vary spatially with vegetation cover [Parsons et al., 1992; Wainwright et al., 1999; Furbish et al., 2009a]. In the case of intermittent surface transport on hillslopes, E(x) may be described by (25).

[27] For reference below, using (17) through (20) with ∣S∣ ≪ Sc, expansion of the average displacement μλ(x) as a binomial series yields

  • equation image

which indicates that, at leading order, μλ(x) is proportional to the local land surface slope S(x) = ∂ζ/∂x. In turn, the variance σλ2(x) of the displacement distance goes as

  • equation image

which indicates that, at leading order, σλ2(x) is constant. We use these points in section 3.3 to clarify predicted forms of land surface profiles under steady state conditions.

3. Behavior of the Convolution

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

3.1. Limiting Cases With Uniform Mobilization Rate

[28] Three limiting cases deserve attention. First, suppose that S(u) = S(v) = S(w) = 0 with a uniform, steady rate of mobilization, E(u) = E0. With qi(x) in (26) given either by (21) or (23), evaluating the integrals gives

  • equation image

illustrating that on this flat surface the flux to the right equals the flux to the left, so the net flux vanishes. Qualitatively, disturbances disperse material over the surface, but with no net motion, as in the quintessential example of grain dispersal by rain splash [Furbish et al., 2009a].

[29] In the second case, consider a symmetrical hillslope under steady state conditions, where the steady rate of uplift is balanced by stream downcutting at the foot of the hillslope, so that ∂ζ(x)/∂t = 0. Assuming E(u) = E0, then taking the derivative of (23) with respect to x and substituting the result into (33) gives

  • equation image

where the prime denotes a derivative. With x = 0 at the hillslope crest, and momentarily setting Ks = 0, numerical solutions of (37) indicate that, whereas the slope S(x) is close to (but not quite) a straight line, the land surface elevation ζ(x) is indistinguishable from a parabola whose curvature C = S′ increases with uplift rate. Upon setting Ks to a finite value, the slope S(x) becomes increasingly linear with increasing Ks. If we therefore assume an approximately constant curvature C, then from (37),

  • equation image

For realistic hillslope relief and small dispersal length scale λ0, the sum of the integrals in (38) varies with x only as a factor β of 2X. And, in the limit of vanishingly small length scale λ0, this sum approaches 2X, whence

  • equation image

Thus, at steady state the curvature C increases with the critical slope Sp and with the ratio of the rate of uplift to sediment mobilization, W/E0. More generally, with uniform mobilization (i.e., negligible influence of land surface slope on the rate of mobilization), the hillslope behaves as a diffusion equation (at steady state) with an apparent diffusivity Ke ∼ −cηW/C, or

  • equation image

which indicates that Ke depends on the system size X, and is not simply an additive function with the bulk diffusivity Ks. (See also section 3.3.) Moreover, as the rate of sediment mobilization E0 goes to zero, Ke goes to Ks.

[30] In the third case, suppose that E(u) = E0 and S(u) = S(v) = S(w) = S (a long hillslope with uniform slope ∣S∣ < Sc). Upon setting the lower limit in the first integral in (16) to zero (the hillslope crest) and the upper limit in the second integral to X (the foot of the hillslope), the result is

  • equation image

where S* = (Sc + S)/(ScS). With small slope ∣S∣ and length scale λ0X, qi(x) is relatively uniform, except near the two boundaries. As ∣S∣ increases, qi(x) begins to approach a linear form, increasing downslope. In the limit of ∣S[RIGHTWARDS ARROW] Sc, the second integral in (16) vanishes and the result is

  • equation image

illustrating that the flux increases linearly with distance x at a rate equal to E0, as it must in steady state. Once mobilized, material immediately travels to the hillslope base, akin to a detachment-limited behavior.

3.2. Effects of Slope-Dependent Mobilization

[31] Here we consider the effects of allowing the rate of mobilization E(u) to vary with land surface slope, as in (25). We numerically solve (33) for steady state conditions, and in the first set of runs we hold the critical slopes constant: Sp = 0.6 (31°) and Sc = 1.5 (56°). We also initially set the length scale λ0 to 1 m. This length scale is much shorter for rain splash, and perhaps significantly longer for processes such as ravel and soil slips, so λ0 ∼ 1 m may be viewed as an average over multiple operative processes. For convenience we set the particle concentration cη = 1.

[32] With E0 = 0.001 m a−1, E1 = 0.01 m a−1 and a relatively small bulk diffusivity Ks = 0.0001 m2 a−1, profiles of land surface elevation ζ(x) are nonparabolic with rounded crests and nearly linear slope forms away from the crest (Figure 5a). Following Foufoula-Georgiou et al. [2010], we plot the fall from the hillslope crest, ζ0ζ(x), versus position x, revealing that the land surface elevation varies as xa with a ∼ 3/2, independently of total relief and uplift rate W = 0.00001 − 0.0001 m a−1 (Figure 5b). The bulk flux qs(x) mimics its linear relation to the slope S(x), whereas the intermittent fluxes in the positive and negative x directions qi+(x) and qi(x) reflect the asymmetry in upslope versus downslope transport associated with slope (Figures 5c and 5d). The total flux q(x) = qs(x) + qi(x) is linear in x, as it must be with a uniform uplift (or exhumation) rate according to (4).

image

Figure 5. Plots of (a) land surface elevation ζ(x), (b) fall from crest ζ0ζ(x) and (c and d) soil flux components versus distance x for three rates of uplift W = 0.00001, 0.00005, and 0.0001 m a−1; parametric values are Sp = 0.6, Sc = 1.5, λ0 = 1 m, Ks = 0.0001 m2 a−1, E0 = 0.001 m a−1, and E1 = 0.01 m a−1; dashed lines in Figure 5b have slopes of 3/2 (top) and 2 (bottom). Bumps in the plotted lines at the hillslope crest in this and subsequent figures reflect a numerical artifact with the change in sign of the slope for a grid spacing of 1 m; this artifact has no effect on the interpretation of the curves.

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[33] Using the example above with W = 0.00005 m a−1, increasing the bulk diffusivity by an order of magnitude (Ks = 0.001 m2 a−1) has the effect of reducing the total hillslope relief and changing the relative magnitude of the flux terms (Figures 6a, 6c, and 6d), but the land surface elevation retains an x3/2 relation (Figure 6b). Only if the intermittent flux qi(x) is negligible relative to the bulk flux does the land surface profile approach a parabolic form.

image

Figure 6. Plots of (a) land surface elevation ζ(x), (b) fall from crest ζ0ζ(x) and (c and d) soil flux components versus distance x for two values of bulk diffusivity Ks = 0.0001 and 0.001 m2 a−1; parametric values are Sp = 0.6, Sc = 1.5, λ0 = 1 m, W = 0.00005 m a−1, E0 = 0.001 m a−1, and E1 = 0.01 m a−1; dashed lines in Figure 6b have slopes of 3/2 (top) and 2 (bottom).

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[34] In the examples above, the rate E0 may be envisioned as a uniform (average) thickness of sediment mobilized per square meter per year, some of which moves downslope, and some upslope, depending on the local slope. Then, the rate of mobilization E(x) increases with the magnitude of the slope such that, for example, E(x) = E0 + E1 when ∣S∣ = 1 (0.011 m a−1 in the examples above). With W = 0.00005 m a−1 and Ks = 0.0001 m2 a−1, increasing E0 (E0 = E1 = 0.01 m a−1) has the effect of reducing the total hillslope relief, and because the relative effect of slope on the rate of mobilization is reduced, E(x) is relatively uniform so that the land surface elevation approaches a parabolic form as described in section 3.1 (Figure 7). Increasing the uplift rate (W = 0.0001 m a−1) in turn increases the relief, so the effect of slope on the rate of mobilization E(x) increases slightly, although the land surface remains nearly parabolic (Figure 7b). In contrast, upon resetting W = 0.00005 m a−1 and E0 = 0.001 m a−1, doubling E1 to 0.02 m a−1 has the effect of reducing the relief where the land surface has an x3/2 relation (Figures 7a and 7b).

image

Figure 7. Plots of (a) land surface elevation ζ(x) and (b) fall from crest ζ0ζ(x) versus distance x for combinations of W, E0, and E1, namely, W = 0.00005 m a−1, E0 = 0.01 m a−1, E1 = 0.01 m a−1 (curves A); W = 0.0001 m a−1, E0 = 0.01 m a−1, E1 = 0.01 m a−1 (curves B); and W = 0.00005 m a−1, E0 = 0.001 m a−1, E1 = 0.02 m a−1 (curves C). Other parametric values are Sp = 0.6, Sc = 1.5, λ0 = 1 m, and Ks = 0.0001 m2 a−1; dashed lines in Figure 7b have slopes of 2 (top) and 3/2 (bottom).

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[35] Decreasing the length scale λ0 (λ0 = 0.1 m) has the effect of increasing the hillslope relief, where the land surface elevation near the crest steepens relative to an x3/2 relation (Figure 8). On the other hand, increasing this length scale (λ0 = 2 m) has the effect of reducing the relief, where the land surface elevation near the crest flattens relative to an x3/2 relation (Figure 8).

image

Figure 8. Plots of (a) land surface elevation ζ(x) and (b) fall from crest ζ0ζ(x) versus distance x for three values of the length scale λ0 = 0.1, 1, and 2 m; parametric values are W = 0.00005 m a−1, Sp = 0.6, Sc = 1.5, Ks = 0.0001 m2 a−1, E0 = 0.001 m a−1, and E1 = 0.01 m a−1; dashed lines in Figure 8b have slopes of 3/2 (top) and 2 (bottom).

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[36] Variations in hillslope length X and the critical slopes Sp and Sc lead to differences in the details of the hillslope profiles and soil material fluxes, but the basic behaviors remain qualitatively similar to the examples summarized above regarding variations in W, Ks, E0, and E1.

[37] We end this section by summarizing important properties of the convolutions in (23) and the kernels in (24), as embodied in the Fokker-Planck equation (33). Near the hillslope crest where ∣S(u)∣ ≪ Sp, the two convolutions on the right side of (23) are of the same magnitude. Dispersal of material mobilized at the rate E(u) mostly occurs over short distances involving the length scale λ0, with slope having a minor role. As ∣S(u)∣ increases, the magnitude of the convolution involving hr(xu, u) increases (in the positive x direction) relative to that involving hs(ux, u) as increasing steepness increases the probability of downslope motion. Then, in the limit of ∣S[RIGHTWARDS ARROW] Sp, the second integral in (23) vanishes and all motion originating at position u is downslope.

3.3. The x3/2 Behavior

[38] The result described above, that the land surface elevation varies as xa with a ∼ 3/2 at steady state when the rate of mobilization of soil material E(x) is slope dependent (as opposed to the “classic” result of a parabolic land surface profile when the soil flux varies linearly with slope or when, in the formulation above, the rate of mobilization E is uniform) has a straightforward explanation. Consider the intermittent flux qi(x) given by (32) and, dropping the partial derivative notation, recall that E(x) = E0 + E1S∣ = E0 + E1∣dζ/dx∣ according to (25). Assuming that the bulk flux qs(x) is much smaller than qi(x), then at steady state, qi(x) = cηWx, or

  • equation image

Momentarily assuming that the rate of mobilization E(x) is uniform (that is, E1 = 0), then

  • equation image

According to (34) and (35), at leading order the advective term in (44) is proportional to the local slope S = dζ/dx and the diffusive term enters at O(S2), which implies that

  • equation image

where k = λ0(1/Sp + 2/Sc) [L]. This yields a parabolic land surface profile with uniform curvature C = −cηW/E0k, where E0k [L2 t−1] is like a diffusivity. Thus, with reference to Figure 7, when E0 is sufficiently large that the rate of mobilization becomes relatively uniform, the land surface profile approaches a parabolic form although effects of finite E1 and the higher-order terms in (34) and (35) flatten the profile relative to an x2 relation (Figure 7b).

[39] Returning to (43) and again assuming that the diffusive term is negligibly small relative to the advective terms, then with μλ ≈ −kdζ/dx at leading order according to (34)

  • equation image

Although this vanishes at the hillslope crest, for E0E1 the linear term on the right side dominates, particularly for small slopes dζ/dx near the crest. And, for E0E1 (as in most of the simulations in section 3.2), the nonlinear term dominates away from the crest. Focusing on the latter case

  • equation image

or

  • equation image

which yields a land surface profile that varies as x3/2, consistent with the simulations above (Figures 58).

4. Comparison With Related Formulations and Field Observations

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

4.1. Related Formulations

[40] Foufoula-Georgiou et al. [2010] propose that the flux q(x) has the form of an upslope convolution of land surface slope. Namely,

  • equation image

where K* is a rate constant. Neglecting the bulk flux qs(x) and comparing (49) with (23), this formulation is equivalent to assuming that the probability p(u) [RIGHTWARDS ARROW] 1, or, that the critical slope Sp in (23) is sufficiently small that all motion is downslope. It also assumes that, with E0 = 0 in (25), E(u) = −K*S(u). In effect this formulation may provide a good description of the flux q(x) starting at a distance from the hillslope crest where the land surface is sufficiently steep that nonlocal effects begin [Foufoula-Georgiou et al., 2010].

[41] The nonlocal formulation in (49) may be interpreted physically in two ways. In the first interpretation, the amount of material mobilized is proportional to slope, thus −K*S(u)du, then dispersed downslope according to h(xu), independently of hillslope geometry (e.g., slope). In the second interpretation, the amount of material mobilized is independent of slope, thus K*du, then dispersed downslope according to −h(xu)S(u), where the slope S(u) locally modulates the magnitude of dispersal. The former says that slope is key in the mobilization process, which is consonant with processes such as ravel, soil slips or landsliding. The latter says that slope figures into how dispersal occurs, which is consonant with transport following mobilization of sediment by biological disturbances. In either case, dispersal of sediment originating from a position u (Figure 3) is not explicitly influenced by downslope land surface geometry (e.g., slope), although this deserves qualification as follows.

[42] The kernel h(xu) [L−1] in (49), as a function that weights the local gradients upslope of position x, may be interpreted as a probability function of the travel distance r = xu. In particular, Foufoula-Georgiou et al. [2010] consider the possibility that h(xu) is a heavy-tailed function that decays with r as hr2−α (with 1 < α < 2, so that a characteristic length scale cannot be defined), in which case (49) takes the fractional form

  • equation image

Substituting this into a statement of mass conservation, in turn, leads to a fractional diffusion equation involving the derivative dαζ/dxα. The units of K* are then determined by the numerical value of α in order to satisfy dimensional homogeneity. Momentarily returning to (13), it is noteworthy that if we assume the fractional rate of deposition P(v) monotonically decreases with v, then one possible form of h(xu) is the Pareto distribution, which is heavy tailed in certain circumstances. This choice of a heavy-tailed function implicitly assumes a decreasing fractional deposition rate with increasing travel distance, albeit not conditioned by the downslope land surface configuration. Nonetheless, where the land surface monotonically steepens with downslope distance, this is consistent with the idea of increasing travel distance with increasing steepness.

[43] A key outcome of the nonlocal flux relation (49) is that it gives rise to a nonlinear relation between the flux and local slope [Foufoula-Georgiou et al., 2010], akin to local nonlinear relations [Andrews and Bucknam, 1987; Roering et al., 1999]. Equally important is that the flux q(x) at any position x in general possesses a nonunique relation to the local slope S(x), as this flux, according to (49), consists of a weighting of all upslope values of S. That is, for a specific slope S(x) at x, different upslope serial configurations of S, depending on the history of evolution, generally will contribute different amounts to the flux at x.

[44] Tucker and Bradley [2010] propose a closely related formulation in which surface particles (representing individual particles or particle clusters, depending on scale) are randomly selected to undergo motion according to a set of probabilistic rules that depend on the configuration of neighboring surface particles, specifically their relative heights. With the onset of downslope motion of a particle, its continued motion (or deposition) is also conditioned by the local surface configuration, with the likelihood of deposition becoming vanishingly small as the surface slope approaches a critical value. When time averaged, the random selection (“disturbance”) of surface particles is entirely akin to a uniform rate of mobilization E0 of particles (absent any bulk motion). And, because the likelihood of continued motion versus deposition is conditioned on local slope, then, like the kernel (22), the ensemble behavior of many particles is explicitly influenced by the serial configuration of surface slope downslope from their positions of origin. Thus the kernel (22) mimics essential ingredients of the probabilistic conditioning of particle motions described by Tucker and Bradley [2010].

[45] Consider the special case where, at steady state, the land surface slope S(x) approaches the critical value Sc over most of x. At the hillslope crest, the effect of particle motions both to the right and left is to round the crest. Away from the crest as S(x) [RIGHTWARDS ARROW]Sc, the kernel hr(x, u) [RIGHTWARDS ARROW] hr(xu, u) [RIGHTWARDS ARROW] 1, and according to (25) the flux qi(x) = E0x = cηWx, or E0 = cηW = cηΓ. That is, a condition where the uplift rate is uniformly balanced by the mobilization rate requires a straight, critical slope S [RIGHTWARDS ARROW]Sc where, once mobilized, material immediately travels to the hillslope base. This is nominally equivalent to the numerical result of Tucker and Bradley [2010] for high uplift rate W and (effectively) unimpeded downslope motions of particles, where the production rate Γ = E0 is set by the (numerical) time average of random particle disturbances.

[46] The description of the flux of soil material as a convolution can be directly mapped to the description of soil grain transport by rain splash as a stochastic advection-dispersion process introduced by Furbish et al. [2009a]. In this case transport is essentially a surface phenomenon so that q(x, t) = qi(x, t), absent bulk transport. Grain motions are certainly intermittent, and because grain dispersal about drop impact sites occurs over a time scale (< 0.1 s) that is much shorter than the time required for significant changes in surface morphology [Furbish et al., 2007], the idea that dispersal occurs “instantaneously” (section 2.2) is readily justified. As described in section 2.3, the probability of downslope motion directly maps to land surface slope S [Furbish et al., 2007; Dunne et al., 2010]. Moreover, as described in section 2.2, the spatial, fractional rates of deposition, P and N, are determined locally by conditions at the position of origin u, that is, P = P(u) and N = N(u), rather than varying with conditions downslope or upslope, inasmuch as these rates of deposition are determined by the e-folding length scales, λr and λs (section 2.5), of exponential-like distributions of displacement distance, fr(r, x) and fs(s, x), consistent with observations [van Dijk et al., 2002; Mouzai and Bouhadef, 2003; Legout et al., 2005; Leguédois et al., 2005; Furbish et al., 2007, Figures 7 and 15].

[47] For these reasons the nonlinear Fokker-Planck (advection-dispersion) equation, namely (33), that derives from the convolution (15) is entirely consistent with the formulation of this equation provided by Furbish et al. [2009a, see equation (16) therein]. The only essential difference is this: Because of the definition of whizzing motions herein, that dispersal is “instantaneous” following mobilization, the time dimension needed to define a rate of dispersal is embedded within the rate of mobilization E in (33) rather than within the mean particle speed or diffusivity. That is, with reference to (33) and to equation (16) of Furbish et al. [2009a], λ = γU and λ2 = γDx, where γ [L] is the volume of grains in motion per unit area (the grain “activity”), U [L t−1] is a grain drift speed, and Dx [L2 t−1] is a diffusivity. Although both formulations are correct, the one provided by Furbish et al. [2009a] is more traditional and does not hinge on the assumption of instantaneous dispersal.

4.2. Field Observations

[48] Here we reproduce comparisons with field observations presented by Foufoula-Georgiou et al. [2010], focusing on the hillslopes described by Roering et al. [1999] and McKean et al. [1993]. The hillslope profile described by Roering et al. [1999] (Figure 9) is in the Oregon Coast Range near Coos Bay, Oregon, humid, forested steeplands developed on Eocene turbidites, with thin soils on ridges and thick soils in colluvial deposits. The hillslope profile described by McKean et al. [1993] (Figure 10) is in the Black Diamond Mines Regional Preserve east of San Francisco, California, grassland hillslopes developed on Eocene marine shales involving seasonal creep of high-plasticity clay soils together with biogenic transport.

image

Figure 9. Plots of (a) land surface elevation ζ(x) and (b) fall from crest ζ0ζ(x) versus distance x for hillslope described by Roering et al. [1999]; solid lines represent steady state version of (33) fitted to the data with parametric values W = 0.000075 m a−1, Sp = 0.6, Sc = 2.2, λ0 = 0.1 m, Ks = 0.00001 m2 a−1, E0 = 0.001 m a−1, and E1 = 0.0125 m a−1.

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image

Figure 10. Plots of (a) land surface elevation ζ(x) and (b) fall from crest ζ0ζ(x) versus distance x for hillslope described by McKean et al. [1993]; solid lines represent steady state version of (33) fitted to the data with parametric values Γ = 0.00026 m a−1, Sp = 0.6, Sc = 2.15, λ0 = 0.16 m, Ks = 0.0001 m2 a−1, E0 = 0.01 m a−1, and E1 = 0.241 m a−1.

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[49] Using reported values of either the uplift rate W or the soil production rate Γ, fitted profiles using the steady state version of (33) reasonably match the measured profiles, notably accommodating the lower parts of the hillslopes relative to a parabolic model (Figures 9a and 10a). Plots of the fall from the hillslope crest, ζ0ζ(x), versus position x, reveal that the profiles indeed decline as x3/2 over the lower parts of the hillslopes (Figures 9b and 10b). For convenience in the numerical fitting we set cη = 1, so the actual value of this concentration is incorporated within the numerical values of E0 and E1. Fitted values of Ks, λ0, E0, E1, Sp and Sc are likely not unique relative to other possible combinations that would give similarly reasonable fits. These values are: For W = 0.000075 m a−1, Ks = 0.00001 m2 a−1, λ0 = 0.1 m, E0 = 0.001 m a−1, E1 = 0.0125 m a−1, Sp = 0.6 and Sc = 2.2 (Figure 9); and for Γ = 0.00026 m a−1, Ks = 0.0001 m2 a−1, λ0 = 0.16 m, E0 = 0.01 m a−1, E1 = 0.241 m a−1, Sp = 0.6 and Sc = 2.15 (Figure 10).

[50] In these examples the computed flux q(x) varies linearly with position x as required by (4), but in relation to land surface slope, computed values of q increase nonlinearly with slope ∣S∣ (Figure 11), where q is dominated by the intermittent flux qi for the parametric values involved. This is consistent with the results of Foufoula-Georgiou et al. [2010]. Note that the “kink” in computed values of q at ∣S∣ = 0.6 (Figure 11a) reflects that the slope is at the critical value Sp, above which all motion is downslope.

image

Figure 11. Plots of computed flux q versus slope ∣S∣ for (a) hillslope described by Roering et al. [1999] and (b) hillslope described by McKean et al. [1993] using fitted parametric values (Figures 9 and 10).

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[51] We also consider three hillslope profiles within the Panther Creek area of the Land-Between-the-Lakes National Recreation Area, northwestern Tennessee [Roseberry, 2009; Furbish et al., 2009b] (Figure 12). These profiles are on hickory-oak-forested soils developed on Mississipian limestones, where disturbances are mostly due to root growth and the activity of worms, beetles, etc., although the profiles also likely reflect remnant effects of tree throw. The profiles are located next to first-order (top), second-order (middle) and third-order (bottom) channels in the same catchment. From the top profile to the bottom profile, the sequence in effect reflects different stages of channel downcutting and hillslope steepening followed by channel abandonment of the hillslope base and hillslope relaxation with floodplain development [Roseberry, 2009]. Due to uncertainty in the soil production rate Γ at this site, rather than numerically fitting the profiles we instead simply plot lines with slope 3/2 on the fall-from-crest plot (Figure 12b). At least for the top and bottom profiles, this fit is reasonable for positions upslope from inflection points (∼50 m) that represent a transition to downslope oversteepening (top hillslope) or deposition (middle and bottom hillslopes). We note that the top portions of hillslopes undergoing soil creep are in general insensitive to variations in downcutting that induce upslope responses (steepening or relaxation) in the hillslope land surface [Furbish and Fagherazzi, 2001; Mudd and Furbish, 2007]. Whereas the top parts of the Panther Creek profiles (Figure 12) are likely undergoing steady, uniform lowering, the lower parts of these profiles are undergoing unsteady steepening or deposition.

image

Figure 12. Plots of (a) land surface elevation ζ(x) and (b) fall from crest ζ0ζ(x) versus distance x for hillslopes described by Roseberry [2009] and Furbish et al. [2009b]; gray dashed lines in Figure 12b have slope of 3/2.

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5. Discussion and Conclusions

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

[52] Our formulation of the intermittent flux qi(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 qi(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 μλ and variance σλ2 of displacements. Thus, in (32) and (33), λ is like the product of an active layer thickness and a drift speed and λ2 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 qi(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 λr = 1/P and λs = 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 Sc 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 Sc 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 E1S∣, 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 = E0, E1 = 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 E1 = 0) appear “diffusive.” With E1 > E0, then as described in sections 3.2 and 3.3, the intermittent flux qi(x) varies with slope as ∣SS as in (47) so that the predicted steady state hillslope profile varies as x3/2. 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 x2 and x3/2 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 x2 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 x3/2 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 x3/2 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 E1E0 with relatively small Ks, 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 x3/2 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 x3/2 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 qi(x, t) given by (32). Namely, substituting (34) and (35) into (32) gives

  • equation image

For a uniform rate of mobilization (E = E0) and nearly uniform slope such that ∂(S2)/∂x is negligibly small, qi(x, t) possesses an approximate one-to-one relation with slope given by the advective terms on the right side of (51). If, however, ∂(S2)/∂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 S2, 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 λ0, and the critical slope Sp 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 Sp 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 Sc, 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 Sc 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.

Appendix A:: Averaging of Land Surface Elevation

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

[62] Letting lx and ly [L] denote lag distances parallel to x and y, we denote the two-dimensional autocorrelation function of elevation as ρ(lx, ly; x), which is assumed to be homogeneous, albeit possibly anisotropic, in the vicinity of x. Integrating ρ(lx, ly; x) over the domain of lx for ly = 0 and then over the domain of ly for lx = 0 provides the decorrelation length scales λx and λy [L] that are characteristic of the roughness created by disturbances. In turn the average of ζ(x, y, t) over y involving a width b is defined by (1) in the main text. Inasmuch as the averaging width bλy with λx > 0, then ζ(x, t) varies smoothly with x so that S(x, t) = ∂ζ(x, t)/∂x also varies smoothly with x. Note, however, that b must not be “too” large if S(x, t) varies systematically along y. That is, envision an averaging window of width b that slides along y (for given x). If S(x, t) varies significantly with y, then a value b > λy averages over this variation. Thus, a value of b, where λyb < LSy, provides an appropriate value of S(x, t) to be associated with position x, where LSy = ∣S∣/∣dS/dy∣ [L] denotes a length scale over which the land surface slope is relatively uniform.

Notation
a

exponent on distance x for fall from crest ζ0ζ(x) [dimensionless].

b

hillslope width along contour [L].

cs

vertically averaged soil particle concentration above η [dimensionless].

cη

soil particle concentration at base of active thickness [dimensionless].

C

land surface curvature [L−1].

Dx

diffusivity associated with rain splash transport [L2 t−1].

E

rate of mobilization of soil material [L t−1].

E0

rate of mobilization of soil material on flat surface [L t−1].

E1

rate of mobilization of soil material with increasing slope ∣S∣ [L t−1].

fr, fs

probability densities of displacement distances r and s [L−1].

F

fraction of sediment splashed downslope [dimensionless].

Fr, Fs

cumulative distributions of displacement distances r and s [dimensionless].

Fu

cumulative probability of deposition of material originating at u [dimensionless].

hr, hs

kernels of displacement distances r and s [dimensionless].

H

active soil thickness [L].

k

first-order modulation of effect of slope on the mean displacement distance μλ [L].

Ke

effective diffusivity at steady state [L2 t−1].

Ki

diffusivity of intermittent motions [L2 t−1].

Ks

diffusivity for bulk soil transport [L2 t−1].

K*

rate constant [units vary].

lx, ly

lag distances parallel to x and y [L].

LS

length scale parallel to x over which slope S is relatively uniform [L].

LSy

length scale parallel to y over which slope S is relatively uniform [L].

p, n

probabilities of motion in positive and negative x directions [dimensionless].

P, N

spatial, fractional rates of deposition [L−1].

q

soil material flux [L2 t−1].

qi

intermittent flux [L2 t−1].

qi+, qi

intermittent flux in positive and negative x directions [L2 t−1].

qs

bulk soil flux [L2 t−1].

Q

total soil flux over b [L3 t−1].

Q+, Q

total soil flux over b in positive and negative x directions [L3 t−1].

r, s

displacement distances in positive and negative x directions [L].

Ru

probability defined by Ru = 1 − Fu [dimensionless].

S

land surface slope [dimensionless].

Sc

critical slope above which motion is not arrested [dimensionless].

Sp

critical slope above which all motion is downslope [dimensionless].

t

time [t].

t0

initial time [t].

T

averaging time scale, mean soil residence time [t].

u, v, w

coordinate positions on x axis [L].

U

downslope drift speed [L t−1].

V

solid volume of sediment crossing position x [L3].

Vi, Vs

solid volume of sediment crossing position x as intermittent and bulk flux [L3].

W

uplift rate [L t−1].

x, y, z

Cartesian coordinates [L].

X

hillslope length [L].

α

exponent of fractional derivative [dimensionless].

β

factor of hillslope length [L].

γ

volume of grains in motion per unit area (grain “activity”) [L].

Γ

soil production rate, or entrainment (or disentrainment) rate [L t−1].

ζ

land surface elevation [L].

ζ0

land surface elevation at hillslope crest [L].

η

elevation of base of active layer [L].

λ

characteristic length scale of soil material dispersal [L].

λr, λs

characteristic length scales of soil material dispersal for displacements r and s [L].

λx, λy

decorrelation length scales of land surface elevation parallel to x and y [L].

λ0

characteristic length scale of soil material dispersal on flat surface [L].

μλ

weighted average displacement distance [L].

ρ

autocorrelation function of land surface elevation [dimensionless].

σλ2

weighted variance of displacement distance [L2].

τ

time [t].

ϕ

porosity [dimensionless].

Acknowledgments

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

[63] We value thoughtful discussions with Efi Foufoula-Georgiou, Josh Roering, John Roseberry, and Greg Tucker and acknowledge support by the National Science Foundation (EAR-0744934). Bob Anderson, Tom Dunne, and Manny Gabet provided valuable reviews.

References

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