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A nonlocal theory of sediment transport on hillslopes

Profile image of Vamsi GantiVamsi Ganti

2010, Journal of Geophysical Research

https://doi.org/10.1029/2009JF001280
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Abstract

1] Hillslopes are typically shaped by varied processes which have a wide range of eventbased downslope transport distances, some of the order of the hillslope length itself. We hypothesize that this can lead to a heavy-tailed distribution of displacement lengths for sediment particles. Here, we propose that such a behavior calls for a nonlocal computation of the sediment flux, where the sediment flux at a point is not strictly a function (linear or nonlinear) of the gradient at that point only but is an integral flux taking into account the upslope topography (convolution Fickian flux). We encapsulate this nonlocal behavior in a simple fractional diffusive model which involves fractional derivatives, with the order of differentiation (1 < a ≤ 2) dictating the degree of nonlocality (a = 2 corresponds to linear diffusion and strictly local dependence on slope). The model predicts an equilibrium hillslope profile which is parabolic close to the ridgetop and transits, at a short downslope distance, to a power law with an exponent equal to the parameter a of the fractional transport model. Hillslope profiles reported in previously studied sites support this prediction. Furthermore, we show that the nonlocal transport model gives rise to a nonlinear dependency on local slope and that variable upslope topography leads to widely varying rates of sediment flux for a given local hillslope gradient. Both of these results are consistent with available field data and suggest that nonlinearity in hillslope flux relationships may arise in part from nonlocal transport effects in which displacement lengths increase with hillslope gradient. The proposed hypothesis of nonlocal transport implies that field studies and models of sediment fluxes should consider the size and displacement lengths of disturbance events that mobilize hillslope colluvium.

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References (65)

  1. Andrews, D. J., and R. C. Buckman (1987), Fitting degradation of shoreline scarps by a nonlinear diffusion model, J. Geophys. Res., 92(B12), 12,857-12,867, doi:10.1029/JB092iB12p12857.
  2. Baeumer, B., D. Benson, M. Meerschaert, and S. Wheatcraft (2001), Sub- ordinated advection-dispersion equation for contaminant transport, Water Resour. Res., 37(6), 1543-1550, doi:10.1029/2000WR900409.
  3. Benson, D. (1998), The fractional advection-dispersion equation: Develop- ment and applications, Ph.D. thesis, Univ. of Nevada, Reno.
  4. Benson, D. A., S. W. Wheatcraft, and M. M. Meerschaert (2000a), Appli- cation of a fractional advection-dispersion equation, Water Resour. Res., 36(6), 1403-1412, doi:10.1029/2000WR900031.
  5. Benson, D. A., S. W. Wheatcraft, and M. M. Meerschaert (2000b), The fractional-order governing equation of Lévy motion, Water Resour. Res., 36(6), 1413-1423, doi:10.1029/2000WR900032.
  6. Berkowitz, B., J. Klafter, R. Metzler, and H. Scher (2002), Physical pictures of transport in heterogeneous media: Advection-dispersion, random- walk, and fractional derivative formulations, Water Resour. Res., 38(10), 1191, doi:10.1029/2001WR001030.
  7. Biler, P., T. Funaki, and W. Woyczynski (1998), Fractal Burgers equations, J. Differential Equations, 148, 9-46, doi:10.1006/jd equation 1998.3458.
  8. Bouchaud, J.-P., and A. Georges (1990), Anomalous diffusion in disor- dered media: Statistical mechanisms, models and physical applications, Phys. Rep., 195(4-5), 127-293, doi:10.1016/0370-1573(90)90099-N.
  9. Bradley, D. N., G. E. Tucker, and D. A. Benson (2010), Fractional disper- sion in a sand-bed river, J. Geophys. Res., 115, F00A09, doi:10.1029/ 2009JF001268.
  10. Campos, D., J. Fort, and V. Mendez (2006), Transport on fractal river net- works: Application to migration fronts, Theor. Popul. Biol., 69, 88-93, doi:10.1016/j.tpb.2005.09.001.
  11. Culling, W. E. H. (1960), Analytical theory of erosion, J. Geol., 68, 336- 344, doi:10.1086/626663.
  12. Cushman, J. H. (1991), On diffusion in fractal porous media, Water Resour. Res., 27(4), 643-644, doi:10.1029/91WR00162.
  13. Cushman, J. H. (1997), The Physics of Fluids in Hierarchical Porous Media: Angstroms to Miles, 467 pp., Kluwer Acad., Norwell, Mass.
  14. Cushman, J. H., and T. Ginn (2000), Fractional advection-dispersion equa- tion: A classical mass balance with convolution-Fickian flux, Water Resour. Res., 36(12), 3763-3766, doi:10.1029/2000WR900261.
  15. Dagan, G. (1997), Subsurface Flow and Transport: A Stochastic Approach, C a m b r i d g e U n i v . P r e s s , C a m b r i d g e , M a s s . , d o i : 10 . 1 0 1 7 / CBO9780511600081.
  16. Darwin, C., (1881), The Formation of Vegetable Mould Through the Action of Worms, John Murray, London.
  17. Deng, Z. Q., J. L. M. P. de Lima, and V. P. Singh (2005), Fractional kinetic model for first flush of stormwater pollutants, J. Environ. Eng., 131(2), 232-241, doi:10.1061/(ASCE)0733-9372(2005)131:2(232).
  18. Deng, Z. Q., J. L. M. P. de Lima, M. I. P. de Lima, and V. P. Singh (2006), A fractional dispersion model for overland solute transport, Water Resour. Res., 42, W03416, doi:10.1029/2005WR004146.
  19. Dietrich, W. E., D. G. Bellugi, L. S. Sklar, J. D. Stock, A. M. Heimsath, and J. J. Roering (2003), Geomorphic transport laws for predicting land- scape form and dynamics, in Prediction in Geomorphology, Geophys. Monogr. Ser., vol. 135, edited by P. Wilcock and R. Iverson, pp. 103- 132, AGU, Washington, D.C., doi:10.1029/135GM09.
  20. Feller, W. (1971), An Introduction to Probability Theory and Its Applica- tions, vol. 2, 2nd ed., Wiley, New York.
  21. Fleming, R. W., and A. M. Johnson (1975), Rates of seasonal creep of silty clay soil, Q. J. Eng. Geol., 8, 1-29, doi:10.1144/GSL.QJEG.1975.008.01.01.
  22. Foufoula-Georgiou, E., V. Ganti and P. Passalacqua (2008), Geomorphic transport laws: Non-local flux with classical mass balance, Rep. UMSI 2008/27, Supercomput. Inst., Univ. of Minn., Minneapolis.
  23. Furbish, D. J., P. K. Haff, W. E. Dietrich, and A. M. Heimsath (2009), Sta- tistical description of slope-dependent soil transport and the diffusion-like coefficient, J. Geophys. Res., 114, F00A05, doi:10.1029/2009JF001267.
  24. Gabet, E. J., D. J. Reichman, and E. W. Seabloom (2003), The effects of bioturbation on soil processes and sediment transport, Annu. Rev. Earth Planet. Sci., 31, 249-273, doi:10.1146/annurev.earth.31.100901.141314.
  25. Ganti, V., M. Meerchaert, E. Foufoula-Georgiou, E. Viparelli, and G. Parker (2010), Normal and anomalous diffusion of gravel tracer particles in rivers, J. Geophys. Res., 115, F00A12, doi:10.1029/2008JF001222.
  26. Gelhar, L. W., and C. L. Axness (1983), Three-dimensional stochastic analysis of macrodispersion in aquifers, Water Resour. Res., 19(1), 161-180, doi:10.1029/WR019i001p00161.
  27. Gilbert, G. K. (1909), The convexity of hilltops, J. Geol., 17, 344-350, doi:10.1086/621620.
  28. Grünwald, A. K. (1867), Uber "begrenzt" derivation und deren anwendung, Z. Angew. Math. Phys., 12, 441-480.
  29. Hack, J. T. (1960), Interpretation of erosional topography in humid temperate regions, Am. J. Sci., 258A, 80-97.
  30. Hack, J. T., and J. C. Goodlett (1960), Geomorphology and forest ecology of a mountain region in the central Appalachians, U.S. Geol. Surv. Prof. Pap., 347, 66 pp.
  31. Heimsath, A. M., W. E. Dietrich, K. Nishilzumi, and R. C. Finkel (1999), Cosmogenics nuclides, topography, and spatial variation of soil depth, Geomorphology, 27, 151-172, doi:10.1016/S0169-555X(98)00095-6.
  32. Heimsath, A. M., W. E. Dietrich, K. Nishiizumi, R. C. Finkel (2001), Stochastic processes of soil production and transport: Erosion rates, topographic variation, and cosmogenic nuclides in the Oregon Coast Range, Earth Surf. Processes Landforms, 26, 531-552.
  33. Howard, A. D. (1994), A detachment-limited model of drainage basin evo- lution, Water Resour. Res., 30(7), 2261-2285, doi:10.1029/94WR00757.
  34. Kirkby, M. J. (1967), Measurement and theory of soil creep, J. Geol., 75, 359-378, doi:10.1086/627267.
  35. Koons, P. O. (1989), The topographic evolution of collisional mountain belts: A numerical look at the southern Alps, New Zealand, Am. J. Sci., 288, 1041-1069.
  36. Lamperti, J. (1962), Semi-stable stochastic processes, Trans. Am. Math. Soc., 104, 62-78, doi:10.2307/1993933.
  37. Lashermes, B., E. Foufoula-Georgiou and W. E. Dietrich (2007), Channel network extraction from high resolution topography using wavelets, Geophys. Res. Lett., 34, L23S04, doi:10.1029/2007GL031140.
  38. McKean, J. A., W. E. Dietrich, R. C. Finkel, J. R. Southon, and M. W. Caffee (1993), Quantification of soil production and downslope creep rates from cosmogenic 10Be accumulations on a hillslope profile, Geology, 21, 343- 346, doi:10.1130/0091-7613(1993)021<0343:QOSPAD>2.3.CO;2.
  39. Meerschaert, M. M., and C. Tadjeran (2004), Finite difference approxima- tions of fractional advection-dispersion flow equations, J. Comput. Appl. Math., 172, 65-77, doi:10.1016/j.cam.2004.01.033.
  40. Meerschaert, M. M., D. A. Benson, and B. Baeumer (1999), Multidimen- sional advection and fractional dispersion, Phys. Rev. E, 59, 5026- 5028, doi:10.1103/PhysRevE.59.5026.
  41. Meerschaert, M. M., D. A. Benson, and B. Baeumer (2001), Operator Le'vy motion and multiscaling anomalous diffusion, Phys. Rev. E, 63, 021112-021117, doi:10.1103/PhysRevE.63.021112.
  42. Metzler, R., and A. Compte (2000), Generalized diffusion-advection schemes and dispersive sedimentation: A fractional approach, J. Phys. Chem. B, 104(16), 3858-3865, doi:10.1021/jp993698f.
  43. Miller, K. S., and B. Ross (1993), An Introduction to the Fractional Calculus and Fractional Differential Equations, J. Wiley, New York.
  44. Norman, S. A., R. J. Schaetzl, and T. W. Small (1995), Effects of slope angle on mass movements by tree uprooting, Geomorphology, 14, 19- 27, doi:10.1016/0169-555X(95)00016-X.
  45. Oldham, K. B., and J. Spanier (1974), The Fractional Calculus, Academic, San Diego, Calif.
  46. Passalacqua, P., F. Porté-Agel, E. Foufoula-Georgiou, and C. Paola (2006), Application of dynamic subgrid-scale concepts from large-eddy simula- tion to modeling landscape evolution, Water Resour. Res., 42, W06D11, doi:10.1029/2006WR004879.
  47. Pekalski, A., and K. Sznajd-Weron (Eds.) (1999), Anomalous Diffusion: From Basics to Applications, Lect. Notes in Phys., vol. 519, Springer, Berlin, doi:10.1007/BFb0106828.
  48. Podlubny, I. (1999), Fractional Differential Equations, vol. 198, Academic, San Diego, Calif.
  49. Reneau, S. L., and W. E. Dietrich (1991), Erosion rates in the southern Oregon Coast Range: Evidence for an equilibrium between hillslope erosion and sediment yield, Earth Surf. Processes Landforms, 16, 307-322, doi:10.1002/esp.3290160405.
  50. Roering, J. J. (2008), How well can hillslope evolution models 'explain' topog- raphy? Simulating soil production and transport using high-resolution topo- graphic data, Geol. Soc. Am. Bull., 120, 1248-1262, doi:10.1130/B26283.1.
  51. Roering, J. J., and M. Gerber (2005), Fire and the evolution of steep, soil- mantled landscapes, Geology, 33, 349-352, doi:10.1130/G21260.1.
  52. Roering, J., J. Kirchner, and W. E. Dietrich (1999), Evidence for nonlinear, diffusive sediment transport on hillslopes and implications for landscape morphology, Water Resour. Res., 35(3), 853-870.
  53. Schey, H. (1992), Div Grad Curl and All That: An Informal Text on Vector Calculus, 2nd ed., W. W. Norton, New York.
  54. Schumer, R., D. Benson, M. Meerschaert, and S. Wheatcraft (2001), Euler- ian derivation of the fractional advection-dispersion equation, J. Contam. Hydrol., 48, 69-88, doi:10.1016/S0169-7722(00)00170-4.
  55. Schumer, R., M. M. Meerschaert, and B. Baeumer (2009), Fractional advection-dispersion equations for modeling transport at the Earth sur- face, J. Geophys. Res., 114, F00A07, doi:10.1029/2008JF001246.
  56. Shlesinger, M. F., G. M. Zaslavsky, and U. Frisch (Eds.) (1995), Lévy Flights and Related Topics in Physics, Lect. Notes in Phys., vol. 450, Springer, Berlin, doi:10.1007/3-540-59222-9.
  57. Stark, C. P., E. Foufoula-Georgiou, and V. Ganti (2009), A nonlocal theory of sediment buffering and bedrock channel evolution, J. Geophys. Res., 114, F01029, doi:10.1029/2008JF000981.
  58. Su, N. (1995), Development of the Fokker-Planck equation and its solu- tions for modeling transport of conservative and reactive solutes in phys- ically heterogeneous media, Water Resour. Res., 31(12), 3025-3032, doi:10.1029/95WR02765.
  59. Tucker, G. E., and D. N. Bradley (2010), Trouble with diffusion: Reasses- sing hillslope erosion laws with a particle-based model, J. Geophys. Res., 115, F00A10, doi:10.1029/2009JF001264.
  60. Voller, V., and C. Paola (2010), Can anomalous diffusion describe deposi- tional fluvial profiles?, J. Geophys. Res., 115, F00A13, doi:10.1029/ 2009JF001278.
  61. Washburn, A. L. (1973), Periglacial Processes and Environments, 320 pp., Edward Arnold, London.
  62. Woyczynski, W. (1998), Burgers-KPZ Turbulence: Gottingen Lectures, Lect. Notes in Math., vol. 1700, Springer, Berlin.
  63. Yoo, K., R. Amundson, A. M. Heimsath, and W. E. Dietrich (2005), Pro- cess-based model linking pocket gopher (Thomomys bottae) activity to sediment transport and soil thickness, Geology, 33, 917-920, doi:10.1130/G21831.1.
  64. Yoo, K., R. Amundson, A. M. Heimsath, and W. E. Dietrich (2006), Spa- tial patterns of soil organic carbon on hillslopes: Integrating geomorphic processes and the biological C cycle, Geoderma, 130, 47-65, doi:10.1016/j.geoderma.2005.01.008.
  65. W. E. Dietrich, Department of Earth and Planetary Science, University of California, Berkeley, CA 94720, USA. (efi@umn.edu) E. Foufoula-Georgiou and V. Ganti, Saint Anthony Falls Laboratory and National Center for Earth-surface Dynamics, Department of Civil Engineering, University of Minnesota, 2 Third Ave. SE, Minneapolis, MN 55414, USA.

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Brian McArdell

Water Resources Research, 1993

Twenty-eight coupled observations of flow, transport, and bed surface grain size distribution were made in a laboratory flume using a wide range of flows and a sediment with a very poorly sorted, bimodal grain size distribution. These observations permit the transport rates of individual size fractions to be scaled by the proportion of each size immediately available for transport on the bed surface. The key to our observations is the use of a sediment in which each size fraction has been painted a different color, which permits reliable, repeatable, and nondestructive measurement of the bed surface grain size distribution from photographs of the bed surface. At a given flow, the fractional transport rates may be divided into two parts: a finer-grained portion within which fractional transport rates are a function only of their proportion on the bed surface and total transport rate, and a coarser-grained portion for which fractional transport rates also depend on the proportion of individual grains within a fraction that remain essentially immobile throughout the experimental run. We define the latter condition as one of partial transport and observe that the grain size separating partial and fully mobilized transport consistently increases with flow strength. Complete mobilization of a size fraction occurs at roughly twice the shear stress necessary for incipient motion of that fraction. Zones of partial and full mobility are quite distinct when fractional transport rates are scaled by the bed surface grain size distribution, although a region of partial transport is evident when these data and other experimental and field observations are scaled by the bulk grain size distribution of the sediment bed. Critical shear stresses for the incipient motion of individual fractions in our experimental sediment vary over an order of magnitude, a result strongly in contrast to many earlier observations, but consistent with our observations of incipient motion in sediments with bimodal grain size distributions. ent sites, or at different times at the same location, require some accounting for these adjustments, which are often complex and poorly understood. The absence of directly correlated observations of transport and bed surface size distribution presents a fundamental obstacle to the unambig-u0us evaluation of the physical processes inherent in any set of transport observations and precludes the availability of a general and consistent set of transport data against which general theory may be tested. A direct approach to this problem can be achieved by correlating individual observations of fractional transport rates with the size distribution of the bed surface from which the transport is entrained. The bed surface size distribution is considerably more difficult to measure accurately than the transport and has been the limiting factor preventing the development of a general and consistent set of surface-based

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Sediment transport mechanisms on soil-mantled hillslopes
Arjun Heimsath

Geology, 2001

Landscape evolution is modeled widely using a simple creep law for complex processes of sediment transport. Here, field data show how a new transport model, combined with an exponential soil production law, better captures spatial variations of soil thickness on hillslopes. We combine parameterizations of simple and depth-dependent creep with overland flow to predict soil thickness and suggest how soil distribution evolves in response to climatic and tectonic forcing. We present an empirical expression for the response time of the system to external forcing that shows strong dependence on relief and is independent of soil production rate. We suggest that this parameterization may be used to quantify upland carbon storage and removal and predict impacts of deforestation or rapid climatic changes.

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A nonlocal theory of sediment buffering and bedrock channel evolution
Colin Stark

Journal of Geophysical Research, 2008

1] Bedrock erosion in mountain river channels ultimately sets the erosion rate of the surrounding hillslopes and the rate of sediment supply to the channels. The supply of coarse bed sediment acts as a dampening effect on further erosion by depositing an alluvial cover that temporarily obscures the bedrock. For landscapes where the residence time of the alluvial bed cover is comparable to the timescale of bedrock incision, coarse sediment supply and transport generate a strong negative feedback on fluvial downcutting and the coupled process of hillslope-channel erosion is inherently self-buffering. Here we study a simple model of self-buffered bedrock channel erosion that incorporates the spreading of bed sediment cover downstream in a way that allows for a broad-tailed, power law probability distribution of transport velocities of bed sediment over the long-term. This leads us to consider a nonlocal transport law (fractional advection) parameterized by a scaling exponent 0 a < 1 which collapses to local advection for a ! 1. For strong sediment buffering, we find that nonlocality 1 À a has a direct control on the power law scaling of channel slope S with upstream area A, giving S $ A À(1Àa)/2 at steady state. Empirical observations of slope-area scaling are consistent with a < 1 and nonlocal transport. In general, the model predicts linear, logarithmic, or power law stream profiles depending on the extent of buffering, the degree of nonlocality, and the scaling of the bedrock erosion law. It also predicts, somewhat counterintuitively, that bed cover should thicken with distance x downstream slower than linearly as x a , i.e., the more nonlocal the bed sediment spreading process (a ! 0), the slower the bed cover increases downstream. We deduce that long-range, heterogeneous transport of coarse sediment in mixed bedrock-alluvial rivers may be a key element of landscape scaling and an important factor in landscape dynamics.

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Related topics

  • Stochastic Process
  • Probability Distribution & Appli...
  • Fractional Derivative

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    Introduction to special section on Stochastic Transport and Emergent Scaling on Earth's Surface: Rethinking geomorphic transport---Stochastic theories, broad scales of motion and nonlocality
    Colin Stark

    Journal of Geophysical Research, 2010

    1] In many geomorphic transport systems, the time and length scales of motion vary widely: particles can be trapped for both short and long periods of time and they can travel large or small distances in very short intervals of time. To model such systems we need fresh conceptual and mathematical formalisms. The goal of this collection of papers is to challenge existing thinking in geomorphic transport by putting forward new ideas and theories for environmental fluxes, from particle transport in a single stream, to landslide debris mobilization, sediment and water transport on hillslopes, dynamic transport on river networks, and interpretation of sedimentary deposits over geologic time. Advanced stochastic theories of transport are proposed, the notion of nonlocal flux is introduced, and fractional advection-diffusion equations are explored as possible models of geomorphic transport.

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    Space-time dynamics of depositional systems: Experimental evidence and theoretical modeling of heavy-tailed statistics
    Vamsi Ganti

    Journal of Geophysical Research, 2011

    1] In depositional systems, channels migrate from one location to another, causing erosion and deposition at any given point in the domain. The durations of depositional and erosional events, as well as their magnitudes, control the structure of the stratigraphic record. In this study, we use high-resolution temporal surface elevation data from a controlled experiment to quantify the probability distributions of the processes that govern the evolution of depositional deltaic systems. Heavy-tailed statistics of erosional and depositional events are documented, indicating that a small but significant chance exists for the occurrence of extreme events. We show that the periods of inactivity, when neither deposition nor erosion occurs, follow a truncated Pareto distribution whose truncation scale is set by the mean characteristic avulsion time scale in the system. Further, we show that the heavy tails in the magnitudes of the erosional and depositional events are not preserved in the stratigraphic record, resulting instead in an exponential distribution for the bed sediment thickness. It is also shown that the temporal evolution of surface elevation exhibits self-similarity with a nonlinear spectrum of scaling exponents (multifractality) quantifying the complex dynamics of the system. Finally, we show how the results of this study can lead to improved diffusional models for surface evolution using the tools of fractional calculus. Citation: Ganti, V., K. M. Straub, E. Foufoula-Georgiou, and C. Paola (2011), Space-time dynamics of depositional systems: Experimental evidence and theoretical modeling of heavy-tailed statistics,

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    Normal and anomalous diffusion of gravel tracer particles in rivers
    Enrica Viparelli

    Journal of Geophysical Research, 2010

    1] One way to study the mechanism of gravel bed load transport is to seed the bed with marked gravel tracer particles within a chosen patch and to follow the pattern of migration and dispersal of particles from this patch. In this study, we invoke the probabilistic Exner equation for sediment conservation of bed gravel, formulated in terms of the difference between the rate of entrainment of gravel into motion and the rate of deposition from motion. Assuming an active layer formulation, stochasticity in particle motion is introduced by considering the step length (distance traveled by a particle once entrained until it is deposited) as a random variable. For step lengths with a relatively thin (e.g., exponential) tail, the above formulation leads to the standard advection-diffusion equation for tracer dispersal. However, the complexity of rivers, characterized by a broad distribution of particle sizes and extreme flood events, can give rise to a heavy-tailed distribution of step lengths. This consideration leads to an anomalous advection-diffusion equation involving fractional derivatives. By identifying the probabilistic Exner equation as a forward Kolmogorov equation for the location of a randomly selected tracer particle, a stochastic model describing the temporal evolution of the relative concentrations is developed. The normal and anomalous advection-diffusion equations are revealed as its long-time asymptotic solution. Sample numerical results illustrate the large differences that can arise in predicted tracer concentrations under the normal and anomalous diffusion models. They highlight the need for intensive data collection efforts to aid the selection of the appropriate model in real rivers.

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    Using hilltop curvature to derive the spatial distribution of erosion rates
    Kyungsoo Yoo

    Journal of Geophysical Research, 2012

    1] Erosion rates dictate the morphology of landscapes, and therefore quantifying them is a critical part of many geomorphic studies. Methods to directly measure erosion rates are expensive and time consuming, whereas topographic analysis facilitates prediction of erosion rates rapidly and over large spatial extents. If hillslope sediment flux is nonlinearly dependent on slope then the curvature of hilltops will be linearly proportional to erosion rates. In this contribution we develop new techniques to extract hilltop networks and sample their adjacent hillslopes in order to test the utility of hilltop curvature for estimating erosion rates using high-resolution (1 m) digital elevation data. Published and new cosmogenic radionuclide analyses in the Feather River basin, California, suggest that erosion rates vary by over an order of magnitude (10 to 250 mm kyr À1 ). Hilltop curvature increases with erosion rates, allowing calibration of the hillslope sediment transport coefficient, which controls the relationship between gradient and sediment flux. Having constraints on sediment transport efficiency allows estimation of erosion rates throughout the landscape by mapping the spatial distribution of hilltop curvature. Additionally, we show that hilltop curvature continues to increase with rising erosion rates after gradient-limited hillslopes have emerged. Hence hilltop curvature can potentially reflect higher erosion rates than can be predicted by hillslope gradient, providing soil production on hilltops can keep pace with erosion. Finally, hilltop curvature can be used to estimate erosion rates in landscapes undergoing a transient adjustment to changing boundary conditions if the response timescale of hillslopes is short relative to channels. Citation: Hurst, M. D., S. M. Mudd, R. Walcott, M. Attal, and K. Yoo (2012), Using hilltop curvature to derive the spatial distribution of erosion rates,

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    Influence of lithology on hillslope morphology and response to tectonic forcing in the northern Sierra Nevada of California
    Rachel Walcott

    Journal of Geophysical Research: Earth Surface, 2013

    1] Many geomorphic studies assume that bedrock geology is not a first-order control on landscape form in order to isolate drivers of geomorphic change (e.g., climate or tectonics). Yet underlying geology may influence the efficacy of soil production and sediment transport on hillslopes. We performed quantitative analysis of LiDAR digital terrain models to examine the topographic form of hillslopes in two distinct lithologies in the Feather River catchment in northern California, a granodiorite pluton and metamorphosed volcanics. The two sites, separated by <2 km and spanning similar elevations, were assumed to have similar climatic histories and are experiencing a transience in landscape evolution characterized by a propagating incision wave in response to accelerated surface uplift c. 5 Ma. Responding to increased incision rates, hillslopes in granodiorite tend to have morphology similar to model predictions for steady state hillslopes, suggesting that they adjust rapidly to keep pace with the incision wave. By contrast, hillslopes in metavolcanics exhibit high gradients but lower hilltop curvature indicative of ongoing transient adjustment to incision. We used existing erosion rate data and the curvature of hilltops proximal to the main channels (where hillslopes have most likely adjusted to accelerated erosion rates) to demonstrate that the sediment transport coefficient is higher in granodiorite (8.8 m 2 ka À1 ) than in metavolcanics (4.8 m 2 ka À1 ). Hillslopes in both lithologies get shorter (i.e., drainage density increases) with increasing erosion rates.

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