A Discrete Random Model Describing Bedrock Profile Abrasion

  • András A. Sipos
  • Gábor Domokos
  • Andrew Wilson
  • Niels Hovius


We use a simple, collision-based, discrete, random abrasion model to compute the profiles for the stoss faces in a bedrock abrasion process. The model is the discrete equivalent of the generalized version of a classical, collision based model of abrasion. Three control parameters (which describe the average size of the colliding objects, the expected direction of the impacts and the average volume removed from the body due to one collision) are sufficient for realistic predictions.

Our computations show the robust emergence of steady state shapes, both the geometry and the time evolution of which shows good quantitative agreement with laboratory experiments.


Firey’s model Shape evolution Bedrock profiles Random model 


  1. Ahnert F (1987) Approaches to dynamic equilibrium in theoretical simulations of slope development. Earth Surf Processes Landf 12:3–15. doi: 10.1002/esp.3290120103 CrossRefGoogle Scholar
  2. Attal M, Lavé J (2006) Changes of bedload characteristics along the Marsyandi River (central Nepal): implications for understanding hillslope sediment supply, sediment load evolution along fluvial networks, and denudation in active orogenic belts. Spec Pap, Geol Soc Am 398:143–171. doi: 10.1130/2006.2398(09) Google Scholar
  3. Carter CL, Anderson RS (2006) Fluvial erosion of physically modeled abrasion-dominated slot canyons. Geomorphology 81:89–113. doi: 10.1016/j.geomorph.2006.04.006 CrossRefGoogle Scholar
  4. Domokos G, Sipos AÁ, Várkonyi PL (2009a) Continuous and discrete models for abrasion processes. Period Polytech, Archit 40:1–6 Google Scholar
  5. Domokos G, Sipos AÁ, Szabó GyM, Várkonyi PL (2009b) Formation of sharp edges and planar areas of asteroids by polyhedral abrasion. Astrophys J 699:L13–L16. doi: 10.1088/0004-637X/699/1/L13 CrossRefGoogle Scholar
  6. Firey WJ (1974) Shapes of worn stones. Mathematika 21:1–11 CrossRefGoogle Scholar
  7. Hartshorn K, Hovius N, Dade WB, Slingerland RL (2002) Climate-driven bedrock incision in an active mountain belt. Science 297:2036–2038. doi: 10.1126/science.1075078 CrossRefGoogle Scholar
  8. Laity JE, Bridges NT (2009) Ventifacts on Earth and Mars: analytical, field and laboratory studies supporting sand abrasion and windward feature development. Geomorphology 105:202–217. doi: 10.1016/j.geomorph.2008.09.014 CrossRefGoogle Scholar
  9. Montgomery DR (2003) Predicting landscape-scale erosion rates using digital elevation models. C R Geosci 335:1121–1130. doi: 10.1016/j.crte.2003.10.005 CrossRefGoogle Scholar
  10. Phillips JD, Lutz JD (2008) Profile convexities in bedrock and alluvial streams. Geomorphology 102:554–566. doi: 10.1016/j.geomorph.2008.05.042 CrossRefGoogle Scholar
  11. Roering JJ, Kirchner JW, Dietrich WE (2001) Hillslope evolution by nonlinear, slope-dependent transport: steady state morphology and equilibrium adjustment timescales. J Geophys Res 106:499–513. doi: 10.1029/2001JB000323 Google Scholar
  12. Sklar L, Dietrich WE (2001) Sediment and rock strength controls on river incision into bedrock. Geology 29:1087–1090. doi: 10.1130/0091-7613 CrossRefGoogle Scholar
  13. Smith TR, Merchant GE, Birnir B (2000) Transient attractors: towards a theory of the graded stream for alluvial and bedrock channels. Comput Geosci 26:541–580. doi: 10.1016/S0098-3004(99)00128-4 CrossRefGoogle Scholar
  14. Schoewe WH (1932) Experiments on the formation of wind faceted pebbles. Am J Sci 24:111–133 CrossRefGoogle Scholar
  15. Tomkin JH (2009) Numerically simulating alpine landscapes: the geomorphologic consequences of incorporating glacial erosion in surface process models. Geomorphology 103:180–188. doi: 10.1016/j.geomorph.2008.04.021 CrossRefGoogle Scholar
  16. Turowski JM, Hovius N, Wilson A, Horng MJ (2008) Hydraulic geometry, river sediment and the definition of bedrock channels. Geomorphology. doi: 10.1016/j.geomorph.2007.10.001 Google Scholar
  17. Wilson A (2009) Fluvial bedrock abrasion by bedload: process and form. Dissertation, University of Cambridge, UK Google Scholar
  18. Wilson A, Hovius N (2010) Upstream facing convex surfaces and the importance of bedload in fluvial bedrock incision: observations from Taiwan. Geophysical Research Abstracts Google Scholar

Copyright information

© International Association for Mathematical Geosciences 2011

Authors and Affiliations

  • András A. Sipos
    • 1
  • Gábor Domokos
    • 1
  • Andrew Wilson
    • 2
  • Niels Hovius
    • 3
  1. 1.Department of Mechanics, Materials and StructuresBudapest University of Technology and EconomicsBudapestHungary
  2. 2.Stratigraphy GroupUniversity of LiverpoolLiverpoolUK
  3. 3.Department of Earth SciencesUniversity of CambridgeCambridgeUK

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