Mathematical Geosciences

, Volume 49, Issue 7, pp 913–942 | Cite as

Numerical Analysis of Fluvial Landscapes

  • David Cattan
  • Björn Birnir


The Smith and Bretherton model for fluvial landsurfaces consists of a pair of partial differential equations: one governing water flow and one governing sediment flow. Numerical solutions of these equations have been shown to provide realistic models of the evolution of fluvial landscapes. Further analysis of these equations shows that they possess scaling laws (Hack’s Law) that are known to exist in nature. The preservation of these scaling laws in simulations is highly dependent on the numerical method used. Two numerical methods, both optimized for overland flow, have been used to simulate these surfaces. The implicit method exhibits the correct scaling laws, but the explicit method fails to do so. These equations, and the resulting models, help to bridge the gap between the deterministic and the stochastic theories of landscape evolution. Slight modifications have been made to this model to make the resulting surfaces more realistic. The most successful of these was the addition of an abrasion term to assist in the channelization of rivers.


Transport-limited erosion Stochastic process Variogram Simulations Explicit versus implicit methods 



We would like to thank Terry Smith for his valuable input throughout this process.

The Upwind/Crank–Nicolson code used to generate these results will be made available at


  1. Bierman RB, Montgomery DR (2014) Key concepts in geomorphology. W. H. Freeman and Company Publishers, New YorkGoogle Scholar
  2. Birnir B (2008) Turbulent rivers. Q Appl Math 66:565–594CrossRefGoogle Scholar
  3. Birnir B, Rowlett J (2013) Mathematical models for erosion and the optimal transportation of sediment. Int J Nonlinear Sci Numer Simul 14:232–337CrossRefGoogle Scholar
  4. Birnir B, Hernández J, Smith TR (2007) The stochastic theory of fluvial landsurfaces. J Nonlinear Sci 17(1):13–57CrossRefGoogle Scholar
  5. Birnir B, Smith TR, Merchant GE (2001) The scaling of fluvial landscapes. Comput Geosci 27(10):1189–1216CrossRefGoogle Scholar
  6. Dingman SL (1984) Fluvial hydrology. W.H. Freeman and Company, United States, New YorkGoogle Scholar
  7. Edwards SF, Wilkinson D (1982) The surface statistics of a granular aggregate. Proc R Soc Lond A Math Phys Eng Sci 381:17–31CrossRefGoogle Scholar
  8. Fennema RJ, Chaudhry MH (1986) Explicit numerical schemes for unsteady free-surface flows with shocks. Water Resour Res 22(13):1923–1930CrossRefGoogle Scholar
  9. Fowler AC, Kopteva N, Oakley C (2007) The formation of river channels. SIAM J Appl Math 67(4):1016–1040CrossRefGoogle Scholar
  10. Hack JT (1957) Studies of longitudinal stream profiles in Virginia and Maryland. U.S. Geological Survey Professional Paper, 294-BGoogle Scholar
  11. Horton RE (1945) Erosional development of streams and their drainage basins; hydrophysical approach to quantitative morphology. Geol Soc Am Bull 56(3):275–370CrossRefGoogle Scholar
  12. Howard AD (1990) Theoretical model of optimal drainage networks. Water Resour Res 26(9):2107–2117CrossRefGoogle Scholar
  13. Howard AD (1994) A detachment-limited model of drainage basin evolution. Water Resour Res 30(7):2261–2285CrossRefGoogle Scholar
  14. Ikeda S, Parker G, Sawai K (1981) Bend theory of river meanders. Part 1. Linear development. J Fluid Mech 112:363–377CrossRefGoogle Scholar
  15. Izumi N (2004) The formation of submarine gullies by turbidity currents. J Geophys Res Oceans 109(C3):1–13Google Scholar
  16. Izumi N, Fujii K (2006) Channelization on plateaus composed of weakly cohesive fine sediment. J Geophys Res Earth Surf 111(F1):1–16Google Scholar
  17. Izumi N, Parker G (1995a) On incipient channels formed at the downstream end of plateaux. J Hydraul Coastal Environ Eng JSCE 521:79–91 (in Japanese)Google Scholar
  18. Izumi N, Parker G (1995b) Inception of channelization and drainage basin formation: upstream-driven theory. J Fluid Mech 283:341–363CrossRefGoogle Scholar
  19. Izumi N, Parker G (2000) Linear stability analysis of channel inception: downstream-driven theory. J Fluid Mech 419:239–262CrossRefGoogle Scholar
  20. Jovanović M, Djordjević D (1995) Experimental verification of the maccormack numerical scheme. Adv Eng Softw 23(1):61–67CrossRefGoogle Scholar
  21. Julien P, Simons D (1985) Sediment transport capacity of overland flow. Trans ASAE 28(3):755–762CrossRefGoogle Scholar
  22. Kramer S, Marder M (1992) Evolution of river networks. Phys Rev Lett 68:205–208CrossRefGoogle Scholar
  23. Mandelbrot BB, Van Ness JW (1968) Fractional brownian motions, fractional noises and applications. SIAM Rev 10(4):422–437CrossRefGoogle Scholar
  24. Mandelbrot BB, Wallis JR (1968) Noah, Joseph, and operational hydrology. Water Resour Res 4(5):909–918. doi: 10.1029/WR004i005p00909 CrossRefGoogle Scholar
  25. Mandelbrot BB, Wallis JR (1969) Computer experiments with fractional gaussian noises: part 1, averages and variances. Water Resour Res 5(1):228–241. doi: 10.1029/WR005i001p00228 CrossRefGoogle Scholar
  26. Moin P (2010) Fundamentals of engineering numerical analysis. Cambridge University Press, New YorkCrossRefGoogle Scholar
  27. Parker G, Izumi N (2000) Purely erosional cyclic and solitary steps created by flow over a cohesive bed. J Fluid Mech 419:203–238CrossRefGoogle Scholar
  28. Perron JT, Dietrich WE, Kirchner JW (2008) Controls on the spacing of first-order valleys. J Geophys Res Earth Surf 113(F4):1–21Google Scholar
  29. Roache PJ (1982) Computational fluid dynamics. Hermosa Publishers, AlbuquerqueGoogle Scholar
  30. Simpson G, Castelltort S (2006) Coupled model of surface water flow, sediment transport and morphological evolution. Comput Geosci 32(10):1600–1614CrossRefGoogle Scholar
  31. Simpson G, Schlunegger F (2003) Topographic evolution and morphology of surfaces evolving in response to coupled fluvial and hillslope sediment transport. J Geophys Res Solid Earth 108(B6):1–16Google Scholar
  32. Smith TR (2010) A theory for the emergence of channelized drainage. J Geophys Res Earth Surf 115(F2):1–32Google Scholar
  33. Smith TR, Bretherton FP (1972) Stability and the conservation of mass in drainage basin evolution. Water Resour Res 8(6):1506–1529CrossRefGoogle Scholar
  34. Smith TR, Merchant GE (1995) Conservation principles and the initiation of channelized surface flows. In: Costa JE, Miller AJ, Potter KW, Wilcock PR (eds) Natural and anthropogenic influences in fluvial geomorphology. American Geophysical Union, Washington, D. C. doi: 10.1029/GM089p0001
  35. Smith TR, Birnir B, Merchant GE (1997a) Towards an elementary theory of drainage basin evolution: I. The theoretical basis. Comput Geosci 23(8):811–822CrossRefGoogle Scholar
  36. Smith TR, Merchant GE, Birnir B (1997b) Towards an elementary theory of drainage basin evolution: II. A computational evaluation. Comput Geosci 23(8):823–849CrossRefGoogle Scholar
  37. Todini E, Venutelli M (1991) Overland flow: a two-dimensional modeling approach. In: Bowles DS, O’Connell PE (eds) Recent advances in the modeling of hydrologic systems, vol 345. Springer Netherlands, Dordrecht, pp 153–166Google Scholar
  38. Voss RF (1989) Random fractals: self-affinity in noise, music, mountains, and clouds. Physica D 38(1):362–371CrossRefGoogle Scholar
  39. Weissel JK, Pratson LF, Malinverno A (1994) The length-scaling properties of topography. J Geophys Res 99:13–997CrossRefGoogle Scholar
  40. Welsh E, Birnir B, Bertozzi A (2006) Shocks in the evolution of an eroding channel. Appl Math Res eXpress 2006:71638Google Scholar
  41. Zhang W, Cundy TW (1989) Modeling of two-dimensional overland flow. Water Resour Res 25(9):2019–2035CrossRefGoogle Scholar

Copyright information

© International Association for Mathematical Geosciences 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of California at Santa BarbaraSanta BarbaraUSA
  2. 2.Department of Mathematics, Center for Complex and Nonlinear ScienceUniversity of California at Santa BarbaraSanta BarbaraUSA
  3. 3.Division of Engineering and the Natural SciencesUniversity of IcelandReykjavikIceland

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