• Andrew Fowler
Part of the Interdisciplinary Applied Mathematics book series (IAM, volume 36)


Of the several geomorphological patterns which occur in rivers, fluvial dunes represent one of the simplest to observe, but the theory to describe their formation is less easy. This chapter discusses models of dune formation in both fluvial and aeolian contexts, and in particular gives detailed discussion of two models based on descriptions of turbulent flow with either a constant eddy viscosity, or one derived from mixing length theory. The former leads to Benjamin’s theory for flow over a bump, the latter leads to the Jackson–Hunt theory. Both produce instability, but the Jackson–Hunt theory has formulation problems. The issue of separation renders the problem difficult. It can be treated using complex variables methods, as described here, but practical models require something more direct, as described in the notes.


Shear Layer Sediment Transport Froude Number Eddy Viscosity Separation Bubble 
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  1. Ahnert F (1996) Introduction to geomorphology. Arnold, London Google Scholar
  2. Allen JRL (1985) Principles of physical sedimentology. Chapman and Hall, London Google Scholar
  3. Bagnold RA (1936) The movement of desert sand. Proc R Soc Lond A 157:594–620 CrossRefGoogle Scholar
  4. Bagnold RA (1941) The physics of blown sand and desert dunes. Methuen, London Google Scholar
  5. Charru F, Hinch EJ (2006) Ripple formation on a particle bed sheared by a viscous liquid. Part 1. Steady flow. J Fluid Mech 550:111–121 MATHCrossRefMathSciNetGoogle Scholar
  6. Cocks D (2005) Mathematical modelling of dune formation. DPhil thesis, Oxford University.
  7. Colombini M (2004) Revisiting the linear theory of sand dune formation. J Fluid Mech 502:1–16 MATHCrossRefMathSciNetGoogle Scholar
  8. Durán O, Herrmann HJ (2006) Vegetation against dune mobility. Phys Rev Lett 97:188001 CrossRefGoogle Scholar
  9. Einstein HA (1950) The bedload function for bedload transportation in open channel flows. Tech Bull No 1026, USDA, Soil Conservation Service, pp 1–71 Google Scholar
  10. Engelund F (1970) Instability of erodible beds. J Fluid Mech 42:225–244 CrossRefGoogle Scholar
  11. Engelund F, Fredsøe J (1982) Sediment ripples and dunes. Annu Rev Fluid Mech 14:13–37 CrossRefGoogle Scholar
  12. Fowler AC, McGuinness MJ, Ellis AS (2011) On an evolution equation for sand dunes. SIAM J Appl Math, submitted Google Scholar
  13. Fredsøe J (1974) On the development of dunes in erodible channels. J Fluid Mech 64:1–16 MATHCrossRefGoogle Scholar
  14. García M, Parker G (1991) Entrainment of bed sediment into suspension. J Hydraul Eng 117:414–435 CrossRefGoogle Scholar
  15. Goudie A (1993) The nature of the environment, 3rd edn. Blackwell, Oxford Google Scholar
  16. Gradshteyn IS, Ryzhik IM (1980) Table of integrals, series and products, Corrected and enlarged edition. Academic Press, New York MATHGoogle Scholar
  17. Hunt JCR, Leibovich S, Richards KJ (1988) Turbulent shear flows over low hills. Q J R Meteorol Soc 114:1435–1470 CrossRefGoogle Scholar
  18. Jackson PS, Hunt JCR (1975) Turbulent wind flow over a low hill. Q J R Meteorol Soc 101:929–955 CrossRefGoogle Scholar
  19. Kennedy JF (1963) The mechanics of dunes and anti-dunes in erodible-bed channels. J Fluid Mech 16:521–544 MATHCrossRefGoogle Scholar
  20. Knighton D (1998) Fluvial forms and processes: a new perspective. Arnold, London Google Scholar
  21. Kroy K, Sauermann G, Herrmann HJ (2002a) Minimal model for sand dunes. Phys Rev Lett 88:054301 CrossRefGoogle Scholar
  22. Kroy K, Sauermann G, Herrmann HJ (2002b) Minimal model for aeolian sand dunes. Phys Rev E 66:031302 CrossRefGoogle Scholar
  23. Meyer-Peter E, Müller R (1948) Formulas for bed-load transport. In: Proc int assoc hydraul res, 3rd annual conference, Stockholm, pp 39–64 Google Scholar
  24. O’Malley K, Fitt AD, Jones TV, Ockendon JR, Wilmott P (1991) Models for high-Reynolds-number flow down a step. J Fluid Mech 222:139–155 MATHCrossRefMathSciNetGoogle Scholar
  25. Parker G (1975) Sediment inertia as a cause of river antidunes. J Hydraul Div ASCE 101:211–221 Google Scholar
  26. Parker G (1978) Self-formed straight rivers with equilibrium banks and mobile bed. Part 1. The sand-silt river. J Fluid Mech 89:109–125 MATHCrossRefGoogle Scholar
  27. Parker G (2004) 1D sediment transport morphodynamics with applications to rivers and turbidity currents.
  28. Parsons DR, Walker IJ, Wiggs GFS (2004) Numerical modelling of flow structures over idealized transverse aeolian dunes of varying geometry. Geomorphology 59:149–164 CrossRefGoogle Scholar
  29. Parteli EJR, Durán O, Herrmann HJ (2007) Minimal size of a barchan dune. Phys Rev E 75:011301 CrossRefGoogle Scholar
  30. Parteli EJR, Durán O, Tsoar H, Schwämmle V, Herrmann HJ (2009) Dune formation under bimodal winds. Proc Natl Acad Sci 106:22085–22089 CrossRefGoogle Scholar
  31. Pye K, Tsoar H (1990) Aeolian sand and sand dunes. Unwin Hyman, London Google Scholar
  32. Reynolds AJ (1965) Waves on the erodible bed of an open channel. J Fluid Mech 22:113–133 CrossRefMathSciNetGoogle Scholar
  33. Richards KJ (1980) The formation of ripples and dunes on an erodible bed. J Fluid Mech 99:597–618 MATHCrossRefGoogle Scholar
  34. Sauermann G, Kroy K, Herrmann HJ (2001) Continuum saltation model for sand dunes. Phys Rev E 64:031305 CrossRefGoogle Scholar
  35. Schlichting H (1979) Boundary layer theory. McGraw-Hill, New York MATHGoogle Scholar
  36. Schwämmle V, Herrmann H (2004) Modelling transverse dunes. Earth Surf Process Landf 29:769–784 CrossRefGoogle Scholar
  37. Shields A (1936) Anwendung der Ähnlichkeits mechanik und der Turbulenzforschung auf die Geschiebebewegung. Mitteilung der Preussischen Versuchanstalt für Wasserbau und Schiffbau, Heft 26, Berlin Google Scholar
  38. Smith JD (1970) Stability of a sand bed subjected to a shear flow of low Froude number. J Geophys Res 75:5928–5940 CrossRefGoogle Scholar
  39. Smith JD, McLean SR (1977) Spatially averaged flow over a wavy surface. J Geophys Res 83:1735–1745 CrossRefGoogle Scholar
  40. Sumer BM, Bakioglu M (1984) On the formation of ripples on an erodible bed. J Fluid Mech 144:177–190 MATHCrossRefGoogle Scholar
  41. Sykes RI (1980) An asymptotic theory of incompressible turbulent boundary-layer flow over a small hump. J Fluid Mech 101:647–670 MATHCrossRefMathSciNetGoogle Scholar
  42. Van Dyke MD (1975) Perturbation methods in fluid mechanics. Parabolic Press, Stanford MATHGoogle Scholar
  43. Van Rijn LC (1984) Sediment transport. Part II. Suspended load transport. J Hydraul Eng 110:1613–1641 CrossRefGoogle Scholar
  44. Vosper SB, Mobbs SD, Gardiner BA (2002) Measurements of the near-surface flow over a hill. Q J Meteorol Soc 128:2257–2280 CrossRefGoogle Scholar
  45. Weng WS, Hunt JCR, Carruthers DJ, Warren A, Wiggs GFS, Livingstone I, Castro I (1991) Air flow and sand transport over sand-dunes. Acta Mech, Suppl 2:1–22 Google Scholar

Copyright information

© Springer-Verlag London Limited 2011

Authors and Affiliations

  1. 1.MACSI, Department of Mathematics & StatisticsUniversity of LimerickLimerickIreland

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