Controls on the height and spacing of eolian ripples and transverse dunes: A numerical modeling investigation

Ripples and transverse dunes in areas of abundant sand supply increase in height and spacing as a function of time, grain size, and excess shear velocity. How and why each of these factors influence ripple and transverse dune size, however, is not precisely known. In this paper, the controls on the height and spacing of ripples and transverse dunes in areas of abundant sand supply are investigated using a numerical model for the formation of eolian bedforms from an initially flat surface. This bedform evolution model combines the basic elements of Werner's [Werner, B.T., 1995. Eolian dunes: Computer simulations and attractor interpretation. Geology 23, 1107-1110.] cellular automaton model of dune formation with a model for boundary layer flow over complex topography. Particular attention is paid to the relationship between bed shear stress and slope on the windward (stoss) side of evolving bedforms. Nonlinear boundary layer model results indicate that bed shear stresses on stoss slopes increase with increasing slope angle up to approximately 20°, then decrease with increasing slope angle as backpressure effects become limiting. In the bedform evolution model, the linear boundary layer flow model of Jackson and Hunt [Jackson, P.S., Hunt, J.C.R., 1975. Turbulent wind flow over a low hill. Quarterly Journal of the Royal Meteorological Society 101, 929-955.], generalized to 3D, is modified to include the nonlinear relationship between bed shear stress and slope. Bed shear stresses predicted by the modified Jackson and Hunt flow model are then used to predict rates of erosion and deposition iteratively through time within a mass-conservative framework similar to Werner [Werner, B.T., 1995. Eolian dunes: Computer simulations and attractor interpretation. Geology 23, 1107-1110.]. Beginning with a flat bed, the model forms ripples that grow in height and spacing until a dynamic steady-state condition is achieved in which bedforms migrate downwind without further growth. The steady-state ripple spacing predicted by this model is approximately 3000 times greater than the aerodynamic roughness length of the initially flat surface, which is a function of grain size and excess shear velocity. Once steady-state ripples form, they become the dominant roughness element of the surface. The increase in roughness associated with ripple formation triggers the same bedform instability that created ripples, causing dunes to form at a larger scale. In this way, the numerical model of this paper suggests that ripples and dunes are genetically linked. Transverse dunes in this model have a steady-state height and spacing that is controlled by the effective roughness length of the rippled surface, which is shown to be on the order of 500 times greater than the original roughness length, but varies significantly with the details of ripple morphology. The model predictions for ripple and dune spacing and their controlling variables are consistent with field measurements from the published literature. The model of this paper provides a preliminary process-based understanding of the granulometric control of ripples and dunes in areas of abundant sand supply and unidirectional prevailing winds, and it argues for a genetic linkage between ripples and dunes via a scaling relationship between eolian bedform size and the aerodynamic roughness length.