Analytical model of propagating sand ripples

We formulate a simple phenomenological model of aeolian sand ripple migration based upon a balance between grain hopping driven by saltation and grain rolling or avalanching under gravity. We develop a set of model equations governing the evolution of the ripple slope. The model has solutions describing steadily Propagating isolated ripples, produced by a horizontal saltation flux, and periodic trains of ripples, which develop when the saltation flux is inclined to the horizontal. In the case of an inclined saltation flux, the ripple wavelength is controlled by the length of the shadow zone, as suggested by R. P. Sharp [J. Geol. 71, 617 (1963)]. Although very simple, our model predicts some of the qualitative features shown by sand ripples in experimental or field studies [R. A. Bagnold, The Physics of Blown Sand and Desert Dunes (Methuen and Co., London, 1941); R. P. Sharp, J. Geol. 71, 617 (1963)]. We find that ripples only develop above a certain threshold value of the saltation flux intensity. Furthermore, at relatively low saltation fluxes, the lee slope of the ripple is a smooth curve, whereas above a critical value of the saltation flux, a slip face develops near the crest. The model predicts a decrease in the speed of propagation as the ripple becomes larger, consistent with observations that smaller ripples are eliminated by ripple merger [IR. P. Sharp, J. Geol. 71, 617 (1963)], and also with numerical simulations [R. S. Anderson, Earth-Sci. Rev. 29, 77 (1990); S. B. Forrest and P. K. Haff, Science 255, 1240 (1992); W. Landry and B. T. Werner, Physica D 77, 238 (1994)].