ANALYTICAL MODEL OF PROPAGATING SAND RIPPLES

We formulate a simple phenomenological model of aeolian sand ripple mi gration based upon a balance between grain hopping driven by saltation and grain rolling or avalanching under gravity. We develop a set of m odel equations governing the evolution of the ripple slope. The model has solutions describing steadily Propagating isolated ripples, produc ed by a horizontal saltation flux, and periodic trains of ripples, whi ch develop when the saltation flux is inclined to the horizontal. In t he case of an inclined saltation flux, the ripple wavelength is contro lled 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 f ield studies [R. A. Bagnold, The Physics of Blown Sand and Desert Dune s (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 saltat ion fluxes, the lee slope of the ripple is a smooth curve, whereas abo ve a critical value of the saltation flux, a slip face develops near t he 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, 61 7 (1963)], and also with numerical simulations [R. S. Anderson, Earth- Sci. Rev. 29, 77 (1990); S. B. Forrest and P. K. Haff, Science 255, 12 40 (1992); W. Landry and B. T. Werner, Physica D 77, 238 (1994)].