Surface instability in windblown sand

We investigate the formation of ripples on the surface of windblown sand ba sed on the one-dimensional model of Nishimori and Ouchi [Phys. Rev. Lett. 7 1, 197 (1993)], which contains the processes of saltation and grain relaxat ion. We carry out a nonlinear analysis to determine the propagation speed o f the restabilized ripple patterns, and the amplitudes and phases of their first, second, and third harmonics. The agreement between the theory and ou r numerical simulations is excellent near the onset of the instability. We also determine the Eckhaus boundary, outside which the steady ripple patter ns are unstable.