On the non-linear evolution of sand dunes

Summary. Adopting a multiple scale method, the non-linear evolution of fully developed dunes of non-cohesive sand is studied using Kennedy's model. It is shown that both the two- and three-dimensional problems are, in general, governed by three coupled second-order partial differential equations. However, for boundary conditions appropriate to localized solutions these equations reduce to one for the two-dimensional problem and to two for the three-dimensional problem. It is also found that the uniform Stokes 'wave' is a solution of the general problem, and the stability of this solution (on the long time-scale of the non-linear theory) is analysed for various values of the parameters of the problem. The significance of the soliton solution of the single equation for the two-dimensional problem is discussed and the relevance of the localized solutions of the two coupled equations to the formation and interaction of fully developed dunes is indicated. The results are in broad agreement with the cumulative effects of dune movement over long periods.