Sand dunes as migrating strings

We develop a reduced complexity model for three-dimensional sand dunes, based on a simplified description of the longitudinal and lateral sand transport. The spatiotemporal evolution of a dune migrating over a nonerodible bed under unidirectional wind is reduced to the dynamics of its crest line, providing a simple framework for the investigation of three-dimensional dunes, such as barchan and transverse dunes. Within this model, we derive analytical solutions for barchan dunes and investigate the stability of a rectilinear transverse dune against lateral fluctuations. We show, in particular, that the latter is unstable only if the lateral transport on the dune slip face prevails over that on the upwind face. We also predict the wavelength and the characteristic time that control the subsequent evolution of an unstable transverse dune into a wavy ridge and the ultimate fragmentation into barchan dunes.