Dynamic scaling of desert dunes

The shapes of shifting sand dunes of different size and under diverse environmental conditions exhibit a remarkably high degree of similarity. On this basis, a reduced shape parametrization of dunes in terms of a few characteristic parameters such as height and length is routinely applied in the geomorphological literature. In view of the a priori extremely high dimensionality of a freely evolving dune's state space, the justification for this common practice is, despite its alluring simplicity, all but obvious. In order to unveil the origin of the apparent reduction of complexity, we study the dynamics of (slices of) isolated dunes within the framework of the recently proposed minimal model of sand dune formation [K. Kroy, G. Sauermann, and H. J. Herrmann, Phys. Rev. Lett. 88, 054301 (2002); Phys. Rev. E 66, 031302 (2002)]. Our numerical solutions complemented by scaling relations derived from the model equations-show that the predicted time evolution of the shape and size of dunes, in response to naturally varying conditions such as wind strength and sand supply, is subject to a similarity law, closely controlled by the instability modes of the steady-state solutions of the model equations. By this dynamical similarity, the multitude of observed shapes and time evolutions of desert dunes is traced back to a unified growth law and to the elementary scales provided by grain size and wind speed.