Free granular surfaces can deform under the shear force exerted by a fluid moving tangentially. A classical example is the deformation of the surface of the sandy desert under the action of the wind. Here in particular, we will consider high Reynolds numbers i.e. turbulent flow conditions. In addition we allow for the smoothening action of gravity. A set of three coupled equations of motion is formulated. One is the integral formulation for the shear stress exerted on a hilly landscape as formulated by Jackson and Hunt. The second one describing the local granular surface flux is based on the saltation mechanism. It is similar to the logistic equation but has additionally strongly non-linear expressions for the saturated flux and for the saturation length as function of the shear stress. The continuum equation closes the system. We will show that using appropriate boundary conditions the numerical solution of these equations reproduces the observed dune shapes quantitatively as well on Earth as on Mars. Solitary wave solutions will also be discussed.
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