THE EQUATIONS OF LANDSCAPE FORMATION: REVIEW AND A NEW MODEL

We start by establishing a list of the main physical and empirical laws that have been proposed for modeling landscape evolution. This discussion distinguishes a priori laws from a posteriori laws governing landscape evolution. While the first kind of law relies on physical principles, the second kind stems from the observation of actual stable landscapes. It emerges from the analysis of several models that the a posteriori laws are often used to create hybrid models combining physical fundamental laws with empirical modeling. Such models can be valid near equilibrium. Nevertheless, this situation is not satisfactory from the mathematical standpoint.

One would like to be able to deduce the a posteriori laws from a mathematical analysis, or at the very least to observe them from numerical simulations using only it a priori physical laws. Such a mathematical and numerical research program is sketched here. We propose the simplest landscape model coping with the main features of all models. This model singles out three spatially distributed scalar state variable, namely the landscape elevation, the water elevation, and the sediment concentration in water. These state variables are linked by three partial differential equations. Two of these equations are mere conservation laws. A third equation copes with the three main features identified in the literature as the main phenomena shaping a landscape: erosion, sedimentation and creep.

Based on these equations, the first numerical simulations confirm that valley formation and drainage network formation can be simulated on virgin artificial landscapes. A first surprising result seems to emerge from these simulations: the valley spacing depends not only a relation between creep and erosion parameters, but is also strongly influenced by the sedimentation rate or by the rainfall intensity. The conjectured mathematical instability and non-uniqueness of landscape evolution is illustrated numerically. On the other hand numerical stability of real landscape topographies under realistic values for their evolution is also observed. The overall modeling and numerical tests suggest that existence and partial regularity results might be obtained under minimal assumptions (such as a positive rain density or a positive creep model).