Application of dynamic subgrid-scale concepts from large-eddy simulation to modeling landscape evolution

[ 1] Landscapes share important similarities with turbulence: both systems exhibit scale invariance (self-similarity) over a wide range of scales, and their behavior can be described using comparable dynamic equations. In particular, modified versions of the Kardar-Parisi-Zhang (KPZ) equation ( a low-dimensional analog to the Navier-Stokes equations) have been shown to capture important features of landscape evolution. This suggests that modeling techniques developed for turbulence may also be adapted to landscape simulations. Using a "toy'' landscape evolution model based on a modified 2-D KPZ equation, we find that the simulated landscape evolution shows a clear dependence on grid resolution. In particular, mean longitudinal profiles of elevation at steady state and bulk erosion rates both have an undesirable dependence on grid resolution because the erosion rate increases with resolution as increasingly small channels are resolved. We propose a new subgrid-scale parameterization to account for the scale dependence of the sediment fluxes. Our approach is inspired by the dynamic procedure used in large-eddy simulation of turbulent flows. The erosion coefficient, assumed exactly known at the finest resolution, is multiplied by a scale dependence coefficient, which is computed dynamically at different time steps on the basis of the dynamics of the resolved scales. This is achieved by taking advantage of the self-similarity that characterizes landscapes over a wide range of scales. The simulated landscapes obtained with the new model show very little dependence on grid resolution.