Two‐dimensional nonlinear diffusive numerical simulation of geomorphic modifications to cinder cones

The temporal evolution of simple landforms such as cinder cones by nonlinear diffusive processes is studied through the use of a new 2D numerical model using well‐established and accurate numerical mathematics and high‐resolution digital elevation models (DEMs). Extending 1D (profile) nonlinear diffusion analyses used in cinder cone, hillslope and fault scarp evolution studies, we have implemented a 2D numerical model with a spatially and temporally varying sediment transport rate coefficient scaled nonlinearly by the ratio of local slope to critical slope. The high accuracy and efficient numerical implementation are documented in the paper and the MATLAB toolkit developed is used to solve for the developmentof an initial 2D cone form. First, we examine the nonlinear transport rule and suggest a refinement that accounts explicitly for flux at threshold slopes. We find that the maximum diffusion (necessarily introduced in the numerical model to avoid infinite rates) at the critical slope controls the final morphology, especially approaching steady state. Secondly, solving the landscape evolution problem in 2D enables a natural accounting for sediment flux convergence or divergence in the profile. Thirdly, the boundary behavior of a given landscape element controls much of what happens in that domain and so we allow for arbitrary flux magnitude or elevation boundary conditions. Fourthly, landscapes are heterogeneous in their surface cover and so we allow for spatially and temporally varying transport rate k and we permit an arbitrary vertical displacement field within the model domain. To test the new formulation for the nonlinear term, the effect of variable diffusivity k and the numerical schemes implemented, we apply the model to cinder cones built on the flanks of Mount Etna in 2001 and 2002–2003. We explore the effects of DEM resolution with data from the 2001 cone and the utility of spatially variable diffusivity to explain the variation in erosion measured by differencing repeat light detection and ranging (LIDAR) surveys gathered in 2004 and 2007 over the 2002–2003 cone complex. Copyright © 2013 John Wiley & Sons, Ltd.