ANALYTICAL AND NUMERICAL SOLUTIONS OF LINEAR AND NONLINEAR DIFFUSION FOR FOUR EXAMPLES OF LANDFORM EVOLUTION

This presentation compares and contrasts analytical and numerical solutions of linear and nonlinear diffusion for four model problems of geomorphologic interest. The first two are the single-event and the multiple-event (continuous faulting) vertical-scarp problems, used extensively in paleoseismologic studies. Both of these are two-dimensional half-space problems, with the right side up (and erosional) and the left side down (and depositional). Relative elevation u is anti-symmetric about horizontal distance x=0 where u=0 is fixed as a boundary condition. The third problem is also a two-dimensional, uniform block uplift/downdrop problem, like the second, but in this case the uplifting block is of finite width and erodes/deposits on both block sides. The fourth problem is yet another uniform block uplift problem, with the block incised by channels that remain at u=0, sweeping all erosional debris from the model domain. This, too, is examined as a two-dimensional problem, along a section perpendicular to the incising drainages. As a set, the solutions to these problems are informative in a variety of ways, as the models progress to greater geometric and physical complexity. The fourth problem allows us to explore the concept of dynamic equilibrium within the context of diffusion-equation representations of landform evolution, for both linear and nonlinear diffusion. The third problem, at first glance an analogue for block uplift/downdrop in an interior-drainage situation similar to the Basin and Range province, presents the most interesting results in that there are no equilibrium solutions for it. At large time, u increases asymptotically with the square root of time, just as is the case for the second problem. Equilibrium solutions of the diffusion equation do not in general exist, and when they do they are the consequence of boundary conditions alone, the choice(s) of which may or may not be reasonable facsimiles of the Earth’s behavior.