Analytic models for orogenic collapse

Orogenic plateaus and collisional mountain belts are formed by tectonic driving forces, such as those associated with a continental collision, and magmatism. When the horizontal driving forces are removed, e.g., at the conclusion of an orogeny, the elevation of the plateau or collisional orogen will relax by one or both of two processes. The first of these is erosion which removes the near surface rocks and allows uplift. The second is the gravitational collapse of elevated topography. In this paper we consider a simplified analysis in order to quantify the relative importance of erosion and gravitational collapse in the removal of topography. We treat the erosion problem using the Culling (diffusion equation) approach. The basic assumption is that the material transport down slope is proportional to the slope with the equivalent coefficient of diffusion, K, the constant of proportionality. The parameter K quantifies the rate of erosion. It is applicable whether or not the topography is compensated. For the gravitational collapse problem, we use the thin viscous sheet approximation and assume a linear viscous rheology. To simplify the analysis, we assume the thickness of the continental crust approximates the thickness of the highly viscous lithosphere, and that the crust overlies a much lower viscosity mantle. We also assume the topography is harmonic with a wavelength, λ.  The viscosity, η quantifies the thinning of the crust due to the buoyancy driven lateral flow. Our results indicate an exponential decay of topography with time and show that the relative role of erosion versus gravitational collapse for compensated topography is controlled by the value of the non-dimensional collapse number C = gh_o λ²ρ_c/16Kη π², where g is the acceleration due to gravity, h_o is a reference thickness of the crust, ρ_c is the density of the crust, and λ, K, and η are as defined above. If C is large compared with unity gravitational collapse dominates, if C is small compared with unity erosion dominates.