The Diffusion Equation for Landscapes
On soil-mantled hillslopes, the slow downslope movement of soil by creep processes can be modeled as a diffusion equation — the same mathematical form that describes heat conduction and chemical diffusion. Sediment flux is proportional to the local slope, and the resulting evolution equation ∂z/∂t = κ·∇²z + U smooths topography over time, rounding hilltops and filling valleys. The diffusivity κ encapsulates all creep processes into a single transport coefficient.
Steady-State Parabolic Profiles
When uplift is spatially uniform and channels at hillslope boundaries maintain fixed elevations, the steady-state solution is a parabola: z(x) = U/(2κ)·(L²/4 − x²). This elegant result means hilltop elevation scales with uplift rate and the square of hillslope length, while inversely proportional to diffusivity. The parabolic form is widely observed on soil-mantled ridges in humid temperate landscapes.
Curvature as an Erosion Rate Proxy
Perhaps the most powerful prediction of hillslope diffusion theory is that hilltop curvature in steady state equals −U/κ. Combined with independent estimates of diffusivity from cosmogenic nuclide measurements, this allows spatially distributed erosion rates to be mapped from topographic curvature measured with LiDAR — a technique that has revolutionized quantitative geomorphology in the past two decades.
Nonlinear Transport on Steep Slopes
The linear diffusion model predicts hillslope gradients proportional to erosion rate, but real slopes cannot exceed the angle of repose (~35-40°). The nonlinear diffusion model addresses this by making flux increase rapidly as slope approaches a critical value, producing planar steep slopes capped by convex hilltops. This transition from convex to planar morphology is a signature of the shift from creep-dominated to threshold-dominated transport.