Eikonal Equation Reproduces Natural Landscapes With Threshold Hillslopes

Many natural landscapes maintain steep planar hillslopes bounded at a typical angle, beyond which shallow landslides or slope failures remove the excess sediment volume by mass wasting. Here we show that the celebrated eikonal equation, derived from a landscape evolution model in conditions of negligible soil diffusion and fluvial erosion, accurately portrays the organization of these topographies. Referred to as “eikonal landscapes,” such solutions feature constant-slope hillslopes originating from downstream boundary conditions and culminating in sharp upstream ridges. We demonstrate that the eikonal landscapes reproduce well a variety of natural landforms, including small islands, a volcano, and an extended mountain ridge. The boundary condition for the eikonal representation is specified through the natural landscape's slope-area relation. Going beyond merely representing landscape statistical features, the present results provide a first-of-kind direct match of mathematical and natural landscapes.