Eikonal Equation Reproduces Natural Landscapes With Threshold Hillslopes

Abstract

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.