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. Natural terrains often feature steep and nearly flat hillslopes, which remain stable up to a threshold angle or the angle of repose. Beyond this angle, mass failure ensues, resulting in the removal of excess sediment volume and preserving the topography. Here we demonstrate the effective use of the eikonal model, originally developed to represent the propagation of light rays two centuries back, in describing the organization of threshold hillslopes in natural landscapes. We directly compare eikonal model-generated topographies with various natural landscapes, including islands, volcanoes, and inland mountains, differing in size, underlying geology, and climatic conditions. The presented model achieves these results with minimal computational cost compared to traditional landscape evolution simulations. The simple eikonal model describes the landscape morphology with threshold hillslopesThe natural landscape's slope-area relation establishes the boundary condition for the modelThe model results faithfully reproduce diverse landforms, such as islands, an inland volcano, and a mountain range