Eikonal approximation for landscapes dominated by threshold hillslopes

Steep landscapes maintain predominantly planar hillslopes over a range of spatial scales. These hillslopes are bounded at a typical angle, beyond which shallow landslides or slope failures remove the excess sediment influx. The evolution of such steep topographies with threshold hillslopes is well approximated by the eikonal equation, which is well-known in problems of geometry, optics, and mechanics. According to this approximation, hillslopes meet upstream to construct a network of sharp ridges and join downstream at the boundary to produce a complementary valley network. We find a good agreement between the proposed approximation and the spatial organization of landscapes with a dominant control of landslide erosion and negligible fluvial erosion. We also show that the eikonal approximation can be utilized to reconstruct the landscapes with threshold hillslopes where fluvial erosion sets the downstream free boundary.