Non-linear Langevin model of geomorphic erosion processes

Abstract

SUMMARY Recent spectral studies of vertical transect profiles of landscapes and mountains have shown them to be self-affine fractals, i.e. the rms height fluctuation Ah(L) averaged over a distance L scales as Ah(L) - Lx with x = 0.5 f 0.1, related to the fractal dimension Df = 2 - x = 1.5 of the horizontal contours. We propose that self-affine rough landscapes are created by the interplay of non-linearity and noise. To illustrate this idea and model the formation of such structures, we suggest a non-linear stochastic equation ahla? = DV*h + A(Vh)' + ~(r, t), which is the generalization of the deterministic Culling's linear equation. The non-linear term h(Vh)* comes from the requirement that erosion is proportional to the exposed area of the landscape; the noise term ~(r, t) accounts for the fact that erosion is locally irregular, as a result of the heterogeneity of soils and distribution of storms. Using this general framework, we recover the scaling law Ah(L) - Lx with x 2 0.4. Several novel avenues of research emerge from this analysis to further quantify geological data.