NONLINEAR LANGEVIN MODEL OF GEOMORPHIC EROSION PROCESSES

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 DELTAh(L) averaged over a distance L scales as DELTAh(L) approximately L(chi) with chi almost-equal-to 0.5 +/- 0.1, related to the fractal dimension D(f) = 2 - chi almost-equal-to 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 partial-derivative h/partial-derivative t = Ddel2h + lambda(delh)2 + eta(r, t), which is the generalization of the deterministic Culling's linear equation. The non-linear term lambda(delh)2 comes from the requirement that erosion is proportional to the exposed area of the landscape; the noise term eta(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 DELTAh(L) approximately L(chi) with chi greater-than-or-equal-to 0.4. Several novel avenues of research emerge from this analysis to further quantify geological data.