Blackwell Publishing Journal Backfiles 1879-2005
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 Δh(L) averaged over a distance L scales as Δh(L)∼Lx with X≈ 0.5 ± 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 ∂h/∂t=DΔ2h+λ(Δh)2+ν(r, t), which is the generalization of the deterministic Culling's linear equation. The non-linear term λ(Δh)2 comes from the requirement that erosion is proportional to the exposed area of the landscape; the noise term ν(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 Δh(L∼Lx with X≥ 0.4. Several novel avenues of research emerge from this analysis to further quantify geological data.
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