The Generation and Scaling of Longitudinal River Profiles

Abstract

The apparent success of inverse modeling of continent-wide drainage inventories is perplexing. An ability to obtain reasonable fits between observed and calculated longitudinal river profiles implies that drainage networks behave simply and predictably at length scales of O(10 2 –10 3 ) km and time scales of O(10 0 –10 2 ) Ma. This behavior suggests that rivers respond in an organized way to large-scale tectonic forcing. On the other hand, stream power laws are empirical approximations since fluvial processes are complex, nonlinear, and probably susceptible to disparate temporal and spatial shocks. To bridge the gap between these different perceptions of landscape evolution, we present and interpret a suite of power spectra for African river profiles that traverse different climatic zones, lithologic boundaries, and biotic distributions. At wavelengths (Formula presented.) 10 2 km, power spectra have slopes of −2, consistent with red noise, demonstrating that profiles are self-similar at these length scales. At wavelengths (Formula presented.) 10 2 km, there is a crossover transition to slopes of −1, consistent with pink noise, for which power scales according to the inverse of wavenumber. Onset of this transition suggests that spatially correlated noise, perhaps generated by instabilities in water flow and by lithologic heterogeneities, becomes more prevalent at wavelengths shorter than ∼100 km. At longer wavelengths, this noise gradually diminishes and self-similar scaling emerges. Our analysis is consistent with the concept that complexities of river profile development can be characterized by an adaptation of the Langevin equation, by which simple advective models of erosion are driven by a combination of external forcing and noise.