A theory for the emergence of channelized drainage

• Published in: Journal of Geophysical Research: Earth Surface 2010-06, Vol.115 (F2), p.n/a
• Main Author: Smith, Terence R.
• Format: Article
• Language: English
• Subjects: advectively-driven diffusion; Aspect ratio; channel formation; channel stability; Channels; Diffusion; Earth sciences; Earth, ocean, space; Emergence; Exact sciences and technology; Geomorphology; Hydrology; Mathematical analysis; Mathematical models; Nonlinearity; Rivers; Sediment transport; Sediments
• ISSN: 0148-0227; 2169-9003; 2156-2202; 2169-9011
• DOI: 10.1029/2008JF001114

A theory of channel formation is derived from mass balance equations for water and sediment, the St. Venant equation for water flowing down an energy surface gradient, and generalized representations of sediment transport. The theory covers transport‐ and detachment‐limited erosion environments. Linearized analysis provides closed form stability criteria for the growth rates of channels in terms of aspect ratios, lateral spacings, and four mechanisms relating to water and sediment flows. Two mechanisms, a nonlinear advective transport of sediment that drives a channel‐cutting instability and a free water surface that selectively neutralizes perturbations, act down the energy surface gradient. Two other mechanisms, advectively driven and slope‐driven diffusion of sediment, act down the land surface gradient and selectively stabilize perturbations. Under subcritical and critical, but not supercritical, flow conditions, the first two mechanisms, together with arbitrary amounts of either diffusive mechanism, select strongly for aspect ratios and lateral spacings of the fastest‐growing channels. Thresholds for channel spacing emerge without requiring critical thresholds for sediment entrainment. Advective sediment transport and a free water surface acting alone select strongly for aspect ratios but weakly for lateral spacings. Numerical solutions and analysis of the nonlinear equations, however, indicate strong selection for spacing through nonlinear amplification and coloring of noise. The aspect ratios and lateral spacings of fastest‐growing channels are determined by six dimensionless ratios and parameters of the constitutive relations. The theory provides a scaling relation between increased lateral spacing of dominant channels and increasing discharge and predicts the emergence of Hack‐type scaling laws with variable exponents for the fastest‐growing channels.