The landscape Reynolds number and other dimensionless measures of Earth surface processes

An analogy between turbulent fluid systems and landscape drainage systems [Parker, G., Haff, P.K., Murray, A.B., 2001, EOS, Transactions of the American Geophysical Union, 82, pp. F564.] is suggested by the observation that transport in both systems can be approximated by diffusion with size-proportional effective diffusivities, with a cross-over at small scales to Fickian diffusion. The “landscape” Reynolds number of a typical fluvial landscape is estimated to be of order Re_L 10⁶ to 10⁹, these large values reflecting the relative efficiency of fluvial transport compared to creep. Re_L is the ratio of the large-scale effective diffusivity of rivers to the small-scale diffusivity of creep processes on hillslopes. The spatial dependence of the effective diffusivity produces rivers with logarithmic long-profiles, similar to the profiles of many rivers in nature, and analogous to the logarithmic dependence of mean fluid velocity on distance from a wall in turbulent flow. The landscape example suggests how other generalized “Reynolds numbers” can be constructed as ratios of large-scale to small-scale diffusivities to measure the efficiencies of complex processes that affect the surface. As an example, the global airline transportation network is estimated to have an efficacy relative to that of direct human mechanisms for transport of similar goods and materials of about 10⁸ as measured by a corresponding “technology” Reynolds number. The appearance of such large dimensionless numbers, pertaining to the consequences of human invention and design, reflects the emergence of the technosphere as an increasingly efficient overlay on the historical domain of biology and surficial geology.