A scale invariant coupling of plants, water, energy, and terrain

Compelling evidence from studies of stream channel density supports the hypothesis that terrain and vegetation are coupled via water and energy fluxes. The slope of a classic power law relation between drainage density and water availability reverses sign and changes value where precipitation equals potential evapotranspiration. The change of slope indicates a "phase transition" from water- to energy-limited vegetation. To initiate a common biophysical theory for these power-law relations and for the phase transition, we partitioned precipitation into an infiltrated fraction available to plants and a fraction available for flow and thus erosion. To estimate infiltration we exploited invariance with respect to spatial scale at the transition. We tested the invariance hypothesis by analyzing the spatial distribution of energy-limited vegetation over length scales between 8,000 and 256,000 m in the Columbia River Basin of the northwestern US, which has a mixture of both phases. We observed a power law relation for the occurrence of energy-limited vegetation based on annual fluxes. We defined two dimensionless parameters that describe excess available energy for photosynthesis and surplus liquid water for terrain formation. Specification of the conditions under which neither parameter changed with spatial scale, in conjunction with a steady-state water balance model, enabled the formulation of an equation of soil infiltration at scales between 8,000 and 256,000 m. In water-limited vegetation, the equation enables the estimation of soil infiltration rates at arbitrary spatial scales for a given plant cover as a function of the ratio of precipitation to potential evapotranspiration. This work represents a first step towards the articulation of a biophysically sound theory about the ecology and hydrology of broad landscapes that respects a conservation law and scale invariance.