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

Compelling evidence from studies of stream channel density supports the hypothesis that terrain vegetation are coupled via water energy fluxes. The slope of a classic power law relation between drainage density water availability reverses sign 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 for the phase transition we partitioned precipitation into an infiltrated fraction available to plants a fraction available for flow 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, 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 surplus liquid water for terrain formation. Specification of the cond tions 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, 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 hydrology of broad landscapes that respects a conservation law scale invariance.