A scale invariant coupling of plants, water, energy, and terrain1

The most compelling, yet unexplained, evidence for the coupled partitioning of water and energy by terrain and vegetation is the nonlinear relationship between drainage density, defined as total stream channel length per unit area, and a measure of surplus water in the landscape (Figure 1). In this classic example, the measure of surplus water is the sum, over 12 months, of ten times the mean monthly precipitation divided by the mean monthly potential evapotranspiration (Thornthwaite, 1933). Thus, a unitless measure, called TI for “Thronthwaite Index”, of 120 results when the A scale invariant coupling of plants, water, energy, and terrain1 Bruce T. MILNE2, Department of Biology, University of New Mexico, Albuquerque, New Mexico 87131, U.S.A. Vijay K. GUPTA, Department of Civil and Environmental Engineering, Cooperative Institute for Research in Environmental Sciences, University of Colorado, Boulder, Colorado 80309, U.S.A. Carla RESTREPO, Department of Biology, University of New Mexico, Albuquerque, New Mexico 87131, U.S.A. Abstract: 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 waterto 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.