A Null Model for Global Root Depth Distributions: Analytical Solution and Comparison to Data

Abstract

To accurately predict earth system response to global change, we must be able to predict the responses of important properties of that system, such as the depths over which plant roots are distributed. In 2008, H. J. Schenk proposed a model for the depth distribution of plant roots based on a simple hydrological scheme and the assumptions that plants will take up the shallowest water available first and will distribute their roots in proportion to long-term mean uptake at each depth. Here, we derive an analytical solution to the Schenk model under an idealised climate (in which infiltration events are treated as a marked Poisson process), explore properties of the result and compare with data. The solution suggests that in very humid and arid climates, the soil wetting and drying cycles induced by root water uptake are generally confined to a characteristic depth below the surface. This depth depends on the typical magnitude of rainfall events (most strongly so in arid climates), the typical total transpiration demand between rainfall events (most strongly in humid climates) and the plant-available water holding capacity of the soil. Root water uptake (and thus predicted root density) in very humid and arid landscapes decreases exponentially with depth at a rate determined by this characteristic depth. However, in a mesic climate, soils may be wet or dry to greater depths below the near-surface, and the duration spent in each state increases with depth. Consequently, root water uptake and root density in mesic climates more closely resemble a power law distribution. When the aridity index is exactly 1, the characteristic depth diverges and the mean rooting depth approaches infinity. This suggests that the most skewed root depth distributions might occur in mesic environments. We compared this model to another analytical solution and a compiled database of root distributions (159 combined locations). For a larger comparison dataset, we also compared 99th percentile rooting depth to rooting depths modeled by two other frameworks and a database of observed rooting depths (1271 combined locations). Results demonstrate that the analytical formulation of the Schenk model performs well as a shallow bound on rooting depths and captures something of the nonexponential form of root distributions, and its error is similar to or less than that of other modeling frameworks. Errors may be partly explained by the deviation of real climate from the idealisations used to obtain an analytical solution (exponentially distributed infiltration events and no seasonality).