Predicting the Water Balance from Optimization of Plant Productivity

How soil-water flows and how fast it moves solutes are important for plant growth and soil formation. The relationship describing the partitioning of precipitation, P, into run-off, Q, and evapotranspiration, ET, is called the water balance. Q incorporates both surface runoff and subsurface flow components, the latter chiefly contributing to soil formation. At shorter time intervals, soil-water storage, S, may change, dS/dt, due to atmosphere-soil water exchange; i.e., infiltrating and evaporating water and root uptake. Over sufficiently long time periods, storage changes are typically neglected (Gentine et al., 2012). Percolation theory from statistical physics provides a powerful tool for predicting soil formation and plant growth (Hunt, 2017) by means of modeling soil pore space as networks, rather than continua. In heterogeneous soils, solute migration typically exhibits non-Gaussian behavior, with statistical models having long tails in arrival time distributions and velocities decreasing over time. Theoretical prediction of solute transport via percolation theory that generates accurate full non-Gaussian arrival time distributions has become possible only recently (Hunt and Ghanbarian, 2016; Hunt and Sahimi, 2017). A unified framework, based on solute transport theory, helps predict soil depth as a function of age and infiltration rate (Yu and Hunt, 2017), soil erosion rates (Yu et al., 2019), chemical weathering (Yu and Hunt, 2018), and plant height and productivity as a function of time and transpiration rates (Hunt, 2017). Expressing soil depth and plant growth inputs to the crop net primary productivity, NPP, permits optimization of NPP with respect to the hydrologic fluxes (Hunt et al., 2020). Some remarkable conclusions also arise from this theory, such as that globally averaged ET is almost twice Q, and that the topology of the network guiding soil-water flow provides limitations on solute transport and chemical weathering. Both plant roots and infiltrating water tend to follow paths of least resistance, but with differing connectivity properties. Except in arid climates (Yang et al., 2016), roots tend to be restricted to the thin topsoil, so lateral root distributions are often considered twodimensional (2D), and root structures employ hierarchical, directional organization, speeding transport by avoiding closed loops. In contrast, infiltrating water (i.e., the subsurface part of Q) tends to follow random paths (Hunt, 2017) and percolates through the topsoil more deeply, giving rise to three-dimensional (3D) flow-path structures. The resulting distinct topologies generate differing nonlinear scaling, which is fractal, between time and distance of solute transport. On a bi-logarithmic space-time plot (Hunt, 2017), optimal paths for the different spatiotemporal scaling laws of root radial extent (RRE) and soil depth, z, are defined by their radial divergence from the same length and time positions. RRE relates to NPP, which is a key determinant of crop productivity, through root fractal dimensionality, df , given by RRE NPP df 1/ ∝ , with predicted values of df of 1.9 and 2.5 for 2D and 3D patterns, respectively (Hunt and Sahimi, 2017). Basic length/time scales are given by the fundamental network size (determined from the soil particle size distribution) and its ratio to mean soil-water flow rate. Yearly average pore-scale flow rates are determined from climate variables (Yu and Hunt, 2017). Each scaling relationship has a spread, representing chiefly the range of flow rates as controlled by P and its partitioning into ET and Q. This conceptual basis makes possible prediction of the dependence of NPP on the hydrologic fluxes, Q (which modulates the soil and root depths), and evapotranspiration, given by ET = P − Q (which modulates RRE). Consider the steady-state soil depth (Yu and