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Predicting the Water Balance from

Optimization of Plant Productivity

A.G. Hunt*, Dept. of Physics, Wright State University, Dayton, Ohio 45435, USA; B. Faybishenko, Energy Geosciences Division,
Lawrence Berkeley National Laboratory, University of California, 1 Cyclotron Road, Berkeley, California 94720, USA; and
B. Ghanbarian, Porous Media Research Lab, Dept. of Geology, Kansas State University, Manhattan, Kansas 66506, USA

How soil-water flows and how fast it moves guiding soil-water flow provides limitations and evapotranspiration, given by ET = P − Q
solutes are important for plant growth and soil on solute transport and chemical weathering. (which modulates RRE).
formation. The relationship describing the Both plant roots and infiltrating water tend Consider the steady-state soil depth (Yu and
partitioning of precipitation, P, into run-off, to follow paths of least resistance, but with 1
Q, and evapotranspiration, ET, is called the differing connectivity properties. Except in Hunt, 2017), z ∝ Q D 1− = Q 1.15 , with D = 1.87,
b
b
water balance. Q incorporates both surface arid climates (Yang et al., 2016), roots tend to governing solute transport, which is the back-
runoff and subsurface flow components, the be restricted to the thin topsoil, so lateral bone fractal dimension of percolation.
P Q)
latter chiefly contributing to soil formation. root distributions are often considered two- Optimization of NPP ∝ RRE ∝ Q ( − df
1.15
At shorter time intervals, soil-water storage, dimensional (2D), and root structures employ with respect to Q by setting d(NPP)⁄d(Q) = 0
S, may change, dS/dt, due to atmosphere-soil hierarchical, directional organization, speed- yields ET = P d ⁄(1.15 + d ) = 0.623P, within
f
f
water exchange; i.e., infiltrating and evaporat- ing transport by avoiding closed loops. In 1–2% of the mean of global estimates (Hunt
ing water and root uptake. Over sufficiently contrast, infiltrating water (i.e., the subsur- et al., 2020).
long time periods, storage changes are typi- face part of Q) tends to follow random paths The ratio ET/P may be represented using
cally neglected (Gentine et al., 2012). (Hunt, 2017) and percolates through the top- the aridity index, AI, often defined as
Percolation theory from statistical physics soil more deeply, giving rise to three-dimen- PET/P (sometimes as its inverse), with PET
provides a powerful tool for predicting soil sional (3D) flow-path structures. The result- being the potential evapotranspiration
formation and plant growth (Hunt, 2017) by ing distinct topologies generate differing (Budyko, 1958). In arid regions, where soil
means of modeling soil pore space as net- nonlinear scaling, which is fractal, between 1
works, rather than continua. time and distance of solute transport. depths are yet increasing, z ∝ Q D b = Q 0.53
In heterogeneous soils, solute migration On a bi-logarithmic space-time plot (Hunt, (Yu and Hunt, 2017). For a bare land area, the
typically exhibits non-Gaussian behavior, 2017), optimal paths for the different spatio- fraction of the surface that plants occupy may
with statistical models having long tails in temporal scaling laws of root radial extent be only P/PET, which is the inverse of the AI.
arrival time distributions and velocities (RRE) and soil depth, z, are defined by their Both tend to increase ET as a fraction of P. For
decreasing over time. Theoretical prediction radial divergence from the same length and high AI, roots are also less confined near the
of solute transport via percolation theory that time positions. RRE relates to NPP, which is surface, searching water more deeply, and
generates accurate full non-Gaussian arrival a key determinant of crop productivity, also increasing ET. Under ideal conditions
time distributions has become possible only through root fractal dimensionality, d f , of neither energy nor water limitation (AI = 1),
recently (Hunt and Ghanbarian, 2016; Hunt given by RRE ∝ NPP 1/ df , with predicted Levang-Brilz and Biondini (2003) determined
and Sahimi, 2017). A unified framework, values of d of 1.9 and 2.5 for 2D and 3D pat- that for 16 grass and 39 Great Plains forb spe-
f
based on solute transport theory, helps pre- terns, respectively (Hunt and Sahimi, 2017). cies the mean d for all forbs was 2.49, but
f
dict soil depth as a function of age and infil- Basic length/time scales are given by the grasses separated into two distinct groups
tration rate (Yu and Hunt, 2017), soil erosion fundamental network size (determined with d = 2.65 and 1.67, in accord with percola-
f
rates (Yu et al., 2019), chemical weathering from the soil particle size distribution) and tion predictions (Hunt and Sahimi, 2017). In
(Yu and Hunt, 2018), and plant height and its ratio to mean soil-water flow rate. Yearly the studied biome, grasses constitute more
productivity as a function of time and tran- average pore-scale flow rates are deter- than 90% of the biomass.
spiration rates (Hunt, 2017). Expressing soil mined from climate variables (Yu and Hunt, Figure 1 shows our predicted upper bound
depth and plant growth inputs to the crop net 2017). Each scaling relationship has a (dotted line) of ET/P as a function of AI. At
primary productivity, NPP, permits optimi- spread, representing chiefly the range of low AI (<1) the known limit ET ≤ PET is
zation of NPP with respect to the hydrologic flow rates as controlled by P and its parti- applied. For large AI, d = 2.5, appropriate for
f
fluxes (Hunt et al., 2020). Some remarkable tioning into ET and Q. This conceptual deeper, more isotropic, root systems. Levang-
conclusions also arise from this theory, such basis makes possible prediction of the depen- Brilz and Biondini’s (2003) experimental d f
as that globally averaged ET is almost twice dence of NPP on the hydrologic fluxes, Q values generate the spread in predicted ET at
Q, and that the topology of the network (which modulates the soil and root depths), selected AI values (though experimental

*Email: allen.hunt@wright.edu

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