A mechanistic derivation of 'alpha-omega' root water uptake models.
Authors: Vanderborght Jan , Schnepf Andrea , Leitner Daniel , Couvreur Valentin , Javaux Mathieu 
Published: February 2023

To describe plant transpiration in drying soil, several models use &#8216;&#945;-stress functions&#8217;, which represent the ratio of the maximal possible water uptake when the plant reaches the wilting point to the transpiration demand or potential transpiration, as a function of the soil water potential. Water potentials vary within the root zone, and the plant &#8216;senses&#8217; with its root system an average root zone water potential and redistributes the uptake from drier to wetter zones in the root zone. This redistribution or root water uptake compensation is accounted for using an average stress index, &#969;, which is a weighted average of the local stress indices &#945; at different depths in the root zone, and a critical stress index &#969;c (Jarvis, 2011; Simunek & Hopmans, 2009). When &#969; > &#969;c, root water uptake is equal to the potential root water uptake or the energy limited potential transpiration. The &#945;-&#969; approach refers to a mechanistic description of water fluxes in the soil-root system but remains semi-empirical missing a direct link with soil and, especially, root hydraulic properties. In this contribution, we derive the &#945;-&#969; approach starting from a mechanistic description of water flow in a hydraulic root architecture assuming that resistance to flow in the soil towards the soil-root interface can be neglected. In a second step, we include the non-linear soil resistance.For relatively wet soil conditions and neglecting the soil resistance, root water uptake functions can be cast in a form that is identical to the &#945;-&#969; approach that was derived by Jarvis (2011), but for opposite conditions, i.e., Jarvis neglected the root resistance compared to soil resistance. Following Jarvis, the &#945;-function should be interpreted as the ratio of the maximal possible uptake by the root system for a certain soil water potential to the maximal possible uptake by the system when the soil is fully saturated, which differs from its common interpretation. This means that the &#945;-function is just a linear function that ranges from zero when the soil water potential is equal to the wilting point to 1 when the soil water potential is zero and that it is independent of the transpiration rate. Another outcome is that the critical stress level &#969;c is inverse proportional to the hydraulic conductance of the root system and is not a constant but a variable parameter that is proportional to the transpiration rate. For dry soil conditions, when soil resistance is important, we find that &#945; and &#969; are non-linear functions of the soil water potential. Using &#945; and &#969; functions that are derived from soil and root hydraulic properties, the uptake distributions can be calculated directly from the soil water potentials without solving a non-linear equation with iterations to derive water potentials in the plant. But, this approach is based on a simplification, which requires further testing.Jarvis, N. J. (2011). Hydrology and Earth System Sciences, 15(11), 3431-3446. doi:10.5194/hess-15-3431-2011Simunek, J., & Hopmans, J. W. (2009). Ecological Modelling, 220(4), 505-521. doi:10.1016/j.ecolmodel.2008.11.004