The shape of the transpiration reduction function under plant water stress

Assuming transpiration to be reduced after a critical pressure head ( usually chosen as -1.5 MPa or -150 m) at the root surface has been reached, transpiration rates in this so-called falling-rate phase were analyzed numerically for soils described by the van Genuchten-Mualem equations ( numerical soils). The analysis was based on the differential equation describing radial flow to a single root. Numerically, the system was simulated by an implicit scheme. It is shown that, at limiting hydraulic conditions, relative transpiration ( ratio between actual and potential transpiration) is equal to relative matric flux potential ( ratio between actual matric flux potential and matric flux potential at the onset of limiting hydraulic conditions). Given this equality, transpiration reduction functions as a function of soil water content and as a function of time are presented for five types of analytical soils: a constant diffusivity, Green and Ampt, Brooks and Corey, versatile nonlinear, and exponential soil. While in the case of constant diffusivity, relative transpiration decreases as a linear function of water content, for the remaining four cases the decrease is a concave function of soil water content. Numerical simulations also result in a concave shape, unless the difference between water content at the onset of limiting hydraulic conditions and at permanent wilting is very small, for example, at high root densities. These discrepancies may be explained by the relative importance of a transition period between the constant- and falling-rate phases.