Formulation of root water uptake in a multi-layer soil-plant model: does van den Honert's equation hold?

Abstract. The withdrawal of water from soil by vegetation, which in steady state conditions is equivalent to the transpiration rate, can be written in terms of water potential in the form of an Ohm's law analogy, known as van den Honert's equation: The difference between an effective soil water potential and the bulk canopy water potential is divided by an effective soil-plant resistance. This equation is commonly used, but little is known about the precise definition of its parameters. The issue of this paper is to bridge the gap between the bulk approach and a multi-layer description of soil-plant water transfer by interpreting the bulk parameters in terms of the characteristics of the multi-layer approach. Water flow through an elementary path within the soil or the root is assumed to follow an Ohm's law analogy, and the soil and root characterisics are allowed to vary with depth. Starting from the basic equations of the multi-layer approach, it is proved that the total rate of transpiration can also be expressed in the form of an Ohm's law analogy. This means that van den Honert's equation holds at canopy scale, insofar as the assumptions made on the physics of root water uptake hold. In the bulk formulation derived, the effective soil-plant resistance appears as a combination of the elementary resistances making up the multi-layer model; and the effective soil water potential is a weighted mean of the water potentials in each soil layer, the weighting system involving the complete set of elementary resistances. Simpler representations of soil-plant interaction leading to Ohm's law type formulations are also examined: a simplified multi-layer model, in which xylem (root axial) resistance is neglected, and a bulk approach, in which soil-root interaction is represented by only one layer. Numerical simulations performed in different standard conditions show that these simpler representations do not provide accurate estimates of the transpiration rate, when compared to the values obtained by the complete algorithm.