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. In the framework of simplified water balance models devoted to irrigation scheduling or crop modelling, the relative transpiration rate (the ratio of actual to maximal transpiration) is assumed to decrease linearly when the soil dries out below a critical available water value. This value is usually expressed as a fraction, F, of the maximal available soil water content. The present work aims to use the basic laws governing water transfer through the plants at a daily time step to compute F dynamically as the crop grows. It can be regarded as an expansion of Slabbers' (1980) approach to crop growing conditions. Starting from the mathematical representation given by single-root models (Gardner, 1960), an analytical expression for F is derived, using simplified hypotheses. This expression accounts for plant attributes such as the mean root radius, the critical leaf water potential for stomatal closure and the root length density profile growing with the crop. Environmental factors such as soil type and atmospheric demand also influence F. The structural influence of soil comes from the required introduction of the bulk soil hydraulic conductivity in the single-root model. The shape of the root length density profile is assumed to be sigmoidal and a new profile is calculated at each value of the rooting depth. A sensitivity analysis of F to all those factors is presented. The first general result is that F decreases as the root system grows in depth. Differences in the shape of the root profile can be responsible for differential water stress sensitivity in the early stages of growth. Yet, low critical leaf water potential can compensate partially for a poor root profile. Conversely, F is relatively insensitive to the average root radius. F sensitivity to soil type seems somewhat artificial: given the bulk soil hydraulic conductivity formula, the soil sensitivity results from F being expressed as a fraction of the maximal available soil water content. The atmospheric demand together with the rooting depth appear as the most important factors. However, when assuming predictable climatic and crop evolution, compensation occurs between those two effects leading to a relative stability of F when the crop is fully developed. Though relying on well-known physical laws, the present approach remains in the framework of single layer models with the same limitations.

An analytical solution for the estimation of the critical available soil water fraction for a single layer water balance model under growing crops