Elementary Mathematics Helps to Shed Light on the Transpiration Budget Under Water Stress

This paper aims to present a methodology for accurately describing transpiration by employing appropriate physical equations. While some simplifications have been made, including the use of a simplified treatment of turbulence and the neglect of the thermal capacity of transpiring leaves, it is argued that the chosen scheme has general validity in identifying the primary mechanisms governing transpiration. To achieve this objective, a traditional treatment involving five equations, including the mass budget, is used. Initially, a simplified approach that does not consider the water budget is introduced to outline the general procedure to explicitly address canopies. Subsequently, the water budget is incorporated to appropriately account for water stress in transpiration. In this context, a novel linearisation of the extended Clausius–Clapeyron equation, incorporating the Kelvin effect, is employed. It is demonstrated that the well-known Penman formula emerges as one of the solutions within a system of equations, providing estimates for temperature (T), vapor content in air (e) and the thermal transport of heat (H). The method, initially conceived for homogeneous canopies, is expanded to encompass sun–shade canopy layers. By employing the water mass balance, the trade-off between atmospheric evaporation demand and the water delivery capacity of the soil and stem is elucidated. Notably, it is revealed that the pressure potential within leaves is not solely determined by capillarity, but rather represents the dynamic outcome of the intricate interactions within the soil–plant–atmosphere continuum. These findings highlight differences from more simplistic approaches commonly employed, particularly concerning canopies. Overall, this study presents a methodological framework to accurately describe transpiration, incorporating key equations and addressing the complex dynamics involved in the soil–plant–atmosphere continuum, and suggests various directions of research in the field.