Elementary Mathematics sheds light on the transpiration budget under water stress

This paper aims to establish a method to accurately describe transpiration by employing appropriate physical equations. Although some simplifications are made, including use of a simplified treatment of turbulence and neglecting 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 ad- dress canopies. Subsequently, the water budget is incorporated to appropriately account for water stress in transpiration. In this context, a novel linearization 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 eluci- dated. 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. The preprint manuscript can be found here.  The reviews and the answer to reviewers are here. The published paper can be found here. 

A Rosetta stone for connecting the various forms of the Darcy-Buckingham law use in Hydrology and Plants Physiology

The information presented here is derived from the study conducted by Carminati and Javaux in 2020, which aimed to provide insights into plant hydraulics. Carminati referred to the work of Kroeber et al. in 2014, who conducted extensive measurements on a variety of plants and reported their data. However, a discrepancy arises between hydrologists and plant physiologists in the units used to measure hydraulic conductivity. While hydrologists measure it in meters per second (m/s), plant physiologists measure it in kilograms per meter per Pascal second [Kg m/(Pa s)].

In their study, Kroeber et al. reported conductivity per unit area, denoted as Kk, measured in kilograms per meter per Pascal second [Kg/(m Pa s)]. This unit might seem unfamiliar or obscure. To bridge the gap between my background and the new papers, Carminati and Javaux provide a clue. They suggest that the relationship between Kk and the commonly used hydraulic conductivity, K_w, expressed in centimeters per day (cm/day), can be established using the enigmatic equation K_w = g * 100 * 10^(-6) * 3600 * 24 * Kk. Now, the question arises: Is 'g' referring to the acceleration due to gravity?

So I dedicated a couple of days of my life to build a Rosetta Stone to translate the units and check the coherence of what done. The result is a short paper by me and Concetta D'Amato that you  can find here. 

For obtaining this I had to walk through the valley of the water potentials expressed in different units, but also this can be interesting for the reader.  

Next step is understand which is the value of the cross section through which the water flow to obtain, at the end, real cubic meter per second or kg per second. 

References

Carminati, Andrea, and Mathieu Javaux. 2020. “Soil Rather Than Xylem Vulnerability Controls Stomatal Response to Drought.” Trends in Plant Science 25 (9): 868–80. https://doi.org/10.1016/j.tplants.2020.04.003.

Kröber, Wenzel, Shouren Zhang, Merten Ehmig, and Helge Bruelheide. 2014. “Linking Xylem Hydraulic Conductivity and Vulnerability to the Leaf Economics Spectrum—A Cross-Species Study of 39 Evergreen and Deciduous Broadleaved Subtropical Tree Species.” PloS One 9 (11): e109211. https://doi.org/10.1371/journal.pone.0109211.