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 address 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 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.

The understating of the dynamics of tracers and water transit times at catchment scale has increasingly grown in the last decades, becoming a consolidated approach in the field of hydrological and ecohydrological research. Recently, a benchmark contribution has been given in the work by Benettin et al. (2022), which reviews the state of art on the topic, also addressing present and future challenge, pointing out some open questions in transit time research. This commentary tries to contextualize the above article, highlighting the most focal points and relating it to a broader context in the field. A brief overview on the main concepts of backward transit times, StorAge selection functions and forward transit time distributions is given in a logical-historical order, giving to the reader the primary instruments for a later comprehensive understating of the Transit Time Theory. Eventually, a numerical example helps to clarify the above concepts in a very simple and effective way.

Let's assume we have a domain \(\Omega\) of boundaries \(\partial\Omega\) as in Figure 1.

This is an informal treatment of the instantaneous unit hydrograph IUH/GIUH \cite{Rigon_2015} estimation under the simplifying assumption that rainfall inputs are given as a discrete sequence of rain impulses (that we will call from now, rain record) This, more than being a simplification, is related to the discrete nature of rain at small time scales of aggregation and has a physical basis (e.g. \cite{Lovejoy_1985}).  This manuscript is distributed according to the CC v4 license. 

We present, by using previous results on extended Petri Nets, the relations of various hydrological dynamical systems ($\mathtt{HDSys}$) derived from the water budget ($\mathtt{DynWB}$). Once $\mathtt{DynWB}$ has been implemented, there exist a consistent way to get the equations for backward travel time distributions ($\mathtt{DynTT}$), for the forward response time distribution ($\mathtt{DynRTD}$) and for the concentration for a solute or a tracer ($\mathtt{DynC}$). We show that the $\mathtt{DynWB}$ has a correspondence one to many with the $\mathtt{DynTT}$. In fact to any one of the $\mathtt{DynWB}$ equation correspond as many equation as the input precipitation events times. The $\mathtt{DynTT}$ is related to $\mathtt{DynRTD}$ by the Niemi’s relationship and, in presence of multiple, $n$ outputs, by the specification of $n-1$ partition functions, which determine which fraction of water volume, injected in the control volume at a specific time $t_{in}$, goes asymptotically into a specific output. The $\mathtt{DynC}$, given $\mathtt{DynTT}$, depends further on the solute/tracer concentration in inputs. The paper clarifies the complicate set of relations above by using an example from literature. Upon the introduction of the appropriate information, it is also shown how these ($\mathtt{HDSys}$) can be solved simultaneously without duplicating calculations. It is also shown that these systems can be solved exactly, under the hypothesis of uniform mixing of water ages inside each reservoir within the system.

This is research material under development. If used, it should be cited properly. Citations to  published papers is scanty and it should be fixed along the way. To account for water movements in a catchment or any other control volume, let us assume that we can record the movements of small group of water molecules, called parcels, inside the control volume (cv) and at its boundaries. We do some assumptions first: