Constraints to transpiration in a simple (but not too simple) model of transpiration

In our collaborative work with Concetta D'Amato  for the WATERSTEM project, we encountered the initial constraint of transpiration imposed by the hydraulic conductance of the stem-root system. Through our research, inspired by Manzoni et al. [2013], we discovered that the sigmoidal form of conductivity leads to an optimum for transpiration. We attempted to reproduce this phenomenon using the data provided by Kroeber et al. [2-13]. After considerable effort, we successfully generated the gray curve in the Figure, which exhibits a peak just before -4 MPa and enables too high transpiration.

However, we realized that the soil resistance was missing from our analysis. To address this, we incorporated the conductivity of a Silt Loam soil using the van Genuchten Mualem parameterization. The resulting brownish curves serve as evidence that the soil plays a crucial role, as anticipated by Carminati and Javaux [2020]. It is important to note that these curves depict the limits imposed by the soil and stem, which determine the potential sapflow rates, but do not reflect the constraints imposed by plant physiology. To account for plant physiology, we introduced the stomatal resistance, represented by the three dashed curves under different working hypotheses whose parameterization was taken from Daly et al. [2004]. The red points in the Figure represent the plant's working points (although the coupling with the atmospheric boundary layer is not depicted). One notable aspect of the Figure is that at typical soil suctions, the sapflow curves appear relatively flat, and the working points result in relatively constant sapflow despite variations in xylem/leaves pressure. The complete story will soon be available in Concetta's Ph.D. thesis, and the detailed process of creating the Figure can be found in its supplemental material notebooks.

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.

Daly, Edoardo, Amilcare Porporato, and Ignacio Rodriguez-Iturbe. 2004. “Coupled Dynamics of Photosynthesis, Transpiration, and Soil Water Balance. Part I: Upscaling from Hourly to Daily Level.” Journal of Hydrometeorology 5 (3): 546–58. https://doi.org/10.1175/1525-7541(2004)005<0546:cdopta>2.0.co;2.

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.

Manzoni, Stefano, Giulia Vico, Gabriel Katul, Sari Palmroth, Robert B. Jackson, and Amilcare Porporato. 2013. “Hydraulic Limits on Maximum Plant Transpiration and the Emergence of the Safety-Efficiency Trade-Off.” The New Phytologist 198 (1): 169–78. https://doi.org/10.1111/nph.12126.

A Fermi's like estimation of water fluxes in a plant (to check some consistencies)

 A Fermi's problem is an order-of-magnitude problem (or order-of-magnitude estimate, order estimation), is an estimation problem designed to teach dimensional analysis or approximation (in this case approximation) of extreme scientific calculations, and such a problem is usually a back-of-the-envelope calculation (cit. Wikipedia)

Let's assume that a plant transpires 1 cm per day (just to exaggerate) per unit of area. Suppose this plant canopy covers an area of 100 m^2. The transpired volume in one day is ET = 0.01 *  100 = 1 m^3 (which is a lot, plants are reported to transpirate "hundred of liters", not cubic meters).

Now let's consider the specific hydraulic conductivity KS in Kg m^{-1} s^{-1} MPa^{-1}. According to Krober et al. (2014) and their database, the maximum hydraulic conductivity of Castanea Henryi is approximately (simplifying the numbers) 10 Kg m^{-1} s^{-1} MPa^{-1}. Skipping some details, the maximum sap flow, E_S, derived from this is of the same order of magnitude, expressed in Kg m^{-1} s^{-1} (hint: you need to calculate K(\psi) \psi, with K varying with psi, and psi being the pressure (in MPa) in the xylem, as in Manzoni et al., 2014).

To compare E_S and ET, I need to multiply E_S by the active trunk cross-sectional area CSA (according to Thurner) and divide it by the plant height (10 m) to account for the gradient. Then, I need to convert from Kg per second to Kg per day (multiplying by 10^5) and divide by the density of water to obtain the result in terms of volume (10^3 kg/m^3). Therefore:

E_S = 10 [ES value] * 10^5 [Seconds in a Day] * CSA [Cross-sectional Area] / 10^4 [Plant Height * Water Density] = 100 CSA

From ES = ET, it follows that:

CSA = 0.01 m^2

which could not  be an unreasonable value (plant physiologists have to tell me). If the density measurement made by Kroeber et al. is actually related to the entire branch/trunk they used, it could mean that in a 1 m^2 stem (if the stem were 1 m^2), 1% contributes to the xylem flow. Unless I have forgotten any factor somewhere (which would be embarrassing, but I'll take the risk) or the measurements made by Kroeber et al. need to be adjusted differently.

According to Lüttschwager's study, this value would imply a much higher specific hydraulic conductivity than the KS observed in the outermost regions of the trunk where the flow is concentrated. Another consequence is that the less conductive species of this Chinese chestnut (38 out of 39 in the study) could only sustain such evaporation demands with much larger stems, which seems unreasonable, or a large percentage of vessels.

I would like to ask if the numbers I presented seem correct and reasonable to you, and if there is anything blatantly wrong in my reasoning or deduction from Kroeber's work (for those familiar with it) or elsewhere. Any comments are welcome.

P.S. - Most species in Kroeber's study have a KS that is 10 times smaller, which would require a CSA 10 times larger for the same evaporative demand.

References

Manzoni, Stefano, Giulia Vico, Gabriel Katul, Sari Palmroth, Robert B. Jackson, and Amilcare Porporato. 2013. “Hydraulic Limits on Maximum Plant Transpiration and the Emergence of the Safety-Efficiency Trade-Off.” The New Phytologist 198 (1): 169–78. https://doi.org/10.1111/nph.12126.

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.

Lüttschwager, Dietmar, and Rainer Remus. 2007. “Radial Distribution of Sap Flux Density in Trunks of a Mature Beech Stand.” Annals of Forest Science 64 (4): 431–38. https://doi.org/10.1051/forest:2007020.

Thurner, Martin, Christian Beer, Thomas Crowther, Daniel Falster, Stefano Manzoni, Anatoly Prokushkin, and Ernst-Detlef Schulze. 2019. “Sapwood Biomass Carbon in Northern Boreal and Temperate Forests.” Global Ecology and Biogeography: A Journal of Macroecology 28 (5): 640–60. https://doi.org/10.1111/geb.12883.

Some papers that discuss tree allometry to obtain biomass and sapwood cross sectional area

In order to accurately determine the water budget of trees (see also yesterday's post), it is crucial to establish a connection between the quantity of sapwood and the transpiration rate from the leaves. One essential factor in this process is obtaining accurate measurements of the sapwood cross-sectional areas (CSA). However, it is important to note that these CSA measurements can vary significantly from one plant to another. Acquiring this data can be challenging, and as a result, researchers have conducted studies aiming to establish allometric relationships as a means to estimate these measurements. To assist me in finding relevant literature on this topic, I reached out to my colleague involved in the WATERSTEM project. Below, you will find the literature they recommended.

References

Berry, Z. Carter, Nathaniel Looker, Friso Holwerda, León Rodrigo Gómez Aguilar, Perla Ortiz Colin, Teresa González Martínez, and Heidi Asbjornsen. 2018. “Why Size Matters: The Interactive Influences of Tree Diameter Distribution and Sap Flow Parameters on Upscaled Transpiration.” Tree Physiology 38 (2): 263–75. https://doi.org/10.1093/treephys/tpx124.

Kubota, Mitsumasa, John Tenhunen, Reiner Zimmermann, Markus Schmidt, Samuel Adiku, and Yoshitaka Kakubari. n.d. “Influences of Environmental Factors on the Radial Profile of Sap Flux Density in Fagus Crenata Growing at Different Elevations in the Naeba.” https://academic.oup.com/treephys/article/25/5/545/1712832.

Lüttschwager, Dietmar, and Hubert Jochheim. 2020. “Drought Primarily Reduces Canopy Transpiration of Exposed Beech Trees and Decreases the Share of Water Uptake from Deeper Soil Layers.” Forests, Trees and Livelihoods 11 (5): 537. https://doi.org/10.3390/f11050537.

Lüttschwager, Dietmar, and Rainer Remus. 2007. “Radial Distribution of Sap Flux Density in Trunks of a Mature Beech Stand.” Annals of Forest Science 64 (4): 431–38. https://doi.org/10.1051/forest:2007020.

Niccoli, Francesco, Arturo Pacheco-Solana, Sylvain Delzon, Jerzy Piotr Kabala, Shahla Asgharinia, Simona Castaldi, Riccardo Valentini, and Giovanna Battipaglia. 2023. “Effects of Wildfire on Growth, Transpiration and Hydraulic Properties of Pinus Pinaster Aiton Forest.” Dendrochronologia 79 (126086): 126086. https://doi.org/10.1016/j.dendro.2023.126086.

Petrík, Peter, Ina Zavadilová, Ladislav Šigut, Natalia Kowalska, Anja Petek-Petrik, Justyna Szatniewska, Georg Jocher, and Marian Pavelka. 2022. “Impact of Environmental Conditions and Seasonality on Ecosystem Transpiration and Evapotranspiration Partitioning (T/ET Ratio) of Pure European Beech Forest.” WATER 14 (19): 3015. https://doi.org/10.3390/w14193015.

Thurner, Martin, Christian Beer, Thomas Crowther, Daniel Falster, Stefano Manzoni, Anatoly Prokushkin, and Ernst-Detlef Schulze. 2019. “Sapwood Biomass Carbon in Northern Boreal and Temperate Forests.” Global Ecology and Biogeography: A Journal of Macroecology 28 (5): 640–60. https://doi.org/10.1111/geb.12883.

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.

Some observations about long rainfall and the generated discharges

 In well-known hydrologic response theories like the IUH, it has been established that for a specific catchment and a constant rainfall, there exists a 'critical rainfall duration' resulting in the maximum discharge for that catchment, which is usually known as concentration time. 

The next step is to associate a return period with the constant rainfall. This allows us to demonstrate that given a precipitation with an assigned return period, there is a critical rainfall duration that yields the highest possible discharge in that river section.This is what has been accomplished in Rigon et al., 2011 (but the research dates back to early 00, which is another interesting story). BTW, In the paper, we have also shown that this time is less or equal to the concentration time. 

The above argument may lead to the misconception that the “maximum discharge” for the catchment cannot be exceeded (keep in mind that the concept of maximum discharge obtainable is incomplete when you do not mention a return period).  Consider doubling the duration of the rainfall while keeping the intensity fixed. The first impulse results in the highest discharge with the assigned return period. Yet, it also has a discharge tail that, depending on the catchment's features, can last quite long. When the second impulse of precipitation arrives with the same intensity, it adds to the recession of the first impulse, usually increasing the discharge beyond the maximum discharge obtained with a single impulse.

In certain cases, like in the kinematic hydrograph model (uniform IUH) the rise of the new impulse discharge may precisely compensate for the decreasing recession of the older impulse, resulting in a constant discharge. However, this is not the general scenario, as simple calculations can show and sticking with this idea can be erroneous. Typically in fact, and especially when there is a marked contrast between the response time of the surface and subsurface storm flow waves, the recession discharge generated of the first impulse decreases more slowly than the increase in the new impulse discharge, effectively acting as additional rainfall. This effect is equivalent to increase the intensity of the effective rainfall to a return period which can be estimated through inverse modelling. In other words, two subsequent rainfall impulses, each with an assigned return period, are equivalent to a precipitation event with a higher return period. While the IUH theory establishes a precise equality between the return period of rainfall and discharge for a single impulse, the two return periods of discharges and rainfall become decoupled when multiple rainfall impulses occur.  

Although real-world precipitations are not constant and uniform, and the response of the catchment may not be time-invariant,  the main qualitative findings described above remain statistically valid and could be tested by generating ensembles of time-variable precipitations with numerical models. Besides, there are additional factors like sediment and vegetation transport that can add volume to the water (see for instance these posts),  increasing more than linearly the return period of discharge with increasing rainfall intensities. 

References 

Rigon, R., P. D’Odorico, and G. Bertoldi. 2011. “The Geomorphic Structure of the Runoff Peak.” Hydrology and Earth System Sciences 15 (6): 1853–63. https://doi.org/10.5194/hess-15-1853-2011.

C3A Six Years Plan

C3A is the Center for Agriculture (Agricoltura), Food (Alimenti), Environment (Ambiente) of the University of Trento (UniTrento).  As you can see in the brief history you can find in the document below, it was established to increase the involvement in the high education and research of the Province of Trento in Agrifood (and environmental field) together with the Edmund Mach Foundation (FEM) six years ago. 

The collaboration was actually not easy but at the same time fruitful and had a change in the recent years that were brought to a new agreement between the FEM and UniTrento. This agreement was the basis for the new six-years plan of the Center (due according to the regulations of the University of Trento) which you can find below. It design the research and educational activities for the next 6 years. 

The plan can be found by clicking on the above Figure. Here you can also find the slides I presented to the board of the University for presenting the Center and the plans.

A Ph.D. position on Po River, DARTHs, Earth Observations

I have an open Ph.D. position which closes at July 6: - Evolution of the system GEOframe/OMS3/CSIP for the building ofa Digital Twin of the Hydrology of river Po - E66E23000170001 

It looks like it is very dedicate to informatics (see also here) but let me say that the candidate should write their  project with a broader view, although it must remain within the scope of what we are doing in the context of the Po River basin project and  related to the exploitation of satellite data to support hydrological modeling. The project that funds it, besides PNRR,  is 4DHYdro, which collects some of the best hydrological modellers in Europe (and from the projects' goal you can find inspiration). 

The general focus of the study are droughts and can contains more computer science-related parts, more conceptual parts, or more applied parts. The themes related to the processes are: snow, plant transpiration, and crop needs. The enabling technology is precisely the systematic use of Earth observation, and the concept paper for the whole system is the one about DARTHs. Further information on DARTHs can be found here. 

If, at this point, you are a little convinced to apply also consider the philosophy of our group that you can find in a sequence of posts, here and and links therein.

Our group is a  crew of international fellows: 2 Indians, 1 Pakistani, 1French, 1 Iranian, 1 Algerian and 8 Italians, including two professors, one researcher (at Eurac), two postdocs, and nine Ph.D. students already.