the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
From hydraulic root architecture models to efficient macroscopic sink terms including perirhizal resistance: Quantifying accuracy and computational speed
Abstract. Root water uptake strongly affects soil water balance and plant development. It can be described by mechanistic models of soilroot hydraulics based on soil water content, soil and root hydraulic properties, and the dynamic development of the root architecture. Recently, novel upscaling methods have emerged, which enable the application of detailed mechanistic models on a larger scale, particularly for land surface and crop models, by using mathematical upscaling.
In this study, we explore the underlying assumptions and the mathematical fundamentals of different upscaling approaches. Our analysis rigorously investigates the errors introduced in each step during the transition from finescale mechanistic models, which considers the nonlinear perirhizal resistance around each root, to more macroscopic representations. Upscaling steps simplify the representation of the root architecture, the perirhizal geometry, and the soil spatial dimension and thus introduces errors compared to the full complex 3D simulations. In order to investigate the extent of these errors, we perform simulation case studies: spring barley as a representative nonrow crop and maize as a representative row crop, and using three different soils.
We show that the error introduced by the upscaling steps strongly differs, depending on root architecture and soil type. Furthermore, we identify the individual steps and assumptions that lead to the most important losses in accuracy. An analysis of the trade off between model complexity and accuracy provides valuable guidance for selecting the most suitable approach for specific applications.
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Status: final response (author comments only)

RC1: 'Comment on egusphere20241319', Anonymous Referee #1, 29 Jul 2024
The manuscript presents a detailed exploration of upscaling methods for modeling root water uptake (RWU) in soilplant systems. The study aims to quantify the accuracy and computational efficiency of these methods, particularly focusing on the transition from finescale mechanistic models to more macroscopic representations. Overall, the study provides valuable insights into the tradeoffs between model complexity and computational speed, making significant contributions to the field of soilplant modeling.
I read the MS and gave the authors suggestions to improve its readability. By addressing these comments and suggestions, the manuscript will be significantly improved in terms of clarity, rigor, and overall impact. This will ensure that the findings are communicated effectively to the scientific community.
General Comments
The manuscript could benefit from a more explicit discussion of the assumptions made in the model, particularly those related to the root architecture and soilroot interface potentials. A dedicated section summarizing these assumptions and their potential impacts on the results would enhance clarity.
Consider adding a table summarizing all model parameters and their units. This would help readers understand the variables used and ensure consistency throughout the manuscript.
Provide more detailed derivations and explanations of key equations. For instance, equations related to perirhizal resistance and root water uptake should be thoroughly explained, including the assumptions and approximations involved.
Specific LinebyLine Comments
Line 3234: Do the authors know of any attempts to solve the NavierStokes Equation across root tissue? Do the authors consider radial or axial flow here? The radial flow seems to be challenging, especially if the cellcell pathway needs to be considered. On the other hand, will the average flow correlate with the average potential gradient? Averaging schemes seem to depend on the arrangement of cells and the composite structure of root tissue, and they are not simple averaging schemes. I suggest that the authors clarify these points in the text.
Line 36: What do the authors mean by "in this way"? Do they refer to 3D architectural models or NavierStokes equations?
Line 37: Replace “potential difference” with “water potential difference”.
Line 56: Replace “we” with “they”. Not all the authors of the mentioned work are coauthors of this manuscript.
Line 87: Replace “Vanderborght et al. (2021) showed that it reproduced the uptake by 3D root architectures quite well” with “Vanderborght et al. (2021) demonstrated that this approach well reproduced the water uptake by 3D root architectures”.
Line 108: What do the authors mean by individual potential? Make it clear that here you are referring to soil water potential.
Line 131138: It is not clear what the authors mean by Axx, Bxx, xAx, and xBx. Provide clear definitions and explanations of these terms in the text. See also the suggestion for Figure 1.
Line 142: Replace the unit of soil water content with cm³/cm³ to indicate that volumetric water content is considered here. Replace “soil total potential” with “total soil water potential” here and elsewhere in the text.
Line 152: What does it mean radial hydraulic conductivity with a unit of 1/day? Typically, it has a unit of a gradient in water potential inside.
Line 157: Did the authors neglect the osmotic potential gradient between soil and root xylem? Comment on this assumption.
Line 247: The authors use the concept of matric flux potential to derive the average hydraulic conductance in the perirhizal zone. Why did they not consider the radial nature of root water uptake in this derivation?
Line 237255: The set of questions described here is hard to follow, particularly Equation 18. Explain the derivation of these equations. To what extent is Equation 19 similar to those presented in Schröder et al. (2008) and Van Lier et al. (2006)? If they are based on similar concepts, I suggest referencing these works.
Line 410413: Do the estimated rootsoil hydraulic conductance and particularly the difference between two selected plants correspond to any literature values?
Line 451454: The reported differences are compared to what? Provide a clear reference for the comparison.
Line 251 & 264: Replace “troug” with “through”.
Figures and Visualizations
Figure 1: Label subplots as A1, A2, and A3 instead of referring to the first, second, and third columns in the caption. This will make it easier for readers to follow the discussion.
Figure 2: Is this really a loop? This figure would benefit from sequential numbering of the steps to clarify the process flow.
Figure 5: Increase the font size to improve readability.
Figure 6: Make the subplots larger and increase the font size to enhance clarity.
Figures 10, 13, 16, 19: These figures are very crowded and hard to distinguish between different lines. I suggest removing the legend from within the subplots. If they share the same legend, specify it once below or at the top of the subplots. Use different colors to separate the variables more clearly.
Figures 11, 12, 17, 18, 21: It is hard to distinguish between solid and dashed lines. Make the xaxis values in Figure 11 more readable.
Citation: https://doi.org/10.5194/egusphere20241319RC1 
AC1: 'Reply on RC1', Daniel Leitner, 21 Sep 2024
Thank you for the positive feedback and for the valuable and constructive comments! We have carefully revised the manuscript to address the suggestions. Below, we provide detailed responses to each of the reviewer's remarks and outline the corresponding revisions made to the manuscript.
General Comments
The manuscript could benefit from a more explicit discussion of the assumptions made in the model, particularly those related to the root architecture and soilroot interface potentials. A dedicated section summarizing these assumptions and their potential impacts on the results would enhance clarity.
We added a section dedicated to summarizing model assumptions to the discussion (L630L637)
Consider adding a table summarizing all model parameters and their units. This would help readers understand the variables used and ensure consistency throughout the manuscript.
The authors added a table summarizing all model parameters and variables in alphabetical order (Table 1)
Provide more detailed derivations and explanations of key equations. For instance, equations related to perirhizal resistance and root water uptake should be thoroughly explained, including the assumptions and approximations involved.
We added the continuum equations for the root hydraulic model, and improved the explanations of the key equations. A new section dedicated to summarizing model assumptions is given in the Discussion.
Specific LinebyLine Comments
Line 3234: Do the authors know of any attempts to solve the NavierStokes Equation across root tissue? Do the authors consider radial or axial flow here? The radial flow seems to be challenging, especially if the cellcell pathway needs to be considered. On the other hand, will the average flow correlate with the average potential gradient? Averaging schemes seem to depend on the arrangement of cells and the composite structure of root tissue, and they are not simple averaging schemes. I suggest that the authors clarify these points in the text.We now mention numerical methods , which solves the NavierStokes Equation across root tissue (Couvreur et a.l 2018), and for soil we mention homogenisation to derive Darcy’s law from NavierStokes equations (Hornung, 1996). We do not go into much detail into averaging or homogenization procedures here, since the effective root radial conductivity and axial conductivities are frequently obtained from experimental measurements (added at L33L36), and these effective hydraulic parameters act as starting point of our upscaling.
Line 36: What do the authors mean by "in this way"? Do they refer to 3D architectural models or NavierStokes equations?
we clarified: By describing the flux balance into the root via a Darcy type flow instead of using the full NavierStokes equations...
Line 37: Replace “potential difference” with “water potential difference”.
done (L46)
Line 56: Replace “we” with “they”. Not all the authors of the mentioned work are coauthors of this manuscript.
we rephrased that sentence
Line 87: Replace “Vanderborght et al. (2021) showed that it reproduced the uptake by 3D root architectures quite well” with “Vanderborght et al. (2021) demonstrated that this approach well reproduced the water uptake by 3D root architectures”.
done
Line 108: What do the authors mean by individual potential? Make it clear that here you are referring to soil water potential.
we rephrased the paragraph
Line 131138: It is not clear what the authors mean by Axx, Bxx, xAx, and xBx. Provide clear definitions and explanations of these terms in the text. See also the suggestion for Figure 1.
We described this new nomenclature in more detail (L126130) and included colored columns to make it easier to follow.
Line 142: Replace the unit of soil water content with cm³/cm³ to indicate that volumetric water content is considered here. Replace “soil total potential” with “total soil water potential” here and elsewhere in the text.
done
Line 152: What does it mean radial hydraulic conductivity with a unit of 1/day? Typically, it has a unit of a gradient in water potential inside.
Yes, typical unit are [cm2/day] being a volumetric flux per gradient [cm3/day 1/cm]. For root radial uptake we divide this value by the segment surface (this is sometimes called intrinsic root radial conductivity). We added the term ‘intrinsic’ for clarification (L162)
Line 157: Did the authors neglect the osmotic potential gradient between soil and root xylem? Comment on this assumption.
This is a model assumption that introduces a certain error. However, the osmotic potential of xylem sap is typically very small compared to other components of water potential in the soilplantatmosphere continuum. More pragmatically, the osmotic potential is frequently neglected in hydraulic models, since relevant processes to formulate the osmotic potential are not modeled in the hydraulic system.
Line 247: The authors use the concept of matric flux potential to derive the average hydraulic conductance in the perirhizal zone. Why did they not consider the radial nature of root water uptake in this derivation?
The derivation uses a cylindrical model, we tried to improve and clarify the description.
Line 237255: The set of questions described here is hard to follow, particularly Equation 18. Explain the derivation of these equations. To what extent is Equation 19 similar to those presented in Schröder et al. (2008) and Van Lier et al. (2006)? If they are based on similar concepts, I suggest referencing these works.
The equations are reformulated from Schröder et al. 2008. We point this out more clearly in (L243).
Line 410413: Do the estimated rootsoil hydraulic conductance and particularly the difference between two selected plants correspond to any literature values?
The authors are not aware of any literature values regarding spring barley. For maize the values are in the range of Meunier et al. (2020).
Félicien Meunier, Adrien Heymans, Xavier Draye, Valentin Couvreur, Mathieu Javaux, Guillaume Lobet, MARSHAL, a novel tool for virtual phenotyping of maize root system hydraulic architectures, in silico Plants, Volume 2, Issue 1, 2020, diz012
Line 451454: The reported differences are compared to what? Provide a clear reference for the comparison.
It is an intermodel comparison, the coarser 1D model is compared to the 3D model, we clarified this in the text.
Line 251 & 264: Replace “troug” with “through”.
done
Figures and Visualizations
Figure 1: Label subplots as A1, A2, and A3 instead of referring to the first, second, and third columns in the caption. This will make it easier for readers to follow the discussion.
We use numbering 13 and colors to make it easier to follow.
Figure 2: Is this really a loop? This figure would benefit from sequential numbering of the steps to clarify the process flow.
We included the numbering
Figure 5: Increase the font size to improve readability. done
Figure 6: Make the subplots larger and increase the font size to enhance clarity. done
Figures 10, 13, 16, 19: These figures are very crowded and hard to distinguish between different lines. I suggest removing the legend from within the subplots. If they share the same legend, specify it once below or at the top of the subplots. Use different colors to separate the variables more clearly. done
Figures 11, 12, 17, 18, 21: It is hard to distinguish between solid and dashed lines. Make the xaxis values in Figure 11 more readable. done
Citation: https://doi.org/10.5194/egusphere20241319AC1

AC1: 'Reply on RC1', Daniel Leitner, 21 Sep 2024

RC2: 'Comment on egusphere20241319', Anonymous Referee #2, 30 Jul 2024
The manuscript presents a nice exploration of methods for modelling rootsoil water acquisition and dynamics. The authors present a comparison between a full 3D model and several techniques for dimensional reduction, including an upscaled model with uniform soil root interface water potentials, parallel resistor (conductance) root model. They also present techniques to estimate the interaction between roots at a given depth by considering Voronoi tessellations vs uniform effective radial distances between roots. The study follows a technical assessment of the tradeoff between the accuracy that one loses by generating geometrically simplified models vs the increase in speed gained from such assumptions. Overall the manuscript is very well written, and despite the technical detail, the authors do a good job of making the material approachable. I have some minor critiques regarding the manuscript mostly with regards to the clarity. I would recommend the manuscript for publication subject to minor changes.
I find the organization of the different models hard to follow. In particular, I’m not sure if I fully understood the definition of the AAA vs AAB vs ABA vs CBB etc. I think there could have been a better naming convention for these that more explicitly state which assumptions are being undertaken in the text. It’s really a cosmetic issue, but I think it would really improve the readability of the manuscript. For example, I’m not sure if I’ve understood very well what the purpose was of the 2D macroscopic grid.
There were minor details regarding the theoretical section that I think could use a bit of elaboration on. I suppose Richards’ equation is also being solved for in order to determine soil matric potential, however, it became a bit confusing whether you needed to after some of the equilibration equations come in (i.e. equilibration between the soil matric potential and the xylem potential). It would be useful to know what boundary conditions are considered for this. I assume that there is some kind of resupply of soil water, but it’s again unclear whether this is the case. This specificity would be useful.
The matrix notation for the root zone radial and xylem water transport is nice, but I think it would be useful to include the differential equations to make it clear what physical phenomena is being considered. I think it’s just an extra two lines to show the equation in and around the plant, the surface Dirchlet and the noflux at the bottom. Something that makes it easy to follow for the reader. Then, I think matrix manipulations are all great.
Getting into some aesthetics, there are so many figures in this manuscript, and I wonder if some of them could be moved to an SI. I would say that 11, 12, 14, 15, 17, 18, 20, and 21 all look quite similar and aren’t adding a lot to the main text. They’re hard to read, and they aren’t really doing a lot of heavy lifting. As such, I would recommend maybe keeping 11 and 12 and then tucking the others into SI if the readers want to see other plots that look similar, but slightly different. Unfortunately, the transpiration figures are all really too important to move to SI. Figure 9 is amazing. I would only recommend to generate a mean value at each given depth for the full model in order to compare it with A and B. This would really show how representative the simplification actually is at capturing the mean behaviour of the full model and how much information we’re loosing by looking at simplifications.
As a last point, I couldn’t help but read this and think, “couldn’t one apply an empirical or semiempirical factor to get the reduced models to match the full models?” I don’t think that’s work that needs to be done here (there’s more than enough in this paper). I just think that it might be useful to think about. In my mind, the discussion about the crop distributions seems to allude to a potential remedy for the discrepancies between the different models. In my mind, I think that there might be some kind of asymptotic homogenization approach that could capture those geometric details and simplify them down to better estimate the dimensionally reduced model. Of course, if you’re certain that this can’t be done, it would be good to include these details in the discussion. I think it would be a strong point to make as well.
I have very small comments also included in an annotated document that I’m trying to attach as well.

AC2: 'Reply on RC2', Daniel Leitner, 21 Sep 2024
Thank you for your encouraging feedback and insightful comments! We have thoroughly revised the manuscript to incorporate the suggestions. Below, we offer detailed responses to each of the reviewer’s points and highlight the corresponding changes made.
RC: I find the organization of the different models hard to follow. In particular, I’m not sure if I fully understood the definition of the AAA vs AAB vs ABA vs CBB etc. I think there could have been a better naming convention for these that more explicitly state which assumptions are being undertaken in the text. It’s really a cosmetic issue, but I think it would really improve the readability of the manuscript. For example, I’m not sure if I’ve understood very well what the purpose was of the 2D macroscopic grid.
We revised the presentation of the nomenclature to make it easier to follow (L111L130).
RC: There were minor details regarding the theoretical section that I think could use a bit of elaboration on. I suppose Richards’ equation is also being solved for in order to determine soil matric potential, however, it became a bit confusing whether you needed to after some of the equilibration equations come in (i.e. equilibration between the soil matric potential and the xylem potential). It would be useful to know what boundary conditions are considered for this. I assume that there is some kind of resupply of soil water, but it’s again unclear whether this is the case. This specificity would be useful.
We clarified in L256263. As a steady rate model, it is independent of time, but only depends on the rate which is determined by the bulk soil total potential Hs and the root xylem potential Hx.
RC: The matrix notation for the root zone radial and xylem water transport is nice, but I think it would be useful to include the differential equations to make it clear what physical phenomena is being considered. I think it’s just an extra two lines to show the equation in and around the plant, the surface Dirchlet and the noflux at the bottom. Something that makes it easy to follow for the reader. Then, I think matrix manipulations are all great.
We added the continuum equations for the root hydraulic model in the beginning of Section 2.3, and show their discretization.
RC: Getting into some aesthetics, there are so many figures in this manuscript, and I wonder if some of them could be moved to an SI. I would say that 11, 12, 14, 15, 17, 18, 20, and 21 all look quite similar and aren’t adding a lot to the main text. They’re hard to read, and they aren’t really doing a lot of heavy lifting. As such, I would recommend maybe keeping 11 and 12 and then tucking the others into SI if the readers want to see other plots that look similar, but slightly different. Unfortunately, the transpiration figures are all really too important to move to SI. Figure 9 is amazing. I would only recommend to generate a mean value at each given depth for the full model in order to compare it with A and B. This would really show how representative the simplification actually is at capturing the mean behaviour of the full model and how much information we’re loosing by looking at simplifications.
We agree and moved 14, 15, 17, 18, 20, and 21 to SI
As a last point, I couldn’t help but read this and think, “couldn’t one apply an empirical or semiempirical factor to get the reduced models to match the full models?” I don’t think that’s work that needs to be done here (there’s more than enough in this paper). I just think that it might be useful to think about. In my mind, the discussion about the crop distributions seems to allude to a potential remedy for the discrepancies between the different models. In my mind, I think that there might be some kind of asymptotic homogenization approach that could capture those geometric details and simplify them down to better estimate the dimensionally reduced model. Of course, if you’re certain that this can’t be done, it would be good to include these details in the discussion. I think it would be a strong point to make as well.
The authors completely agree. We added a section on model limitations to the discussion and mention homogenisation approach as valuable tool for model development (L630L637)
Citation: https://doi.org/10.5194/egusphere20241319AC2

AC2: 'Reply on RC2', Daniel Leitner, 21 Sep 2024

AC3: 'Comment on egusphere20241319', Daniel Leitner, 23 Sep 2024
I had problems uploading the revised manuscript and asked the editorial service for support. I hope it will be available for everyone soon.
Citation: https://doi.org/10.5194/egusphere20241319AC3
Model code and software
CPlantBox Daniel Leitner, Andrea Schnepf, and Jan Vanderborght https://github.com/PlantRootSoilInteractionsModelling/CPlantBox
DumuxRosi Daniel Leitner, Andrea Schnepf, and Jan Vanderborght https://github.com/PlantRootSoilInteractionsModelling/dumuxrosi
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