the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 13 Sep 2023
A rigorous approach to the specific surface area evolution in snow during temperature gradient metamorphism
Abstract. Despite being one of the most fundamental microstructural parameters of snow, the specific surface area (SSA) dynamics during temperature gradient metamorphism (TGM) have so far been addressed only within empirical modeling. To surpass this limitation, we propose a rigorous modeling of SSA dynamics using an exact equation for the temporal evolution of the surface area, fed by pore-scale finite element simulations of the water vapor field coupled with the temperature field on X-ray computed-tomography images. The proposed methodology derives from physics' first principles and thus does not rely on any empirical parameter. Since the calculated evolution of the SSA is highly sensitive to fluctuations in the experimental data, we address the impact of these fluctuations within a stochastic error model. In our simulations, the only poorly constrained physical parameter is the vapor attachment coefficient α onto ice. We address this problem by simulating the SSA evolution for a wide range of α and estimate optimal values by minimizing the differences between simulations and experiments. This methodology suggests that α lies in the intermediate range 10-3 < α < 10-1 and slightly varies between experiments. Also, our results suggest a transition of the value of α in one TGM experiment, which can be explained by a transition in the underlying surface morphology. Overall, we are able to reproduce very subtle variations in the SSA evolution with correlations of R2 = 0.95 and 0.99, respectively, for the two considered TGM time series. Finally, our work highlights the necessity of including kinetics effects and of using realistic microstructures to comprehend the evolution of SSA during TGM.
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Notice on discussion status
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
(10097 KB)
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
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- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2023-1947', Z.R. Courville, 12 Oct 2023
The manuscript presents a very interesting physically-based modeling approach to the evolution of snow specific area during temperature gradient metamorphism, notably presenting results constraining the kinetic attachment coefficient, which is difficult to measure and has proved an elusive parameter in overall efforts at understanding temperature gradient metamorphism. I found the result that the kinetic coefficient varied between the two samples, and then over the course of one of the experiments particularly interesting, but intuitively makes sense in terms of the dependence of the kinetic coefficient on the morphology of snow grains. I also find the SSA modeling results presented in Fig 6 compelling with respect to the microCT data, with the match between model and experimental results remarkable. The manuscript is very well written. Below, I offer a few minor suggestions for consideration to improve clarity for a reader. The main suggestion I have is to use a consistent definition of alpha throughout the text. I also had a few questions about the specifics of the model mentioned that I think might warrant clarification.
Line 64: “at the downside” is not quite the right phrase, I would suggest “at the expense” instead
Line 73: I would suggest writing: “While the interfacial curvature is a geometrical quantity, the interface growth velocity must be computed from a physical model.” That is only a suggestion to make that sentence clearer vs. “first” and “second” term since that sentence has a lot of terms in it, and that’s if I’ve interpreted the sentence correctly.
Line 88: I’m not sure “motion” is the best term for the interface, and would suggest “evolution” or maybe “migration” instead.
Line 89: How was the size of the representative snow volume determined? (or is that in the Pinzer article? If they do discuss how the representative volume was determined, I would mention that briefly.)
Line 115: Throughout the text, there are several definitions/names of the parameter alpha (or at least I think they are all referring to alpha). As a suggestion, I recommend either being more consistent, or explaining at the first instance that alpha has been called different things. The first time it happened, I was wondering why the change from “vapor attachment coefficient” to “condensation coefficient”, and recommended defining alpha as “the vapor attachment coefficient, or the condensation coefficient” at the first definition of alpha, but then I noted that there are several different forms of the definition used throughout the manuscript, including “kinetic coefficient” (line 293) and “attachment kinetics coefficient” (line 297). Again, I **think** these are all referring to alpha, but I am not sure.
Line 119: Ditto that last comment for the definition of alpha in this instance (I stopped noting all the different terms used for alpha as I went on in my review, see the above comment. I think either calling it the same thing or discussing all the different variations is warranted to alleviate confusion.)
Line 119: Suggest rewriting as “the kinetic coefficient α is defined as the probability of a water molecule sticking to an impinging surface.” (this is only a minor grammar/usage suggestion)
Line 123: Is (7) referring to a reference in bibtex or some other citation managing software? Or is it referring to equation 7? Might be clearer if it said “eq. 7”
Line 145: By “shorter” does that mean the sample is physically smaller, or that the time was shorter (I mean, I think I know the answer since the hours are greater for Series 2)? Suggest rewriting to clarify, maybe “Series 1 lasted 384 h and had a shorter sample height…” if that is what is meant. Also seems like the sample thicknesses/heights should be included as a well as the temperature gradients, even if the details are in the Pinzer paper.
Line 146: Does mean T refer to the average air/ambient temperature for the experiment or the average temp throughout the sample?
Line 159: How was “a reasonable volumetric division” determined or quantified? Specify the requirements.
Line 165: Likewise, define “small air padding” quantitatively, or if dependent on the size of the volume of interest/SSA or sample grain size, describe how that was determined.
Line 189: For readers not familiar with Elmer, it would be good to add a brief description of what an ILU preconditioner is or does. I will note, though, that in general the authors have done a very good job of describing what the different functions in Elmer are for a non-Elmer user.
Line 263: I would put in the length scale of the test case (0.9 mm) so the reader doesn’t have to do the math, i.e., “In this way, the length scales of the test case (0.9 mm for the outer radius) are a similar order of magnitude…”
Figure 1. Suggest putting a scale bar in for the sphere (in mm) if it doesn’t clutter the figure too much since that will help a reader compare to typical snow grain sizes, or adding the outer sphere dimension to the caption.
Figure 1. For b) is the blue the “air padding” similar to what was added to the microCT volume?
Figure 1. For c) what are the red and blue dashed lines showing? I’m guessing that is the (sim-theo)/theo for values of alpha, but that should be called out in the legend, and which axis those values are plotted on should be indicated for easy of reading.
Line 308: what does the RMSE minimum “is deeper” mean? That the RMSE minimum is lower?
Line 311: Should be “a time step refined down to the time interval between two microCT images…” or something (seems like there is a missing preposition after “down”).
Citation: https://doi.org/10.5194/egusphere-2023-1947-RC1 - AC2: 'Reply on RC1', Anna Braun, 22 Dec 2023
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RC2: 'Comment on egusphere-2023-1947', Thomas Kaempfer, 16 Oct 2023
The paper presents a novel approach to model the evolution of the specific surface area (SSA) during temperature gradient metamorphism (TGM). It uses X-ray micro-computed tomography (mu-CT) images of snow in combination with a numerical solution of steady-state energy and mass conservation equations at the micro-structural scale and a surface area equation based on physics first principles. The only "free" parameter in the model is the vapor attachment coefficient alpha and it is proposed that SSA evolution can be predicted using an adequately chosen "effective" alpha.
The paper is generally well written with a strong emphasis on the numerical solution and analysis of error propagation. The strength and limitations of the approach are clearly presented.
Clarity could be improved by using more concise language and unified terminology (see minor comments in annotated PDF-file). The paper could further benefit by considering the following specific remarks. I suggest the paper to be revised accordingly before accepting it for publication.
Specific remarks
- Introduction, line 36ff: While it is OK to quickly concentrate on the relevant mechanisms for the TGM situation studied in this paper (energy and mass conservation, attachment kinetics), I suggest justifying why other processes are of second order instead of "boldly" postulating that it is all about alpha.
- Presentation of the approach (e.g., Introduction, lines 70ff; or beginning of section 3, lines 159ff – possibly add a new sub-subsection at the beginning of sub-section 3.2): The coupling of the mu-CT images and the numerical models (Finite Elements and surface equation) should somewhere carefully be explained. If I understood it correctly, you have resp. do:
- time-laps mu-CT images of snow (4D)
- use a (sub-)set of 3D images as input to your numerical model
- for each chosen 3D image, pre-process and discretize (e.g., image processing, triangulation)
- determine some parameters from the geometry directly (e.g., surface curvature)
- determine other parameters from numerical modeling (e.g., growth velocity)
- compute SSA evolution using above; fit alpha to the experimental results
- Unclear to me are in particular:
- how many 3D images do you use for the modeling? When and how do you decide to "update" the micro-structure in your models (e.g., using a new image from the 4D series)? Or do you never do this (and always use the 1st image only)? In general, the discretization in time of the numerical model is unclear to me.
- selection of appropriate (sub)volumes from the 3D images, pre-processing and volumetric averaging: how exactly are the volumes that feed into the numerical model selected? For several parameters, volumetric averaging is performed (e.g., SSA, curvature, alpha). Is the averaging volume always the same (e.g., the entire domain)? Are we sure to have a size large enough to be representative? For the kinetic coefficient, it is very late in the paper that the concept of "effective coefficient" is introduced. Maybe the concept(s) could be introduced and justified early in an overall context.
- reason for "numerical (?)" tricks like the "air padding" and iterative (2-times) solution of the energy conservation equation
- Error analysis (sections 3.4 / 4.3 / 5.3): While I do like this systematic error analysis, I find the focus very much on "time" related errors; however, I think that other errors (e.g., discretization in space) are at least equally important. These are captured and a bit discussed in section 5.3. Clearer statements – already in earlier sections – would help to better understand the strengths and weaknesses of the approach chosen.
- AC1: 'Reply on RC2', Anna Braun, 22 Dec 2023
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2023-1947', Z.R. Courville, 12 Oct 2023
The manuscript presents a very interesting physically-based modeling approach to the evolution of snow specific area during temperature gradient metamorphism, notably presenting results constraining the kinetic attachment coefficient, which is difficult to measure and has proved an elusive parameter in overall efforts at understanding temperature gradient metamorphism. I found the result that the kinetic coefficient varied between the two samples, and then over the course of one of the experiments particularly interesting, but intuitively makes sense in terms of the dependence of the kinetic coefficient on the morphology of snow grains. I also find the SSA modeling results presented in Fig 6 compelling with respect to the microCT data, with the match between model and experimental results remarkable. The manuscript is very well written. Below, I offer a few minor suggestions for consideration to improve clarity for a reader. The main suggestion I have is to use a consistent definition of alpha throughout the text. I also had a few questions about the specifics of the model mentioned that I think might warrant clarification.
Line 64: “at the downside” is not quite the right phrase, I would suggest “at the expense” instead
Line 73: I would suggest writing: “While the interfacial curvature is a geometrical quantity, the interface growth velocity must be computed from a physical model.” That is only a suggestion to make that sentence clearer vs. “first” and “second” term since that sentence has a lot of terms in it, and that’s if I’ve interpreted the sentence correctly.
Line 88: I’m not sure “motion” is the best term for the interface, and would suggest “evolution” or maybe “migration” instead.
Line 89: How was the size of the representative snow volume determined? (or is that in the Pinzer article? If they do discuss how the representative volume was determined, I would mention that briefly.)
Line 115: Throughout the text, there are several definitions/names of the parameter alpha (or at least I think they are all referring to alpha). As a suggestion, I recommend either being more consistent, or explaining at the first instance that alpha has been called different things. The first time it happened, I was wondering why the change from “vapor attachment coefficient” to “condensation coefficient”, and recommended defining alpha as “the vapor attachment coefficient, or the condensation coefficient” at the first definition of alpha, but then I noted that there are several different forms of the definition used throughout the manuscript, including “kinetic coefficient” (line 293) and “attachment kinetics coefficient” (line 297). Again, I **think** these are all referring to alpha, but I am not sure.
Line 119: Ditto that last comment for the definition of alpha in this instance (I stopped noting all the different terms used for alpha as I went on in my review, see the above comment. I think either calling it the same thing or discussing all the different variations is warranted to alleviate confusion.)
Line 119: Suggest rewriting as “the kinetic coefficient α is defined as the probability of a water molecule sticking to an impinging surface.” (this is only a minor grammar/usage suggestion)
Line 123: Is (7) referring to a reference in bibtex or some other citation managing software? Or is it referring to equation 7? Might be clearer if it said “eq. 7”
Line 145: By “shorter” does that mean the sample is physically smaller, or that the time was shorter (I mean, I think I know the answer since the hours are greater for Series 2)? Suggest rewriting to clarify, maybe “Series 1 lasted 384 h and had a shorter sample height…” if that is what is meant. Also seems like the sample thicknesses/heights should be included as a well as the temperature gradients, even if the details are in the Pinzer paper.
Line 146: Does mean T refer to the average air/ambient temperature for the experiment or the average temp throughout the sample?
Line 159: How was “a reasonable volumetric division” determined or quantified? Specify the requirements.
Line 165: Likewise, define “small air padding” quantitatively, or if dependent on the size of the volume of interest/SSA or sample grain size, describe how that was determined.
Line 189: For readers not familiar with Elmer, it would be good to add a brief description of what an ILU preconditioner is or does. I will note, though, that in general the authors have done a very good job of describing what the different functions in Elmer are for a non-Elmer user.
Line 263: I would put in the length scale of the test case (0.9 mm) so the reader doesn’t have to do the math, i.e., “In this way, the length scales of the test case (0.9 mm for the outer radius) are a similar order of magnitude…”
Figure 1. Suggest putting a scale bar in for the sphere (in mm) if it doesn’t clutter the figure too much since that will help a reader compare to typical snow grain sizes, or adding the outer sphere dimension to the caption.
Figure 1. For b) is the blue the “air padding” similar to what was added to the microCT volume?
Figure 1. For c) what are the red and blue dashed lines showing? I’m guessing that is the (sim-theo)/theo for values of alpha, but that should be called out in the legend, and which axis those values are plotted on should be indicated for easy of reading.
Line 308: what does the RMSE minimum “is deeper” mean? That the RMSE minimum is lower?
Line 311: Should be “a time step refined down to the time interval between two microCT images…” or something (seems like there is a missing preposition after “down”).
Citation: https://doi.org/10.5194/egusphere-2023-1947-RC1 - AC2: 'Reply on RC1', Anna Braun, 22 Dec 2023
-
RC2: 'Comment on egusphere-2023-1947', Thomas Kaempfer, 16 Oct 2023
The paper presents a novel approach to model the evolution of the specific surface area (SSA) during temperature gradient metamorphism (TGM). It uses X-ray micro-computed tomography (mu-CT) images of snow in combination with a numerical solution of steady-state energy and mass conservation equations at the micro-structural scale and a surface area equation based on physics first principles. The only "free" parameter in the model is the vapor attachment coefficient alpha and it is proposed that SSA evolution can be predicted using an adequately chosen "effective" alpha.
The paper is generally well written with a strong emphasis on the numerical solution and analysis of error propagation. The strength and limitations of the approach are clearly presented.
Clarity could be improved by using more concise language and unified terminology (see minor comments in annotated PDF-file). The paper could further benefit by considering the following specific remarks. I suggest the paper to be revised accordingly before accepting it for publication.
Specific remarks
- Introduction, line 36ff: While it is OK to quickly concentrate on the relevant mechanisms for the TGM situation studied in this paper (energy and mass conservation, attachment kinetics), I suggest justifying why other processes are of second order instead of "boldly" postulating that it is all about alpha.
- Presentation of the approach (e.g., Introduction, lines 70ff; or beginning of section 3, lines 159ff – possibly add a new sub-subsection at the beginning of sub-section 3.2): The coupling of the mu-CT images and the numerical models (Finite Elements and surface equation) should somewhere carefully be explained. If I understood it correctly, you have resp. do:
- time-laps mu-CT images of snow (4D)
- use a (sub-)set of 3D images as input to your numerical model
- for each chosen 3D image, pre-process and discretize (e.g., image processing, triangulation)
- determine some parameters from the geometry directly (e.g., surface curvature)
- determine other parameters from numerical modeling (e.g., growth velocity)
- compute SSA evolution using above; fit alpha to the experimental results
- Unclear to me are in particular:
- how many 3D images do you use for the modeling? When and how do you decide to "update" the micro-structure in your models (e.g., using a new image from the 4D series)? Or do you never do this (and always use the 1st image only)? In general, the discretization in time of the numerical model is unclear to me.
- selection of appropriate (sub)volumes from the 3D images, pre-processing and volumetric averaging: how exactly are the volumes that feed into the numerical model selected? For several parameters, volumetric averaging is performed (e.g., SSA, curvature, alpha). Is the averaging volume always the same (e.g., the entire domain)? Are we sure to have a size large enough to be representative? For the kinetic coefficient, it is very late in the paper that the concept of "effective coefficient" is introduced. Maybe the concept(s) could be introduced and justified early in an overall context.
- reason for "numerical (?)" tricks like the "air padding" and iterative (2-times) solution of the energy conservation equation
- Error analysis (sections 3.4 / 4.3 / 5.3): While I do like this systematic error analysis, I find the focus very much on "time" related errors; however, I think that other errors (e.g., discretization in space) are at least equally important. These are captured and a bit discussed in section 5.3. Clearer statements – already in earlier sections – would help to better understand the strengths and weaknesses of the approach chosen.
- AC1: 'Reply on RC2', Anna Braun, 22 Dec 2023
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Anna Braun
Kévin Fourteau
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
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