the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 25 Apr 2025
On unifying carbonate rheology
Abstract. We review the results from twenty three experimental works conducted on the rheology of carbonates from the last fifty years to revisit the long-noted discordance in the experimental results from a range of limestones and marbles. Such an exercise is needed to bring together the various datasets generated in the twenty three years since the last major review, as many of them observe relationships that run contrary to existing rheological models. By revisiting the large data set, we find that most low and high stress experimental measurements can be explained by the combined effect of grain size and the molar fraction of magnesium carbonate (XMgCO3). Our results highlight that much of the calcite-dolomite series exists in a continuum of strength that changes with XMgCO3. In contrast to previous findings, we establish that diffusion creep in calcite is sensitive to both grain size and magnesium content, showing that an increase in XMgCO3 acts to weaken a rock. While in dislocation creep we confirm the observation that XMgCO3 has a strengthening effect but extend it beyond synthetic Mg-calcite samples to natural starting materials . Most notably our results suggest that when the composition of a carbonate is factored in then grain size can be shown to have a weakening effect in dislocation creep for fine grained rocks. This is the opposite finding to the currently accepted flow law for calcite rocks in the dislocation creep regime where a decrease in grain size strengths a rock. We contextualise these new results by combining them with data from natural shear zones to show that carbonates are much weaker than would be expected from previous flow laws in a crustal section. Ultimately our review provides new pragmatic flow laws for carbonates in the calcite-dolomite series for diffusion and dislocation creep.
Competing interests: The contact author has declared that neither of the authors has any competing interests.
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RC1: 'Comment on egusphere-2025-1718', Anonymous Referee #1, 27 May 2025
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AC1: 'Reply to RC1,2 and 3', James Gilgannon, 28 Jul 2025
Dear Editor,
We kindly thank the three reviewers for their comments on our manuscript.
One common thread in their different discussions of the text is that we were a bit too quick in the step regarding our method and the description of figures. The primary changes in the manuscript focused on addressing these aspects and the issues with clarity that they brought.
Beyond this, Referee 2 noted some additional limitations to the interpretations we make and we now include those in the discussion. Additionally, Referee 3 drew our attention to a comparison to magnesite (the Mg end member of the carbonate system we investigate) and we have now also incorporated data from relevant experiments and discuss them. In both case, we thank the reviewers for drawing attention to literature we missed.
Lastly, during the review of the initial submission, Mario Ebel noted that figure 11, now figure 12, was problematic for readers with colour vision deficiencies. We have changed this figure to now label the lines with the model grain sizes. This should now cover all of the colour vision deficiencies, including Monochromacy/Achromatopsia.
We hope that the changes to the manuscript increase its clarity and scope.
Best,
James Gilgannon
In the follow rebuttal we have addressed each reviewer in turn. First we address the general points they raised and then the specific comments below that. We only included the relevant sections of the referee’s comments and have coloured them in a light grey to make it easier to read our replies.
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AC1: 'Reply to RC1,2 and 3', James Gilgannon, 28 Jul 2025
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RC2: 'Comment on egusphere-2025-1718', Brian Evans, 26 Jun 2025
Review of the paper: “On unifying carbonate rheology” by Gilganon and Herwegh
Brief comments from Brian EvansScientific significance: Excellent
Scientific quality: Good – (The analysis is useful and creative.)
Presentation quality: Good – (The explanation of the analytical technique could be
improved.)This is a creative, thoughtful, and, in some ways, courageous reanalysis of existing data. The authors point out the inter-relationships of composition and grain size that affect the mechanical behavior of calcite rocks. Based on interpretation of mechanical behavior of metals, it seems that the effect of grain size on creep strength, for fixed chemical composition, probably depends on the conditions, including T and strain rate, [Figueiredo et al., 2023]. Most of the analyses given by Gilganon and Herwegh were done on experiments performed in the temperature range 900-1100K. In that region, they separate the behavior into high temperature-low stress and high stress-low temperature regions. For the higher temperatures and lower strain rates, the strain rate sensitivity to stress (n) is about 1, suggesting that diffusion and grain boundary sliding are more important. It is completely consistent with the understanding of grain boundary sliding mechanics that creep strength will be directly proportional to grain size [Langdon, 1994]. G&H suggest that, in this region, increased Mg content will weaken the rock, an important conclusion.
For the experiments at lower T and high rates, G&H’s analyses suggest that n increases to 8, indicating a transition to deformation dominated by dislocation creep. At yet lower temperatures and higher strain rates in metals, strength is indirectly proportional to grain size [Figueiredo et al., 2023; Zhu and Langdon, 2005]. Often this relation is given as , i.e., the Hall-Petch relation, but deviations from this relation are sometimes noted [Li et al., 2016]. There is evidence that the grain size also ihas this effect on calcite rocks at low temperatures, e.g., [Harbord et al., 2023]. Of course the effect of Mg content was not considered in that study. In metals, at low T, grain size can be altered while chemistry remains fixed. Hall-Petch type behavior is often recorded. In calcite rocks (and metals), twinning causes a similar effect by dividing the grains into separate slip domains. This inverse relation between twinning dimension and strength (the TWIP effect) is well-established (although the details are still a topic of discussion) [Rowe and Rutter, 1990; Rutter et al., 2022; Rybacki et al., 2021]. In the case of twinning, there is no change in chemistry during the deformation, and so, the strength effects are due solely to structural changes in the grains. Thus, I would be reluctant to say that reduction in grain (twin) size has a weakening effect on calcite rocks under all conditions.
Similarly, in the discussion, you have extended the findings to conditions of T and rate in the Earth that are quite different from the experimental conditions (line 385 and following). This seems risky, particularly given that at least some of the data are derived from the transition between grain boundary sliding/ diffusion creep and dislocation creep. If the dependence on grain size changes with changes in T and rate as it does in metals, then it is more likely that there are two regions with different behavior, much the same as is proposed for deformation maps.
Specific comments:
- In abstract: Sentence beginning with “Most notably our results…can be shown to have a weakening effect in dislocation creep…” change to “in high temperature creep where diffusion is important…” Change sentence with “This is the opposite finding to the currently accepted flow law” to acknowledge the fact that we might expect there to be two regimes with different dependences on grain size.
- Figure 1 and Table 2: The data sets are identified by codes (H03) and so on. But the codes are hidden in table 2; make a new column or include them in the first column so that they are more prominent. In the figure captions (Particularly figure 1) provide a reference to table 2 so that the reader will know what the codes mean.
- The description of the normalization techniques wasn’t at all clear. Was the success in normalization judged solely by observing a decreased dispersion of the data points? Did you quantify the normalization by some statistical measure? I am assume that the normalization technique robustly eliminates any cross-correlation between the grain size and the magnesium content.
- Is it correct that there are 6 components (Afg, nfg, Qfg, Acg, ncg, Qcg)) to be determined for fine-grained samples, and three more to be determined for the coarse-grained samples? Given this number of variables can you be assured that you will achieve accuracy? I apologize if this question is ill-informed, but perhaps a comment in the text is in order.
References:
Figueiredo, R. B., M. Kawasaki, and T. G. Langdon (2023), Seventy years of Hall-Petch, ninety years of superplasticity and a generalized approach to the effect of grain size on flow stress, Progress in Materials Science, 137, 101131, doi:https://doi.org/10.1016/j.pmatsci.2023.101131.
Harbord, C., N. Brantut, and D. Wallis (2023), Grain-size effects during semi-brittle flow of calcite rocks, Journal of Geophysical Research - Solid Earth, in press.
Langdon, T. G. (1994), A unified approach to grain boundary sliding in creep and superplasticity, Acta Metallurgica et Materialia, 42(7), 2437-2443, doi:https://doi.org/10.1016/0956-7151(94)90322-0.
Li, Y., A. J. Bushby, and D. J. Dunstan (2016), The Hall-Petch effect as a manifestation of the general size effect, Proceedings of the Royal Society, A: Math. Phys. Eng. Sci., 472(2190), 20150890, doi:10.1098/rspa.2015.0890.
Rowe, K. J., and E. Rutter (1990), Palaeostress estimation using calcite twinning: Experimental calibration and application to nature Journal of Structural Geology, 12, 1-17.
Rutter, E., D. Wallis, and K. Kosiorek (2022), Application of electron backscatter diffraction to calcite-twinning paleopiezometry, Geosciences, 12, 222 doi:10.3390/geosciences12060222.
Rybacki, E., L. Niu, and B. Evans (2021), Semi‐Brittle Deformation of Carrara Marble: Hardening and Twinning Induced Plasticity, Journal of Geophysical Research: Solid Earth, 126(12), doi:10.1029/2021jb022573.
Zhu, Y. T., and T. G. Langdon (2005), Influence of grain size on deformation mechanisms: An extension to nanocrystalline materials, Materials Science and Engineering a-Structural Materials Properties Microstructure and Processing, 409(1-2), 234-242.
Citation: https://doi.org/10.5194/egusphere-2025-1718-RC2 -
AC1: 'Reply to RC1,2 and 3', James Gilgannon, 28 Jul 2025
Dear Editor,
We kindly thank the three reviewers for their comments on our manuscript.
One common thread in their different discussions of the text is that we were a bit too quick in the step regarding our method and the description of figures. The primary changes in the manuscript focused on addressing these aspects and the issues with clarity that they brought.
Beyond this, Referee 2 noted some additional limitations to the interpretations we make and we now include those in the discussion. Additionally, Referee 3 drew our attention to a comparison to magnesite (the Mg end member of the carbonate system we investigate) and we have now also incorporated data from relevant experiments and discuss them. In both case, we thank the reviewers for drawing attention to literature we missed.
Lastly, during the review of the initial submission, Mario Ebel noted that figure 11, now figure 12, was problematic for readers with colour vision deficiencies. We have changed this figure to now label the lines with the model grain sizes. This should now cover all of the colour vision deficiencies, including Monochromacy/Achromatopsia.
We hope that the changes to the manuscript increase its clarity and scope.
Best,
James Gilgannon
In the follow rebuttal we have addressed each reviewer in turn. First we address the general points they raised and then the specific comments below that. We only included the relevant sections of the referee’s comments and have coloured them in a light grey to make it easier to read our replies.
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RC3: 'Comment on egusphere-2025-1718', Andreas Kronenberg, 02 Jul 2025
This manuscript presents an overview of the many papers published on high temperature flow laws for carbonates, to a great extent made up of natural and synthetic polycrystalline calcite, and the compiled results are used to evaluate a global flow law that is consistent with all of the experimental data. While earlier studies attributed changes in rheology to grain size or to composition (primarily XMgCO3) alone, this re- evaluation of the entire dataset shows that polycrystalline calcite depends on both parameters, including deformation dominated by dislocation creep and diffusion creep. That is, deformation due to intracrystalline and inter-crystalline mechanisms depend on these parameters. This result is new and significant, particularly as increasing numbers of grain-size sensitive flow laws for geologically important lithologies (carbonates and silicates) are published and assumed to be due to grain boundary diffusion and sliding, and dislocation processes are typically thought to be independent of grain size (when internal climb and recover preclude increases in defect density) or depend on grain size when glide (Peierls law deformation) occurs with limited diffusion (and climb), giving rise to Hall-Petch strengthening by grain boundaries, not the weakening determined by this analysis.
One detail that might help the reader follow the development here is to state which carbonate minerals are included in the analysis. I find the inclusion of dolomite in this paper interesting, as some would not expect a single flow law to apply to both calcite aggregates/rocks (of varying but relatively low solid solution XMgCO3 values) and dolomite (at large XMgCO3 = 0.5). At the same time, I think this is an interesting perspective brought out by including both carbonate phases, calcite and dolomite, and attempting a fit to the entire dataset. At the outset, the different slip systems of these two carbonates would be expected to lead to different dislocation creep laws. However, without further information about the grain boundary structures and properties, I see the authors’ point of including both fine-grained calcite and dolomite data that are dominated by grain-size- sensitive processes, including grain boundary diffusion and sliding. Why not compare these results normalizing by parameters of grain size and XMgCO3? It is worth stating the beginning premise(s) of this analysis (and why you include dolomite data). Taking this further, it might be interesting to include magnesite deformation data (with XMgCO3 nearly equal to 1.0; Holyoke et al., 2014). Indeed, the mineral symmetry and structure of magnesite is essentially the same as calcite, while dolomite differs due to its ordered cation structure. Thus, any trends that go beyond creep of an individual carbonate phase might be more likely to hold for calcite and magnesite than for calcite and dolomite.
Of course, when applications are made to crustal rocks, the deformation behavior of calcite (of varying Mg content) and dolomite will be most useful. Thus, the ability to formulate a flow law that fits such a large dataset for crustal carbonates at varied conditions, grain sizes, and compositions is impressive and will serve as a reference in applications to natural deformations of carbonates. This contribution should therefore be of great interest to readers of EGUsphere, including geologists studying deformation in the laboratory, natural shear zones, and theoreticians evaluating physical processes of deformation.
This manuscript is well written and shouldn’t take much revision for final publication. In the following, I offer some minor points that may improve the manuscript. However, I leave much of this to the authors’ discretion.
Early in Section 3.4 (Normalization of data), it would be worth stating more explicitly that all mechanical data sets where calcite compositions have been reported and dolomite have been used in the normalization. Or I may have misunderstood how this was done. The dolomite data may throw off a best-fit to calcite aggregate data. In any case, please state exactly how this was done.
In Figure 11, it would be worth stating what the assumed tectonic setting is. These are classic lithosphere-dimensioned yield envelopes for carbonate deformation including frictional and plastic flow laws for an assumed, fixed strain rate, geothermal gradient and tectonic stress state (compressional, extensional, strike-slip/transform boundaries). Please include the assumed tectonic setting in the caption. Similarly, please include the assumed strain rate in figures 12b and 12c.
I find the comparisons of flow laws for non-zero-XMgCO3 calcite aggregates and dolomite very interesting, and I wonder if the authors might add some possible explanations of the differences in the discussion of this manuscript. At the outset, differences in crystal symmetry and structure of calcite and dolomite, and the different slip systems of these two minerals offer obvious reasons that their flow laws for coarse-grained aggregates associated with intracrystalline deformation cannot readily be combined into one simple relation. However, it is fair to wonder why this doesn’t work well for grain-size-sensitive deformation and processes at grain boundaries (grain boundary diffusion and sliding). We know little about the atomic structures of grain boundaries, so a relationship that includes grain size and XMgCO3 might be expected to work for inter-crystalline creep mechanisms in relatively “unstructured” carbonate grain boundaries. It could be that mean jump distances or frequencies might differ for dolomite grain boundaries (where dolomite grains are ordered in Ca and Mg) from those at calcite grain boundaries. This might mean that grain boundary diffusion depends more strongly on XMgCO3 than captured by grain-size-sensitive calcite deformation results. Alternatively, nucleation (or reaction to add to one of the dolomite grains adjacent to a grain boundary) during diffusion creep and grain boundary sliding may differ at XMgCO3 values near 0.5. I don’t think these potential differences can be evaluated without further data or information, but they are interesting possibilities. I wonder if grain-size-sensitive deformation of magnesite (at
XMgCO3 =1) with the same mineral structure as calcite might fit the universal carbonate deformation law presented here better than flow laws reported for dolomite.
Throughout, this manuscript is well written and I have only very minor editorial suggestions for rewording, which I will list below by line number:
Line 2 (abstract) – “... limestones and marbles. Such an”
Line 100 – Was this normalization done just for calcite samples? Or both calcite and dolomite samples? Please specify explicitly (“... of natural calcite samples... “ or “... of natural calcite and synthetic (?) dolomite samples ...”)
Line 108 – again please specify if just calcite samples or both calcite and dolomite. The max XMgCO3 value of 0.17 certainly suggests that dolomite data were not included in the global fit.
Line 115 – “... sample data are transformed to:”
Line 117 – “and results for coarse grained ...”
Line 120 – “... exponents to collapse data to a one-to-one curve ...”
Line 128 – “... regimes that we think contribute to deformation and can be used in investigation and normalization. In our data there are two”
Line 129 – “... different deformation mechanisms are apparent: a low stress domain in which data have a slope of 1 in”
Line 130 – “... a high stress domain characterized by a slope of 7 ...”
Line 147 – delete “then”
Line 155 – suggest deleting “that”
Line 156 – “... mechanisms; (2) XMgCO3 has two opposing effects ...”
Line 157 – “... coarse grained natural samples is only half that of the compositional sensitivity of fine grained”
Line 160 – suggest replacing “visualized” by “illustrated” Line 171 – “... and coarse grained samples in sections 5.2 and 5.3.”
Line 174 – “The effect of magnesium on diffusion creep was first noted in the original work ...”
Line 177 – “... magnesium can be separated empirically from grain size. Therefore,”
Lines 184-185 – “the observation of the same or broadly similar activation energies across Mg-carbonates (Herwegh et al., 2003). This observation suggests that mechanisms activated are thermodynamically equivalent, possibly the exact same mechanism for all chemistries.”
Line 189 – “2008)) as reported for cation species self diffusion or diffusion creep of calcite (Herwegh ...”
Line 190 – “... one should expect that local spatial gradient are reduced as XMgCO3 increases, and magnesium”
Line 191 – “... This is what we observe. ...” Line 192 – delete “which are derived from linearly spaced XMgCO3 values”
Lines 196-197 – “... (fig. 4a and c). Indeed, our results suggest that this compositional effect on flow law extended beyond synthetic Mg-carbonates ...”
Line 198 – “strengthening by XMgCO3 may be due to solute-drag ...”
Line 199 – “to the progress of linear defects. This second possibility is not expected if chemical diffusion is substantial as discussed earlier”
Line 235 – delete “the framing of” Line 238 – “0.3). Notably the newly defined XMgCO3-sensitive diffusion creep law ...”
Lines 245-246 – “... Lochseiten mylonite data. It is important to ask how well the equations fit carbonates ...”
Line 248 – “Equations 5-6 were compared ...”
Lines 251-253 – “... data appear to be fit well. In the case of diffusion creep of the Lochseiten mylonite, the data fit equation 5 at higher temperatures but not at lower temperatures (fig. 8c). Figure 8d shows that, as the second phase content rises to 10% and above, the observed rheologies depart from our model.”
Line 255 – “The normalization of experimental results of coarse-grained natural sample results was limited to ...”
Line 258 – “When compared to other coarse grained calcite experiments, the data fit the model relatively well (9a-c). However, they do”
Line 261 – “the spread of data points. ....”
Line 262 – delete “will” Line 263 – “there is a marked increase in strain. ‘the fit clearly overestimates ...” Line 264 – “... with in the range of Carrara marble experimental results. Regardless” Line 269 – “... result of differences in sample ...” Line 271 – “... synthetic and natural for XMgCO3 values that” Line 272 – “... Data plotted for SMgCO3 = 0.07 come from a fine grained ...” Line 275 – “While data plotted for XMgCO3 = 0.03 comes from a fine grained ... Line 277 – “... is not restricted to synthetic samples alone – it is not an artifact”
Lines 278-280 – “... there is a transition to a grain size sensitive dislocation creep field. We conclude from this that grain size sensitive and grain size insensitive dislocation creep fields are real.”
Line 292 – “a modified Hall-Petch relation has been proposed; the smaller the grain size, the stronger ...”
Lines 294-296 – “in the dislocation field appears to be similar to the d-3 dependence of diffusion creep. Moreover, figure 10 implies that there is a transition as grain size increases to a grain size insensitive rheology.”
Line 298 – “To explain the ever diminishing ...” Line 300 – “... could transition from grain size strengthening to weakening and ...”
Lines 301-302 – “deformation as grain size increased. In this model the grain size sensitivity of dislocation accommodated deformation changes according to the storage and recovery of dislocations, and the dependence of these processes on grain size. Breithaupt et al. (2023)”
Lines 305-309 – “... Breithaupt, 2022). Thus, the apparent results for diffusion creep and grain size sensitive deformation of polycrystalline calcite might be linked to the storage and recovery in grains, and how these change with size of grains. The model of Breithaupt et al. (2023) also explains the variation (hardening and weakening) in the effect of grain size for different calcite experiments. ...”
Lines 312- 313 – “et al., 2002). This is most likely for lower temperature experiments of Sly et al. (2019). While the grain sizes of Renner et al. (2002) and Herwegh et al. (2003) seem to cover a similar range ...”
Lines 315-316 – “... vs area-weighted); grain size values of Renner et al. (2002) might be half those measured by Herwegh et al. (2003; cf. table 5 in Berhger et al., 2011).”
Line 319 – “... sensitivity is needed. Short of adapting”
Lines 320-321 – “... the effect of XMgCO3, a pragmatic guide to modeling different rheologies based on grain size domains (Fig. 10).”
Line 323 – replace “if” by “for” Line 325 – replace “if” by “for” Line 326 – replace “if” by “for” Line 328 – “... 1987), figure 11 illustrates this approach ...”
Lines 329-331 – “it to a case for fine grained dislocation creep rheology that shows no grain size effect (dashed lines in fig. 11b), and the published flow laws for calcite (Schmid et al. 1980; Herwegh et al., 2003) fig. 11c). Regardless of which composite ...”
Lines 333-336 – “when modelling crustal strength. Figure 11 also shows that the fine grained dislocation creep rheology predicts low strengths in the crust with depth.”
Line 338 – “Using temperature and grain size measurements for mylonites ...” Line 339 – “... Figure 12 shows data from several thrusts of the Helvetic”
Lines 340-341 – “... Naxos (Herwegh et al., 2011) deformed at greenschist to amphibolite facies conditions. As before, the calcite-dolomite solvus ...”
Line 353 – “... Using the same data as previously, figure ...” Line 355 – “... shear zones deform by a grain size sensitive ...” Line 359 – “... a shear zone would show more strength evolution ...”
Line 367 – I’m a bit confused by the use of “grain size has a weakening effect”. This would seem to state that larger grain sizes weaken the flow law, but I think you mean to state that fine-grain sizes (or larger grain boundary densities) weaken the flow law. Please revise. How about “grain boundaries have a weakening effect”?
Line 382 – “An. Important conclusion of this work is that ...”
Line 385 – “If true, this calls into question the predictions of carbonate paleopiezometry for crustal strength. More importantly, these”
Lines 386-387 – “... also been shown that flow strength depends on iron concentration (Zhao et al., 2009-2018). ...”
I look forward to seeing this paper in print, Andreas Kronenberg
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AC1: 'Reply to RC1,2 and 3', James Gilgannon, 28 Jul 2025
Dear Editor,
We kindly thank the three reviewers for their comments on our manuscript.
One common thread in their different discussions of the text is that we were a bit too quick in the step regarding our method and the description of figures. The primary changes in the manuscript focused on addressing these aspects and the issues with clarity that they brought.
Beyond this, Referee 2 noted some additional limitations to the interpretations we make and we now include those in the discussion. Additionally, Referee 3 drew our attention to a comparison to magnesite (the Mg end member of the carbonate system we investigate) and we have now also incorporated data from relevant experiments and discuss them. In both case, we thank the reviewers for drawing attention to literature we missed.
Lastly, during the review of the initial submission, Mario Ebel noted that figure 11, now figure 12, was problematic for readers with colour vision deficiencies. We have changed this figure to now label the lines with the model grain sizes. This should now cover all of the colour vision deficiencies, including Monochromacy/Achromatopsia.
We hope that the changes to the manuscript increase its clarity and scope.
Best,
James Gilgannon
In the follow rebuttal we have addressed each reviewer in turn. First we address the general points they raised and then the specific comments below that. We only included the relevant sections of the referee’s comments and have coloured them in a light grey to make it easier to read our replies.
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AC1: 'Reply to RC1,2 and 3', James Gilgannon, 28 Jul 2025
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2025-1718', Anonymous Referee #1, 27 May 2025
The manuscript by Gilgannon and Herwegh presents a compilation of experimental data on carbonate rheology and their unified interpretation. To fit the compiled data, they suggest new flow laws that account for the effect of magnesium on the rheology. In their study, they extensively demonstrate the applicability of the new flow laws and discuss their potential limitations.
The proposed flow laws may be widely used in future research, which definitely makes them worth publication. However, I have concerns about the quality of presentation of the method and results. While I generally appreciate concise papers, I think that in this case, the brevity of the text sometimes makes it difficult to understand. I suggest careful rereading and rewriting of the text with an emphasis on its clarity and completeness.
Specifically, I would like to learn more about the central methodological part of the manuscript - the normalization of the data (Sections 3.4 and 4). The procedure of normalization is the key part of the study, but its description is very short. Where do the values of exponents used in the normalization come from? Did you test different values? How did you choose these particular values? What is the estimated error range?
I also often miss information in figure captions. Namely:
Fig. 1 - what is the meaning of different colors? Is the same color coding also used in other figures in the manuscript? Abbreviations (H11, Eb08…) are not explained in the caption - a reference to Tables would help.
Fig. 2c - description of this panel is not clear. What is in the inset panel? How is it related to the panel c)?
Fig. 3! - The figure is busy and important at the same time, but the caption is practically lacking.
Fig. 4 - What are the individual lines/curves? What is their spacing?
Fig. 10 - What is the relationship between the inset panel and the main panel? The caption is, again, too brief.
Fig. 11a - Can you plot the depth-XMgCO3(T) relationship directly in the main panel (having a secondary x-axis)?
Fig. 11b,c - References to the flow laws are missing in the caption.
Fig. 12 - Which existing rheologies? Line 342 can be moved into the caption.
Minor comments:
Within captions, panels can be referred to as a) or b) and not Figure 1a or Figure 1b.
Some grey lines in the figures are barely visible.
Equations in sections 3.2, 3.3 - colons before the equations are missing?
line 217 - Why do you write pure “Cc” here? Is it explained anywhere?
line 228 - This heading is not clear to me
Lines 345-355 are difficult to follow
There are a lot of typos and small grammatical mistakes - a more careful reading and writing is needed.
Citation: https://doi.org/10.5194/egusphere-2025-1718-RC1 -
AC1: 'Reply to RC1,2 and 3', James Gilgannon, 28 Jul 2025
Dear Editor,
We kindly thank the three reviewers for their comments on our manuscript.
One common thread in their different discussions of the text is that we were a bit too quick in the step regarding our method and the description of figures. The primary changes in the manuscript focused on addressing these aspects and the issues with clarity that they brought.
Beyond this, Referee 2 noted some additional limitations to the interpretations we make and we now include those in the discussion. Additionally, Referee 3 drew our attention to a comparison to magnesite (the Mg end member of the carbonate system we investigate) and we have now also incorporated data from relevant experiments and discuss them. In both case, we thank the reviewers for drawing attention to literature we missed.
Lastly, during the review of the initial submission, Mario Ebel noted that figure 11, now figure 12, was problematic for readers with colour vision deficiencies. We have changed this figure to now label the lines with the model grain sizes. This should now cover all of the colour vision deficiencies, including Monochromacy/Achromatopsia.
We hope that the changes to the manuscript increase its clarity and scope.
Best,
James Gilgannon
In the follow rebuttal we have addressed each reviewer in turn. First we address the general points they raised and then the specific comments below that. We only included the relevant sections of the referee’s comments and have coloured them in a light grey to make it easier to read our replies.
-
AC1: 'Reply to RC1,2 and 3', James Gilgannon, 28 Jul 2025
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RC2: 'Comment on egusphere-2025-1718', Brian Evans, 26 Jun 2025
Review of the paper: “On unifying carbonate rheology” by Gilganon and Herwegh
Brief comments from Brian EvansScientific significance: Excellent
Scientific quality: Good – (The analysis is useful and creative.)
Presentation quality: Good – (The explanation of the analytical technique could be
improved.)This is a creative, thoughtful, and, in some ways, courageous reanalysis of existing data. The authors point out the inter-relationships of composition and grain size that affect the mechanical behavior of calcite rocks. Based on interpretation of mechanical behavior of metals, it seems that the effect of grain size on creep strength, for fixed chemical composition, probably depends on the conditions, including T and strain rate, [Figueiredo et al., 2023]. Most of the analyses given by Gilganon and Herwegh were done on experiments performed in the temperature range 900-1100K. In that region, they separate the behavior into high temperature-low stress and high stress-low temperature regions. For the higher temperatures and lower strain rates, the strain rate sensitivity to stress (n) is about 1, suggesting that diffusion and grain boundary sliding are more important. It is completely consistent with the understanding of grain boundary sliding mechanics that creep strength will be directly proportional to grain size [Langdon, 1994]. G&H suggest that, in this region, increased Mg content will weaken the rock, an important conclusion.
For the experiments at lower T and high rates, G&H’s analyses suggest that n increases to 8, indicating a transition to deformation dominated by dislocation creep. At yet lower temperatures and higher strain rates in metals, strength is indirectly proportional to grain size [Figueiredo et al., 2023; Zhu and Langdon, 2005]. Often this relation is given as , i.e., the Hall-Petch relation, but deviations from this relation are sometimes noted [Li et al., 2016]. There is evidence that the grain size also ihas this effect on calcite rocks at low temperatures, e.g., [Harbord et al., 2023]. Of course the effect of Mg content was not considered in that study. In metals, at low T, grain size can be altered while chemistry remains fixed. Hall-Petch type behavior is often recorded. In calcite rocks (and metals), twinning causes a similar effect by dividing the grains into separate slip domains. This inverse relation between twinning dimension and strength (the TWIP effect) is well-established (although the details are still a topic of discussion) [Rowe and Rutter, 1990; Rutter et al., 2022; Rybacki et al., 2021]. In the case of twinning, there is no change in chemistry during the deformation, and so, the strength effects are due solely to structural changes in the grains. Thus, I would be reluctant to say that reduction in grain (twin) size has a weakening effect on calcite rocks under all conditions.
Similarly, in the discussion, you have extended the findings to conditions of T and rate in the Earth that are quite different from the experimental conditions (line 385 and following). This seems risky, particularly given that at least some of the data are derived from the transition between grain boundary sliding/ diffusion creep and dislocation creep. If the dependence on grain size changes with changes in T and rate as it does in metals, then it is more likely that there are two regions with different behavior, much the same as is proposed for deformation maps.
Specific comments:
- In abstract: Sentence beginning with “Most notably our results…can be shown to have a weakening effect in dislocation creep…” change to “in high temperature creep where diffusion is important…” Change sentence with “This is the opposite finding to the currently accepted flow law” to acknowledge the fact that we might expect there to be two regimes with different dependences on grain size.
- Figure 1 and Table 2: The data sets are identified by codes (H03) and so on. But the codes are hidden in table 2; make a new column or include them in the first column so that they are more prominent. In the figure captions (Particularly figure 1) provide a reference to table 2 so that the reader will know what the codes mean.
- The description of the normalization techniques wasn’t at all clear. Was the success in normalization judged solely by observing a decreased dispersion of the data points? Did you quantify the normalization by some statistical measure? I am assume that the normalization technique robustly eliminates any cross-correlation between the grain size and the magnesium content.
- Is it correct that there are 6 components (Afg, nfg, Qfg, Acg, ncg, Qcg)) to be determined for fine-grained samples, and three more to be determined for the coarse-grained samples? Given this number of variables can you be assured that you will achieve accuracy? I apologize if this question is ill-informed, but perhaps a comment in the text is in order.
References:
Figueiredo, R. B., M. Kawasaki, and T. G. Langdon (2023), Seventy years of Hall-Petch, ninety years of superplasticity and a generalized approach to the effect of grain size on flow stress, Progress in Materials Science, 137, 101131, doi:https://doi.org/10.1016/j.pmatsci.2023.101131.
Harbord, C., N. Brantut, and D. Wallis (2023), Grain-size effects during semi-brittle flow of calcite rocks, Journal of Geophysical Research - Solid Earth, in press.
Langdon, T. G. (1994), A unified approach to grain boundary sliding in creep and superplasticity, Acta Metallurgica et Materialia, 42(7), 2437-2443, doi:https://doi.org/10.1016/0956-7151(94)90322-0.
Li, Y., A. J. Bushby, and D. J. Dunstan (2016), The Hall-Petch effect as a manifestation of the general size effect, Proceedings of the Royal Society, A: Math. Phys. Eng. Sci., 472(2190), 20150890, doi:10.1098/rspa.2015.0890.
Rowe, K. J., and E. Rutter (1990), Palaeostress estimation using calcite twinning: Experimental calibration and application to nature Journal of Structural Geology, 12, 1-17.
Rutter, E., D. Wallis, and K. Kosiorek (2022), Application of electron backscatter diffraction to calcite-twinning paleopiezometry, Geosciences, 12, 222 doi:10.3390/geosciences12060222.
Rybacki, E., L. Niu, and B. Evans (2021), Semi‐Brittle Deformation of Carrara Marble: Hardening and Twinning Induced Plasticity, Journal of Geophysical Research: Solid Earth, 126(12), doi:10.1029/2021jb022573.
Zhu, Y. T., and T. G. Langdon (2005), Influence of grain size on deformation mechanisms: An extension to nanocrystalline materials, Materials Science and Engineering a-Structural Materials Properties Microstructure and Processing, 409(1-2), 234-242.
Citation: https://doi.org/10.5194/egusphere-2025-1718-RC2 -
AC1: 'Reply to RC1,2 and 3', James Gilgannon, 28 Jul 2025
Dear Editor,
We kindly thank the three reviewers for their comments on our manuscript.
One common thread in their different discussions of the text is that we were a bit too quick in the step regarding our method and the description of figures. The primary changes in the manuscript focused on addressing these aspects and the issues with clarity that they brought.
Beyond this, Referee 2 noted some additional limitations to the interpretations we make and we now include those in the discussion. Additionally, Referee 3 drew our attention to a comparison to magnesite (the Mg end member of the carbonate system we investigate) and we have now also incorporated data from relevant experiments and discuss them. In both case, we thank the reviewers for drawing attention to literature we missed.
Lastly, during the review of the initial submission, Mario Ebel noted that figure 11, now figure 12, was problematic for readers with colour vision deficiencies. We have changed this figure to now label the lines with the model grain sizes. This should now cover all of the colour vision deficiencies, including Monochromacy/Achromatopsia.
We hope that the changes to the manuscript increase its clarity and scope.
Best,
James Gilgannon
In the follow rebuttal we have addressed each reviewer in turn. First we address the general points they raised and then the specific comments below that. We only included the relevant sections of the referee’s comments and have coloured them in a light grey to make it easier to read our replies.
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RC3: 'Comment on egusphere-2025-1718', Andreas Kronenberg, 02 Jul 2025
This manuscript presents an overview of the many papers published on high temperature flow laws for carbonates, to a great extent made up of natural and synthetic polycrystalline calcite, and the compiled results are used to evaluate a global flow law that is consistent with all of the experimental data. While earlier studies attributed changes in rheology to grain size or to composition (primarily XMgCO3) alone, this re- evaluation of the entire dataset shows that polycrystalline calcite depends on both parameters, including deformation dominated by dislocation creep and diffusion creep. That is, deformation due to intracrystalline and inter-crystalline mechanisms depend on these parameters. This result is new and significant, particularly as increasing numbers of grain-size sensitive flow laws for geologically important lithologies (carbonates and silicates) are published and assumed to be due to grain boundary diffusion and sliding, and dislocation processes are typically thought to be independent of grain size (when internal climb and recover preclude increases in defect density) or depend on grain size when glide (Peierls law deformation) occurs with limited diffusion (and climb), giving rise to Hall-Petch strengthening by grain boundaries, not the weakening determined by this analysis.
One detail that might help the reader follow the development here is to state which carbonate minerals are included in the analysis. I find the inclusion of dolomite in this paper interesting, as some would not expect a single flow law to apply to both calcite aggregates/rocks (of varying but relatively low solid solution XMgCO3 values) and dolomite (at large XMgCO3 = 0.5). At the same time, I think this is an interesting perspective brought out by including both carbonate phases, calcite and dolomite, and attempting a fit to the entire dataset. At the outset, the different slip systems of these two carbonates would be expected to lead to different dislocation creep laws. However, without further information about the grain boundary structures and properties, I see the authors’ point of including both fine-grained calcite and dolomite data that are dominated by grain-size- sensitive processes, including grain boundary diffusion and sliding. Why not compare these results normalizing by parameters of grain size and XMgCO3? It is worth stating the beginning premise(s) of this analysis (and why you include dolomite data). Taking this further, it might be interesting to include magnesite deformation data (with XMgCO3 nearly equal to 1.0; Holyoke et al., 2014). Indeed, the mineral symmetry and structure of magnesite is essentially the same as calcite, while dolomite differs due to its ordered cation structure. Thus, any trends that go beyond creep of an individual carbonate phase might be more likely to hold for calcite and magnesite than for calcite and dolomite.
Of course, when applications are made to crustal rocks, the deformation behavior of calcite (of varying Mg content) and dolomite will be most useful. Thus, the ability to formulate a flow law that fits such a large dataset for crustal carbonates at varied conditions, grain sizes, and compositions is impressive and will serve as a reference in applications to natural deformations of carbonates. This contribution should therefore be of great interest to readers of EGUsphere, including geologists studying deformation in the laboratory, natural shear zones, and theoreticians evaluating physical processes of deformation.
This manuscript is well written and shouldn’t take much revision for final publication. In the following, I offer some minor points that may improve the manuscript. However, I leave much of this to the authors’ discretion.
Early in Section 3.4 (Normalization of data), it would be worth stating more explicitly that all mechanical data sets where calcite compositions have been reported and dolomite have been used in the normalization. Or I may have misunderstood how this was done. The dolomite data may throw off a best-fit to calcite aggregate data. In any case, please state exactly how this was done.
In Figure 11, it would be worth stating what the assumed tectonic setting is. These are classic lithosphere-dimensioned yield envelopes for carbonate deformation including frictional and plastic flow laws for an assumed, fixed strain rate, geothermal gradient and tectonic stress state (compressional, extensional, strike-slip/transform boundaries). Please include the assumed tectonic setting in the caption. Similarly, please include the assumed strain rate in figures 12b and 12c.
I find the comparisons of flow laws for non-zero-XMgCO3 calcite aggregates and dolomite very interesting, and I wonder if the authors might add some possible explanations of the differences in the discussion of this manuscript. At the outset, differences in crystal symmetry and structure of calcite and dolomite, and the different slip systems of these two minerals offer obvious reasons that their flow laws for coarse-grained aggregates associated with intracrystalline deformation cannot readily be combined into one simple relation. However, it is fair to wonder why this doesn’t work well for grain-size-sensitive deformation and processes at grain boundaries (grain boundary diffusion and sliding). We know little about the atomic structures of grain boundaries, so a relationship that includes grain size and XMgCO3 might be expected to work for inter-crystalline creep mechanisms in relatively “unstructured” carbonate grain boundaries. It could be that mean jump distances or frequencies might differ for dolomite grain boundaries (where dolomite grains are ordered in Ca and Mg) from those at calcite grain boundaries. This might mean that grain boundary diffusion depends more strongly on XMgCO3 than captured by grain-size-sensitive calcite deformation results. Alternatively, nucleation (or reaction to add to one of the dolomite grains adjacent to a grain boundary) during diffusion creep and grain boundary sliding may differ at XMgCO3 values near 0.5. I don’t think these potential differences can be evaluated without further data or information, but they are interesting possibilities. I wonder if grain-size-sensitive deformation of magnesite (at
XMgCO3 =1) with the same mineral structure as calcite might fit the universal carbonate deformation law presented here better than flow laws reported for dolomite.
Throughout, this manuscript is well written and I have only very minor editorial suggestions for rewording, which I will list below by line number:
Line 2 (abstract) – “... limestones and marbles. Such an”
Line 100 – Was this normalization done just for calcite samples? Or both calcite and dolomite samples? Please specify explicitly (“... of natural calcite samples... “ or “... of natural calcite and synthetic (?) dolomite samples ...”)
Line 108 – again please specify if just calcite samples or both calcite and dolomite. The max XMgCO3 value of 0.17 certainly suggests that dolomite data were not included in the global fit.
Line 115 – “... sample data are transformed to:”
Line 117 – “and results for coarse grained ...”
Line 120 – “... exponents to collapse data to a one-to-one curve ...”
Line 128 – “... regimes that we think contribute to deformation and can be used in investigation and normalization. In our data there are two”
Line 129 – “... different deformation mechanisms are apparent: a low stress domain in which data have a slope of 1 in”
Line 130 – “... a high stress domain characterized by a slope of 7 ...”
Line 147 – delete “then”
Line 155 – suggest deleting “that”
Line 156 – “... mechanisms; (2) XMgCO3 has two opposing effects ...”
Line 157 – “... coarse grained natural samples is only half that of the compositional sensitivity of fine grained”
Line 160 – suggest replacing “visualized” by “illustrated” Line 171 – “... and coarse grained samples in sections 5.2 and 5.3.”
Line 174 – “The effect of magnesium on diffusion creep was first noted in the original work ...”
Line 177 – “... magnesium can be separated empirically from grain size. Therefore,”
Lines 184-185 – “the observation of the same or broadly similar activation energies across Mg-carbonates (Herwegh et al., 2003). This observation suggests that mechanisms activated are thermodynamically equivalent, possibly the exact same mechanism for all chemistries.”
Line 189 – “2008)) as reported for cation species self diffusion or diffusion creep of calcite (Herwegh ...”
Line 190 – “... one should expect that local spatial gradient are reduced as XMgCO3 increases, and magnesium”
Line 191 – “... This is what we observe. ...” Line 192 – delete “which are derived from linearly spaced XMgCO3 values”
Lines 196-197 – “... (fig. 4a and c). Indeed, our results suggest that this compositional effect on flow law extended beyond synthetic Mg-carbonates ...”
Line 198 – “strengthening by XMgCO3 may be due to solute-drag ...”
Line 199 – “to the progress of linear defects. This second possibility is not expected if chemical diffusion is substantial as discussed earlier”
Line 235 – delete “the framing of” Line 238 – “0.3). Notably the newly defined XMgCO3-sensitive diffusion creep law ...”
Lines 245-246 – “... Lochseiten mylonite data. It is important to ask how well the equations fit carbonates ...”
Line 248 – “Equations 5-6 were compared ...”
Lines 251-253 – “... data appear to be fit well. In the case of diffusion creep of the Lochseiten mylonite, the data fit equation 5 at higher temperatures but not at lower temperatures (fig. 8c). Figure 8d shows that, as the second phase content rises to 10% and above, the observed rheologies depart from our model.”
Line 255 – “The normalization of experimental results of coarse-grained natural sample results was limited to ...”
Line 258 – “When compared to other coarse grained calcite experiments, the data fit the model relatively well (9a-c). However, they do”
Line 261 – “the spread of data points. ....”
Line 262 – delete “will” Line 263 – “there is a marked increase in strain. ‘the fit clearly overestimates ...” Line 264 – “... with in the range of Carrara marble experimental results. Regardless” Line 269 – “... result of differences in sample ...” Line 271 – “... synthetic and natural for XMgCO3 values that” Line 272 – “... Data plotted for SMgCO3 = 0.07 come from a fine grained ...” Line 275 – “While data plotted for XMgCO3 = 0.03 comes from a fine grained ... Line 277 – “... is not restricted to synthetic samples alone – it is not an artifact”
Lines 278-280 – “... there is a transition to a grain size sensitive dislocation creep field. We conclude from this that grain size sensitive and grain size insensitive dislocation creep fields are real.”
Line 292 – “a modified Hall-Petch relation has been proposed; the smaller the grain size, the stronger ...”
Lines 294-296 – “in the dislocation field appears to be similar to the d-3 dependence of diffusion creep. Moreover, figure 10 implies that there is a transition as grain size increases to a grain size insensitive rheology.”
Line 298 – “To explain the ever diminishing ...” Line 300 – “... could transition from grain size strengthening to weakening and ...”
Lines 301-302 – “deformation as grain size increased. In this model the grain size sensitivity of dislocation accommodated deformation changes according to the storage and recovery of dislocations, and the dependence of these processes on grain size. Breithaupt et al. (2023)”
Lines 305-309 – “... Breithaupt, 2022). Thus, the apparent results for diffusion creep and grain size sensitive deformation of polycrystalline calcite might be linked to the storage and recovery in grains, and how these change with size of grains. The model of Breithaupt et al. (2023) also explains the variation (hardening and weakening) in the effect of grain size for different calcite experiments. ...”
Lines 312- 313 – “et al., 2002). This is most likely for lower temperature experiments of Sly et al. (2019). While the grain sizes of Renner et al. (2002) and Herwegh et al. (2003) seem to cover a similar range ...”
Lines 315-316 – “... vs area-weighted); grain size values of Renner et al. (2002) might be half those measured by Herwegh et al. (2003; cf. table 5 in Berhger et al., 2011).”
Line 319 – “... sensitivity is needed. Short of adapting”
Lines 320-321 – “... the effect of XMgCO3, a pragmatic guide to modeling different rheologies based on grain size domains (Fig. 10).”
Line 323 – replace “if” by “for” Line 325 – replace “if” by “for” Line 326 – replace “if” by “for” Line 328 – “... 1987), figure 11 illustrates this approach ...”
Lines 329-331 – “it to a case for fine grained dislocation creep rheology that shows no grain size effect (dashed lines in fig. 11b), and the published flow laws for calcite (Schmid et al. 1980; Herwegh et al., 2003) fig. 11c). Regardless of which composite ...”
Lines 333-336 – “when modelling crustal strength. Figure 11 also shows that the fine grained dislocation creep rheology predicts low strengths in the crust with depth.”
Line 338 – “Using temperature and grain size measurements for mylonites ...” Line 339 – “... Figure 12 shows data from several thrusts of the Helvetic”
Lines 340-341 – “... Naxos (Herwegh et al., 2011) deformed at greenschist to amphibolite facies conditions. As before, the calcite-dolomite solvus ...”
Line 353 – “... Using the same data as previously, figure ...” Line 355 – “... shear zones deform by a grain size sensitive ...” Line 359 – “... a shear zone would show more strength evolution ...”
Line 367 – I’m a bit confused by the use of “grain size has a weakening effect”. This would seem to state that larger grain sizes weaken the flow law, but I think you mean to state that fine-grain sizes (or larger grain boundary densities) weaken the flow law. Please revise. How about “grain boundaries have a weakening effect”?
Line 382 – “An. Important conclusion of this work is that ...”
Line 385 – “If true, this calls into question the predictions of carbonate paleopiezometry for crustal strength. More importantly, these”
Lines 386-387 – “... also been shown that flow strength depends on iron concentration (Zhao et al., 2009-2018). ...”
I look forward to seeing this paper in print, Andreas Kronenberg
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AC1: 'Reply to RC1,2 and 3', James Gilgannon, 28 Jul 2025
Dear Editor,
We kindly thank the three reviewers for their comments on our manuscript.
One common thread in their different discussions of the text is that we were a bit too quick in the step regarding our method and the description of figures. The primary changes in the manuscript focused on addressing these aspects and the issues with clarity that they brought.
Beyond this, Referee 2 noted some additional limitations to the interpretations we make and we now include those in the discussion. Additionally, Referee 3 drew our attention to a comparison to magnesite (the Mg end member of the carbonate system we investigate) and we have now also incorporated data from relevant experiments and discuss them. In both case, we thank the reviewers for drawing attention to literature we missed.
Lastly, during the review of the initial submission, Mario Ebel noted that figure 11, now figure 12, was problematic for readers with colour vision deficiencies. We have changed this figure to now label the lines with the model grain sizes. This should now cover all of the colour vision deficiencies, including Monochromacy/Achromatopsia.
We hope that the changes to the manuscript increase its clarity and scope.
Best,
James Gilgannon
In the follow rebuttal we have addressed each reviewer in turn. First we address the general points they raised and then the specific comments below that. We only included the relevant sections of the referee’s comments and have coloured them in a light grey to make it easier to read our replies.
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AC1: 'Reply to RC1,2 and 3', James Gilgannon, 28 Jul 2025
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James Gilgannon
Marco Herwegh
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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The manuscript by Gilgannon and Herwegh presents a compilation of experimental data on carbonate rheology and their unified interpretation. To fit the compiled data, they suggest new flow laws that account for the effect of magnesium on the rheology. In their study, they extensively demonstrate the applicability of the new flow laws and discuss their potential limitations.
The proposed flow laws may be widely used in future research, which definitely makes them worth publication. However, I have concerns about the quality of presentation of the method and results. While I generally appreciate concise papers, I think that in this case, the brevity of the text sometimes makes it difficult to understand. I suggest careful rereading and rewriting of the text with an emphasis on its clarity and completeness.
Specifically, I would like to learn more about the central methodological part of the manuscript - the normalization of the data (Sections 3.4 and 4). The procedure of normalization is the key part of the study, but its description is very short. Where do the values of exponents used in the normalization come from? Did you test different values? How did you choose these particular values? What is the estimated error range?
I also often miss information in figure captions. Namely:
Fig. 1 - what is the meaning of different colors? Is the same color coding also used in other figures in the manuscript? Abbreviations (H11, Eb08…) are not explained in the caption - a reference to Tables would help.
Fig. 2c - description of this panel is not clear. What is in the inset panel? How is it related to the panel c)?
Fig. 3! - The figure is busy and important at the same time, but the caption is practically lacking.
Fig. 4 - What are the individual lines/curves? What is their spacing?
Fig. 10 - What is the relationship between the inset panel and the main panel? The caption is, again, too brief.
Fig. 11a - Can you plot the depth-XMgCO3(T) relationship directly in the main panel (having a secondary x-axis)?
Fig. 11b,c - References to the flow laws are missing in the caption.
Fig. 12 - Which existing rheologies? Line 342 can be moved into the caption.
Minor comments:
Within captions, panels can be referred to as a) or b) and not Figure 1a or Figure 1b.
Some grey lines in the figures are barely visible.
Equations in sections 3.2, 3.3 - colons before the equations are missing?
line 217 - Why do you write pure “Cc” here? Is it explained anywhere?
line 228 - This heading is not clear to me
Lines 345-355 are difficult to follow
There are a lot of typos and small grammatical mistakes - a more careful reading and writing is needed.