From Strong Plates to Weak Boundaries: Strain Localization in the Lithospheric Mantle with Low- to High-Temperature Dislocation Creep

Van Broeck, Etienne; Garel, Fanny; Thoraval, Catherine; Arcay, Diane; Davies, D. Rhodri

doi:10.5194/egusphere-2025-5615

Preprints

https://doi.org/10.5194/egusphere-2025-5615

Preprints

26 Nov 2025

| 26 Nov 2025

From Strong Plates to Weak Boundaries: Strain Localization in the Lithospheric Mantle with Low- to High-Temperature Dislocation Creep

Etienne Van Broeck, Fanny Garel, Catherine Thoraval, Diane Arcay, and D. Rhodri Davies

Abstract. Plate-like behavior may be reproduced in numerical experiments of mantle convection if rheology combines a temperature-dependent viscosity capped with a low yield stress. Other processes limiting the mantle ductile strength have been proposed, among which low-temperature plasticity. We propose to investigate how such a rheology can promote deformation localization at the lithospheric scale. We design a 2-D thermo-mechanical model of plate extension to compare the modes of strain localization as a function of the rheological parameterization. Various experimental flow laws for olivine (diffusion creep and dislocation creep at low- and/or high- temperature) are tested, sometimes combined with a yield-stress formulation. We quantify the evolution of the deformation pattern throughout the lithosphere, and define new diagnostics to assess whether the decrease in plate viscosity is due to an increase in strain rate ('mechanical weakening') and/or in temperature ('thermal weakening'). Deformation localization leads to a new extensional plate boundary in two successive stages: (i) narrowing of the deforming zone, (ii) rapid thinning of the highly deformed lithosphere. Here, a rheology combining diffusion creep and yield stress results in a mantle weakening that is either fully mechanical, respectively fully thermal, for temperatures lower, respectively higher, than ~1300 K. Accounting for dislocation creep enables additional feedbacks between mechanical and thermal weakening within the deforming plate, thereby enhancing the efficiency of strain localization. We demonstrate the consistency of using a yield-stress approximation to cap the mantle strength for temperature below ~1000 K. We also show that dislocation creep and yield-stress are not interchangeable for most of the lithospheric mantle (1000–1500 K), and using only the latter may overestimate the duration of plate break-up. Finally, we discuss how our results compare to natural continental rifting cases.

Received: 14 Nov 2025 – Discussion started: 26 Nov 2025

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Journal article(s) based on this preprint

06 Aug 2026

From strong plates to weak boundaries: strain localization in the lithospheric mantle with low- to high-temperature dislocation creep

Etienne Van Broeck, Fanny Garel, Catherine Thoraval, Diane Arcay, and D. Rhodri Davies

Solid Earth, 17, 947–977, https://doi.org/10.5194/se-17-947-2026,https://doi.org/10.5194/se-17-947-2026, 2026

Short summary

Etienne Van Broeck, Fanny Garel, Catherine Thoraval, Diane Arcay, and D. Rhodri Davies

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-5615', Antonio Manjon Cabeza Cordoba, 02 Jan 2026

Dear editor,
this technical paper presents a series of numerical experiments to address the role of rheology on deformation localization during extensional tectonics. While several detailed models of extension and rifting exist, evaluation of localization and adequate rheological mechanism is mostly absent in modern literature. In addition, they add the role of low-temperature plasticity, a continuation of their own work (Gouriet 2019; Garel 2020)
They show that considering multiple rheological mechanisms, in particular non-linear rheologies, is important to reproduce localization processes as we imagine them on Earth. They could specify a bit more on whether this was ever called into question, or what are the consequences of ignoring one rheology or the other, not just for their models, but for the wider community that will be forced to take simplifying assumptions regardless.
The manuscript is quite technical but the methodology is well explained. I have little to say there. Because Solid Earth has a wide focus - inasmuch as the deep Earth can be called a wide field – the article would benefit for a slightly more careful contextualization (e.g. what advances or new evidence motivate this study? How does this work compare with other studies? What are the implications for other geophysicists/geologists? etc). But this is not a criticism of the science of the manuscript itself.
Overall, I have few important comments to make and the majority of them are oriented to increase the reader interest on the manuscript rather than a criticism of the science of the manuscript. As a matter of taste, I find that they describe too many details of their results; these are simplified far-from-reality numerical experiments, and describing too many details may have little relevance. Perhaps summarizing a bit would improve the manuscript (i.e. in some sections there are some details that would be obvious to most people, like with faster extension comes faster thinning).
Attached, you can find the main comments to be addressed by the authors, then you can find a series of minor suggestions that the authors are free to address or not, or even ignore during review/reply process.

Citation: https://doi.org/10.5194/egusphere-2025-5615-RC1
- AC2:
  'Reply on RC1', Etienne Van Broeck, 19 Mar 2026
  Dear Reviewers, dear Editor,
  Following the reviewers’ constructive and insightful comments, we have substantially revised and reorganized the manuscript. The main modifications are summarized below, and detailed point-by-point responses are provided afterwards.
  The Introduction has been revised to clarify more clearly the motivation, scope, and strategy of the study, including the rationale for the mantle-only configuration and for the weakening diagnostics introduced here. We also rebalanced and expanded the reference framework.
  
  The Results section has been substantially reworked. We added a short preamble to clarify the structure of the section; introduced a new appendix figure defining the criteria used to classify localization outcomes across simulations; revised the figure and presentation of the two-stage localization analysis; added a new appendix figure showing the temporal evolution of thermo-mechanical vertical profiles, which complements the theoretical profiles discussed in the main text; moved the former Fig. 11 to the Appendix; and reduced repetition in the final Results subsection. We also corrected a mistake in the report of “D-Y_{(500 MPa, 950 K)}” simulation results in Table 3, where the final state outcome was “inc. loc” instead of “distr def”.
  
  Following the Referees’ pieces of advise, additional simulations have been performed to strengthen the interpretation and test the robustness of our conclusions, including simulations with depth-dependent yield stress and with temporally variable extension boundary conditions. Some technical material has been moved from the Appendix to the Supplementary Material, which now also includes further analyses supporting our responses to the reviewers’ comments, for example regarding boundary conditions, dynamic pressure, and a posteriori estimates of shear heating.
  
  The Discussion has been extensively restructured. Section 5.1 now begins with the two-stage localization pathway and its comparison with previous numerical studies and selected natural rift systems. Sections 5.2 and 5.3 then compare localization feedbacks across rheological parameterizations and discuss the interpretation of models without dislocation creep or with vertically uniform yield stress, including whether yield stress can be regarded as a proxy for low-temperature plasticity and how the results might change with a more realistic shallow brittle rheology. Section 5.4 emphasizes the two main implications of the study: first, that localization timescales may be overestimated in the absence of dislocation creep; and second, that the new weakening diagnostics (F_T and F_SR) may be useful in other geodynamical contexts and for rheologies with additional state-dependence. We also discuss the principal limitation of the present study, namely the absence of crustal layering.
  
  We have refined the language throughout the manuscript, with particular attention to the Abstract, and we now include in the Appendix the simulations featuring depth-dependent yield stress.
  
  Citation: https://doi.org/10.5194/egusphere-2025-5615-AC2
RC2:
'Comment on egusphere-2025-5615', Vojtěch Patočka, 05 Jan 2026

General comments:

The manuscript by Van Broeck et al. studies the effects of rheological parameterization on the efficiency of strain localization in models of plate extension. Systematic tests of several rheological parameterizations are performed, particularly also the low-temperature dislocation creep is included (Gouriet et al., 2019), and the relative importance of yielding, dislocation, and diffusion creep is quantified during a plate boundary formation. Despite the vast existing literature on numerical models of extensional tectonics, it seems like a useful task: There is a well known discrepancy between the yield stress values adopted in geodynamical models with plate tectonics-like behavior and the laboratory-derived values. The discrepancy stems from the simplifications that need to be adopted when brittle deformation is modelled at large spatial and temporal scales, but the yield stress values are often tuned ad hoc, while the present manuscript clarifies the interplay between dislocation creep parameters and the adopted yield stress values.

The manuscript is well written and the figures are clear. However, there is a certain discrepancy between Introduction and Results (see the first two Specific comments below) and the Results section is a bit exhaustive, with too many observables describing essentially quite similar behavior. The results report a robust, two-stage process of plate-boundary formation, obtained in nearly all models with plastic yielding. My main objection is that, in all the performed simulations, the extension is driven by a prescribed velocity that mimics Couette flow, computed for a vertical viscosity profile following only the diffusion creep rheology (Appendix A). As a result, mechanical weakening below the plate is restricted near the vertical boundaries of the box (velocity is prescribed there), probably narrowing the range of extensional scenarios that are obtained (and limiting the feedback between surface plate velocity and the amount of below-plate mechanical weakening, Patočka et al., 2024). Given the Discussion (lines 520-525 and 555-560), the manuscript would benefit from testing also a constant tectonic force at the vertical sides, acting on the plate under extension.

In summary, the manuscript is a contribution to the literature, providing a diagnostic tool for quantifying the relative amount of thermal and mechanical weakening, and demonstrating that dislocation creep and plastic yielding result in different efficiency of strain localization (although the obtained overall patterns of deformation are similar). The model set-up is a bit simplistic (2D, no shear heating, single-composition material, pressure approximated by the lithostatic pressure), but given the general focus of the study it can be justified. Nevertheless, the employed simplifications should be discussed in a more quantitative manner - for instance, the amount of shear heating and thus thermal weakening can be estimated from the obtained strain rates and viscosities; dynamic pressure can be compared to the lithostatic one. The role of boundary conditions at the vertical sides should be investigated more thoroughly and it should be clarified why a crustal layer is missing.

Specific comments:

The term intra-plate deformation is used in Introduction in the context of distributed deformation occurring outside plate boundaries. Later, when discussing the performed 2D models, "intra-plate" is used to describe deformation within certain depths of the plate (lines 380-385). Are these two phenomena linked? Generally, a stronger link between Introduction and the obtained results would help, Section 5.4 is mostly providing comparison to numerical studies rather than to natural observations, and in many ways the paragraphs there describe further "limitations of the present study" rather than "comparison with natural cases". Also, the Introduction mentions depth-dependent yield stress as the common modeling approach (line 37) and then a depth-constant yield stress is assumed.

Lines 45-55 indicate that low-temperature dislocation creep is an important mechanism that will have a significant effect on the obtained results. However, from Table 3 it seems that the opposite is true and that yielding mostly overrules the low-temperature dislocation creep (see particularly the comparison of models D-d_HT-Y_500 vs ref-1; both models seem to perform nearly identically). If I read that correctly, the effect of including LT-HT is smaller than anticipated (a negative result) and it should be admitted in the Introduction (or Abstract). See also lines 290-291, where a model without LT-HT is present in group 2, and where dislocation creep is found responsible only for the strain localization speed rather than the overall style of extension.

The upper boundary is a free surface, but viscoelasticity is neglected. As shown in a different tectonic context, including a free surface while neglecting elasticity leads to a significant overestimation of shallow stresses (Patočka et al., 2019). In a compressional setting, the effects of viscoelasticity were demonstrated by Jaquet et al., 2016. In an extensional setting, the effect of neglecting viscoelasticity is probably not so dramatic, assuming the resulting topography is small, but it should be discussed.

Line 79, the boundary condition for the central inflow at the base of the model domain should be given. Even in Appendix B, it only says that the velocity is "left free to adjust", but since it is not one of the basic boundary conditions, the governing equation should be formulated.

Similarly, in Section 2.2.2, Eq. C3 should be included to make the rheological description complete. Also, in Eq. 6, the total strain-rate is denoted as edot_creep, but later, in Eq. 7 and 13, it is labeled simply as edot. I assume that only one total strain-rate is involved, this being attributed either fully to yielding or to creeping, with the distinction being based on whether eta_creep or eta_yield is bigger (Eq. 8 and line 131). These details should be clarified. Note also that, generally, edot is not a very good label for strain-rate. Although widely used in geodynamical literature, it is not used in mathematical literature, because the symmetric part of the velocity gradient (i.e. strain-rate) is not the material derivative of any standard measure of strain (such a relation is only approximative, valid in the linearized theory of deformations).

Line 139: Defining 500 MPa as "the upper bound for the validity of the LT-HT dislocation creep" is not very clear. This value is larger than the typical values needed to obtain plate-like behavior in geodynamical models, and so it is promising that localized plate boundaries are obtained with this parameterization, but the motivation behind the particular value could be elaborated a bit more (given that most of the presented results employ this value).

Lines 195-200: It would help to clarify how Eq. 11 is related to the traction acting on the vertical boundary and why the employed stresses are evaluated 100 km away from the boundary.

If I understand it correctly, Fig. 9 is computed not from the performed simulations, but solely from the rheological equations (i.e. from the viscosity as a function of temperature and strain-rate). This should be stated in the text and not only in the figure caption. It is not fully clear how the time-derivative of temperature and strain-rate is determined in that figure (i.e. what exactly is meant by the employed 50 to 20 Myr plate age scenario, over what time is this change prescribed?).

Line 448-450: The nearly identical behavior in models that differ only in the dislocation creep parameters does not imply that "Low-temperature dislocation creep and yield stress seem to play analogous roles in reducing the high lithospheric strength in the temperature range 800-1100K". Instead, it indicates that the low-temperature dislocation creep does not get activated very much, perhaps because yield stress dominates the region, or am I missing something? The sentence would be true if yield stress was not included in the d_LT-HT simulation, but it is.

Given the extensive discussion of the absence of a crustal layer (lines 505-535), why isn't a crustal layer prescribed in the simulations? Is it because Fluidity does not allow it, or because it was desired to reduce the number of tested parameters (with crustal rheology introducing new ones)? This should be clarified.

Technical comments:

Line 3: among which low-temperature plasticity... a verb seems to be missing.
Table 1: Volumic mass, isn't it more common to simply say density?
Table 2: Activation volume, not energy (in the third rows)
Line 221: the word "homogeneous" is often incorrectly used where "uniform" should be used instead.
Line 236: "Highly deformed zone" is used for regions with strain-rate amplification > 1. But that is not a very strict criterion, if the number is not at least 2, calling the zone highly deformed may be misleading.
Line 250: Why depth-invariance of the weakening rate results in small lateral viscosity contrast is not clear to me.
Fig. 4 caption: We represent only values where >60% ... but the color maps seem full, so it is not clear what this sentence is referring to.
Line 265: superimposed boundaries? Perhaps reformulate simply as "The dominant def. mech. are delimited by the dashed dark blue lines..."
Fig. 5b: The initial evolution (<1 Myr) is not commented on. Is it perhaps an effect of the imposed BCs?
Fig 6 caption: White regions represent hardening?
Line 318: even if -> even though
Line 355: The relative change in the width of the deformed zone does not seem so small, so perhaps giving rough numbers would help.
Line 535: Two @ symbols

References:
Patočka, V., Čížková, H., Tackley, P. (2019): Do elasticity and a free surface affect lithospheric stresses caused by upper mantle convection?, Geophys. J. Int., 216(3), 1740-1760.
Jaquet Y., Duretz T., Schmalholz S.M. (2016): Dramatic effect of elasticity on thermal softening and strain localization during lithospheric shortening, Geophys. J. Int., 204(2), 780–784.

Citation: https://doi.org/10.5194/egusphere-2025-5615-RC2
- AC1:
  'Reply on RC2', Etienne Van Broeck, 19 Mar 2026
  Dear Reviewers, dear Editor,
  Following the reviewers’ constructive and insightful comments, we have substantially revised and reorganized the manuscript. The main modifications are summarized below, and detailed point-by-point responses are provided afterwards.
  The Introduction has been revised to clarify more clearly the motivation, scope, and strategy of the study, including the rationale for the mantle-only configuration and for the weakening diagnostics introduced here. We also rebalanced and expanded the reference framework.
  
  The Results section has been substantially reworked. We added a short preamble to clarify the structure of the section; introduced a new appendix figure defining the criteria used to classify localization outcomes across simulations; revised the figure and presentation of the two-stage localization analysis; added a new appendix figure showing the temporal evolution of thermo-mechanical vertical profiles, which complements the theoretical profiles discussed in the main text; moved the former Fig. 11 to the Appendix; and reduced repetition in the final Results subsection. We also corrected a mistake in the report of “D-Y_{(500 MPa, 950 K)}” simulation results in Table 3, where the final state outcome was “inc. loc” instead of “distr def”.
  
  Following the Referees’ pieces of advise, additional simulations have been performed to strengthen the interpretation and test the robustness of our conclusions, including simulations with depth-dependent yield stress and with temporally variable extension boundary conditions. Some technical material has been moved from the Appendix to the Supplementary Material, which now also includes further analyses supporting our responses to the reviewers’ comments, for example regarding boundary conditions, dynamic pressure, and a posteriori estimates of shear heating.
  
  The Discussion has been extensively restructured. Section 5.1 now begins with the two-stage localization pathway and its comparison with previous numerical studies and selected natural rift systems. Sections 5.2 and 5.3 then compare localization feedbacks across rheological parameterizations and discuss the interpretation of models without dislocation creep or with vertically uniform yield stress, including whether yield stress can be regarded as a proxy for low-temperature plasticity and how the results might change with a more realistic shallow brittle rheology. Section 5.4 emphasizes the two main implications of the study: first, that localization timescales may be overestimated in the absence of dislocation creep; and second, that the new weakening diagnostics (F_T and F_SR) may be useful in other geodynamical contexts and for rheologies with additional state-dependence. We also discuss the principal limitation of the present study, namely the absence of crustal layering.
  
  We have refined the language throughout the manuscript, with particular attention to the Abstract, and we now include in the Appendix the simulations featuring depth-dependent yield stress.
  
  Citation: https://doi.org/10.5194/egusphere-2025-5615-AC1

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

AR by Etienne Van Broeck on behalf of the Authors (19 Mar 2026) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (23 Mar 2026) by Philip Heron

RR by Vojtěch Patočka (08 Apr 2026)

RR by Antonio Manjon Cabeza Cordoba (05 May 2026)

Suggestions for revision or reasons for rejection

Second review of "From Strong Plates to Weak Boundaries: Strain Localization in the Lithospheric Mantle with Low- to High-Temperature Dislocation Creep" by Etienne Van Broeck, Fanny Garel, Catherine Thoraval, and D. Rhodri Davies.
By Antonio Manjón-Cabeza Córdoba

The authors have done a substantial re-write of the manuscript to address mine and the other reviewer's comments. I do not think this was needed, and the authors have introduced a few technical typos that could have been avoided had they done a more localized, surgical and subtractive revision. Nonetheless, the conclusions remain robust and I do not think that delaying the publication of this article any longer should be done on the sole base of my opinion.
Unfortunately, when I accepted the second review I was not aware of the level of work put into the new version and I needed to ask for an extension once I realised I could not just check a few boxes. Due to personal reasons, it has been a bit frustrating to follow through with this review and I am not convinced of my objectivity (i.e., I may have been overly picky). For that I apologise to the authors and the editor.

Overall, the article is good and has improved. I still find the article overly long. In some parts it is repetitive and in others discusses issues that do not serve the conclusions. The manuscript features > 600 lines of the main text, 5 Appendixes and supplementary material; all this for conclusions that do not need that much argument (i.e., the more robust the conclusions the less discussion they need). However, some parts are definitely clearer now, and the figures are more synthetic and to the point. At the moment, I debate myself between recommending a technical review or a minor revision, and I pass the responsibility to the editor. Regardless, I think that I have contributed what I could contribute and, should another review be needed, I recommend someone else to do it.

ABSTRACT

I warn against the use of the word ductile altogether. Ductile is a rather phenomenological term that in numerical models can (and probably should) be changed by the word "viscous" whenever possible. If a distinction needs to be made, then I suggest to refer to the specific mechanism. In what the field scientists would classically call the "ductile regime", a combination of viscous creeps may apply (and even combined with some micro-brittle behavior). Note that later in the abstract dislocation creep is addressed explicitly (good).

INTRODUCTION

Plate tectonics describes the motion of rigid lithospheric plates separated by narrow, weak boundaries (e.g. Le Pichon, 1968).
-> I don't understand this sentence. Plate tectonics describes? Maybe add "The hypothesis of plate tectonics"? Or "The theory of plate tectonics"? Or even "The law of plate tectonics"? Perhaps more technically "The paradigm of plate tectonics involves the motion of rigid lithospheric plates ..."?

"The lithosphere is expected to be strong under cold, ductile conditions". Ok so, at the very least, please define ductile. I have a rough idea of what the authors mean but a lot of people, particularly those working on the lithosphere will imagine that cold and ductile are an oxymoron. Hot and cold are relative, ok, but brittle is colder. Again another case for abandoning the word "ductile" altogether.

Line 49, low with respect to what? Maybe better, "lower than those predicted at x depth by assuming that the yield stress represents brittle deformation".

Line 53,54 "Low-temperature ductile deformation has been proposed as an alternative route to limiting lithospheric strength without imposing
an ad-hoc stress cap throughout the ductile regime." This is true but even better with a citation.
Also, low-temperature ductile deformation... is the low-temperature relative to other ductile deformation mechanisms or is ductile used indistinctively from dislocation? I am intentionally being obnoxious precisely to point out the problem with using "ductile" within the context of geodynamic models.

Line 67 "Weinstein and Olson 1992: Thanks for this reference! Did not know it and it is indeed useful.

Line 91. Please be consistent, either sects 5 and 6 or sects Discussion and Conclusions

METHODS

Table 1. "Density" -> Reference density
Figure 1. "Freesurface" -> Free surface? Why? BTW?

Line 162-163. References are inline, probably unintentional.

Table 2. Adisl ^{-7}? Is this correct? Please revise

The tanh label makes sense within the context of the previous publications (Garel 2020), but I'm not sure it is clear here as only one formulation is described in the methods

Line 198. I am a bit confused. Some lines above the authors include an adiabatic correction for the viscosity, but now they claim they solve the problem in the Boussinesq approximation. I'm sorry I missed this in the previous review, but I don't think this is clear in the text.

RESULTS

Line 254. I'm sorry if I missed this in the previous review, but F here is confusing. Earlier, a tectonic force F with units of N m^{-1} is defined but now these ratios seem dimensionless. I would expect the same capital letter should correspond to the same units.

4D evolution actually means 5 D (i.e. 4D in time). I recommend either 3D evolution or 4D models.

Line 309 Formatting cmyr^{-1} -> cm yr^{-1}. And around this paragraph, the space between label terms disappear please be consistent throughout the text (Fig. 5 or Fig.5 but not both).

Figure 4: There may be a label missing in the vertical axis of panel b (km). Maybe this was intentional?

Line 352 Fig.9 -> Fig. 9: Line 355 Fig.9d -> Fig. 9d
Line 370 Fig.9 -> Fig 9: Table3 -> Table 3.

Lines ~360-374. Rather than a friction coefficient, the authors impose a temperature limit. Fine. But then they should not analyse a stiff layer that arises precisely because of this cutoff! A friction coefficient should indeed result on a stiffer part of the lithosphere, but the one in this manuscript is simply imposed. I think the authors overinterpret their results here (which they don't need at all).

Figure 9: Notice that HT and LT-HT are not equivalent formulations at high t. The viscosity scale is logarithmic and therefore the difference is noticeable. Not sure I agree with the authors both cases are comparable (this does not hurt the general conclusions of the paper, but does put a limit on the strength of their remarks on LT-HT vs HT, which admittedly, they tuned down). Just to be clear, I do like the author's LT-HT formulation better than just HT, my comment is just on what's shown in the manuscript

DISCUSSION

"Taken together, our results show that lithospheric-scale strain localization in this extensional setting is controlled not simply
by the integrated strength of the plate, but by where weakening occurs and how rheology couples deformation focusing
to asthenospheric upwelling through rheology-modulated feedbacks linking deformation focusing, plate weakening and asthenospheric
upwelling." -> This sentence is key and it is too hard to read. I would suggest dividing in 2-3 sentences. Otherwise readers may miss the main point of this work. What about... [...] our results show that [...] strain localization [...] is controlled by the integrated strength of the plate and by where weakening occurs. In addition, a key controlling effect is the rheological coupling of deformation focusing and asthenospheric upwelling [...] ...? (I'm sorry if I did not capture the original meaning, but at the very least I should have shown this wasn't easy to understand).

Line 443 Potentially unintentional inline citation (Ulvrova, 2019).

Line 474 Fig.9 -> Fig. 9

Line 483 Fig.6c -> Fig. 6c

Line 574 "[...] approaches should distinguish [...]". Should distinguish for what? I understand that there is something missing when not including dislocation creep or a "ductile yield stress", but are the authors suggesting that all models that do not include this should be automatically scrapped? I would not agree with that, nor I think the authors have the results to make such statement. Instead, this sentence would be better written as "Models that would like to accurately estimate the absolute timescale of localization should distinguish [...]". That way this statement is precise and with real actionable implications.
I apologise to the authors if this comment seems pedantic, but this sort of comments are too common and of little help in our community (For example, I work on melting calculations in this sort of models, I could argue that extension tectonics of this sort usually happens under strong magmatism, therefore casting doubt on the authors' results because "Models should include the effect of melting". But I honestly think that the authors' models are better because they are simplified and therefore could properly study creep-based localisation without the effect of melting.

Line 577 "result" -> aspect (not technically a result).

Line 585. I understand that the authors acted to address a reviewer's comment. But I wanted to highlight that by no means I believe this is the main limitation of the study. No action needed here.

CONCLUSIONS

Line 627 "In all simulations that successfully localize" Localize what? I may be overly picky here, the sentence should be understandable for most readers, but an object is missing.

GENERAL COMMENTS

I suggest you review the order of the citations. This is not important, but at the moment, they are not ordered neither chronologically nor alphabetically in some parts of the text. When they do, they do so chronologically, which I like, so I suggest to be consistent throughout the text. Perhaps does not make that much of a change, but easier on the reader.

Many of the labels for the Figures do not have a space between the "Fig." and the number. Others do. I have pointed some out, but I am sure I missed some. This is just a technical issue but appears throughout the manuscript, please revise.

Hide

ED: Publish subject to minor revisions (review by editor) (14 May 2026) by Philip Heron

AR by Etienne Van Broeck on behalf of the Authors (21 May 2026) Author's response Author's tracked changes Manuscript

ED: Publish as is (23 May 2026) by Philip Heron

ED: Publish as is (23 May 2026) by Susanne Buiter (Executive editor)

AR by Etienne Van Broeck on behalf of the Authors (27 May 2026) Author's response

Interactive discussion

Status: closed

RC1:
'Comment on egusphere-2025-5615', Antonio Manjon Cabeza Cordoba, 02 Jan 2026

Dear editor,
this technical paper presents a series of numerical experiments to address the role of rheology on deformation localization during extensional tectonics. While several detailed models of extension and rifting exist, evaluation of localization and adequate rheological mechanism is mostly absent in modern literature. In addition, they add the role of low-temperature plasticity, a continuation of their own work (Gouriet 2019; Garel 2020)
They show that considering multiple rheological mechanisms, in particular non-linear rheologies, is important to reproduce localization processes as we imagine them on Earth. They could specify a bit more on whether this was ever called into question, or what are the consequences of ignoring one rheology or the other, not just for their models, but for the wider community that will be forced to take simplifying assumptions regardless.
The manuscript is quite technical but the methodology is well explained. I have little to say there. Because Solid Earth has a wide focus - inasmuch as the deep Earth can be called a wide field – the article would benefit for a slightly more careful contextualization (e.g. what advances or new evidence motivate this study? How does this work compare with other studies? What are the implications for other geophysicists/geologists? etc). But this is not a criticism of the science of the manuscript itself.
Overall, I have few important comments to make and the majority of them are oriented to increase the reader interest on the manuscript rather than a criticism of the science of the manuscript. As a matter of taste, I find that they describe too many details of their results; these are simplified far-from-reality numerical experiments, and describing too many details may have little relevance. Perhaps summarizing a bit would improve the manuscript (i.e. in some sections there are some details that would be obvious to most people, like with faster extension comes faster thinning).
Attached, you can find the main comments to be addressed by the authors, then you can find a series of minor suggestions that the authors are free to address or not, or even ignore during review/reply process.

Citation: https://doi.org/10.5194/egusphere-2025-5615-RC1
- AC2:
  'Reply on RC1', Etienne Van Broeck, 19 Mar 2026
  Dear Reviewers, dear Editor,
  Following the reviewers’ constructive and insightful comments, we have substantially revised and reorganized the manuscript. The main modifications are summarized below, and detailed point-by-point responses are provided afterwards.
  The Introduction has been revised to clarify more clearly the motivation, scope, and strategy of the study, including the rationale for the mantle-only configuration and for the weakening diagnostics introduced here. We also rebalanced and expanded the reference framework.
  
  The Results section has been substantially reworked. We added a short preamble to clarify the structure of the section; introduced a new appendix figure defining the criteria used to classify localization outcomes across simulations; revised the figure and presentation of the two-stage localization analysis; added a new appendix figure showing the temporal evolution of thermo-mechanical vertical profiles, which complements the theoretical profiles discussed in the main text; moved the former Fig. 11 to the Appendix; and reduced repetition in the final Results subsection. We also corrected a mistake in the report of “D-Y_{(500 MPa, 950 K)}” simulation results in Table 3, where the final state outcome was “inc. loc” instead of “distr def”.
  
  Following the Referees’ pieces of advise, additional simulations have been performed to strengthen the interpretation and test the robustness of our conclusions, including simulations with depth-dependent yield stress and with temporally variable extension boundary conditions. Some technical material has been moved from the Appendix to the Supplementary Material, which now also includes further analyses supporting our responses to the reviewers’ comments, for example regarding boundary conditions, dynamic pressure, and a posteriori estimates of shear heating.
  
  The Discussion has been extensively restructured. Section 5.1 now begins with the two-stage localization pathway and its comparison with previous numerical studies and selected natural rift systems. Sections 5.2 and 5.3 then compare localization feedbacks across rheological parameterizations and discuss the interpretation of models without dislocation creep or with vertically uniform yield stress, including whether yield stress can be regarded as a proxy for low-temperature plasticity and how the results might change with a more realistic shallow brittle rheology. Section 5.4 emphasizes the two main implications of the study: first, that localization timescales may be overestimated in the absence of dislocation creep; and second, that the new weakening diagnostics (F_T and F_SR) may be useful in other geodynamical contexts and for rheologies with additional state-dependence. We also discuss the principal limitation of the present study, namely the absence of crustal layering.
  
  We have refined the language throughout the manuscript, with particular attention to the Abstract, and we now include in the Appendix the simulations featuring depth-dependent yield stress.
  
  Citation: https://doi.org/10.5194/egusphere-2025-5615-AC2
RC2:
'Comment on egusphere-2025-5615', Vojtěch Patočka, 05 Jan 2026

General comments:

The manuscript by Van Broeck et al. studies the effects of rheological parameterization on the efficiency of strain localization in models of plate extension. Systematic tests of several rheological parameterizations are performed, particularly also the low-temperature dislocation creep is included (Gouriet et al., 2019), and the relative importance of yielding, dislocation, and diffusion creep is quantified during a plate boundary formation. Despite the vast existing literature on numerical models of extensional tectonics, it seems like a useful task: There is a well known discrepancy between the yield stress values adopted in geodynamical models with plate tectonics-like behavior and the laboratory-derived values. The discrepancy stems from the simplifications that need to be adopted when brittle deformation is modelled at large spatial and temporal scales, but the yield stress values are often tuned ad hoc, while the present manuscript clarifies the interplay between dislocation creep parameters and the adopted yield stress values.

The manuscript is well written and the figures are clear. However, there is a certain discrepancy between Introduction and Results (see the first two Specific comments below) and the Results section is a bit exhaustive, with too many observables describing essentially quite similar behavior. The results report a robust, two-stage process of plate-boundary formation, obtained in nearly all models with plastic yielding. My main objection is that, in all the performed simulations, the extension is driven by a prescribed velocity that mimics Couette flow, computed for a vertical viscosity profile following only the diffusion creep rheology (Appendix A). As a result, mechanical weakening below the plate is restricted near the vertical boundaries of the box (velocity is prescribed there), probably narrowing the range of extensional scenarios that are obtained (and limiting the feedback between surface plate velocity and the amount of below-plate mechanical weakening, Patočka et al., 2024). Given the Discussion (lines 520-525 and 555-560), the manuscript would benefit from testing also a constant tectonic force at the vertical sides, acting on the plate under extension.

In summary, the manuscript is a contribution to the literature, providing a diagnostic tool for quantifying the relative amount of thermal and mechanical weakening, and demonstrating that dislocation creep and plastic yielding result in different efficiency of strain localization (although the obtained overall patterns of deformation are similar). The model set-up is a bit simplistic (2D, no shear heating, single-composition material, pressure approximated by the lithostatic pressure), but given the general focus of the study it can be justified. Nevertheless, the employed simplifications should be discussed in a more quantitative manner - for instance, the amount of shear heating and thus thermal weakening can be estimated from the obtained strain rates and viscosities; dynamic pressure can be compared to the lithostatic one. The role of boundary conditions at the vertical sides should be investigated more thoroughly and it should be clarified why a crustal layer is missing.

Specific comments:

The term intra-plate deformation is used in Introduction in the context of distributed deformation occurring outside plate boundaries. Later, when discussing the performed 2D models, "intra-plate" is used to describe deformation within certain depths of the plate (lines 380-385). Are these two phenomena linked? Generally, a stronger link between Introduction and the obtained results would help, Section 5.4 is mostly providing comparison to numerical studies rather than to natural observations, and in many ways the paragraphs there describe further "limitations of the present study" rather than "comparison with natural cases". Also, the Introduction mentions depth-dependent yield stress as the common modeling approach (line 37) and then a depth-constant yield stress is assumed.

Lines 45-55 indicate that low-temperature dislocation creep is an important mechanism that will have a significant effect on the obtained results. However, from Table 3 it seems that the opposite is true and that yielding mostly overrules the low-temperature dislocation creep (see particularly the comparison of models D-d_HT-Y_500 vs ref-1; both models seem to perform nearly identically). If I read that correctly, the effect of including LT-HT is smaller than anticipated (a negative result) and it should be admitted in the Introduction (or Abstract). See also lines 290-291, where a model without LT-HT is present in group 2, and where dislocation creep is found responsible only for the strain localization speed rather than the overall style of extension.

The upper boundary is a free surface, but viscoelasticity is neglected. As shown in a different tectonic context, including a free surface while neglecting elasticity leads to a significant overestimation of shallow stresses (Patočka et al., 2019). In a compressional setting, the effects of viscoelasticity were demonstrated by Jaquet et al., 2016. In an extensional setting, the effect of neglecting viscoelasticity is probably not so dramatic, assuming the resulting topography is small, but it should be discussed.

Line 79, the boundary condition for the central inflow at the base of the model domain should be given. Even in Appendix B, it only says that the velocity is "left free to adjust", but since it is not one of the basic boundary conditions, the governing equation should be formulated.

Similarly, in Section 2.2.2, Eq. C3 should be included to make the rheological description complete. Also, in Eq. 6, the total strain-rate is denoted as edot_creep, but later, in Eq. 7 and 13, it is labeled simply as edot. I assume that only one total strain-rate is involved, this being attributed either fully to yielding or to creeping, with the distinction being based on whether eta_creep or eta_yield is bigger (Eq. 8 and line 131). These details should be clarified. Note also that, generally, edot is not a very good label for strain-rate. Although widely used in geodynamical literature, it is not used in mathematical literature, because the symmetric part of the velocity gradient (i.e. strain-rate) is not the material derivative of any standard measure of strain (such a relation is only approximative, valid in the linearized theory of deformations).

Line 139: Defining 500 MPa as "the upper bound for the validity of the LT-HT dislocation creep" is not very clear. This value is larger than the typical values needed to obtain plate-like behavior in geodynamical models, and so it is promising that localized plate boundaries are obtained with this parameterization, but the motivation behind the particular value could be elaborated a bit more (given that most of the presented results employ this value).

Lines 195-200: It would help to clarify how Eq. 11 is related to the traction acting on the vertical boundary and why the employed stresses are evaluated 100 km away from the boundary.

If I understand it correctly, Fig. 9 is computed not from the performed simulations, but solely from the rheological equations (i.e. from the viscosity as a function of temperature and strain-rate). This should be stated in the text and not only in the figure caption. It is not fully clear how the time-derivative of temperature and strain-rate is determined in that figure (i.e. what exactly is meant by the employed 50 to 20 Myr plate age scenario, over what time is this change prescribed?).

Line 448-450: The nearly identical behavior in models that differ only in the dislocation creep parameters does not imply that "Low-temperature dislocation creep and yield stress seem to play analogous roles in reducing the high lithospheric strength in the temperature range 800-1100K". Instead, it indicates that the low-temperature dislocation creep does not get activated very much, perhaps because yield stress dominates the region, or am I missing something? The sentence would be true if yield stress was not included in the d_LT-HT simulation, but it is.

Given the extensive discussion of the absence of a crustal layer (lines 505-535), why isn't a crustal layer prescribed in the simulations? Is it because Fluidity does not allow it, or because it was desired to reduce the number of tested parameters (with crustal rheology introducing new ones)? This should be clarified.

Technical comments:

Line 3: among which low-temperature plasticity... a verb seems to be missing.
Table 1: Volumic mass, isn't it more common to simply say density?
Table 2: Activation volume, not energy (in the third rows)
Line 221: the word "homogeneous" is often incorrectly used where "uniform" should be used instead.
Line 236: "Highly deformed zone" is used for regions with strain-rate amplification > 1. But that is not a very strict criterion, if the number is not at least 2, calling the zone highly deformed may be misleading.
Line 250: Why depth-invariance of the weakening rate results in small lateral viscosity contrast is not clear to me.
Fig. 4 caption: We represent only values where >60% ... but the color maps seem full, so it is not clear what this sentence is referring to.
Line 265: superimposed boundaries? Perhaps reformulate simply as "The dominant def. mech. are delimited by the dashed dark blue lines..."
Fig. 5b: The initial evolution (<1 Myr) is not commented on. Is it perhaps an effect of the imposed BCs?
Fig 6 caption: White regions represent hardening?
Line 318: even if -> even though
Line 355: The relative change in the width of the deformed zone does not seem so small, so perhaps giving rough numbers would help.
Line 535: Two @ symbols

References:
Patočka, V., Čížková, H., Tackley, P. (2019): Do elasticity and a free surface affect lithospheric stresses caused by upper mantle convection?, Geophys. J. Int., 216(3), 1740-1760.
Jaquet Y., Duretz T., Schmalholz S.M. (2016): Dramatic effect of elasticity on thermal softening and strain localization during lithospheric shortening, Geophys. J. Int., 204(2), 780–784.

Citation: https://doi.org/10.5194/egusphere-2025-5615-RC2
- AC1:
  'Reply on RC2', Etienne Van Broeck, 19 Mar 2026
  Dear Reviewers, dear Editor,
  Following the reviewers’ constructive and insightful comments, we have substantially revised and reorganized the manuscript. The main modifications are summarized below, and detailed point-by-point responses are provided afterwards.
  The Introduction has been revised to clarify more clearly the motivation, scope, and strategy of the study, including the rationale for the mantle-only configuration and for the weakening diagnostics introduced here. We also rebalanced and expanded the reference framework.
  
  The Results section has been substantially reworked. We added a short preamble to clarify the structure of the section; introduced a new appendix figure defining the criteria used to classify localization outcomes across simulations; revised the figure and presentation of the two-stage localization analysis; added a new appendix figure showing the temporal evolution of thermo-mechanical vertical profiles, which complements the theoretical profiles discussed in the main text; moved the former Fig. 11 to the Appendix; and reduced repetition in the final Results subsection. We also corrected a mistake in the report of “D-Y_{(500 MPa, 950 K)}” simulation results in Table 3, where the final state outcome was “inc. loc” instead of “distr def”.
  
  Following the Referees’ pieces of advise, additional simulations have been performed to strengthen the interpretation and test the robustness of our conclusions, including simulations with depth-dependent yield stress and with temporally variable extension boundary conditions. Some technical material has been moved from the Appendix to the Supplementary Material, which now also includes further analyses supporting our responses to the reviewers’ comments, for example regarding boundary conditions, dynamic pressure, and a posteriori estimates of shear heating.
  
  The Discussion has been extensively restructured. Section 5.1 now begins with the two-stage localization pathway and its comparison with previous numerical studies and selected natural rift systems. Sections 5.2 and 5.3 then compare localization feedbacks across rheological parameterizations and discuss the interpretation of models without dislocation creep or with vertically uniform yield stress, including whether yield stress can be regarded as a proxy for low-temperature plasticity and how the results might change with a more realistic shallow brittle rheology. Section 5.4 emphasizes the two main implications of the study: first, that localization timescales may be overestimated in the absence of dislocation creep; and second, that the new weakening diagnostics (F_T and F_SR) may be useful in other geodynamical contexts and for rheologies with additional state-dependence. We also discuss the principal limitation of the present study, namely the absence of crustal layering.
  
  We have refined the language throughout the manuscript, with particular attention to the Abstract, and we now include in the Appendix the simulations featuring depth-dependent yield stress.
  
  Citation: https://doi.org/10.5194/egusphere-2025-5615-AC1

Peer review completion

AR – Author's response | RR – Referee report | ED – Editor decision | EF – Editorial file upload

AR by Etienne Van Broeck on behalf of the Authors (19 Mar 2026) Author's response Author's tracked changes Manuscript

ED: Referee Nomination & Report Request started (23 Mar 2026) by Philip Heron

RR by Vojtěch Patočka (08 Apr 2026)

RR by Antonio Manjon Cabeza Cordoba (05 May 2026)

Suggestions for revision or reasons for rejection

Hide

ED: Publish subject to minor revisions (review by editor) (14 May 2026) by Philip Heron

AR by Etienne Van Broeck on behalf of the Authors (21 May 2026) Author's response Author's tracked changes Manuscript

ED: Publish as is (23 May 2026) by Philip Heron

ED: Publish as is (23 May 2026) by Susanne Buiter (Executive editor)

AR by Etienne Van Broeck on behalf of the Authors (27 May 2026) Author's response

Journal article(s) based on this preprint

06 Aug 2026

From strong plates to weak boundaries: strain localization in the lithospheric mantle with low- to high-temperature dislocation creep

Etienne Van Broeck, Fanny Garel, Catherine Thoraval, Diane Arcay, and D. Rhodri Davies

Solid Earth, 17, 947–977, https://doi.org/10.5194/se-17-947-2026,https://doi.org/10.5194/se-17-947-2026, 2026

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Etienne Van Broeck, Fanny Garel, Catherine Thoraval, Diane Arcay, and D. Rhodri Davies

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https://doi.org/10.5194/egusphere-2025-5615-supplement

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From Strong Plates to Weak Boundaries: Strain Localization in the Lithospheric Mantle with Low- to High-Temperature Dislocation Creep - Data set Etienne Van Broeck et al. https://zenodo.org/records/17606932

Etienne Van Broeck, Fanny Garel, Catherine Thoraval, Diane Arcay, and D. Rhodri Davies

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Earth’s solid shell is made of stiff tectonic plates which can break apart to form new weak plate boundaries. We use numerical models of plate extension to investigate how mechanical properties control the concentration of deformation leading to such a boundary as hot mantle rises underneath. We highlight feedbacks, and quantify how and why, under various deformation mechanisms, the plate weakens as temperature and strain rate increase. We compare the model predictions to natural rifting cases.


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