The submitted revised manuscript by Dohmen and Schmeling systematically investigates magma ascent dynamics in order to capture the transition from the solitary wave regime to diapirism. The authors explore this transition by varying the relative compaction length of the system - here by changing the model extend while keeping the compaction length constant.
The authors addressed majority of the concerns raised during the first round of revisions. However, the new version of the submitted manuscript still suffers from major design issues, both in the content and form. Rather than "time investment and good enough-ness", focus should be on scientific approach and accuracy. To the point, the main story of the manuscript -channelling- is not receivable as such. The authors motivate their revised study by unveiling apparent channelling mechanism occurring while transitioning from the solitary-wave to the Stokes regime. Their argumentation would only be receivable if following a scientific approach, i.e., including more than wishful thinking.
Simple words for simple things; Assuming there is a not yet discovered channelling mechanisms, natural steps would be following: (1) provide a parameter accounting for it; (2) test and report the influence of this parameter in a systematic study; (3) prove the robustness of the suggested results by providing (numerical) convergence tests (physical results should no longer vary with further increase in numerical resolution - proof of a robust numerical implementation) targeting the configuration of interest. To date none of these steps are successfully implemented.
Now, and unless proven otherwise, the underlying equation do not contain any channelling mechanism as such. Thus, the reported channels may rather be the expression of a lack in numerical resolution. This conclusion still confirms the outcome of previous reviews.
Channelling ultimately requires an asymmetry in compaction versus decompaction regimes, obtained upon nonlinear bulk rheology by mechanisms such as e.g., decompaction weakening (Connolly and Podladchikov, 1998; Räss, 2018; Räss, 2019) or brittle failure (Keller, 2013; Yarushina, 2015). Moreover, including the full shear stress tensor for the mixture velocities and total pressure won't produce extra focussing and asymmetry; neither would porosity dependent and even strain-dependent shear rheology. Both may impact the compaction length which may further influence the relative inclusion size, at most.
Finally, the effort spent in providing further insights into the underlying physics and mathematical model (Section 2.1) is very much appreciated. However, this new section reports inconsistent derivations. Equation (3) reports de analogy of the fluid momentum equations as a generalised Darcy law that contains de gradient of the fluid pressure P minus the buoyant fluid force ρfg. Equation (4) reports the total force or momentum balance, where the viscous stresses and total (mixture) pressure P equilibrate the total buoyancy force ρ̄g. The pressure term in equation (3) represents as such the fluid pressure Pf, while the pressure in equation (4) stands for the total pressure Ptot. Equation (10) and line 96 is thus wrong. Fluid pressure ≠ total pressure (Pf≠Ptot) and CANNOT be eliminated.
While the original study needed some revision, the here submitted revised version addresses none of the early design issues. Instead of providing scientifically robust proofs about potential new transient regimes, it further motivates wishful thinking instead of results.
To accept the claims made by the authors about the existence of an intermediate regime leading to flow channelling while transitioning from solitary waves to diapirism, following steps should be included:
1) identification of a physical and testable parameter accounting for focusing
2) systematically testing and reporting of the influence of this physical parameter
3) numerical convergence test to support the robustness of the numerical results (independent of the chosen numerical implementation)
-- Further detailed comments (line numbers refer to the manuscript version 4):
l.9: Not only size but related to compaction length. Size could be kept constant but change in compaction length may lead to similar results
l.19-22: No channels will form. Results seem to report a lack of resolution here. To form channels, one needs an asymmetry in compaction versus decompaction rheology. This asymmetry one does not get with the shear rheology. Including porosity dependence in bulk and shear rheology may induce a change in compaction length but no asymmetry.
l.45-47: It's the same, as compaction length and radius are interconnected. Changing compaction length using Rt may be the same than changing the bulk to shear viscosity ratio, which will ultimately also impact the compaction length.
l.68-72: Boussinesq approximation. There is no need for abbreviation since you only use "BA" twice. Also, the wording could be improved here as it is not very clear in the current form.
---- Section 2.1
Equations (3) and (4) have a pressure issue. How can the same P both be used in the Darcy flow and in the total momentum balance, once relating to fluid density, once relating to total density? Needs revision, modification and clarification.
Minor notation issue: this section could be enhanced with notation homogenisation. Either adopting the ∇ or ∂/∂x notation. Also, some i,j,k may be missing if including those.
---- Section 2.3
What linear and nonlinear absolute and or relative tolerances are used (criterion to stop iteration and accept the current solution before starting the next physical time step)?
---- Results Section:
l.233-234: What is observed in Fig.1a-d is simply the evidence of the problem's internal length scale, the compaction length. Although the initial melt anomaly becomes larger, flow still re-organises within a blob of characteristic size given by the compaction length.
l.244-245: There is no channelling. You may see focusing of melt from an original distribution into a new circular one, but the channels you refer to are numerical artefacts. A model including at least >10 grid points to resolve the channel width will be needed to validate your statement.
l.245-267: If you can both model Stokes and porosity waves, why do you need analytical velocity formulation for your lines in Fig. 2? Basically, Fig. 2 should be obtained by tracking your results, or at least some of those. How does one know whether your results match the curves on Fig. 2? One does not need to run any numerical model to reach your conclusions if they're based upon evaluating semi-analytical solutions from literature.
l.319-325: Be careful with statement. You still do not resolve the small instabilities and those may just be small blobs if properly resolved.
l.339-341: Receivable statement upon successful proof that those are not numerical artefacts.
---- Numerical issues:
As long as no convergence test neither benchmark (available in e.g., the Appendix of Räss (2019) and Keller (2013)) is provided, the reported results, especially channels, could well be under-resolved and thus numerical artefacts.
4.1 channelling: As long as bulk rheology is linear; no literature reports any growth of instabilities besides splitting of original wave into new size owing to dynamical change in compaction length. Richardson (1998) shows minor impact of external stresses on blob's shape, but no channelling as such is presented.
l.404-405: No channelling here, changes in compaction length will change the characteristic diameter of the spherical wave which needs to be resolved.
l.411-414: Unfounded claims. Connolly & Podladchikov (1998) do not suggest following "this upward weakening might not be strong enough to lead to the focusing needed for the nucleation of dykes".
l.422-428: No channelling. As long as there is no asymmetry in viscous compaction versus decompaction, you won't get channels out of blobs. Taking full stress tensor into account and having porosity dependent viscosity will just impact compaction length, nothing else (Räss, 2019).
l.439: "the velocities fit quite nicely to the observed model velocities" where does one sees this? You report analytical solution from other authors and your analytical solutions, but nowhere your modelled results. Since your model includes the stress tensor and velocities, it would be very interesting to report those to support your statement and make them receivable.
l. 447-448: A mechanism needs a testable physical parameter, and a verification that this parameter delivers robust and resolution independent results.
l. 451: (2) see previous comment.
l. 459-463: Good point. Apply it; re-run the suggested simulations with 10 times higher resolutions and longer travel path to convince the reader that you won't get blobs but some real channels being resolved at least with more than 10 grid points.
As of the current state, major revisions are warmly suggested.
-- References
Connolly, J. A. D., & Podladchikov, Y. Y. (1998). Compaction-driven fluid flow in viscoelastic rock. Geodinamica Acta, 11(2-3), 55-84.
https://www.tandfonline.com/doi/abs/10.1080/09853111.1998.11105311
Räss, L., Simon, N. S., & Podladchikov, Y. Y. (2018). Spontaneous formation of fluid escape pipes from subsurface reservoirs. Scientific reports, 8(1), 1-11.
https://www.nature.com/articles/s41598-018-29485-5
Räss, L., Duretz, T., & Podladchikov, Y. Y. (2019). Resolving hydromechanical coupling in two and three dimensions: spontaneous channelling of porous fluids owing to decompaction weakening. Geophysical Journal International, 218(3), 1591-1616.
https://academic.oup.com/gji/article/216/1/365/5140152?casa_token=ffbIm7VK8EsAAAAA:L7LuROOcMXTBgosYEdylrae-1rhNCS2E_kVvfn9aOpM3-LnRn5RFtmHEvvFOLvpPlCRssxARrVVa4Zg
Keller, T., May, D. A., & Kaus, B. J. (2013). Numerical modelling of magma dynamics coupled to tectonic deformation of lithosphere and crust. Geophysical Journal International, 195(3), 1406-1442.
https://academic.oup.com/gji/article-abstract/195/3/1406/2874184
Yarushina, V. M., & Podladchikov, Y. Y. (2015). (De) compaction of porous viscoelastoplastic media: Model formulation. Journal of Geophysical Research: Solid Earth, 120(6), 4146-4170.
https://agupubs.onlinelibrary.wiley.com/doi/abs/10.1002/2014JB011258 |