General Comments

This manuscript presents a compelling exploration of the effects of branching and curved phase transitions on the stagnation of downgoing plates or cold anomalies in the upper and lower mantle. The study investigates the hypothesis that these transitions influence the likelihood of slab stagnation at various depths. The findings suggest that while these transitions may exert stagnation-supporting forces, the magnitude of these forces is insufficient to produce a discernible impact in global convection models. As such, the manuscript qualifies as a null-result paper—a less common but nonetheless important contribution to the field.

The manuscript is well-written, and the results are presented in a clear and logical manner. However, some figures could benefit from refinement (detailed suggestions are provided below). The structure of the paper is somewhat unconventional, with Sections 2 and 3 each resembling standalone studies, while the discussion and conclusions synthesise findings from both sections.

My primary concern is the limited motivation provided for conducting and publishing this study. While the authors cite two references that propose the ‘branching’ mechanism (Cottaar and Deuss, 2016; Chanyshev et al., 2022) and one for the ‘curving’ mechanism (Ishii et al., 2023) as contributors to slab stagnation, the rationale for exploring these mechanisms further is not sufficiently emphasised. I would encourage the authors to elaborate on why these mechanisms are worth investigating and, even if they are shown to have minimal relevance for Earth-like conditions in their models, to identify the conditions under which they might play a more significant role. The authors briefly address this for the ‘branching’ mechanism, suggesting relevance for stagnating flat slabs, but do not provide a similar discussion for the ‘curving’ mechanism. At this stage, it is unclear whether addressing this issue would require additional experiments (which would constitute a major revision) or could be achieved using existing results (a minor revision).

Specific Comments

1. Lines 120–123:

“For a downgoing body whose temperature is 500 K below the critical temperature of the reactions ‘A’, ‘B’, and ‘Z’, and with Clapeyron slopes of γ_A = +1.5 and γ_B = −6 MPa/K, we estimate a maximum separation between the phase transition surfaces inside the downgoing body as on the order of 100 km”.

Could you include this calculation, perhaps in the supplementary materials?

1. Line 198:

Given the limitations of the models discussed in Section 3, it might be helpful to note at the beginning of Section 2 that the initial set of models is not intended to be Earth-realistic. A cross-reference to the explanation in Section 3 would be beneficial for readers who may skim certain sections.

Figure-Specific Comments

1. Figure 1:
   
   Consider marking the upper material (density ρ₁) with a colour to make its presence more apparent. Additionally, reposition the text boxes for ρ₁ and ρ₂ to clearly associate them with the bulk material, avoiding any potential confusion with density variations along the dotted line.
2. Figures 2 and 3:
   
   These figures effectively summarise the phenomena under investigation. You might consider merging them into a single figure with two or three panels for better visual coherence.
3. Figures 5, 6, and 7:
   • Merge these figures into a single composite figure with six panels. Labelling each panel (e.g., with text in the red centre or a corner) would facilitate direct comparison between simulations, particularly since the text frequently refers to differences between Figures 6 and 7.
   • To address the local versus global nature of stagnation phenomena, consider adding supplementary material showing the 3D variations of your results. Options include additional slices, volume elements (similar to Figure 10), or a video of a rotating cross-section.
4. Figure 8:
   
   The diagonal line separating the two regimes does not appear to be well-supported by the data. A more accurate representation might involve marking the region between γ_cool = -17 and -13 as a transitional zone for any T_710c value. Please clarify in the text how you inferred the slope of the line and why it is presented as such.
5. Figure 9:

• Increase the font size for axes and labels.
• Provide a rationale for the behaviour of the radial viscosity factor, presumably designed to replicate Earth’s mantle structure.
• Clarify the factor’s values at the bottom of the lower mantle and in the upper mantle, as the graph suggests these may approach zero, which would imply η = 0 according to Equation 7.

Minor Typographical Errors

1. Line 46: “consider these morphologies and to consider and in particular” – likely an extra “and”
2. Line 105: “the model is run”
3. Line 199: “the simulated mid-mantles”
4. Line 281: “the full depth of the Mantle” – unnecessary capitalisation of “Mantle”

I hope these suggestions help refine and strengthen your manuscript.

This is an interesting and generally well-written contribution. There is a lot of discussion about phase transformations so this work is timely. I noted the presentation as “good” because I struggled with some of the figures and felt the author’s need to do better. The text is clear and concise—almost terse—but fully understandable. With some revision to the figures the presentation would be excellent.

I struggle because the authors neglect temperature-dependent rheology until the last section, especially because we have shown that it is important in layering 20+ years ago (King and Ita, 1995) but, I recognize that the authors are trying to do a clean series of calculations and keep things as simple as possible. At the very least, they need to acknowledge the important role temperature-dependent rheology can play.

The work uses the Bousinessq approximation and does not include the latent heat associated with the phase changes. That is the correct approximation (latent heat has the same non-dimensional terms as adiabatic compression and shear heating, so if you include it you should at least use extended Bousinessq). My question is have you thought about the role of latent heat? Is it secondary? You should alert the reader to this upfront. An old but useful ? reference might be Ita and King, 1994 where we found the formulation of the equations wasn’t a major factor suggesting that is right. I don’t know whether this 30 year old work stands the test of time or not.

Line 37: This is odd coming from me—because I am a big fan of non-dimensional formulations—but it would help you communicate to the non-geodynamics deep earth reader if you listed the related Clapeyron slopes along with values of P here.

Line 40: You should list Table 1 before listing Table 2 (or reverse the order)

Line 48: Actually, Ita and King, 1994 and 1998 did what you describe in the much more distance past… at least for the olivine system reactions, there wasn’t enough pyroxene/garnet data to do that part.

Lines 54: I’m not sure Branching or Curving should be capitalized… that’s a copy editor thing.

Line 78: Post-Garnet -> post-garnet

Lines 84-85: Curving -> curving; Branching -> branching

Figure 3: Post-Garnet -> post-garnet

Line 106: Because you are focused on slabs/downwellings, this seems to be a real short coming. You should call attention to it for the reader. Here I think about Christensen, 1984 “In almost all cases power-law rheology leads to considerably different flow patterns and heat transfer properties than those predicted for Newtonian convection.”

Line 108: Olivine-out -> olivine-out

Figure 5, lines 144-146: presenting two slices through a 3D model is not very intuitive as we know (e.g., Tackley et al., 1993) that different behavior can be happening in different parts of the sphere. I was going to suggest showing radial correlation functions or maps at the 660 km depth but, I saw later that you present radial temperature histograms (Fig. 11) and so you must have the code to do this. I would find that more persuasive. I realize the challenge is that when you have some slabs stagnating and some not this might be misleading but, I think those would be more reliable than the single slices.

 Lines 160-164: I wonder if you can come up with a way to quantify stagnation or not a bit better. I think we used to use things like the reduction in radial velocity near the transformation. I admit it might suffer the challenges I brought up regarding radial correlation functions but, I find the reliance on patterns—especially when not shown as in this passage—to be unsatisfying.

Line 169-168: This wording is a bit cumbersome. Maybe something more like, “As we increase the proportion of the donwelling subject to counter-convective forcing, stagnation becomes more likely.

Figures 6 and 7: Here I found the images and the text unsatisfying. I believe you would make a stronger case if you had a more quantitative measure. It took a lot of flipping back and forth to try and see the difference between these.

Figure 8: The slope of the grey band (not rough region) is not constrained by the calculations. I assume the authors are using theory to guide the slope.  It is unfortunate that they have so many calculations with Pcool greater than -0.025 but, I would not suggest they leave any off. It appears that those calculations could have been used to better determine the line (ahh, hindsight is 20/20). Plotting the change of regime suggested by Ishii et al. (2023) would help to make the point in lines 184-185. My impression is that the slope is not the important point of the figure, the point is that the change happens well short of where Ishii et al. predict.

Lines 212-216: There is a problem with the Frank-Kamenetskii rheology (equation 7) when used for slabs. Because of the exponential it is too weak (see Javaheri et al, 2024 or King, 2009). It has been shown some time ago that rheology matters (King and Ita 2005).

Section 3.2 I find this section to be more persuasive because of Figure 11. Adding one more calculation without the temperature-dependent rheology would be useful and would mitigate some of my concerns over rheology above.

References:

Ita, J. J. and S. D. King, The influence of thermodynamic formulation on simulations of subduction zone geometry and history, Geophysical Research Letters, 125, 1463-1466, 1998.

Ita, J. J., and S. D. King, The sensitivity of convection with an endothermic phase change to the form of governing equations, initial conditions, aspect ratio, and equation of state, Journal of Geophysical Research, 99, 15,919-15,938, 1994.

King, S. D., On topography and geoid from 2D stagnant-lid convection calculations, Geochemistry, Geophysics, Geosystems, 10, Q3002, 2009.  doi:10.1029/2008GC002250

King, S. D., and J. J. Ita, The effect of slab rheology on mass transport across a phase transition boundary, Journal of Geophysical Research, 100, 20,211-20,222, 1995.

Pejvak Javaheri, Julian P. Lowman, Paul J. Tackley, 2024. Spherical geometry convection in a fluid with an Arrhenius thermal viscosity dependence: The impact of core size and surface temperature on the scaling of stagnant-lid thickness and internal temperature, Physics of the Earth and Planetary Interiors, 349, 107157, https://doi.org/10.1016/j.pepi.2024.107157.

Tackley, P.J., Stevenson, D.J., Glatzmaier, G.A. and Schubert, G., 1993. Effects of an endothermic phase transition at 670 km depth in a spherical model of convection in the Earth's mantle. Nature, 361(6414), pp.699-704.