the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 14 Apr 2026
Modeling thermodynamically consistent phase transitions in multi-component assemblages: An entropy method for geodynamic models
Abstract. Phase transitions strongly influence mantle convection as their effects on buoyancy can hinder or accelerate slabs and plumes. In a heterogeneous mantle, different mineral assemblages undergo phase transitions at different depths, leading to lateral buoyancy variations that can cause specific compositions to stagnate or accumulate within characteristic depth ranges. However, complex phase relations, abrupt changes in material properties, and the release and absorption of latent heat pose significant challenges for modeling phase transitions. Our previous work addressed these challenges by formulating the energy equation in terms of entropy rather than temperature, but remained limited to chemically homogeneous models.
Here we extend the entropy formulation to multiple components. By solving one entropy advection equation for each chemical component and then thermally equilibrating all components, our method enables a thermodynamically consistent treatment of phase transitions in multi-component systems. Our tests demonstrate that the method accurately conserves energy, and remains robust even for degenerate cases. We show its applicability in a series of global convection models, which reveal that small differences in phase relations between a pyrolitic equilibrium assemblage and a basalt–harzburgite mechanical mixture with the same composition can lead to major differences in convection patterns. Our results highlight the importance of accurately capturing the full effects of phase transitions in a chemically heterogeneous mantle, and our approach enables new investigations into how planetary interiors evolve.
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-1972', Anonymous Referee #1, 20 May 2026
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RC2: 'Comment on egusphere-2026-1972', Anonymous Referee #2, 01 Jul 2026
Dear editor, I have read the work of Li and co-authors regarding the thermodynamic approach they developed to solve the case of a convecting, chemically heterogeneous mantle. I have no major objections with the results of the authors, and I find that this work is needed since it presents a substantial contribution to the field. However, as you will see in my specific comments below, I have some major concerns regarding some of the statements of the authors. Perhaps I may be too pedantic, but the authors tackle a very challenging subject with a very long history. I hope that the authors find my comments useful.
Specific comments:
Abstract. L.2: Do you mean phase transitions in the strict sense or also mineral reactions in general. It is common in the geoscience community to confuse mineral reactions to phase transitions s.s. – especially for a system with many components that involves solid solutions.
L6: Your previous work did not formulate the energy equation in terms of entropy – this is a textbook result. Your work used the entropy formulation (which admittedly has advantages to the classic temperature forms under certain conditions).
L.9: Although your method shows potential and has advantages, it is not thermodynamically consistent in the strict sense of the term. My reasoning is as follows: the projected density approximation is still an approximation compared to the true density (as you have in equations 1-3). I am aware of your previous work and arguments (following Gassmoller et. al. 2020) against using the true density and pressure oscillations. The arguments used there are related to computational limitations rather than physics. This is not mentioned as a criticism to the authors for their numerical schemes or abilities – “soundproofing” of numerical, compressible codes has been a problem in many communities (i.e. atmospheric models). Therefore, I agree that this is a good approximation but calling it just “fully consistent” is a bit misleading. More consistent to previous attempts yes, "fully consistent” I would disagree.
L:20: citations to early works that brought attention to this matter in the framework of convection simulations would be needed here (e.g. Schubert & Turcotte, 1971 – Schubert et al. 1975 and goes on).
L:25: As before, the role of phase transitions was known before Faccenda and Dal Zilio, (2017). Please use “e.g.” or cite relevant earlier papers.
L:39 (citation to Billen, 2008) – As before. I understand that the authors need to cite something, but all these are general statements that are known for decades, please use “e.g.”. Otherwise, the reader who is not familiar to this literature may think that all these phenomena became known after the 2000s… I will stop making comments of the same kind because this is not my role to add references. However, I encourage the authors to give more credit to original literature (when general statements are made) or use “e.g.” to emphasize that these are examples. For example, all the work in compressible convection from Trubitsyn and co-workers, Ricard and co-workers has been ignored.
L:62: Simplified expressions are not necessarily bad. In fact, simple systems will have sharper phase transitions and “jumps” compared to more complex chemical systems. Thus, in such cases the hydrodynamic instabilities may be even more severe and produce more complex results in the hydrodynamic sense. Thus, it is not a matter of complexity, it’s the fact that you aim to be more realistic.
L:69: By the use of the word “transitions” in this sentence you mean mineral reactions in general. Please be specific to thermodynamic terminology.
L:71: Please add a reference to that statement. Which kind of solution you are referring to? There are numerous works on the Stefan problem that solve such things.
L:74: You didn’t introduce the entropy method; you used it and implemented it in this community. This formulation is known in textbooks.
L:103: Please add a reference on how the effective properties are obtained.
L:117: This is the heart of my criticism regarding thermodynamic consistency. The use of a hydrostatic pressure instead of the true pressure is a gross simplification that neglects important physics such as the delta V of reactions (and phase transitions) as a response to dynamic pressure variations. I understand why you do it and I do not object to that, but please change the wording the statements regarding “consistency” and accuracy. It is a better (or improved) approximation compared to what you had before.
L: 144-150: Your assumptions on the reactivity are a bit difficult to justify in general (or I am missing something here). How would you reconcile the case where the modal amounts of minerals change during P-T changes?
Eq. 9: If I understand correctly by “component” you mean mineral phases as mechanical mixtures. Correct? If yes, please state it – I had to go at the end of the manuscript to understand it. If not, please specify what you mean by “component” (i.e. chemical component)? If it is a chemical component, then your advection equation needs further justification (i.e. negligible diffusivity across phases etc).
L: 209: “Smoothing” the thermodynamic lookup tables is not always a consistent approach. It depends on resolution and on the accuracy of capturing the “step” in case of phase transitions. For example, in your figure 8 (left side) it is actually very poorly resolved at the boundaries around 24 and 10 GPa (i.e. it is not sure if you “hit” the transition or if your material goes through the “pixels” that indicate the transition).
L:297: In the problem of mass conservation, I noticed that your advection equation of the “component” lacks density and it is given in the non-conservative form. Thus, your “component” approach is prone to problems with mass conservation, how do you monitor it?
Citation: https://doi.org/10.5194/egusphere-2026-1972-RC2
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- 1
The manuscript under review refines and tests a method for coupling mantle (fluid) dynamics with mineralogical, solid-state phase change. The method casts the conservation of energy equation in terms of entropy and couples this evolution equation to mass and momentum conservation. The manuscript extends the method, which was previously described in Dannberg et al 2022, to handle multiple chemical components, and hence to enable the modelling of mantle heterogeneity. Phase-change in captured through a thermochemical look-up method whereby pressure and entropy enable the calculation of thermodynamic properties including phase stability. The method is verified in a large number of well-described and relevant tests, and is then applied to two scenarios where phase-change affects the dynamics.
On the whole, this is a useful scientific contribution that is presented clearly. The demonstration application is intriguing. There are various points that the authors should address before this is published, but I think they are minor. So I see no significant barriers to the publication of this work.
- The first issue that I see is clarity in the meaning of key words. The manuscript deals with phase-change --- where one mineral undergoes a change in atomic structure to form another mineral (or pair of minerals). But the manuscript repeatedly refers to "components" in a way that is confusing. Thermochemical components are chemical building blocks of phases that behave as inseparable units. They can recombine in reactions and the comprise the phases. The phases are the minerals, which may undergo phase change without changes in composition (i.e., the mole fractions of components). So sentences in the manuscript like "multiple compositionally-distinct components" (line 80) are confusing and suggest that these words aren't used in the standard way. Since accepted definitions are what makes words useful, the usage of "phase" and "component" should be clarified. Because the theory assumes no chemical reaction, the distinction is somewhat blurred here --- phase change doesn't fractionate chemistry. But the words should be used correctly anyway.
- The manuscript invokes an entropy equation and cites a previous work where it was derived, but doesn't emphasise that it represents conservation of energy. And yet that's the real underlying physics here. Neither temperature nor entropy are conserved quantities. Although the derivation of the entropy equation need not be repeated here, it is essential that its physical basis be discussed. Furthermore, the conservation of energy should probably play a larger role in the manuscript elsewhere. For example, in re-equilibrating the phase temperatures after a phase-change has occurred, the approach is to conserve the total entropy. To do this, heat is added (or removed) from the phases until they are all at the same temperature but the bulk entropy is constant. This leaves open the possibility that the sum of the heat increments is not zero --- and hence that energy is not conserved. Of course there are other violations of energy conservation that we ignore (work associated with density changes during equilibration, surface energy differences of mineral phases, etc) and this is reasonable. But I think that the equilibration step should use energy conservation as a constraint rather than entropy. The sum of heat increments should equilibrate the temperature and sum to zero -- even if this changes the total entropy.
- The entropy approach that is used here is very similar in motivation and in implementation to the enthalpy method that has been used in geodynamic models of melting. It would seem appropriate to make and discuss that link in the introduction.
- Advection of heterogeneity in numerical models would seem to be fundamental to the success of this theory, and yet there is almost no discussion of the numerical implementation of the advective terms in the equations. Do any of the verifications test whether the advection is working properly, and whether it can handle sharp(ish) gradients? Is energy still conserved when transport is two-dimensional and velocities are non-constant?
- The two demonstration calculations at the end of the manuscript are interesting, though the difference between them is arguably not very large. The authors are attempting to make an important point about how phase change affects the dynamics, and hence the overall stirring of the mantle. But they limit themselves to a very restricted set of only two scenarios that differ only in the number of "components" which I think means the number of phases. But an advantage of numerical models is that parameters can be varied at whim. So it would be nice to look at the effect of a single parameter (say A) that controls the dynamic importance of the phase change and test the sensitivity of the model to that parameter. The the authors could plot a relevant metric, say the Nusselt number Nu, versus A. What is d Nu / d A? Or something along those lines that enables a more vivid depiction of the sensitivity. I love figures 12 and 10, but to me the leading-order observation is that the Mechanical Mixture and the Single Component Pyrolite are essentially the same.
- line 13: I don't see major differences in convection patterns. I think this is an overstatement.
- line 31 "compositional differences in phase relations" this phrase is confusing.
- equations should be treated as part of sentences in terms of punctuation. Use of colons before equations is not good style; commas following equations are needed where the sentence continues.
- line 98: this is where the enthalpy method and the entropy formulation are very closely aligned in terms of motivation.
- line 123: better not to start sentences with mathematical symbols
- line 124: probably you mean provoking or instigating or something else. "invoking" doesn't make sense here.
- eqn (7): if Xi is a mass fraction and the theory has non-zero divergence, then I think this equation is missing the term X_i div(u)
- eqn (8): why incorporate q_eq,i into this equation and then solve the equation without this term? a bit misleading.
- line 144: "individual chemical components do not chemically react with each other" is ambiguous but important and should be clarified. I think that what this means is that phases can undergo phase-change, but there is no exchange of mass between, co-existing phases that have differing compositions. Or something like that.
- line 164: odd to call the energy conservation equation the "advection equation" because it is much more than just advection.
- line 168: what is the convergence criterion?
- It is somewhat unclear to me how Figure 2 shows that the "partitioning of entropy between the compositions becomes undefined" and generally it is surprising that the entropy approach doesn't distribute the phase change over a range of entropy such that there is no step. So I think a bit more explanation is needed here. If a Gaussian filter is really needed, then it is imperative to note the parameters and their values that are entailed in the smoothing, and how those parameter values are chosen.
-line 228 and elsewhere: why should you need a boundary condition on stress when the velocity is prescribed?
- line 243: "both sides" should probably be both ends, as a cylindrical pipe doesn't have two sides. And since this model has variations in density, it is surprising that the velocity (ie the volume flux) is identical at both ends of the pipe. The mass rate should be identical by conservation.
- line 264. The thermal diffusion problem can be solved analytically by periodic extension and Fourier series.
- line 207. What is the order of accuracy of the time-stepping scheme?
- line 313. thermal equilibrium with no density change is hardly an "extreme" case -- perhaps an endmember case.
- FIgure 8. I didn't see where this figure was described or used.
- line 482. hot thermal instabilities doesn't really make sense. I think you mean hot thermal plumes. But hot thermal is redundant. So how about just hot plumes? An instability is a bifurcation in state space. In this case, the instability leads a laterally uniform system to produce upwelling plumes and downwelling plumes.
- line 534, "even small variations in phase transitions can lead to substantial differences in mantle convection style" -- this is entirely subjective at present, and I think is an overstatement of the actual results. But what is really needed is to quantify the sensitivity so that adjectives aren't needed. Small compared to what? Significant in what sense? There should be a way to answer these questions quantitatively.
- Throughout. For a paper as long as this one, it would be easier to read if the figure references were at the start of sentences, and the main text actually described what the figure shows.