the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
Quantifying mantle mixing through configurational Entropy
Abstract. Geodynamic models of mantle convection provide a powerful tool to obtain insights into the structure and composition of the Earth’s mantle that resulted from a long history of differentiating and mixing. Comparing such models with geophysical and geochemical observations is challenging as these datasets often sample entirely different temporal and spatial scales. Here, we explore the use of configurational entropy, based on tracer and compositional distribution on a global and local scale. We show means to calculate configurational entropy in a 2D annulus and find that these calculations may be used to quantitatively compare longterm geodynamic models with each other. The entropy may be used to analyze, with a single measure, the mixed state of the mantle as a whole and may also be useful to validate numerical models against local anomalies in the mantle that may be inferred from seismological or geochemical observations.
Status: open (extended)

CC1: 'Review by Nicolas Coltice', Nicolas Coltice, 13 Feb 2024
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The manuscript presents a geodynamic study that investigate the use of configurational entropy to characterise the mixing property of a model, and comparing a variety of models. One expressed goal is to use this measure to understand some geophysical and geochemical observations. Maybe it is because it is a paper about mixing, but I have a mixed analysis of the work. What I appreciate the most is the quality of the geodynamics models, of the data analysis and how the results are presented. It is impressive to have that in a paper, and it provides a lot of confidence in the outcomes. The shortcomings to me come from the target of the study, which is not yet found. My feeling is that it comes from the lack of inclusion of the literature about mantle mixing (maybe because most of it was published before 2010), and the duration of the simulation being 1 Gy that is too short to discuss primordial heterogeneities. The first studies of mantle mixing date back to the 1980’s and one of the difficulty was to deal with scales. So 2 approaches were taken: one dealing explicitly with scales, such as the authors do here with configurational entropy. It was in relationship with the observations of heterogeneities at all scales (marble cake like at first, more comprehensive nowadays with midocean ridge heterogeneities etc…). The other approach is to work with the theory of dynamic systems, because mantle convection is chaotic. The use of Lyapunov exponent characterize the local and global properties of stretching of heterogeneities. Mixing happens when an heterogeneity is stretched at a scale at which chemical diffusion operates. This scale for the Earth is about 1cm I would say (some papers talk about it). Lyapunov exponent are a form of stretching rate when mixing is chaotic. I think Ferrachat and Ricard introduced it it 1998. Configurational entropy and Lyapunov exponent can be related with Ergodicity theory. It is fundamental to acknowledge the literature more extensively here and situate the innovation of the study. As the authors say, Naliboff and Kellogg in 2007 already used this measure, so the paper cannot really be the introduction of this measure. It has to be more than this (which I think it can be, but more explicitly). And it is also important to relate this measure to the Lyapunov exponent studies. At the end of the day, the stretching rate scales with the velocity in mantle convection. So faster velocities imply better mixing, which is what is observed here in the models as well. There is an issue I think with the duration of the simulation. They are 1Gy long. It seems long, but it is not for a mixing study because (1) radiogenic geochemical heterogeneities needs time to develop and they mostly show preservation of >2Gy, (2) 1Gy is a little more than one mantle overturn today if we consider a slab takes 200 My to reach the bottom of the mantle, (3) primordial mantle means it is almost as old as the Earth and (4) mixing/erasing heterogeneities at melting regions rates in the ancient Earth seemed much faster than today. For the geodynamic community, 1Gy of preservation at the slow mantle convection rate of today is no surprise and does not necessarily teaches us about preservation of primordial heterogeneities. When working with Lyapunov exponent, it is important to run the simulation for a time sufficient to obtain finite time Lyapunov exponent. I am not sure 1 Gy would be sufficient here. So in this study, the setup and the methods are great, as good as they can be, but my appreciation is that the target of the study is not completely mature. I suggest refining the research objective, as aligning the study with previous work and integrating observations could significantly enhance its relevance. Ultimately, this approach could elevate the study, making it a substantial contribution to the field. A minor detail: in cylindrical geometry the volume of the lower mantle is larger than in spherical geometry. To compare with Earth, maybe the geometry regions that mix in the study could be scaled by volume instead of depth? For the literature, I suggest to integrate papers by D. Turcotte, with L. Kellogg, also works of M. Gurnis and G. Davies, works of S. Ferrachat, Y. Ricard, of H. Samuel, of U. Christensen, P. Olson, P. Van Keken.
Citation: https://doi.org/10.5194/egusphere20232889CC1 
RC1: 'Comment on egusphere20232889', Anonymous Referee #1, 15 Feb 2024
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General Comments
This is a compact contribution that reintroduces the idea of configurational entropy to quantify mixing to the geosciences. The concept of configurational entropy to quantify mixing has been discussed in the geoscience literature before, and citations to such work are provided in the text. Where this extends on earlier work is that it extends the equations to cases where one has more than two types of objects / particles. This is a potentially very useful extension. They also go through some simple cases in the appendix which can help solidify the reader’s understanding. The manuscript is concise and well written.
The work, after defining the equations for configurational entropy in such cases, applies it to 2D annulus mantle convection simulations. These simulations have already been published elsewhere. These examples show how the global entropy measure quantifies mixing in a single number – and shows visually that both the local and global measure of configurational entropy correspond qualitatively with the amount of mixing observed. The work is not sufficiently extensive to talk in detail about the implications for mantle mixing, and the authors do not attempt that, but it does allow them to speculate about future uses.
It is unclear how useful this measure will be though. We will need to wait to see how researchers use it. It is also unclear what the absolute value means, and it is dependent on spatial resolution (as mentioned by the authors).
The contribution could have taken the opportunity discuss how the work here relates to the wider body of mantle mixing studies (e.g. Kellogg and Turcotte, McKenzie, Davies and Gurnis, Ferrachat and Ricard, Samuel and Farnetani, Tackley, van Keken, Olson, etc) . This literature is extensive, and some of it relates (indirectly) to configurational entropy. While these earlier studies do need to be better recognised here, careful consideration should be given as to the extent of additional description. While significant additional description of earlier mixing studies would make for a more rounded contribution, especially if the discussion section tried to draw relationships from this work with earlier work introduced in an extended introduction. Such additional material though could potentially detract the reader from what is now a tight and focussed contribution. Therefore if the authors cannot see a way to make this a much more significant contribution by relating it to earlier work, then I would suggest that they restrict themselves to succinctly acknowledging and summarising the earlier work in this general field of mantle mixing to maintain its clear and concise form.
Overall, my assessment is that this is a useful contribution that deserves to be in the literature but only time will tell how significant it really will be. In this context we note that uses to date of earlier versions of configurational entropy have been limited in the geosciences, but maybe the additional flexibility of the measures presented here will encourage greater use.
Specific Comments
L 11 – Unclear how the measure can ‘validate’ a numerical model? Also unclear how can a model be ‘validated’ against local anomalies in the mantle inferred from other observations?
L63/64 – While Shannon brought Entropy from a data perspective to people’s attention, he did not talk about Configurational Entropy in that reference – nor how fast information on compositional particle distribution is lost through flow. I accept that there is a relationship between standard configurational entropy and Shannon’s information entropy. What is “fast information”? I think this sentence and reference needs a bit of work.
L79/80 – a bit strange to talk about – conditional probability – for a deterministic system. Maybe it is the conditional probability of finding this group of particles of composition c in cell j out of all other possible configurations. I appreciate that entropy related work is frequently discussed in terms of probability. Maybe it could instead be described as just something like the local proportion of particles of composition c (measured in terms of density) in cell j, relative to the total number of particles (measured in terms of density) in cell j.
L82 – again – maybe rather than the probability for the cellsum – maybe a more deterministic description can be given here also. Is it just the proportion of all particles (again measured in terms of particle density) in cell j?
If this suggestion is taken up for describing these terms, I think it would also be OK to later or before include a statement pointing out that in statistical physics similar terms would be considered probabilities.
L116/117 – The reference quoted states average RMS plate velocities over past 200 Ma of around 4 cm/yr, but your model R presents mean average surface velocities of around 2 cm/yr. I am not sure that is really close enough to say that it is in the range of reconstructed values. Maybe the sentence should be more specific – “the mean surface velocities in the model were x cm/yr, which can be compared with y cm/yr reconstructed in Zahirovic et al., 2015.”
L288 – I feel that ‘primordial’ could be an emotive word here. For most whole Earth geoscientists, primordial suggests something that has survived since Earth’s formation. I appreciate that ‘primordial’ here is taken to mean from the start of the simulation, but I think a more straightforward expression (with less ‘baggage’) could be used. Maybe ‘original’. Speed readers might think that you have demonstrated that large regions of the lower mantle are likely to survive from Earth formation, not just 1000 Myr.
Technical Corrections
L 16 – ‘stooled’?
L37 – ‘entirely spatial’ – missing word? Different?
L 48 – ‘model the’– missing word? with?
L69 – not sure if this is a general definition of entropy. I think it is a definition of configurational entropy.
L75 ‘amount particles’ > amount of particles
L257 – ‘spherical’ resolution – unclear what you mean by spherical here? Do you mean lateral, or …?
L273 – “that has stays”?
L276278 – As regards the “a local entropy of 1”, it reads as if you mean the lower mantle composition  where? Anywhere? but that does not make sense  maybe you mean  " and therefore 'the local entropy above 660km' cannot have a local entropy of 1"?
L308309 – ‘illustrates ….. successfully quantifies mixing states’. While this might be suggested visually in a qualitative sense, I am not convinced that it has been shown in a quantitative way. I think it deserves a more accurate and weaker statement. Something that talks to the fact that this was a visual comparison that supports that configuration entropy gives the right ranking of mixing states.
Line 311. From what time in the simulation is this? I presume at the end?
Start of Appendix A – with the equations, I wonder whether n_c,j, N_c, M and C can be defined again here. Everything else is in the equations. It would then be complete.
Line 435  “which is the global entropy is”
Citation: https://doi.org/10.5194/egusphere20232889RC1
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