the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 08 Jun 2022
Influence of heterogeneous thermal conductivity on the long-term evolution of the lower mantle thermochemical structure: implications for primordial reservoirs
Abstract. The long-term evolution of the mantle is simulated using 2D spherical annulus geometry to examine the effect of heterogeneous thermal conductivity on the stability of reservoirs of primordial material. In numerical models, mantle conductivity is often emulated using purely depth-dependent profiles (taking on values between 3 and 9 Wm-1 K-1). This approach is meant to synthesize the mean conductivities of mantle materials at their respective conditions in-situ. However, because conductivity depends also on temperature and composition, the effects of these dependencies in mantle conductivity is masked. This issue is significant because dynamically evolving temperature and composition introduce lateral variations in conductivity, especially in the deep-mantle. Minimum and maximum variations in conductivity are due to the temperatures of plumes and slabs, respectively, and depth-dependence directly controls the amplitude of the conductivity (and its variations) across the mantle depth. Our simulations allow assessing the consequences of these variations on mantle dynamics, in combination with the reduction of thermochemical pile conductivity with iron composition, which has so far not been well examined. We find that the temperature- and depth- variations combined characterize the mean conductivity ratio from top-to-bottom. For the mean conductivity profile to be comparable to the conductivity often assumed in numerical models, the depth- dependent ratio must be at least 9 times the surface conductivity. When the conductivity profile is underestimated, the imparted thermal buoyancy (from heat-producing element (HPE) enrichment) destabilizes the reservoirs and influences core-mantle boundary (CMB) coverage configuration and the onset of entrainment. The compositional correction for conductivity only plays a minor role that behaves similarly to conductivity reduction due to temperature. Nevertheless, this effect may be amplified when depth- dependence is increased. For the cases we examine, when the lowermost mantle's mean conductivity is greater than the surface conductivity, reservoirs can remain stable for periods exceeding the age of the solar system.
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RC1: 'Comment on egusphere-2022-418', Anonymous Referee #1, 30 Jul 2022
This study uses mantle convection simulations to address the influence of pressure-, temperature- and composition-dependence of thermal conductivity on the fate of a dense primitive layer sitting at the bottom of the mantle, mimicking the LL(S)VPs on the Earth. While previous studies have investigated the effect of those various dependencies of thermal conductivity have been investigated individually, the novelty of this work is to consider consider the interplay between them in the context of the Earth’s lower mantle structure. While I think that the results are worth being published, their presentation in the current manuscript is hard to follow and lacks a clear narrative thread, and more importantly, lacks in interpretation. Therefore, I think the paper would benefit from a thorough re-organization of the results section.
General
The study adopts a rather complex baseline set up (mixed basal + internal heating, phase transition, yield stress) to study the effect of pressure, temperature and composition on thermal conductivity. While my preference goes to simpler models when trying to unravel such systematic trends, one can argue that the current set-up is relevant for an Earth-like case, and I think it is fair.
I find the introduction a bit too light. I would expect some more background on the physics of the various dependencies of thermal conductivity, and in particular that the trends be clearly announced: increase of pressure results in increase of thermal conductivity, increase in temperature results in decrease in thermal conductivity, increase in iron content results in decrease in thermal conductivity (it seems). Also, it seems that considering a variable thermal conductivity is particularly relevant for compressible convection. If this is indeed the case, I would expect more time to be spent discussing why.
In my opinion, the main weakness of the paper in its current state is the lack of apparent organization in its exploration of the parameter space, which makes it pretty confused and fails at highlighting general tendencies in the effect of the different dependences of the thermal conductivity. This is probably due in part to the simultaneous changes in investigated parameters and diagnostic quantities from one section to the other (first time-evolution of the heat flux varying KC and n, then instantaneous mantle structures varying everything, and finally time-evolution of entrainment again varying KC and n). While I seem to get the idea in the progression, I had a really hard time getting a clear picture out of this section. Maybe starting by isolating the effect of each dependencies, and then considering their correlations could make things simpler. Maybe also distinguishing more systematically between the effect on mantle flow (pile structure) and on the time-evolution (e.g. of the CMB and surface heat fluxes) could help. Another possibility would be to add dependencies successively to a fiducial model, for which a preferred value of KD would be selected before considering the effect of n, and in turn a preferred value of n would be selected before adding a third layer with KC. That could spare some cases of the parameter space if there are reason to think they are less relevant. Thse are suggestion, hopefully that can help. Anyway, one important thing which is lacking in my opinion is also some theoretical speculation on the expected effect of the parameters that should come before presenting the results, and would help their interpretation (e.g. we expect the effect of increaing KD to increase the thermal conductivity in the lower mantle and thus to make it more stable to convection (it decreses Ra) by homogenizing its temperature, potentially building heat… etc.). I think it is better for the reader to be incited to think in advance of seeing the results.
Specific
This article will probably mainly be read by people familiar with the equations of mantle convections. Nevertheless, I think it would be beneficial to write down the conservation equations (they are not even in the supplementary materials!), at least the heat conservation where the thermal conductivity appears, as it would make clear where the supplementary mechanism induced by varying thermal conductivity operate.
A plot of the thermal conductivity profile corresponding to the reference state for the various KD, KC and n (not all of them but the few most relevant combinations, e.g. the cases presented in Figure 2) would be insightful. It could be a supplementary panel in Figure 1, or a stand-alone figure.
I don’t really understand why simulations run until 11 Gyr, but some values are avergaed aroung 4.5 Gyr. Anyway, the averaged values reported in Table 1 do not seem to be used.
Dimensional and non-dimensional quantities are often mixed, which lacks a bit of rigor.
Technical
L. 48: “compositional- dependent” ← “composition-dependent”
L. 84: “a quadratic that smoothly” ← “a quadratic curve that smoothly”
Figure 1: the markers for “<200 ppm water” are hard to see, please change for a more contrasting colour.
L. 118: “We first defined a purely depth- dependent reference case characterized by depth- dependence, K_D = 2.5, with lower mantle conductivities comparable to current estimates” in Figure 1, it seems that K_D = 10 is the closest match to lower mantle estimates.
L. 121: “by approximately 75%” it seems less than that in Figure 2.
L. 122: “in agreement with Li et al. (2022) findings” ← “in agreement with Li et al. (2022)’s findings”
L. 150: “We observed that T_prim increased with greater temperature dependence (top-to-bottom rows in Figure 3).” I don’t see it.
For the snapshot figures, when a parameter is held constant, write it in the caption rather than for each snapshot. That will lighten a bit the figures.
Figure 5: Please alternate colormaps between the different quantities plotted. In particular, don’t take a “divergent” colormap for the primordial mantle one which only has two extremal values, which are both very dark and hard to distinguish in the current plot.
L. 172: “and is equivalent to” ← “and which is equivalent to” ?
Figure 6: Same as figure 5: it is very hard to know which colormap corresponds to the snapshot and which corresponds to the time-evolution plots. Also please change colours for the onset of instability (the magenta is pretty hard to see).
L. 189: “may not compensate (or be exceeded) by” ← “may not compensate (or be exceeded by)”
Supplementary Materials
L. 43: “and the depth variation of thermal expansivity imply a depth average” ← “and the depth variation of thermal expansivity implies a depth average”
L. 176: “The density anomalies [...] is calculated” ← “The density anomalies [...] are calculated”
Figure S1: the y-axis is non-dimensional height, not depth.
Figure S4: What do line styles correspond to?
Citation: https://doi.org/10.5194/egusphere-2022-418-RC1 -
AC1: 'Reply on RC1', Joshua Guerrero, 22 Aug 2022
The authors appreciate the comments made by R1 and we address each below. We have decided to expand the introduction and
rearrange the results section so that the manuscript is easier to follow.General Comments
-Yes, the mantle convection model set-up we consider is intended to be Earth-like. However, we do not intend on reproducing
the Earth mantle's history. Within this framework we have been able to examine the effect of the thermal conductivity model
on the evolution of the primordial layer at the bottom of the mantle.-We agree with R1 that the introduction can be expanded. Regarding the thermal conductivity, each of the component (dependency)
of the conductivity model will be discussed. In addition, we will now address why conductivity is relevant for compressible mantle
convection.-We consider the suggestions that R1 outlines for reorganizing the results section. As the manuscript stands, introducing the
reference case K_D = 2.5 first is too much too quickly for the reader. We now separate all the effects so that the progression
of the results is easier to digest. The subsections of the results are now:1) Effect of a purely depth- dependent conductivity
2) Effect of a temperature- and depth- dependent conductivity
3) Including the effect of composition- dependent conductivity
4) Long-term stability of thermochemical reservoirs featuring mineral physics derived conductivitiesSpecific Comments
-The heat conservation equation will now be added to the Methods section to help the reader understand where the conductivity
model influences mantle dynamics.-A new figure with a plot of the initial thermal conductivity profiles for cases featuring mineral physics defined
conductivities are now included so that the conductivity reductions from different temperature- and composition- dependencies
is clarified.-The simulation time is long (~11 Gyr) to allow the system to reach a statistically steady state and to investigate how the
heterogeneous conductivity will affect the long-term evolution of the primordial layer. The simulation averages are taken
towards the start of simulations' statistically steady state (approximately 4.5 Gyr). Again, we do not intend to model the exact
evolution of the Earth's mantle. The conditions for simulating Earth's history (i.e., initial conditions, decaying heat sources,
and cooling bottom boundary) are not included in our model setup. In the current version of the manuscript, the averaged values
were inset within the annulus snapshots and their trends were discussed within the results subsections. Specific values were not
used often so that the text would not be inundated with numbers. We now incorporate more references to the Table 1 averages into
the manuscript.-The values presented in the manuscript will be presented as dimensional.
Technical Comments:
-L. 48: for clarity, "composition- dependent" will now be used throughout the manuscript.
-L. 84: inserted the word 'curve'.
-Figure 1: markers for "< 200 ppm water" are changed to bright green to be more contrasting with the background.
-L. 118: The case with a purely depth- dependent thermal conductivity and K_D = 2.5 is considered as an analogue
for models that already account for the combined depth- and temperature- dependences. In the latter (hypothetical)
model of thermal conductivity, the depth- dependent component would have conductivity values that are much greater
than those defined by K_D = 2.5. To clarify, the sentence has been changed to
"As an analogue for a thermal conductivity model that accounts for depth- and temperature- dependence, we first
consider a purely depth- dependent reference case characterized by depth- dependence, K_D = 2.5. Our reference case
produces lower mantle conductivities that are comparable to current estimates (e.g., Deschamps & Hsieh, 2019; Geballe et al., 2020)."
-L. 121: "by approximately 75%", this refers to a comparison of heat flow values averaged about 4.5 Gyr for cases #4 and #8.
This comparison is between a purely depth- dependent conductivity case (#4) and heterogeneous conductivity case (#8).
The sentence has now been revised to read:
"Comparing the purely depth- dependent conductivity model to the completely heterogeneous conductivity model, the latter greatly
reduced both CMB and surface heat flows (by approximately 76% and 22%, respectively), as less heat can be extracted from the
base."
*The subsection including the timeseries figure has now been moved to the final subsection of the results section. It has now
been updated so that the cases are relevant to the mineral physics derived conductivity profile.
-L. 122: fixed the citation to read "... Li et al., (2022)'s findings".
-L. 150: The mean temperature of primordial material is indicated by T_{prim} (not to be confused with the global mean
temperature of the system <T>). T_{prim} is inset within the annulus and is indicated for each case. T_{prim} can be seen
decreasing for cases with the same K_D value as n is increased. In addition, the temperature field is offset with respect to the
CMB temperature so that the differences in hotter temperatures within the piles is easier to see. The boundary between regular
mantle material and primordial material is indicated by a green contour. When n is increased, the red colour within these
contours becomes more saturated.
-Snapshot figures have been revised so that superfluous conductivity model labels are omitted.
-Figure 5: The colormaps for each different field are now alternated.
-L. 172: inserted the word 'which'.
-Figure 6: The colormap is changed. The onset of instability is now indicated by a dashed cyan line.
-L. 189: The parantheses has been moved.
Supplemental Materials:
-L. 43: "imply" -> "implies".
-L. 176: "is" -> "are".
-Figure S1: "depth" -> "height".
-Figure S4: The case numbers have been added to the legend so that the line styles are now clear.Citation: https://doi.org/10.5194/egusphere-2022-418-AC1
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AC1: 'Reply on RC1', Joshua Guerrero, 22 Aug 2022
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RC2: 'Comment on egusphere-2022-418', Anonymous Referee #2, 02 Aug 2022
In this paper the authors carried out numerical experiments of thermochemical mantle convection by varying the spatial changes in thermal conductivity to investigate the temporal changes in the distributions of dense "primordial" materials initially imposed above the bottom boundary. My honest impression is, unfortunately, that the manuscript is very hard to follow because of its poor organization and description from the reasons summarized below. I therefore strongly suggest that the authors should thoroughly revise the manuscript before the reconsideration.
1. One of my major criticism is that the main issues in this study were not well described either in the abstract or introduction section. In my understanding the major intention of the authors is to reduce the thermal conductivity in the deep mantle to much lower levels than those simply expected from the dependence on temperature and pressure (or depth), in order to induce an "instability" from the initial layer of dense materials within a sufficiently short period of time. The authors should clearly indicate their ultimate motivation in earlier parts of the manuscript.
2. I am not well convinced of the meaningfulness of the onset time of "instability" from the initial layer of dense materials. It seems to me that the authors had assumed that the deformation of the basal layer of dense materials occurs only in an "intrinsic" manner owing to the thermal buoyancy. However, several earlier studies had demonstrated the ultimate importance of "extrinsic" deformation induced by the cold subduction from the top surface. I do not therefore think that the onset time of "instability" can be a good measure to investigate the influence the thermal conductivity at depth on the thermal buoyancy in the basal layer of dense materials.
3. I felt quite odd with the authors' exaggerated claim on the overall profiles of thermal conductivity including "We find that the temperature- and depth- variations combined characterize the mean conductivity ratio from top-to-bottom" in the abstract. It is quite obvious from the assumptions made by the authors themselves.
4. In Section 2.1 the authors should state the reason why the phase change from perovskite (pv) to post-perovskite (ppv) is ignored in their numerical model. If they believe that the pv-ppv phase change is of little importance on the dynamic behaviors of the basal layers of dense materials, the authors should make clear the reason why.
5. I was quite disgusted to see that the profile given by equation (2) is denoted by "KD=9.185". Such a denotation should be used only for the profiles given by equation (1) !!
6. Near equation (3) the authors should indicate the magnitude of the reduction in thermal conductivity due to the increase in temperature within the modeled domain by the choice of n=0.5 and n=0.8.
7. Near equation (4) and later, "compositional correction" should be rephrased with "compositional dependence". The word "CORRECTION" could imply that the numerical experiments without the compositional dependence in thermal conductivity are meaningless.
8. In Section 3.1 "QCMB" is used without explicitly defined.
9. To show the 2-D distributions of thermal conductivity in Figures 4 and 5, the authors should show the ratio of thermal conductivity to its surface value (KS) rather than the values of conductivity itself.
10. The paper by Marzotto et al. (2020) has not been cited anywhere in the main text.
11. Near equation (8) of the Supplement, the spatial coordinates should not be in Cartesian (x,y,z) but in 2-D polar (r,phi) in this study. Similarly in equation (12) of the Supplement, the coordinate is not in 3-D polar (r,theta,phi).
12. In Section 3.1 of the Supplement, I think that the authors can use the potential temperature instead of the "adiabatic correction" a(z).
13. The paper by Xu et al. (2004) has not been cited anywhere in the Supplement.Citation: https://doi.org/10.5194/egusphere-2022-418-RC2 -
CEC1: 'Reply on RC2', Susanne Buiter, 12 Aug 2022
Dear reviewer, dear authors,
I thank the reviewer for their feedback and suggestions for improvement of the manuscript. As a general note, i would like to use this opportunity to re-iterate that Solid Earth strives for polite and constructive language in all communications, including reviews. Specifically regarding this manuscript, i would ask the authors to ignore the wording that was used in comment 5 and to try to focus on the content of the feedback given. I hope this will work for everyone.
Susanne Buiter
Citation: https://doi.org/10.5194/egusphere-2022-418-CEC1 -
AC2: 'Reply on RC2', Joshua Guerrero, 22 Aug 2022
The authors appreciate the comments made by R2 and we address each below. In addition to the commments made by R1, we have
decided to thoroughly revise the manuscript so that it is easier to follow.1. In alignment with the comments of R1, we will expand the introduction section and make the intentions of our study clearer.
2. We calculate the onset time for instability from the temporal variations in the average height of dense material. It is true
that these variations in average height do not discriminate between 'intrinsic' (i.e., thermal buoyancy) or 'extrinsic'
(i.e., downwellings) deformation. From examining the average height timeseries and animations of the fluid flow we can see the
influence of downwellings. The first downwellings impringe on the initial dense layer but do not result in a rapid uplift of
material (sufficient to eject blobs of dense material). Once the initial dense layer has organized into piles, downwellings tend
to move dense material laterally over the CMB but not rapidly increasing the pile height. Furthermore, we find that it is easier
for downwelling currents to push primordial material that has been made lighter due to their retained heat. We agree that
downwellings are important in deforming the dense primordial layer. This mechanism will now be discussed in addition to the
thermal effect regarding instability.3. Right. This statement is a bit of a tautology. The intention for this statement is that for Earth-like models that consider a
variable thermal conductivity, it may be ill advised to isolate for just one dependency. On one hand, to simulate a 2-pile
configuration, a purely depth- dependent conductivity may be employed, but its top-to-bottom ratio should implicitly emulate the
temperature and composition effects. On the other hand, a purely temperature- dependent conductivity will result in the
entrainment of a dense primordial layer. By assuming a parametrized conductivity model, it is predictable what the mean
conductivity ratio from top-to-bottom will be (having specifed the temperature contrast and the dependencies of the conductivity
model).4. We will now discuss the reasons why the phase change from perovskite (pv) to post-perovskite (ppv) is ignored in our numerical
model. The effects of the pv-ppv transition properties on the stability and structure of primordial reservoirs has been
investigated previously by Li et al., (2015). There are many controlling parameters for the pv-ppv transition including the
temperature of the CMB, the viscosity contrast between pv and ppv, and the viscosity contrast between ppv and primordial material
that can affect the stability of piles. For instance, weak ppv (i.e., low viscosity contrast between pv and ppv) and a low Tcmb
(i.e., Tcmb ~ 3350 K) can result in entrainment of primordial reservoirs. Because of the model setup we consider in our study,
the inclusion of a pv-ppv transition will result in the entrainment of a dense primordial reservoir. Thus, the pv-ppv phase
transition will mask the effect of thermal conductivity on the stability of primordial reservoirs that we are examining. This
discussion will be added to the new text in the introduction as requested by R1.5. All usage of the label "K_D = 9.185" will be replaced with "K_{DH}" to indicate that this depth profile was defined by
parameterizations published in Deschamps and Hsieh, (2019).6. The magnitude of the reduction in thermal conductivity depends on temperature (which changes with depth). A simple
calculation can be added to show how much the conductivity is reduced at CMB temperatures for different n values. The
reduction in conductivity will also be shown by the inclusion of a new figure which shows the initial conductivity profiles for cases featuring K_{DH} with different temperature- and composition- dependence.7. In alignment with the comments of R1, all usage of "compositional correction" will be rephrased to "composition- dependence".
8. Q_{CMB} and Q_{SURF} will be clearly defined in Section 3.1.
9. As suggested by R1, we will keep consistent with dimensional values. We currently present 2D distributions of thermal
conductivity in Figures 4 and 5. By plotting the conductivity fields relative to the surface value k_{S}, it would be converted
to the non-dimensional conductivity field. We will include the non-dimensional conductivity (the ratio of local conductivity to
surface conductivity) field to the Supplement.10. Marzotto et al., (2020) will be removed from the references. Conductivity data points included in Figure 1 are from the
references cited within the caption.11. Coordinates for 3D geometry (x,y,z) or (r,theta,phi) will be replaced by 2D spherical annulus coordinates (r,phi).
12. The potential temperature definition will also be stated in addition to the adiabatic correction.
13. Xu et al., (2004) will be removed from the references. This reference was included in the methods section discussion on the
thermal conductivity model (Section 2.2) and had been moved from the supplement to the main text. The reference to Klemens,(1960)
in the supplement will also be removed for the same reason.Citation: https://doi.org/10.5194/egusphere-2022-418-AC2
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CEC1: 'Reply on RC2', Susanne Buiter, 12 Aug 2022
Interactive discussion
Status: closed
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RC1: 'Comment on egusphere-2022-418', Anonymous Referee #1, 30 Jul 2022
This study uses mantle convection simulations to address the influence of pressure-, temperature- and composition-dependence of thermal conductivity on the fate of a dense primitive layer sitting at the bottom of the mantle, mimicking the LL(S)VPs on the Earth. While previous studies have investigated the effect of those various dependencies of thermal conductivity have been investigated individually, the novelty of this work is to consider consider the interplay between them in the context of the Earth’s lower mantle structure. While I think that the results are worth being published, their presentation in the current manuscript is hard to follow and lacks a clear narrative thread, and more importantly, lacks in interpretation. Therefore, I think the paper would benefit from a thorough re-organization of the results section.
General
The study adopts a rather complex baseline set up (mixed basal + internal heating, phase transition, yield stress) to study the effect of pressure, temperature and composition on thermal conductivity. While my preference goes to simpler models when trying to unravel such systematic trends, one can argue that the current set-up is relevant for an Earth-like case, and I think it is fair.
I find the introduction a bit too light. I would expect some more background on the physics of the various dependencies of thermal conductivity, and in particular that the trends be clearly announced: increase of pressure results in increase of thermal conductivity, increase in temperature results in decrease in thermal conductivity, increase in iron content results in decrease in thermal conductivity (it seems). Also, it seems that considering a variable thermal conductivity is particularly relevant for compressible convection. If this is indeed the case, I would expect more time to be spent discussing why.
In my opinion, the main weakness of the paper in its current state is the lack of apparent organization in its exploration of the parameter space, which makes it pretty confused and fails at highlighting general tendencies in the effect of the different dependences of the thermal conductivity. This is probably due in part to the simultaneous changes in investigated parameters and diagnostic quantities from one section to the other (first time-evolution of the heat flux varying KC and n, then instantaneous mantle structures varying everything, and finally time-evolution of entrainment again varying KC and n). While I seem to get the idea in the progression, I had a really hard time getting a clear picture out of this section. Maybe starting by isolating the effect of each dependencies, and then considering their correlations could make things simpler. Maybe also distinguishing more systematically between the effect on mantle flow (pile structure) and on the time-evolution (e.g. of the CMB and surface heat fluxes) could help. Another possibility would be to add dependencies successively to a fiducial model, for which a preferred value of KD would be selected before considering the effect of n, and in turn a preferred value of n would be selected before adding a third layer with KC. That could spare some cases of the parameter space if there are reason to think they are less relevant. Thse are suggestion, hopefully that can help. Anyway, one important thing which is lacking in my opinion is also some theoretical speculation on the expected effect of the parameters that should come before presenting the results, and would help their interpretation (e.g. we expect the effect of increaing KD to increase the thermal conductivity in the lower mantle and thus to make it more stable to convection (it decreses Ra) by homogenizing its temperature, potentially building heat… etc.). I think it is better for the reader to be incited to think in advance of seeing the results.
Specific
This article will probably mainly be read by people familiar with the equations of mantle convections. Nevertheless, I think it would be beneficial to write down the conservation equations (they are not even in the supplementary materials!), at least the heat conservation where the thermal conductivity appears, as it would make clear where the supplementary mechanism induced by varying thermal conductivity operate.
A plot of the thermal conductivity profile corresponding to the reference state for the various KD, KC and n (not all of them but the few most relevant combinations, e.g. the cases presented in Figure 2) would be insightful. It could be a supplementary panel in Figure 1, or a stand-alone figure.
I don’t really understand why simulations run until 11 Gyr, but some values are avergaed aroung 4.5 Gyr. Anyway, the averaged values reported in Table 1 do not seem to be used.
Dimensional and non-dimensional quantities are often mixed, which lacks a bit of rigor.
Technical
L. 48: “compositional- dependent” ← “composition-dependent”
L. 84: “a quadratic that smoothly” ← “a quadratic curve that smoothly”
Figure 1: the markers for “<200 ppm water” are hard to see, please change for a more contrasting colour.
L. 118: “We first defined a purely depth- dependent reference case characterized by depth- dependence, K_D = 2.5, with lower mantle conductivities comparable to current estimates” in Figure 1, it seems that K_D = 10 is the closest match to lower mantle estimates.
L. 121: “by approximately 75%” it seems less than that in Figure 2.
L. 122: “in agreement with Li et al. (2022) findings” ← “in agreement with Li et al. (2022)’s findings”
L. 150: “We observed that T_prim increased with greater temperature dependence (top-to-bottom rows in Figure 3).” I don’t see it.
For the snapshot figures, when a parameter is held constant, write it in the caption rather than for each snapshot. That will lighten a bit the figures.
Figure 5: Please alternate colormaps between the different quantities plotted. In particular, don’t take a “divergent” colormap for the primordial mantle one which only has two extremal values, which are both very dark and hard to distinguish in the current plot.
L. 172: “and is equivalent to” ← “and which is equivalent to” ?
Figure 6: Same as figure 5: it is very hard to know which colormap corresponds to the snapshot and which corresponds to the time-evolution plots. Also please change colours for the onset of instability (the magenta is pretty hard to see).
L. 189: “may not compensate (or be exceeded) by” ← “may not compensate (or be exceeded by)”
Supplementary Materials
L. 43: “and the depth variation of thermal expansivity imply a depth average” ← “and the depth variation of thermal expansivity implies a depth average”
L. 176: “The density anomalies [...] is calculated” ← “The density anomalies [...] are calculated”
Figure S1: the y-axis is non-dimensional height, not depth.
Figure S4: What do line styles correspond to?
Citation: https://doi.org/10.5194/egusphere-2022-418-RC1 -
AC1: 'Reply on RC1', Joshua Guerrero, 22 Aug 2022
The authors appreciate the comments made by R1 and we address each below. We have decided to expand the introduction and
rearrange the results section so that the manuscript is easier to follow.General Comments
-Yes, the mantle convection model set-up we consider is intended to be Earth-like. However, we do not intend on reproducing
the Earth mantle's history. Within this framework we have been able to examine the effect of the thermal conductivity model
on the evolution of the primordial layer at the bottom of the mantle.-We agree with R1 that the introduction can be expanded. Regarding the thermal conductivity, each of the component (dependency)
of the conductivity model will be discussed. In addition, we will now address why conductivity is relevant for compressible mantle
convection.-We consider the suggestions that R1 outlines for reorganizing the results section. As the manuscript stands, introducing the
reference case K_D = 2.5 first is too much too quickly for the reader. We now separate all the effects so that the progression
of the results is easier to digest. The subsections of the results are now:1) Effect of a purely depth- dependent conductivity
2) Effect of a temperature- and depth- dependent conductivity
3) Including the effect of composition- dependent conductivity
4) Long-term stability of thermochemical reservoirs featuring mineral physics derived conductivitiesSpecific Comments
-The heat conservation equation will now be added to the Methods section to help the reader understand where the conductivity
model influences mantle dynamics.-A new figure with a plot of the initial thermal conductivity profiles for cases featuring mineral physics defined
conductivities are now included so that the conductivity reductions from different temperature- and composition- dependencies
is clarified.-The simulation time is long (~11 Gyr) to allow the system to reach a statistically steady state and to investigate how the
heterogeneous conductivity will affect the long-term evolution of the primordial layer. The simulation averages are taken
towards the start of simulations' statistically steady state (approximately 4.5 Gyr). Again, we do not intend to model the exact
evolution of the Earth's mantle. The conditions for simulating Earth's history (i.e., initial conditions, decaying heat sources,
and cooling bottom boundary) are not included in our model setup. In the current version of the manuscript, the averaged values
were inset within the annulus snapshots and their trends were discussed within the results subsections. Specific values were not
used often so that the text would not be inundated with numbers. We now incorporate more references to the Table 1 averages into
the manuscript.-The values presented in the manuscript will be presented as dimensional.
Technical Comments:
-L. 48: for clarity, "composition- dependent" will now be used throughout the manuscript.
-L. 84: inserted the word 'curve'.
-Figure 1: markers for "< 200 ppm water" are changed to bright green to be more contrasting with the background.
-L. 118: The case with a purely depth- dependent thermal conductivity and K_D = 2.5 is considered as an analogue
for models that already account for the combined depth- and temperature- dependences. In the latter (hypothetical)
model of thermal conductivity, the depth- dependent component would have conductivity values that are much greater
than those defined by K_D = 2.5. To clarify, the sentence has been changed to
"As an analogue for a thermal conductivity model that accounts for depth- and temperature- dependence, we first
consider a purely depth- dependent reference case characterized by depth- dependence, K_D = 2.5. Our reference case
produces lower mantle conductivities that are comparable to current estimates (e.g., Deschamps & Hsieh, 2019; Geballe et al., 2020)."
-L. 121: "by approximately 75%", this refers to a comparison of heat flow values averaged about 4.5 Gyr for cases #4 and #8.
This comparison is between a purely depth- dependent conductivity case (#4) and heterogeneous conductivity case (#8).
The sentence has now been revised to read:
"Comparing the purely depth- dependent conductivity model to the completely heterogeneous conductivity model, the latter greatly
reduced both CMB and surface heat flows (by approximately 76% and 22%, respectively), as less heat can be extracted from the
base."
*The subsection including the timeseries figure has now been moved to the final subsection of the results section. It has now
been updated so that the cases are relevant to the mineral physics derived conductivity profile.
-L. 122: fixed the citation to read "... Li et al., (2022)'s findings".
-L. 150: The mean temperature of primordial material is indicated by T_{prim} (not to be confused with the global mean
temperature of the system <T>). T_{prim} is inset within the annulus and is indicated for each case. T_{prim} can be seen
decreasing for cases with the same K_D value as n is increased. In addition, the temperature field is offset with respect to the
CMB temperature so that the differences in hotter temperatures within the piles is easier to see. The boundary between regular
mantle material and primordial material is indicated by a green contour. When n is increased, the red colour within these
contours becomes more saturated.
-Snapshot figures have been revised so that superfluous conductivity model labels are omitted.
-Figure 5: The colormaps for each different field are now alternated.
-L. 172: inserted the word 'which'.
-Figure 6: The colormap is changed. The onset of instability is now indicated by a dashed cyan line.
-L. 189: The parantheses has been moved.
Supplemental Materials:
-L. 43: "imply" -> "implies".
-L. 176: "is" -> "are".
-Figure S1: "depth" -> "height".
-Figure S4: The case numbers have been added to the legend so that the line styles are now clear.Citation: https://doi.org/10.5194/egusphere-2022-418-AC1
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AC1: 'Reply on RC1', Joshua Guerrero, 22 Aug 2022
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RC2: 'Comment on egusphere-2022-418', Anonymous Referee #2, 02 Aug 2022
In this paper the authors carried out numerical experiments of thermochemical mantle convection by varying the spatial changes in thermal conductivity to investigate the temporal changes in the distributions of dense "primordial" materials initially imposed above the bottom boundary. My honest impression is, unfortunately, that the manuscript is very hard to follow because of its poor organization and description from the reasons summarized below. I therefore strongly suggest that the authors should thoroughly revise the manuscript before the reconsideration.
1. One of my major criticism is that the main issues in this study were not well described either in the abstract or introduction section. In my understanding the major intention of the authors is to reduce the thermal conductivity in the deep mantle to much lower levels than those simply expected from the dependence on temperature and pressure (or depth), in order to induce an "instability" from the initial layer of dense materials within a sufficiently short period of time. The authors should clearly indicate their ultimate motivation in earlier parts of the manuscript.
2. I am not well convinced of the meaningfulness of the onset time of "instability" from the initial layer of dense materials. It seems to me that the authors had assumed that the deformation of the basal layer of dense materials occurs only in an "intrinsic" manner owing to the thermal buoyancy. However, several earlier studies had demonstrated the ultimate importance of "extrinsic" deformation induced by the cold subduction from the top surface. I do not therefore think that the onset time of "instability" can be a good measure to investigate the influence the thermal conductivity at depth on the thermal buoyancy in the basal layer of dense materials.
3. I felt quite odd with the authors' exaggerated claim on the overall profiles of thermal conductivity including "We find that the temperature- and depth- variations combined characterize the mean conductivity ratio from top-to-bottom" in the abstract. It is quite obvious from the assumptions made by the authors themselves.
4. In Section 2.1 the authors should state the reason why the phase change from perovskite (pv) to post-perovskite (ppv) is ignored in their numerical model. If they believe that the pv-ppv phase change is of little importance on the dynamic behaviors of the basal layers of dense materials, the authors should make clear the reason why.
5. I was quite disgusted to see that the profile given by equation (2) is denoted by "KD=9.185". Such a denotation should be used only for the profiles given by equation (1) !!
6. Near equation (3) the authors should indicate the magnitude of the reduction in thermal conductivity due to the increase in temperature within the modeled domain by the choice of n=0.5 and n=0.8.
7. Near equation (4) and later, "compositional correction" should be rephrased with "compositional dependence". The word "CORRECTION" could imply that the numerical experiments without the compositional dependence in thermal conductivity are meaningless.
8. In Section 3.1 "QCMB" is used without explicitly defined.
9. To show the 2-D distributions of thermal conductivity in Figures 4 and 5, the authors should show the ratio of thermal conductivity to its surface value (KS) rather than the values of conductivity itself.
10. The paper by Marzotto et al. (2020) has not been cited anywhere in the main text.
11. Near equation (8) of the Supplement, the spatial coordinates should not be in Cartesian (x,y,z) but in 2-D polar (r,phi) in this study. Similarly in equation (12) of the Supplement, the coordinate is not in 3-D polar (r,theta,phi).
12. In Section 3.1 of the Supplement, I think that the authors can use the potential temperature instead of the "adiabatic correction" a(z).
13. The paper by Xu et al. (2004) has not been cited anywhere in the Supplement.Citation: https://doi.org/10.5194/egusphere-2022-418-RC2 -
CEC1: 'Reply on RC2', Susanne Buiter, 12 Aug 2022
Dear reviewer, dear authors,
I thank the reviewer for their feedback and suggestions for improvement of the manuscript. As a general note, i would like to use this opportunity to re-iterate that Solid Earth strives for polite and constructive language in all communications, including reviews. Specifically regarding this manuscript, i would ask the authors to ignore the wording that was used in comment 5 and to try to focus on the content of the feedback given. I hope this will work for everyone.
Susanne Buiter
Citation: https://doi.org/10.5194/egusphere-2022-418-CEC1 -
AC2: 'Reply on RC2', Joshua Guerrero, 22 Aug 2022
The authors appreciate the comments made by R2 and we address each below. In addition to the commments made by R1, we have
decided to thoroughly revise the manuscript so that it is easier to follow.1. In alignment with the comments of R1, we will expand the introduction section and make the intentions of our study clearer.
2. We calculate the onset time for instability from the temporal variations in the average height of dense material. It is true
that these variations in average height do not discriminate between 'intrinsic' (i.e., thermal buoyancy) or 'extrinsic'
(i.e., downwellings) deformation. From examining the average height timeseries and animations of the fluid flow we can see the
influence of downwellings. The first downwellings impringe on the initial dense layer but do not result in a rapid uplift of
material (sufficient to eject blobs of dense material). Once the initial dense layer has organized into piles, downwellings tend
to move dense material laterally over the CMB but not rapidly increasing the pile height. Furthermore, we find that it is easier
for downwelling currents to push primordial material that has been made lighter due to their retained heat. We agree that
downwellings are important in deforming the dense primordial layer. This mechanism will now be discussed in addition to the
thermal effect regarding instability.3. Right. This statement is a bit of a tautology. The intention for this statement is that for Earth-like models that consider a
variable thermal conductivity, it may be ill advised to isolate for just one dependency. On one hand, to simulate a 2-pile
configuration, a purely depth- dependent conductivity may be employed, but its top-to-bottom ratio should implicitly emulate the
temperature and composition effects. On the other hand, a purely temperature- dependent conductivity will result in the
entrainment of a dense primordial layer. By assuming a parametrized conductivity model, it is predictable what the mean
conductivity ratio from top-to-bottom will be (having specifed the temperature contrast and the dependencies of the conductivity
model).4. We will now discuss the reasons why the phase change from perovskite (pv) to post-perovskite (ppv) is ignored in our numerical
model. The effects of the pv-ppv transition properties on the stability and structure of primordial reservoirs has been
investigated previously by Li et al., (2015). There are many controlling parameters for the pv-ppv transition including the
temperature of the CMB, the viscosity contrast between pv and ppv, and the viscosity contrast between ppv and primordial material
that can affect the stability of piles. For instance, weak ppv (i.e., low viscosity contrast between pv and ppv) and a low Tcmb
(i.e., Tcmb ~ 3350 K) can result in entrainment of primordial reservoirs. Because of the model setup we consider in our study,
the inclusion of a pv-ppv transition will result in the entrainment of a dense primordial reservoir. Thus, the pv-ppv phase
transition will mask the effect of thermal conductivity on the stability of primordial reservoirs that we are examining. This
discussion will be added to the new text in the introduction as requested by R1.5. All usage of the label "K_D = 9.185" will be replaced with "K_{DH}" to indicate that this depth profile was defined by
parameterizations published in Deschamps and Hsieh, (2019).6. The magnitude of the reduction in thermal conductivity depends on temperature (which changes with depth). A simple
calculation can be added to show how much the conductivity is reduced at CMB temperatures for different n values. The
reduction in conductivity will also be shown by the inclusion of a new figure which shows the initial conductivity profiles for cases featuring K_{DH} with different temperature- and composition- dependence.7. In alignment with the comments of R1, all usage of "compositional correction" will be rephrased to "composition- dependence".
8. Q_{CMB} and Q_{SURF} will be clearly defined in Section 3.1.
9. As suggested by R1, we will keep consistent with dimensional values. We currently present 2D distributions of thermal
conductivity in Figures 4 and 5. By plotting the conductivity fields relative to the surface value k_{S}, it would be converted
to the non-dimensional conductivity field. We will include the non-dimensional conductivity (the ratio of local conductivity to
surface conductivity) field to the Supplement.10. Marzotto et al., (2020) will be removed from the references. Conductivity data points included in Figure 1 are from the
references cited within the caption.11. Coordinates for 3D geometry (x,y,z) or (r,theta,phi) will be replaced by 2D spherical annulus coordinates (r,phi).
12. The potential temperature definition will also be stated in addition to the adiabatic correction.
13. Xu et al., (2004) will be removed from the references. This reference was included in the methods section discussion on the
thermal conductivity model (Section 2.2) and had been moved from the supplement to the main text. The reference to Klemens,(1960)
in the supplement will also be removed for the same reason.Citation: https://doi.org/10.5194/egusphere-2022-418-AC2
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CEC1: 'Reply on RC2', Susanne Buiter, 12 Aug 2022
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Joshua Martin Guerrero
Frédéric Deschamps
Wen-Pin Hsieh
Paul James Tackley
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