Multidimensional Simulations of Core Convection
Round 1
Reviewer 1 Report
The authors reviewed the current status of multidimensional simulations of stellar core convection and their applications to understand the processes which govern stellar evolution. I only have some minor revisions to suggest, perhaps the most important being about a couple of additional works to be mentioned about convective mixing in the core regions of low-mass AGB stars, which in my opinion would make the final manuscript definitely more complete. Find my suggestions in the following lines:
Section 2.2.3. Overshoot length
The authors mentioned "Anders & Pedersen 2023, Galaxies", but I'm not sure why mentioning the journal in the main text is necessary. Especially considering how this is not done for all the other references in the main text, where the standard form "Author, X et al. (year)" is used.
3.1. Equations describing the problem
The formal conditions for their applicability are... (incomplete)
Section 3.2.2. Temporal discretization
I guess PDE stands for Partial Derivative Equation, but it's never defined as such.
Section 4.2.3. Chemical mixing by waves
This section could definitely be more complete by citing, at its end, another multi-D study by Falk Herwig for convection-induced IGW mixing within the H-free core of AGB stars, during the He-flash (Herwig et al.a 2007, http://aspbooks.org/custom/publications/paper/378-0043.html). Most importantly, they provided the convective boundary mixing parameters in order to capture the IGW-induced mixing in 1D. When this has been done (see e.g. Battino et al. 2016; MNRAS Volume 489, Issue 1, October 2019, Pages 1082–1098, https://doi.org/10.1093/mnras/stz2158 and Battino et al 2016 ApJ 827 3, DOI 10.3847/0004-637X/827/1/30), key-observational constraints of AGB evolution, like the enriched He, C and O surface abundance of post-AGB stars, are successfully reproduced. This would also be an additional example of the importance of using the insights gained in hydrodynamic simulations in stellar evolution codes, as stated in the closing lines of the paper
I list here some minor suggestions to improve the quality of English language:
2.1.2. Buoyancy, page 4
The shortEST possible period...
2.1.3. Convection
"Thus, convection produces..." (without "the"?)
Section 2.2.3. Overshoot length
Here we estimate the overshoot length by equating the kinetic energy in the convectIVE ZONE...
Section 3.1. Equations describing the problem
"The formal conditions for their applicability are..." (incomplete sentence)
Page 11
If "In the former case" refers to the explicit method case, then there's no need for a new paragraph after "computationally very expensive".
Section 3.2.3. Spatial discretization
"This a is function"... I guess this should be "This is a function"
No need for a new paragraph before "Toro [57] and LeVeque [49] give an overview of the various aspects of finite-volume methods"
Author Response
We thank the reviewer for the very constructive remarks. We have addressed them in the manuscript and are briefly replying to the individual points here.
“Section 2.2.3. Overshoot length
The authors mentioned "Anders & Pedersen 2023, Galaxies", but I'm not sure why mentioning the journal in the main text is necessary. Especially considering how this is not done for all the other references in the main text, where the standard form "Author, X et al. (year)" is used.”
We have fixed these references.
“3.1. Equations describing the problem
The formal conditions for their applicability are... (incomplete)”
This has been fixed.
“Section 3.2.2. Temporal discretization
I guess PDE stands for Partial Derivative Equation, but it's never defined as such.”
The definition has been added.
“Section 4.2.3. Chemical mixing by waves
This section could definitely be more complete by citing, at its end, another multi-D study by Falk Herwig for convection-induced IGW mixing within the H-free core of AGB stars, during the He-flash (Herwig et al.a 2007, http://aspbooks.org/custom/publications/paper/378-0043.html). Most importantly, they provided the convective boundary mixing parameters in order to capture the IGW-induced mixing in 1D. When this has been done (see e.g. Battino et al. 2016; MNRAS Volume 489, Issue 1, October 2019, Pages 1082–1098, https://doi.org/10.1093/mnras/stz2158 and Battino et al 2016 ApJ 827 3, DOI 10.3847/0004-637X/827/1/30), key-observational constraints of AGB evolution, like the enriched He, C and O surface abundance of post-AGB stars, are successfully reproduced. This would also be an additional example of the importance of using the insights gained in hydrodynamic simulations in stellar evolution codes, as stated in the closing lines of the paper”
We added the Battino references to the end of the article. Herwig et al. (2007) does not include simulations of core convection, so we have decided to not include it in section 4.2.3.
2.1.2. Buoyancy, page 4
The shortEST possible period...
Fixed
2.1.3. Convection
"Thus, convection produces..." (without "the"?)
Fixed
Section 2.2.3. Overshoot length
Here we estimate the overshoot length by equating the kinetic energy in the convectIVE ZONE...
We have switched to “kinetic energy of convective flows”. If we wrote “in the convection zone”, we are worried the readers may think of the volume-integrated energy over the whole convection zone, where here we are thinking of the kinetic energy density.
Section 3.1. Equations describing the problem
"The formal conditions for their applicability are..." (incomplete sentence)
This has been completed.
Page 11
If "In the former case" refers to the explicit method case, then there's no need for a new paragraph after "computationally very expensive".
Fixed
Section 3.2.3. Spatial discretization
"This a is function"... I guess this should be "This is a function"
Fixed
No need for a new paragraph before "Toro [57] and LeVeque [49] give an overview of the various aspects of finite-volume methods"
Fixed
Reviewer 2 Report
This review paper discusses the theoretical aspects and latest strategies in simulating core convection. There is a challenge in simulating core convection due to the wide range of temporal and spatial scales involved in the problem. The review mentions some latest simulations of core convection and their main findings, discusses the connection to asteroseismology, and highlights problems with multidimensional simulations and methods for overcoming these problems. The authors mention in this context the methods of accelerated stellar evolution and artificially increasing the stellar luminosity by a factor of 1000 to decrease the thermal time.
There are two outstanding problems with multidimensional simulations: it is not always straightforward to extrapolate from simulation parameters to the parameters of real stars, and simulations that use different methods arrive at contradictory results. To address these issues, the authors propose that next-generation simulations should address how their results depend on stellar luminosity, dimensionality, and turbulence intensity. Code comparison projects will be essential to establish robust parameterizations that will become the new standard in stellar modeling.
In my opinion, the paper reviews the topic very well. I think any reader in the field or in a close field would learn from it. There are a few things that are relevant for a more general audience than those interested in working on stellar convection. For example, the discussion on the differences between 2d and 3d simulations and the limitations of the former is relevant to many other simulations and objects. The same for technical numerical methods for solving PDEs such as the Navier-Stokes equation. I, therefore, think this paper has a large merit. I also agree that a code comparison project is essential in this field, as it has proved itself in other fields, most notably in cosmological and galaxy evolution simulations.
Below I give some comments that can significantly improve the paper (in my opinion). I hope the authors will adopt at least some of my suggestions, especially the one about adding figures (see below).
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(1) The paper does not cite the very relevant review by Kupka & Muthsa (2017). Their review is extremely detailed and discusses many of the topics discussed in the present review and more. I think it MUST be cited.
(Kupka, F., Muthsam, H.J. Modelling of stellar convection. Living Rev Comput Astrophys 3, 1 (2017). https://doi.org/10.1007/s41115-017-0001-9 ; https://link.springer.com/article/10.1007/s41115-017-0001-9)
That being said, the present paper discusses many new works from the past ~5 years (since Kupka & Muthsa 2017), so it is very helpful to anyone who wishes to be updated with recent developments.
(2) A thing I found a little confusing is the lack of clear distinction when switching between discussions of "older" theory and "modern" work. The authors usually do not give references to what they consider (and I agree) well-known theory, and when references appear, it is often when mentioning newer methods or conclusions. This is not always the case (e.g., Courant et al. 1928 is referenced in section 3.2.2). I suggest making the difference clearer throughout the paper by separating between the two. This could be done by using proper wording or having separate paragraphs for discussions of "older" theory and "modern" work. A good example where this is done is the second paragraph of section 2.3.3 and most of section 4.1.
(3) The paper does not include any figures. I don't know if there is a particular reason for this, but I found it to be a significant omission. As this is a review paper, adding figures from other papers is, as far as I know, allowed and is an accepted practice. Some suggested figures could include the structure of the star showing the convection zone and the different spatial scales, Kipenhahn diagrams, results from numerical simulations, comparisons of 2d and 3d numerical results, comparison figure of the different results spatial spectrum scaling of convection. I think the problem is not to find figures but to filter the most important ones for this review. I believe that adding figures would not be a demanding task and would significantly enhance the paper.
(4) Introduction: I suggest to add a few lines about MLT including the theory and the way it is implemented in modern 1D astrophysical codes (MLT++ prescription for example).
(5) I suggest to add a reference to Mao et al. 2023 (arXiv:2304.10470) who performed simulations of a 25 Msol star and studied the effect of radiative diffusion on convective and thermal timescales on convection.
(6) I suggest to change the order of section 2.1 time scales and 2.2. Length scales, as quantities that appear in the time scales section are introduced in the length scale section (pressure scale height, convection radius).
(7) Section 2.1: Discussing the timescales is very important. However, I don't think that the expressions given for the timescales are useful, at least not most of them. For example, the convective luminosity is not a straightforward quantity to calculate, and giving an expression for t_th with the radiative diffusivity without providing more formulae is not very helpful. While this is obviously a review paper and not a textbook that can provide all equations, maybe the authors could provide more detailed expressions.
(8) Are all timescales given consistently for a 10Msun star? I wonder if there are available scalings for other stellar masses, so each timescale will be given by its value times (Mstar/10Msun)^a, where a is some power. If such scaling is not possible, I suggest calculating the values for 2-3 other stellar masses and presenting them in a list or table.
(9) Section 2.2.5: I think it is worthwhile to mention that viscosity is often implemented using an artificial viscosity coefficient.
What is the typical Reynolds number in the convective core? what is the importance of its value?
(10) Section 3.2.4: I think it is a very useful section. While there are many codes and not all can be included I think the FLASH code must be added, and better also to add ENZO.
(11) Section 4.1: "It is surprising there remain order-of-magnitude discrepancies in the convective overshoot length when using similar analyzes on 2D simulations solving similar equations (with filtered sound waves), with similar background stellar models." Can the authors explain why it is surprising? All the other discussions in the paper did not prepare me to be surprised that 2d simulations give different results.
(12) I think sections 4 and 5 are very useful and highlight new developments and present problems. They are written very well in my opinion. Again, adding figures would make them better.
Minor comments:
The head title ia "article", but should be "review".
Section 2, first paragraph: I think that stellar rotation should be added to the list of what influences convection (it is later mentioned).
Section 2.2.4 : How does the convective turnover time relate to t_c in 2.1.3 ? I suppose the authors mean 0.5t_c, but it is not defined.
Section 3, 1st paragraph. The last line reads "The formal conditions for their applicability are...". The continuation seems missing.
Equation 18: I think there are typos in the fifth term of the equation: should start with: uτ^{zx} + vτ^{zy} + wτ^{zz}
Section 3.2.3 has this one sentence paragraph : "Toro [57] and LeVeque [49] give an overview of the various aspects of finite-volume methods. " It should be attached to he previous paragraph.
Section 4.1, first paragraph. Reference for "Anders & Pedersen 2023, Galaxies", is missing.
Author Response
We thank the reviewer for the very constructive remarks. We have addressed them in the manuscript and are briefly replying to the individual points here.
(1) The paper does not cite the very relevant review by Kupka & Muthsa (2017). Their review is extremely detailed and discusses many of the topics discussed in the present review and more. I think it MUST be cited.
(Kupka, F., Muthsam, H.J. Modelling of stellar convection. Living Rev Comput Astrophys 3, 1 (2017). https://doi.org/10.1007/s41115-017-0001-9 ; https://link.springer.com/article/10.1007/s41115-017-0001-9)
That being said, the present paper discusses many new works from the past ~5 years (since Kupka & Muthsa 2017), so it is very helpful to anyone who wishes to be updated with recent developments.
Kupka & Muthsa (2017) focus their discussion on convective envelopes, e.g., in main sequence solar-type stars. Here we only discuss core convection. Core convection and convective envelopes share many similarities, but there are some important differences (e.g., much less density stratification in core convection, numerical issues associated with r=0). It is a valuable reference that we have added to the beginning of section 2 and in section 3.
(2) A thing I found a little confusing is the lack of clear distinction when switching between discussions of "older" theory and "modern" work. The authors usually do not give references to what they consider (and I agree) well-known theory, and when references appear, it is often when mentioning newer methods or conclusions. This is not always the case (e.g., Courant et al. 1928 is referenced in section 3.2.2). I suggest making the difference clearer throughout the paper by separating between the two. This could be done by using proper wording or having separate paragraphs for discussions of "older" theory and "modern" work. A good example where this is done is the second paragraph of section 2.3.3 and most of section 4.1.
This is a valid point and we have tried to break the text up between the "older" and "modern" work, especially in Section 3.
(3) The paper does not include any figures. I don't know if there is a particular reason for this, but I found it to be a significant omission. As this is a review paper, adding figures from other papers is, as far as I know, allowed and is an accepted practice. Some suggested figures could include the structure of the star showing the convection zone and the different spatial scales, Kipenhahn diagrams, results from numerical simulations, comparisons of 2d and 3d numerical results, comparison figure of the different results spatial spectrum scaling of convection. I think the problem is not to find figures but to filter the most important ones for this review. I believe that adding figures would not be a demanding task and would significantly enhance the paper.
Thank you for your very good suggestion to add figures. The difficulty here is that when considering convective boundary mixing and internal gravity waves, most papers present different analyzes which are difficult to directly compare. We have attempted to make some comparisons in the text, but do not think showing the underlying data from the different papers would be illuminating. Instead, to help organize the different simulations, we have added a figure showing simulation visualizations from nine representative works (even in that figure each subplot is of a different quantity, making it challenging to directly compare between simulations). We also added a table of simulations studying internal gravity waves from core convection.
(4) Introduction: I suggest to add a few lines about MLT including the theory and the way it is implemented in modern 1D astrophysical codes (MLT++ prescription for example).
While MLT & its various implementations in 1D stellar evolution codes is very important for surface convection zones, it is irrelevant for core convection. This is because MLT only affects stellar structure in regions with inefficient convection. Core convection is always efficient, so MLT does not matter. We have added a footnote mentioning this to the introduction.
(5) I suggest to add a reference to Mao et al. 2023 (arXiv:2304.10470) who performed simulations of a 25 Msol star and studied the effect of radiative diffusion on convective and thermal timescales on convection.
Thanks for mentioning this very interesting new work. It was uploaded to the arxiv after we submitted this manuscript for review. We have added a discussion of several new papers that were uploaded to the arxiv since our original submission: Mao et al. (2023), Ratnasingam et al. (2023), Le Saux et al. (2023), and Anders et al. (2023). For this reason, we have added a new Table 2 summarizing the different simulations.
(6) I suggest to change the order of section 2.1 time scales and 2.2. Length scales, as quantities that appear in the time scales section are introduced in the length scale section (pressure scale height, convection radius).
The referee is correct that our estimates of timescales depend on the convection zone radius, but that is the only length scale they depend on. Meanwhile, one of the length scales depends on a timescale.
On the other hand, we originally discussed the timescales before the length scales because we feel the time scale separation is less widely appreciated/discussed than the timescale separation. Given that the time scales only depend on a single length scale (R_c), we do not think it is so confusing to present time scales first in order for this pedagogical purpose.
(7) Section 2.1: Discussing the timescales is very important. However, I don't think that the expressions given for the timescales are useful, at least not most of them. For example, the convective luminosity is not a straightforward quantity to calculate, and giving an expression for t_th with the radiative diffusivity without providing more formulae is not very helpful. While this is obviously a review paper and not a textbook that can provide all equations, maybe the authors could provide more detailed expressions.
We have recalled that L_conv = L_tot - L_rad, and have defined the radiative diffusivity.
(8) Are all timescales given consistently for a 10Msun star? I wonder if there are available scalings for other stellar masses, so each timescale will be given by its value times (Mstar/10Msun)^a, where a is some power. If such scaling is not possible, I suggest calculating the values for 2-3 other stellar masses and presenting them in a list or table.
At the beginning of section 2 we state that the precise ratio of time scales and length scales vary as a function of mass and age, but that the order of magnitude is similar for all intermediate- and high-mass stars. We refer the reader to Jermyn et al (2022) for more details.
(9) Section 2.2.5: I think it is worthwhile to mention that viscosity is often implemented using an artificial viscosity coefficient.
What is the typical Reynolds number in the convective core? what is the importance of its value?
We mention artificial viscosity in this section. The typical Reynolds number is 10^11, and we do not think the value is important, which is why we do not include it in the review.
(10) Section 3.2.4: I think it is a very useful section. While there are many codes and not all can be included I think the FLASH code must be added, and better also to add ENZO.
The codes have been added along with PROMPI.
(11) Section 4.1: "It is surprising there remain order-of-magnitude discrepancies in the convective overshoot length when using similar analyzes on 2D simulations solving similar equations (with filtered sound waves), with similar background stellar models." Can the authors explain why it is surprising? All the other discussions in the paper did not prepare me to be surprised that 2d simulations give different results.
We have removed the surprise from the text.
(12) I think sections 4 and 5 are very useful and highlight new developments and present problems. They are written very well in my opinion. Again, adding figures would make them better.
As described above, we have added a figure showing snapshots from different simulations of core convection.
The head title ia "article", but should be "review".
Section 2, first paragraph: I think that stellar rotation should be added to the list of what influences convection (it is later mentioned).
We have added it to the list.
Section 2.2.4 : How does the convective turnover time relate to t_c in 2.1.3 ? I suppose the authors mean 0.5t_c, but it is not defined.
We have added more description of this estimate to clarify the argument.
Section 3, 1st paragraph. The last line reads "The formal conditions for their applicability are...". The continuation seems missing.
This has been completed.
Equation 18: I think there are typos in the fifth term of the equation: should start with: uτ^{zx} + vτ^{zy} + wτ^{zz}
This has been fixed.
Section 3.2.3 has this one sentence paragraph : "Toro [57] and LeVeque [49] give an overview of the various aspects of finite-volume methods. " It should be attached to he previous paragraph.
Fixed
Section 4.1, first paragraph. Reference for "Anders & Pedersen 2023, Galaxies", is missing.
This has been fixed.
