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% Reply to Report 1 %
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We thank the referee for a very useful and constructive report on our paper.
In the following, we address all of the concerns and criticisms and comment on
changes in the paper.
All changes in the text are marked by blue font.
> I would urge the authors to pay attention to the following issues, however:
> 1. The implications of the two major findings are not properly discussed.
> - The chiral alpha effect is proportional to the initial chemical potential,
> mu_0, and that determines how much magnetic energy will be available to be
> transformed into kinetic one by the Lorentz force. In that sense, instead
> of the ratio of kinetic to magnetic energy (Upsilon), a more relevant quantity
> to define and discuss would be an energy transfer rate that takes into account
> this inevitable dependence on mu_0. The energy ratio being independent on mu_0
> does not imply that the amount of energy that ends up in turbulent form would
> not. This is the impression that one gets now.
We agree with the referee that the energy transfer rate is a very interesting
quantity to analyze. Therefore, in the new equation (20), we define the
dimensionless ratio
\Phi \equiv \frac{\langle U \cdot ( J \times B)\rangle}{|v_\mu B \cdot \nabla \times B|},
e.g. the ratio of the production rate of kinetic energy via the Lorentz force
and the production rate of magnetic energy via the chiral instability.
We present the time evolution of \Phi in the new Figure 8 for runs with
different values of v_mu and in the upper right panel of Figure 9 for runs
with different values of Pm.
Moreover, to highlight the flow of energy from the chemical potential to turbulent
kinetic energy, we added a sketch as Figure 1.
> - The Prandtl number dependence and the estimated values of Pr_M in the early
> universe imply that the energy ratio would drop to values of the order of
> 0.01. The implications of such small energy ratios for the chiral dynamo
> scenario are not clearly discussed, but are hidden in Section 4.2, where
> the authors have decided to estimate a magnetic Reynolds number for the
> early universe instead, also introducing a free parameter vartheta to "explore
> different initial conditions". It is totally unclear which parameter choices
> resulted in the final estimate of Re_M being O(3), nor are all the parameter
> values given in 4.2 to enable the computation of the numbers (e.g. value of
> g_100?), if the reader would be interested in repeating the exercise.
* The value of g_100 has been introduced below Equation (26).
* We have reformulated the last sentence of Section 4.2, to clarify the
parameter choices leading to our estimate for Rm.
> Also, one remains wondering, whether Eq. (31) can be correct at all.
> Moreover, Eq. (32) gives, if one estimates the coherence scale of the
> IGM mgf to be about a Megapc, value of 10(-41) for vartheta.
Here seems to be a misunderstanding: The scales k_mu and k_lambda (the latter
is estimated in the former Equation 33) are much smaller than the comoving
1 Mpc scale, where IGM magnetic fields could be present today.
If one is interested in the co-moving magnetic field at present day that
result from chiral dynamos, it is crucial to include the evolution of the
fields from to "chiral length scales" to larger scales in the inverse cascade.
This inverse cascade it mentioned in the left panel of the current Figure 2,
here named "decaying helical MHD turbulence". Magnetic fields generated in
chiral MHD are fully helical, implying that, without additional energy input,
they decay while \approx \xi*B^2 remains constant. If B decreases,
the correlation length \xi increases by many orders of magnitude.
> I strongly urge the authors to check, correct, and clarify Sect 4.2, and more
> clearly discuss the relevance of the chiral dynamo scenario given the small
> energy ratios obtained with the expected, large Pr_M.
We have re-checked the calculation in Sec. 4.2 again and improved the text for
clarification.
> - The models have different values of lambda_mu (hence different k_lambda)
> and are, therefore, not directly comparable (different phases last for
> a different time, are not reached at all, and so on). Now this heterogeneity
> issue is dealt with computing averages over different tresholds of Upsilon.
> A better way would definitely have been to divide each simulation by their
> phases (defined in section 2.3, and appear easily distinguishable from
> the simulations), and compute the energy ratios separately for each. This
> would have, in addition, given much more information about the system than
> collapsing the data into a tresholded time average smearing the phases.
One of our main results is the measurement of Upsilon during the mean-field
dynamo phase for different values of mu and lambda. Therefore we perform
a time average during phase 2 (mean-field dynamo phase).
Of course, there are different possible criteria for the time range during which
the averaging is performed. For example, one could consider the growth rate
of the magnetic field and average for times, after gamma has dropped
significantly. Such a criterion will, however, result into similar conclusions
as our current criterion, since Upsilon becomes constant once the plasma enters
phase 2.
Our current criterion, based on a fraction of 0.5 or 0.9 of the maximum of
Upsilon, has the advantage that the averaging is only performed as long as
the peak of the magnetic energy spectrum k_M is inside the numerical domain.
As discussed in the manuscript, numerical artifacts occur, once k_M reaches the
minimum wavenumber of the box k_1. In our analysis times for which k_M = k_1
are automatically excluded.
> 2. The paper is really hard to read.
> - One reason is the overwhelming amount
> of different parameters given, not properly defined, or defined much later
> in the text than when used for the first time, and their meanings
> not properly explained. I strongly recommend that the authors make a table
> of all parameters and their meanings and typical values. Even more, I would
> also suggest that the authors check whether all these parameters are needed.
> If their number can be reduced, all the better.
* We added a table summarizing the main parameters discussed in the paper (Table A1).
* We have skipped the definition of the chiral flipping rate, $\Gamma_f$,
which is neglected in this work anyways.
> - The other reason for the poor readibility are illegible axis labels (Fig 3)
Figure 4 (former Figure 3) has been increased.
> or heterogeneous axis definitions (e.g. the usage different time axis in
> between Fig 2., 3, 6-8).
Figure 3 (former Figure 2) is a sketch, where we do units are omitted for simplicity.
Figure 5 (former Figure 4) was taken from Schober et al. 2017, where time was given in terms
of the diffusive time. Since, in the following figures runs with very different
dynamo time scales are compared directly, it is more convenient to normalize
to the inverse dynamo timescale gamma_\mu^{-1}. For heterogeneous in the
current manuscript, we decided to change the normalization of all time
axes to gamma_\mu^{-1}.
> The authors make an attempt to clarify their
> complicated plots by indicating some parameter values to mark some key
> transition points (t_box, lambda_mu, k_mu, k_lambda, k_1, Bsat, B1->2,
> C_lambda mu_0/lambda ....), but this does not work taken that the parameters
> are not explained clearly enough.
The new Table A1, should help the reader to navigate through the different
parameters in the paper.
Also, we have improved the text, making sure that all parameters are defined
at their first occurrence.
> - The caption of Table 1 is confusing, if not completely wrong. The tabulated
> values are hard to relate to the actual model parameters.
> Is the purpose of this table to list only input quantities? It is not
> completely
> clear whether k_mu and k_lambda are measured as outputs from the simulations,
> or are they computed from the analytical/empirical formulae (11)/(12)? This
> should be clarified.
Table 1 lists all input parameters of our simulations. We have updated the
caption to make this clear.
Also, we have corrected a typo: It was indicated in the caption, that runs C
have been performed for different Pm, however, there is only one run C with
Pm=1.
> What does it mean if k_lambda/k_1 is smaller than one - the
> large-scale
> dynamo instability is not properly captured?
If k_lambda is smaller than k_1, the saturation stage of the dynamo instability
is not captured accurately by the box and we expect artificial numerical effects
as discussed in the paper.
Regarding the real large-scale dynamo instability scale, k_lambda: This one is
never captured in our simulations.
> It would have been a good idea to
> list some output quantities as well. urms? brms? Upsilon? Bsat? B_1->2?
> .......
The output parameters depend strongly on time, and we present them in our
figures.
> - Please avoid repetition of the definitions of what is plotted in the figures
> in their caption and the text (e.g. Fig 1 caption and the text on page 6, last
> full
> paragraph). The paper is exhaustively long in any case, and extending the text
> this way does not help.
We have shortened the caption of Figure 1 and refer to the discussion in
the main text instead.
> - Please consider to what extent Section 2.3 material has been represented
> in your earlier papers on the topic, and whether such an extensive review is
> really necessary here. This section should, at the least, be made much more
> compact.
We have reviewed our Section 2.3 and deleted parts that were not crucial for
understanding the remaining part of the paper. This includes the plots
showing the time evolution of the mean magnetic field and the helicity spectra.
> 3. The text is not comprehensible/is misleading/appears irrelevant at places.
> Abstract:
> Here you only talk about the small-scale chiral dynamo, but then refer to the
> large-scale dynamo by talking about "MHD dynamos". This is confusing.
We have rephrased this part of the abstract, to avoid the confusion.
> "we estimate the properties of chiral magnetically driven turbulence..."
> is an overstatement and vague. Values of magnetic Prandtl and Reynolds
> number are estimated.
We agree, of course, that our analysis does not cover all aspects of
chiral magnetically driven turbulence. Therefore we have reformulated this
sentence to "we analyze the energetics of chiral magnetically driven turbulence".
> Intro: please reformulate the following sentences to contain
> a clear, sensible message:
> 1st para
> "MHD dynamos are often caused by ..., so for example in the cases of the
> small-scale ...."
> 2nd para
> "In recent years, the nature .... has been more and more constrained."
> "The lower limits on the strength .... might be due to possible remains..."
> The chiral instability is by no means the only proposed scenario for
> primordial
> mgfs, as the authors now claim. Please modify to better reflect the reality:
> "Cosmological seed fields, however,..., and have been connected to a
> microphysical effect....
This paragraph has been reformulated, according to the suggestions of the
referee.
> Section 2.2 is overall quite confusing. This section, indeed, is
> comprehensible to only
> those readers that are familiar with MFE, and as such, it is difficult to see
> the
> rationale to include it in this form. Please restrict the discussion to the
> relevant
> points for the chiral dynamo here, do not dwell into the delicacies of
> turbulent dynamos.
> The text starting from "In MFE, it implies that ... " until the end of the
> section seems
> all dispensable.
The other referees and colleagues have appreciated this section, but
we have now specifically addressed "Readers unfamiliar with dynamical
quenching may skip forward to section 2.3."
> 4. Please check the language carefully. At least correct
> "\Upsilon^2" in the abstract, "analytical analysis" and " conversation law" in
> the Intro, and "size of the inertial range" in 4.1
"\Upsilon^2" in the abstract is correct because we are talking about energy
ratios.
We have corrected "analytical analysis" and "conversation law" in the
introduction.
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% Reply to Report 2 %
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We thank the referee for reading our manuscript and sending a very positive
report.
> I have only one recommendation. Because the value of the magnetic Prandtl number plays
> the central role in the present study, I think that it would be instructive to mention in the
> manuscript (at an appropriate place) the existence of recent nontrivial theoretical calculations
> of the turbulent magnetic Prandtl number in turbulent MHD systems with and without
> vilation of spatial parity (PHYSICAL REVIEW E 84, 046311 (2011) and
> PHYSICAL REVIEW E 87, 043010 (2013)), where it was shown that 1) it seems that the
> magnetic Prandtl number is stable perturbatively and 2) that the presence of spatial
> parity violation decreases its value. Note that theoretical prediction for the
> turbulent magnetic magnetic Prandtl number is approximatelly 0.71 and is perfectly in the
> interval of magnetic Prandtl numbers used in the manuscript.
We thank the referee for mentioning the turbulent magnetic Prandtl number.
However, we would like to stress that we are probing with our simulations the
microscopic magnetic Prandtl number.
We have now added a corresponding comment on page 17 after Equation (28).
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% Reply to Report 3 %
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We thank the referee for a positive report and for pointing our several
typos in the text. In the following, we respond to all the issues raised.
> p.2 l.37: this is the first time that the quantity μ appears, while it
> has not been introduced yet.
Corrected.
> p.4 l.21 Total chirality is defined only later (in Eq. 16).
We have added a definition of the total chirality at this place.
> p.10 Eq. (20): is this equation correct for non-uniform ρ?
The former Equation (20) is a definition, so it is valid also for a
non-uniform density.
> p.10 l.15 effected → affected.
Corrected.
> p.11 caption of Fig. 11: verses → versus.
Corrected.
> p.11 l.53: this Reynolds number is the _magnetic_ Reynolds number.
Corrected.
> p.12 3.3.2: the first reference to Fig. 9 seems rather to be a reference
> to Fig. 8 (time evolution of Υ and ReM).
This has been corrected. Now it is referred to Figure 9 (previous Figure 8).
> p.14 caption of Fig.10: vertical bars → horizontal bars
Corrected.
> p.15 Eq. (25): g_{100} is not introduced.
Corrected.
> p.16 l.30: the reference to Eq. (34) (the next equation) is probably wrong.
Corrected.
> p.16 l.59: "Υ, the square root of which is the ratio ..." → "Υ, the
> square root of the ratio ..."
Corrected.