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> Referee #1
> The present work is devoted to the investigation of influence of
> turbulent mixing on the decay rate of temperature perturbations. It is
> an interesting fundamental problem relevant for many applications.
> The research is based on numerical simulations, so they play a crucial
> role and have to be well described. But there is still no full problem
> setup: the domain size is not provided, value T1 of perturbation amplitude
> is missing. Also please specify the units for \nu after eq. (16).
> It is not clear for what quantity the energy spectra EK is calculated?
> Please specify for what series of runs Fig. 2 is plotted?

We thank the referee for their comments. We have responded to all of them
and displayed the resulting changes to the paper in blue fonts. We now
specify T1 and L just after Eq.(15) on page 4. The units of \nu are now
added; see the text just after Eq.(16). The energy spectra are computed
from the velocity; see the new text in the third sentence of the first
paragraph of Sec.IV.B. Figure 2 is for Series A, as is now said in the
lower part of Sec.IV.A.

> Thus authors should make some amendments to the text to be ready for
> publication.

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> Referee #2

We thank the referee for their comments. We have responded to all of
them and displayed the resulting changes to the paper in blue fonts.

> I must preface my review here by saying that this work is well outside
> of my field of expertise, which is in experimental environmental and
> industrial aerodynamics. I would therefore rate my own confidence in this
> review as "low", and cannot offer any assessment of either the novelty or
> the importance of this work. I will try, however, to offer some feedback.

We were aware of the mixed audience and appreciate the feedback.

> First- incidentally the reason why I accepted the review- neither the
> title nor the abstract makes it clear that the subject of the paper
> is astrophysics. Because of the generality of the description (and the
> ambiguity of some of the discipline-specific terminology used), I had
> thought the paper would be about a novel technique for characterizing
> the performance of heat exchangers. It may be helpful to include in the
> (open access) abstract or title that the work is specific to the extreme
> environment inside stars.

We have now added a sentence right in the beginning of the abstract.
This should now clarify a possible contact point.

> The only other feedback I can offer is that I found the description of
> the numerical methodology to be somewhat lacking in detail. This may
> be a discipline-specific convention, but I would have expected to see
> an explicit description of all the boundary and initial conditions, a
> discussion of the sensitivity of the solutions to these, as well as of of
> the statistical convergence of the results. I also would have expected
> Fig 1 to appear later in the paper- this (it seems) is what has been
> used to validate the simulation, and would fit better into the logical
> chain of reasoning with the description of the methodology. It might
> also help the non-specialist to highlight that this was (I believe) a
> direct numerical simulation and didn't use any spatial or spectral models
> (which are almost universal in engineering) for small-scale structures.

At the end of the last paragraph of the new section III.E, we have
now stated that our simulations are direct numerical simulations. This
section has been assembled from text that was previously near the end
of the introduction. Regarding Figures 1, we should say that it is not
a validation of our turbulence results, but mainly an illustration of
the cooling curve in the absence of turbulence. We also show that, in
the absence of turbulence, the numerical simulations agree better with
the results form the Eddington approximation, but that the difference
to Spiegel's solution is small. The referee's suggestion regarding
validation is in principle already realized in our Figure 4, where we
overplot the non-turbulent result in a plot of our simulation results.

> Finally, a couple of trivial typos: the authors have omitted the units
> from viscosity on P4 under eq. 16, and although perhaps redundant,
> formally fig. 2 should include axes and scales on the spatial dimensions.

We have now added the units of the viscosity, and we have added axis
labels and a scale to Fig.2. Clarifications on this have been added at
the end of the caption.

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> Referee #4
> This is a solid paper reporting recent calculational results in radiative
> diffusion in turbulent flows. It should be published once the following
> minor changes are made to make it more accessible to the non-specialist:

We thank the referee for their comments. We have responded to all of
them and displayed the resulting changes to the paper in blue fonts.

> I. Newtonian cooling should be defined when it is first used. This
> is an elementary concept, but it is easy to forget the definition,
> and some readers will be many decades past their introductory physics
> courses.

We have now clarified the meaning of Newtonian cooling in the middle
of the third paragraph of the Introduction.

> II. \ell, as defined here, is the absorption length, not the optical
> depth. The optical depth is \int dx/\ell(x). In IV.D. k\ell is
> dimensionless. Eddington approximation should be defined where it
> is first used.

This is right; we now write "photon mean-free path" at the end of the
paragraph after Eq.(3). Yes, k\ell is dimensionless, and it wouldn't
be otherwise, but this seems to be already said correctly in IV.D.
The Eddington approximation is now explained just before Eq.(6).

> III.A. "automatically" isn't quite the right word; it implies a
> process. "Implicitly" would be better, but deleting the word entirely
> would probably be best; "precluded" is enough.

We have now omitted the word "automatically".

> III.B. A brief explanation or derivation of Eq. 11 would be useful.
> What does "double logarithmic" mean?

We have now added a derivation of Eq.(11) in terms of the first law of
thermodynamics, and we have added the definition of the double-logarithmic
temperature gradient in the following paragraph.

> III.C. What does "\delta-correlated" mean? That its temporal
> autocorrelation function is \delta(t)? If so, say so.  Following Eq. 14:
> Don't the authors mean |\tilde{f}|^2=1? f contains dimensional factors
> in {\cal N} that are not unity but depend on the strength of the forcing.

We have now clarified the meaning of \delta-correlated in the beginning
of Sec.III.C and corrected the normalization of f just after Eq.(14).