Dear Chunxiao Xu Please find attached the response to referees #2 and #3. To strengthen our case for novelty and relevance, we have now also added more papers of the last 5 years. Out of the now 41 papers, we count 14 of them as published 2016 or later, and 27 from before that time. On behalf of the authors, Axel Brandenburg ---------------------------------- X ------------------------------------ > Referee #2 (Comments to Both Author and Editor): > The work concerns with test of the thermal transfers inside a turbulent > box submitted to temperature wave and under radiative conditions for > different optical absorption conditions (from thin to thick optical > depth). The paper tends to show how decay of scalar fluctuations > evolved as a function of optical depth and turbulence levels. > I couldn't support the publication of this work before clearing the > followings points: We thank the referee for their critical remarks. They have helped to improve the presentation of our work. A lot of time has passed owing to other commitments, but we hope the referee is willing to review our improved version again. All changes are marked in blue. > 1) Precisions have to be given to explain the numerical forcing for > turbulence : > a. What is the form of this forcing ? We have now added subsections to Section III. The forcing function is now explained in Section III.C. > b. Is the way this forcing is obtained susceptible to modify the > results ? We share the worry of the referee that the nature of the forcing may well affect our results. Irrotational forcing, for example, is known to produce new effects, for example a strongly enhanced bottleneck effect. We have now discussed some caveats in the new Section III.A, where we also motivate the case for extended future studies. > 2) Precision have to be given concerning the convergence of > the results. What are the criteria chosen to obtain convergence of > computational results ? What are the computational time duration versus > integral scalar scales ? We appreciate this question and have now added a discussion on this in the new Section III.E, where we explain that this question is not a priori obvious, but why we believe that the present study is not affected by convergence issues. > 3) The size of the box have to be related to the different physical > scales (turbulence, temperature ... ) to show, for example, how the > kinematic turbulence spectrum is accounted for. We recall that the initially sinusoidal modulation of the temperature is to model effects on the scale of the system. Its wavenumber is k=k_1. The wavenumber of the forcing is kf, so the ratio kf/k1 is the scale separation. In the paper, we discuss cases with 10 and 1.5. We now discuss and present kinetic energy spectra in the new Section IV.B, where we compare runs from Series A and C with kf/k1=10 and 1.5, respectively. > 4) Some hypothesis have to be justified : > a) Is the viscosity u constant ? > b) If yes is it valid for the present temperature range ? > c) If no what is the relationship you use to account for ? In our simulations it is constant. In the Sun, nu varies only little, but it is many orders of magnitude smaller than what can be simulated even in future; the Reynolds number is at least 1e12. We have now added a corresponding discussion in the new Section III.A. We hope that this addresses the concerns of the referee. We recall that our goal is not to reproduce a realistic model of the Sun, but rather to study turbulent Newtonian cooling as a new physical effect. > 5) On P. 5 you have some comments on temperature fluctuation evolutions > as: "We clearly see that the temperature equilibrates..." We have now rephrased the wording to say "We see that the large-scale temperature contrast (wavenumber $k=k_1$) decreases the fastest for kl=1, and more slowly for kl=0.1 and 1. The title of the relevant subsection now reads "Range of simulations and qualitative aspects" so as to clarify that the quantitative assessment will be done later (in IV.C). > could you support this point of view with a clear criteria linked > for example with temperature fluctuation levels in the flow and as a > function of the different depth cases ? This was only a qualitative statement. The temperature contrast is then measured, and those results are presented in the following plots. > 6) Results plotted on figure 6 are not very clear to follow, even > after reading the reference paper 6, as situation is not the same for > the present work : > a) Could you explain clearly the condition of the test ? We have now added additional text to clarify to differences and similarities; see the first paragraph of Sect.IV.E. > b) For the scalar what is the diffusivity, the temperature > conditions, ... We have now explained it in the text by saying: "In these papers, the microphysical and turbulent diffusivities were referred to as kappa and kappa_t for the passive scalar (BSV09) (not to be confused with the opacity in the present paper) and eta and eta_t for the magnetic diffusivity (BRS08). They have the same meaning at chi and chi_t in the present paper. In these cases, we plot the time scale on which a large-scale sinusoidal profile of the passive scalar or the magnetic field gets diffused away." > 7) Figure 6 compare results from the present work with those of ref > 6 in which there is simulated flow with rotation ... is it similar > situation and in what way this could be compared ? In the present paper, we didn't talk about rotation, except once in the introduction. It is true that in that paper, they *also* studied the effects of rotation and magnetic fields. To avoid confusion, we have therefore now stated "They also studied the effects of rotation and magnetic fields, but those results were not used for the present comparison." > From my point of view, this work is interesting but some hypothesis > and computational conditions are difficult to follow probably due to > lack of explanations. This needs to be improve for the reader. We hope that the new clarifications added to the paper have helped improving the readability. ---------------------------------- X ------------------------------------ > Referee #3 (Comments to Both Author and Editor): > Review of the > manuscript POF20-AR-IFM2020-03460 > The present work is devoted to the investigation of influence of > turbulent mixing on the decay rate of the temperature perturbations. It > is interesting fundamental problem relevant for many applications. > Authors should make some amendments to the text or give comments to > the following questions before the publishing this work. We thank the referee for their constructive remarks. They have helped to improve the presentation of our work. A lot of time has passed owing to other commitments, but we hope the referee is willing to review our improved version again. All changes are marked in blue. > Generally, speaking about the turbulence one keeps in mind the necessity > of the broad band spectrum of initial perturbations but in your work > only the growth of single mode perturbations are considered. Why do you > think that it is enough for investigation of suggested problem? The > situation is worse by the usage of very low Reynolds numbers that > restricts perturbations development. Please comment this and clarify > the corresponding parts in the text. We now show kinetic energy spectra (see the new Fig.3 on page 6) to illustrate the level of turbulence that we are dealing with. We see that there is a broad spectrum with an inertial range covering about an order of magnitude, but it is already affected by the bottleneck effect. This is a physical effect and also seen in atmospheric turbulence, but it is weaker in the 1D spectra, as was explained in a 2003 paper, which we now also quote, in addition to more recent ones. > Section II. > On Fig 1 please use different line styles for curves (not only > color). It is difficult to use the text in the white-black view. We do already use different line types, but have then also additional groups that are shown in different colors. We have now added extra labels on the different lines. > Section III. > The section III should be supplemented by a short description of the > numerical code. It would be useful to understand what approach do you > use for solving basic set of equations. Also the clarification of the > model is needed. It is not clear do you solve the radiation transport > equation or just an equation of radiative heat conduction. We have now added a bit more text about the code and have now also referenced a recent publication in the Journal for Open Source Software. Regarding the radiation transfer, it is not the radiative heat conduction. This should have become clear when we state the transfer equation and talk about optically thick and thin in many places. In the introduction on page 2, we now explain the parallelization technique and write "This is done by splitting the calculation into parts that are local and nonlocal with respect to each processor. The two local parts are compute-intensive and the nonlocal does not require any computation and is therefore fast." > Please provide explicit expression for forcing function f. Its physical > meaning is unclear. We have now added this in the new Sec.III.C. > And finally what do you expect from using more realistic EOS? We have mentioned this now in the two paragraphs of the new Sect.III.A. In a stratified system system, it makes the photosphere sharper, but we don't have this here. One could imagine specific experiments where it plays a role, but this would seem to be then fairly independent of our goal of studying turbulent Newtonian cooling.