Diagnostic Relations between Pressure and Entropy Perturbations for Acoustic and Entropy Modes
Round 1
Reviewer 1 Report
I have no serious comments or objections to the manuscript after resubmission. I believe that the article should be accepted in the present form
Author Response
Please see attached.
Author Response File: 
Author Response.pdf
Reviewer 2 Report
The paper has been modified, but my main concerns (especially about the vorticiry mode and a clear brdiging with the general model decompostion in compressible turbulent flows) have not been takenb into account in a satisfactory way.
Since the authors obviously don"t want to revise the paper accordingly, I recommend to reject this manuscript
Author Response
Please see attached.
Author Response File: Author Response.pdf
Reviewer 3 Report
This manuscript presents a theoretical study of the dynamics of perturbations of an ideal atmospheric gas using a diagnostic approach. In this work a diagnostic equation is obtained. This solution gives the vertical profile of the contribution of the acoustic mode to the perturbation of entropy. Possibility of estimating the relative weight of the mode contribution is considered. The main result of the research is a model of a diagnostic algorithm in a one-dimensional exponential atmosphere. The comparison of the results obtained in the course of the analytical study with the results of numerical calculations is made. This article presents a fundamental study and has major importance for the further progress of atmospheric physics. However, in the section “Conclusions” I would like to see a text about the possible application of the developed method to theoretical and experimental studies of waves in the atmosphere and problems of detecting waves caused by earthquakes and tsunamis using Earth remote sensing facilities. Despite the fact that the scientific part looks convincing, the text of the article was prepared inaccurately. The analytical part is described in too much detail, some images are in fact duplicated and there are not always references to them in the text, formulas are sometimes also duplicated or their line-by-line transfer leads to excessive numbering, and more. This part needs to be optimized for the convenience of the readers. After the work is brought to its normal form, I can recommend it for publication.
Minor comments
in text:
Page 1, line 10: “Hence” is not needed here.
Page 2, line 39: replace "at the bottom of the Atmosphere" with "at the lower boundary in the atmosphere"
Page 2, line 46: There is a missing point before The
Page 2, line 63: It is not clear what is meant by "geophysical impacts"
Page 2, line 85: Remove "named as" heating "in laboratory acoustics"
Page 3, line 97: It is necessary to clarify the necessity of the condition of non-positive energy density with reference to the formulas.
Page 4, first line: instead of "defined in this way (Eqs, (4,5,6))" define verbally the meanings of P, F, U
Page 5, four line: “would” to “will”
Page 7, 5 line from the bottom of the page: remove H and alpha as formulas and just include them in the text.
Page 7, line 151: there is no need to write the equality for H (z) - it is already obvious. In the sentence, reference is made to the red graph of Figure 3, which is actually green.
Page 12, line 190: It is not clear which drawing Fig.5 is compared with. With figure 7? But in Figure 8, the graphs shown in Figure 5 and Figure 7. There is no need to repeat yourself. One Figure is enough 8.
Page 14, line 190: Lines 209-213 look like a summary. Therefore, it would be more logical to place them at the very end of this section.
Page 15, line 236: typo for functional parameter υ
in equations:
Page 4: Equations 7,8,9 can be combined into one system and referenced in the text to avoid redundant numbering of formulas.
Page 4:Equations 13,14 can be combined under one number and so refer to them further.
Page 4: Equations 15,16 can be combined under one number.
Page 4-5: It is not necessary to label the two-component vector and the column representing the state vector as separate formulas. You can try to start with the section with equation 19, to which to explain vectors in more detail, without labeling these explanations as separate formulas.
Page 9: Equations 41, 42 are identical to formulas 45, 46.
Page 9: Remove formula 48.
Page 12: Formulas 70 and 71 must have the same designation.
Hereinafter, it would be more convenient if the axis for height (z (km)) were the OY axis, not the OX axis. Also in Figure 2 you can see that the number of points on the graph is uneven. I would like to clarify why the number of points obtained for H (zi) is different at different ranges of heights.
The figure is located before its first mention in the text in another section.
Figure 3: The figure is positioned before its first mention in the text.
Figure 4: For Figures (a) and (b) make the same axes for f0 and fa.
Figure 6-7: These figures are not referenced in the body of the article.
Figure 8: This figure is a concatenation of fig. 5 and 7. Leave either fig.5,7 or fig.8.
Figure 8-9: The location of the figures should correspond to the references in the text.
Author Response
Please see attached.
Author Response File: Author Response.pdf
Reviewer 4 Report
Dear Authors, I would recommend checking English a little more. Although in general it is not bad. However, for example, from my point of view, there should be "gravity field" and not "gravitational field", because this is a stable phrase. It also seems to me that some of the calculations from Sections 4.3 and 5.2 maight be possible transferred to the appendices. This is a delicate question, and it have to be dealt with carefully. The function f _0 (z) appeared in formula (21), and nothing was said about it. At least a few words could be dedicated to her. It also seems to me that the formulation of the problem, formulas (7) - (10), perhaps, could be described in more detail. For example, do you mean an initial problem, which is further analyzed?
Author Response
Please see attached.
Author Response File: Author Response.pdf
Round 2
Reviewer 3 Report
The new version of the article is much better. I recommend the article for publication.
This manuscript is a resubmission of an earlier submission. The following is a list of the peer review reports and author responses from that submission.
Round 1
Reviewer 1 Report
In the present manuscript the new method for analyzing variations in the observed atmospheric parameters with the possibility of separating the contributions of the acoustic and entropy modes is proposed. The latter corresponds to the dissipation of propagating acoustic waves in the atmosphere. The calculations of the parameters of the acoustic and entropy modes have been carried out under the conditions of the linear dependence of the height of a homogeneous atmosphere on the altitude. The results of calculations within the framework of the linear approximation model for the height scale are in good agreement with ones using the standard atmosphere model at altitudes from 120 up to 160 km that proves the validity of the theory. This approach is proposed for the first time and is promising for practical application.
To summarize, I think that this paper could be interesting and useful for a potential reader of the "Atmosphere" journal. The article may be accepted for publication after minor revision which takes into account the comments specified in attached file.
Comments for author File: Comments.pdf
Author Response
Dear Reviewer, thank You very much for the attentive reading of our manuscript!
Especially for the our mistake You had noticed and poined!
below - our comments, pdf - attached.
General remark for all referees.
By the decision of the editor, we have made the major revision of the manuscript.
- We have corrected mistakes and gave more explanatory details.
- More essential accents are given with respect to the physical sense and applicability of the model under consideration.
- We do restrict ourselves by the 1D model because it is the first step to the novel methodology elaboration. The methodology demonstrates a possibility to extract the entropy mode (sometimes named as stationary or the heat one) and wave modes at the atmosphere with a more realistic equilibrium height structure, compared with the so-called "exponential."
- Four citations are added; two related to the origin of the method (pioneer Kovasznay papers), and two recent publications on Earth surface hazard phenomena detection via transmission of acoustic waves to produce ionosphere disturbances.
- One of the referees suggested changing the axis of the plot, directing the abscissa variable "z" up. We, however, left the plots as it was, because of the rest referees opinion, thinking also about potential readers from applied mathematics community.
Next, we comment on the referee's remarks"
1) Authors use incorrect reference order. For example:
line 43: [2,10] should be replaced by [2,6]
line 47: [5,8,10] should be replaced by [5-7]
line 51: [15] should be replaced by [8]
and so on. This makes it difficult to read the text as a whole. The numbering
should be done in the traditional order [1], [2], [3], ... .
- Done. Thank You.
2) The axes in Figures 2-9 should be swapped. It is more familiar to the reader to indicate the altitude z along the vertical axis. Moreover, units of measurement must be specified in the axis captions.
We, with apologies, would leave the axis z direction as it was, because of the rest referees opinion, thinking also about potential readers from the applied mathematics community.
3) A short explanatory text should be given after Figures 3, 4, and 6. What conclusions can be drawn from looking at them? Specific references to Figures 3, 4
- Yes, thank You. The following paragraph
«Note, that the difference between the dependence of H(z) taken from the dataset and the linear approximations, given at Fig. \ref{liner H(z_i)} within 120 km to 180 km range, is almost invisible at such scale. This gives as an argument to use such linear approximation in further modeling.»
Is inserted under the Fig. 3;
«The oscillations of the RHSs of the diagnostic equations for $P_{0,a}$ (Figs \ref{4}), apart from a small variation of the functional parameter $H(z)$, appear due to the application of differentiation operation to the dataset components as in \eqref{Dai}, which scale of coordinate differences and errors are noticeable.
It is seen in Figs 5, that the result of the diagnosis as vertical structure of the contributions P0,a in the pressure perturbation P looks much more smooth because its definition contains integration.»
Is inserted under Fig. 4, related to Fig. 5;
«We see, that the plots look as smooth as ones at Fig. \ref{P0Pa}. It is the result of the integration that acts as a "smoothing" operation, as opposite to differentiation. Such phenomena are well-known in the theory of inverse problems.
The plots of the Fig \ref{comp} represent one of the principal results of this work: it shows that there is a discrepancy between the profile obtained by the direct dataset processing and handling by means of the apparatus built by the analytical approximation of the theory elements. The difference, however, is not so big, the linear model allows to estimate the entropy mode profile. The transition to energy distribution leads to the results for which the difference almost disappear, see Fig. \ref{ene}.»
Is written after Eq. (75).
4) The equations of the theory may be presented in a more compact form. For example, the relations (53)-(57) contain too many details. , and 6 should also be made in the text of the article.
- Done.
Specific comments:
1) line 13: heights should be replaced by height
- Thanks, changed.
2) line 30: The abbreviation PDE
should be deciphered, i.e. partial differential
equation
- Yes, inserted.
3) line 60-61: The term constant mass force is unfortunate. It should be replaced or removed
- OK, removed, changed to "gravitational field".
4) line 81: sound should be replaced by acoustic
- Done.
5) line 103: contributions should be replaced by contribution
- Done.
6) line 109: It is necessary to determine the values g, Cp, and Cv in the formula (2)
and the value \gamma in the formula (3)
- Done, thank You. Inserted the sentence:
Here the conventional gas parameters are used: g - gravity acceleration, $C_{p,v}$ are the molar heat capacities at constant p,v correspondingly.
7) line 121: It is not clear why on the right side of the first line of expression (23) it is assumed that P0+D0 0=0, while according to equation (15) 0+D0P0=0. It is necessary to check carefully expressions (13), (15), and (23) once again
-Thank You! It is important! We have made a mistake, sorry. It leads to the impossibility to use the first order differential equation, so we were forced to return to the second order equation (22), which we name the first diagnostic equation. We also introduce the second one (24), which we use for a solid test because the sum of the solution should give the pressure perturbation.
8) figure 1: The axis labels in Figure 1 are too small. It is necessary to increase
their size
-OK.
9) figures 7 and 8: Obviously, the information presented in Figure 8 duplicates the data in Figure 7. In my opinion, it is enough to leave only the Figure 8, and delete the Figure 7.
-OK, corrected.
Author Response File: Author Response.pdf
Reviewer 2 Report
The paper deals with a modal 1D decomposition of fluctions in a stratified atmosphere into acoustic and entropic modes. The topic might be interestng, but I don't recommend the paper for publication. My main concerns are the following:
- the idea of projecting fluctuations onto a modal basis made of acoustic, entropy and vortical mode is not new (pioneering works were done by Chu and Kovasznay in the 1950s). The list of reference is severly truncated and mostly oriented toward russian authors, completly missing other works
- Here, the framework is very limited: the 1D approximation for the fluctuations preclude the vortical mode. This is a very restrictive approximation. 3D perturbations around a 1D base flow should be considered, since vortical modes are present in experimental data. The present projection completely miss that.
- A more general framework should be considered, including roration effects (which are important for geophysical flows), and the combination of stratification, rotation and shear yields the definition of much more complex fluctuating modes
Therefore, eventhough the present paper is mathematically correct, the physical limitations are so strong that the results are of poor interest for the international scientific community
Author Response
Dear Reviewer, thank You very much for attentive reding of our manuscript and profound analysis of it. Our comments are below, pdf attached.
General remark for all referees. By the decision of the editor, we have made the major revision of the manuscript.
- We have corrected mistakes and gave more explanatory details.
- More essential accents are given with respect to the physical sense and applicability of the model under consideration.
- We do restrict ourselves by the 1D model because it is the first step to the novel methodology elaboration. The methodology demonstrates a possibility to extract the entropy mode (sometimes named as stationary or the heat one) and wave modes at the atmosphere with a more realistic equilibrium height structure, compared with the so-called "exponential
- Four citations are added; two related to the origin of the method (pioneer Kovasznay papers), and two recent publications on Earth surface hazard phenomena detection via transmission of acoustic waves to produce ionosphere disturbances.
- One of the referees suggested changing the axis of the plot, directing the abscissa variable "z" up. We, however, left the plots as it was, because of the rest referees opinion, thinking also about potential readers from applied mathematics community.
Next, we comment on the referee's remarks:
- the idea of projecting fluctuations onto a modal basis made of acoustic, entropy and vortical mode is not new (pioneering works were done by Chu and Kovasznay in the 1950s).
- You are right, the idea of the mode content/separation first appears in the paper of Kovasznay [1] and Chu and Kovasznay [2] in the 1950s, refs we focus on the application/realization of this idea (see ref [10], Leble, S.; Perelomova, A. Dynamical projectors method in hydro- and electrodynamics; CRC Press, Taylor and Frensis group, 2018, where the history of the approach is described in details.)
- The list of reference is severely truncated and mostly oriented toward Russian authors, completely missing other works
The list of refs looks truncated because we suggest a reader review [10]. It is important to say that most of the published papers (we know) relate either to solid state physics or nonlinear acoustics. The problem of diagnostics is scarcely touched. Now, we have added the refs [1,2] to Kovasznay and Chu. We also have included references to [7.9] in the context of possible atmosphere perturbation origin and structure. The ref. [3] is excluded for its difficult access.
- Here, the framework is very limited: the 1D approximation for the fluctuations preclude the vortical mode. This is a very restrictive approximation
- Such investigation that we provide is, perhaps, the first one, that realizes the idea in such practical aspect as determination of acoustic and entropy mode contribution in a non-exponential atmosphere perturbation. It explains its 1D realization. A more complicated 2D case consideration is in our plans, but it needs a serious development of the mathematical groundwork. We believe, however, that in diagnostics namely 1D case gives a reasonable estimation of the contributions at least in the case of rather plane wavefront, that is presented in papers that study the impact of atmosphere perturbations from earthquakes and tsunami phenomena [Zettergren, M.D.; Snively, J.B. Ionospheric response to infrasonic- acoustic waves generated by natural hazard events, J. Geophys. Res. Space Phys. 2015, 120, 8002Ц8024, doi:10.1002/2015JA021116.]. The authors stress, that the acoustic component of a perturbation is the first that reach the ionosphere.
- 3D perturbations around a 1D base flow should be considered, since vortical modes are present in experimental data. The present projection completely miss that.
- The rotational mode account is also in our plans, the case with constant H(z) is considered in [ Leble, S.B. \textit{Nonlinear waves in waveguides with stratification; -Verlag, Berlin, 1990]
and recent [Ivan Karpov and Sergey Leble.On Problem of Rossby and Poincarіe Atmospheric Waves Separation and Spectra.TASK Quarterly, v. 221 (2) P. 85-96 2016]. Its generalization to the height dependent H needs serious mathematical efforts.
Generally, we believe that such steps in the proper diagnosis theory and practice development are necessary, and it could be interesting for geophysicists that work in pure and applied aspects of atmosphere problems, as well as for applied mathematicians, that would investigate the statement (and solution) of direct and inverse problems of the class we formulate.
Author Response File: Author Response.pdf
Reviewer 3 Report
See attached file.
Comments for author File: Comments.pdf
Author Response
Dear Reviewer, thank You very much for attentive reading of our manuscript/profound remarks. Our comments are below, its pdf is attached.
General remark for all referees. By the decision of the editor, we have made the major revision of the manuscript.
- We have corrected mistakes and gave more explanatory details.
- More essential accents are given with respect to the physical sense and applicability of the model under consideration.
- We do restrict ourselves by the 1D model because it is the first step to the novel methodology elaboration. The methodology demonstrates a possibility to extract the entropy (sometimes named the heat one) and wave modes at the atmosphere with a more realistic equilibrium height structure, compared with the so-called "exponential."
- Four citations are added; two related to the origin of the method (pioneer Kovasznay papers), and two recent publications on Earth surface hazard phenomena detection via transmission of acoustic waves to produce ionosphere disturbances.
- One of the referees suggested changing the axis of the plot, directing the abscissa variable "z" up. We, however, left the plots as it was, because of the rest referees opinion, thinking also about potential readers from the applied mathematics community
The main referee’s comments:
- The authors should better explain differences between their “entropy modes” and traditional stationary background solution.
- We name the case, of our model corresponding \omega=0, as «entropy mode», following celebrated Chu, Kovasznay [1,2] papers, see also Leble, Perelomova [11], Its wave character (propagation presence) may manifest in a theory that accounts dissipation and nonlinearity, when the dispersion equation root is nonzero, but small [Leble, Perelomova, 11] In physics it exhibits also as heating, energy transfer from wave such mode. . In a 3D problem, we have also a component which evolution is of rotational type (streaming in laboratory nonlinear acoustics), also with a frequency close to 0.
- The model of background temperature profile used in the study needs more
descriptions and clarifications.
- The following phrase "The height scale profile is built directly by the table for background temperature $T(z_i)$, taken from \cite{SA}." is added under Fig. 2. The reference to the \cite{SA} is added to the Fig. 2 caption.
- The equations contain overlapping notations, which should be corrected.
- Done.
On specific comments:
- Line 12; Replace “[120;180] km” with “120 – 280 km”.
- OK, corrected.
- Line 30. Abbreviation PDE is not defined.
- Now defined.
- Lines 39 – 41. Strange classification. Usually, waves in the “exponential atmosphere” are classified as acoustic waves, gravity waves and stationary background solution. Authors should explain how their “entropy waves” correspond to conventional classification. Or, this is
a completely new branch of waves?
- We use the same classification for the linear problem solution for the "exponential atmosphere", taking the term "entropy mode" for the general case, corresponding to the "stationary solution". Its origin goes to Kovasznay nomenclature [1,2] suitable for dissipation and nonlinearity account. In a sense, we trace the diagnosis of the phenomenon of the "heating" or "warming"
at the Atmosphere.
We, now avoid the term "wave entropy mode", using simply "entropy mode' in our manuscript.
- Lines 93, 102. Replace [120;180] by [120,180].
- OK, replaced.
- Lines 110-112. Authors claim that perturbations represent deviations from the adiabatic state, but their equations for the perturbations (7) – (10) are also adiabatic. Therefore, the term “entropy perturbations” require further clarifications.
- We use again the notation from old [1,2], it is simply perturbation of the entropy of the ideal gas, It, naturally, could be adiabatic, You are right.
We rewrite the sentence:
"We will name it the entropy perturbation, because in a limit with $g=0$m and constant background temperature $\overline T$,
$\varphi^{\prime}$ represents the deviation of the thermodynamic process occurring in fluid from the adiabatic one. "
as
"We will name it the entropy perturbation, because in a limit with $g=0$ and constant background temperature $\overline T$,
$\varphi^{\prime}$ represents the deviation of the ideal gas entropy from the equilibrium one [28]. "
- L 113. It is not defined what is the “acoustic mode” for eq. (7) – (10)? What is the “entropy
mode”?
- The definitions are the equations (12) for the acoustic mode (the explanation is given by the sentence above (12)), next (13) for the "entropy mode" with the explanation above.
From Eq. (13), it seems that the entropy mode is equivalent to the stationary hydrostatic atmosphere. Usually, such solutions do not consider waves at all. This is the background atmosphere.
See the comment on Lines 110-112. You are right, in the limit H(z)-> const in non-dissipative linear equations of atmosphere gas, the "entropy mode" is equivalent to the "stationary one, \omega=0" mode. We prefer to name the mode "entropy mode" because we apply the diagnostic tool to the numerical solution of the nonlinear dissipative system, in which the phenomenon of "heating" is implied.
- L 121. The abbreviation “LHS” is not defined.
- Ok, done.
- L. 131 – 133. Comparisons of the 1D model with numerical solutions for the 2D model require more explanations.
- Fig. 2. Why so big drop in H(z) at z ~ 300 km is exits? Which temperature model was used in the study?
- We have used the ‘standard atmosphere”, see the last ref. [30], that was the base in the numerical simulation we analyze.
- Fig. 3. What is g(z)? In the Eq. (7) – (10) g is the gravity acceleration and its height dependence should be different from Fig.3.
- The notation "g" is changed to "f_{0,a}". Thank You.
- Fig. 5. Red line is not visible.
- OK, changed to blue and green.
- Fig. 6. What are reasons for not smooth line g(z) at smooth temperature profile in Fig. 5 and smooth Φ(z) in the right panel of Fig.6?
- The explanations are added between Fig 4 and 5 (new numeration)
The oscillations of the RHSs of the diagnostic equations for $P_{0,a}$ (Figs 4), apart from small variation of the functional parameter $H(z)$, appear due to application of differentiation operation to the dataset components as in \eqref{Dai}, which scale of coordinate differences and errors
- after Fig 5:
It is seen at the Figs \ref{P0Pa}, that the result of the diagnosis as vertical structure of the contributions $P_{0,a}$ in the pressure perturbation $P$ looks much more smooth because its definition contains integration.
- after Fig 7:
We see, that the plots look as smooth as ones at Fig. \ref{P0Pa}. It is the result of the integration that acts as a "smoothing' operation, as opposite to differentiation. Such phenomena are well-known in the theory of inverse problems.
- Fig. 8. It is still unclear, which “standard model” was used for temperature? Dependence H(z) in Fig. 2 looks strange above 300 km.
- We were forced to use the same background temperature z-profile as the authors of the simulations under consideration. The behaviour above 180 km does not influence our calculations.
- Also, blue lines in Fig. 8 are obtained not from 1D, but from the 2D numerical model.
- No, we use the 2D data field (say, functions of x,z) from a numerical model, but analyze the set of values at $z_i$ for the fixed point x=0, observing rather a plane wavefront at Fig. 1.
- L 176-178. It is still unclear advantages in calling the background stationary solution as “entropy wave mode”? The paper does not contain any evidences that the background fields have wave structures.
- Yes, the eventual wave structure of the "entropy mode" is out of scope in the one-moment diagnostics we develop here, see the comments above.
- We also have added the following comment under Fig. 8
The plots of the Fig \ref{comp} represent one of the principal results of this work: it shows that there is a discrepancy between the profile obtained by the direct dataset processing and handling using the apparatus built by the analytical approximation of the theory elements. The difference, however, is not so big, the linear model allows to estimate the entropy mode profile. The transition to energy distribution leads to the results for which the difference almost disappear, see Fig. \ref{ene}.
- The following sentence is added in the Sec. 6:
"The addition of independent results of calculations ofP0 and a gives the curve closely matching with the graph of a function P represented by formula (4), which is consistent with the main idea of the expansion into modes P=P0+Pa."
Author Response File: Author Response.pdf
