Questions and answers

1. Where is the source for the density data used in the scale height calculation?

The isothermal scale height is defined as H = p₀ (0)/ρ₀ (0) = v_s² /γg = const. Knowing that sound velocity is defined as v_s²=γRT, it can be seen that H=RT/g. Therefore, we only need to know temperature to calculate H. The source of the temperature data in articles:

Atmospheric Layers in Response to the Propagation of Gravity Waves under Nonisothermal, Wind-shear, and Dissipative Conditions John Z. G. Ma,

in Gavrilov, N. M., Kshevetskii, S. P., and Koval, A. V.: Propagation of non-stationary acoustic-gravity waves at thermospheric temperatures corresponding to different solar activity, Journal of Atmospheric and Solar-Terrestrial Physics, 172, 100-106, https://doi.org/10.1016/j.jastp.2018.03.021, 2018,

in Kshevetskii, S.P.; Kurdyaeva, Y.A.; Gavrilov, N.M. Spectra of Acoustic-Gravity Waves in the Atmosphere with a Quasi-Isothermal Upper Layer. Atmosphere 2021, 12, 818. https:// doi.org/10.3390/atmos12070818

is NRLMSISE-00 (ref. Picone, J.M.; Hedin, A.E.; Drob, D.P.; Aikin, A.C. NRLMSISE-00 Empirical model of the atmosphere: statistical comparisons and scientific Isssues. J. Geophys. Res. 2002, 107, 1468.).

I prefer the newest source NRLMSIS 2.0, ref. Emmert, J. T., Drob, D. P., Picone, J. M., Siskind, D. E., Jones, M. Jr., Mlynczak, M. G., et al. (2020). NRLMSIS 2.0: A whole‐atmosphere empirical model of temperature and neutral species densities. Earth and Space Science, 7, e2020EA001321. https://doi.org/10.1029/2020EA001321, fig. 1, page 4.

2. While the derivation of the equations is thorough, adding more detailed explanations in some intermediate steps would enhance reader comprehension. For instance, it would be beneficial to clearly explain why the conditions for K_p²​ and V_h^2​ differ between evanescent and vertically propagating waves, ensuring readers can easily follow along.

Answer: In page 4 is: The AGWs propagate in the vertical direction if K _z² > 0. This is fulfilled when Eqs. 11 and 12 are satisfied. The next sentence will be changed in this way:  "The AGWs become evanescent if K_z² < 0, i.e. when..."

3. Please ensure that the labels and legends in the figures are larger and more distinguishable for better clarity.

Answer: I make the figures larger and changed legends in Figs. 5 and 7.

4. Many readers are likely to learn substantial knowledge from your article, particularly due to the well-organized Discussion section. It would be beneficial to address the limitations of the study (e.g., not only the primary generated acoustic-gravity waves reach the ionosphere, but also secondary waves due to nonlinear interactions) and suggest future research directions to address these shortcomings.

For example, in this article, the reflection coefficient is set with the lower boundary condition strictly at z=0z=0z=0, which limits the discussion to waves propagating in the 60-100 km range. According to Gavrilov and Kshevetskii, secondary waves can emerge from nonlinear interactions in this region. Given that high-frequency AGWs are likely to undergo significant nonlinear interactions, discussing these secondary effects would provide valuable insights to researchers in the field of acoustic-gravity waves and gravity waves.

Answer: I added new paragraph at the end of the Discussion  section:

"AGWs driven from the Earth’s surface or troposphere are typically characterized as primary or higher order (e.g., secondary)

depending on how they propagate to thermospheric altitudes (Zawdie et al. , 2022). Primary AGWs propagate directly through

the thermosphere and can be modeled using linear theory. Klymenko et al. (2021) proposed a method for recognizing the types

of linear AGWs in the atmosphere from satellite measurements. Higher order AGWs are created when primary AGWs break

in the upper atmosphere; nonlinear propagation theory is required to simulate them (Vadas and Crowley, 2010; Gavrilov and

Kshevetskii , 2014; Gavrilov, Kshevetskii, and Koval , 2018; Dong et al. , 2022). Considerable attention has recently been paid

to the study of so-called secondary AGWs that arise as a result of instability and nonlinear interactions of primary wave modes

propagating from atmospheric sources, among themselves, and with the mean flow. Gavrilov and Kshevetskii (2023) separated

the horizontal spatial spectra of primary and secondary AGWs at fixed altitude levels in the middle and upper atmosphere at

different time moments using a three-dimensional nonlinear high-resolution model AtmoSym. This separation of the spectra of

primary and secondary AGWs makes it possible to estimate the relative contribution of secondary AGW at different altitudes,

at different times, and with different stability of background temperature and wind profiles in the atmosphere. These issues are

important for future research, and numerical models could be a good tool for them."

The  references mentioned above are:

Zawdie, K., Belehaki, A., Burleigh, M., Chou, M-Y., Dhadly, M. S., Greer, K., Halford, A. J., Hickey, D., Inchin, P., Kaeppler, S. R., Klenzing,

J., Narayanan, V. L., Sassi, F., Sivakandan, M., Smith, J. M., Zabotin, N., Zettergren, M. D., and Zhang, S-R.: Impacts of acoustic and

gravity waves on the ionosphere, Front. Astron. Space Sci. 9:1064152, doi: 10.3389/fspas.2022.1064152, 2022.

Klymenko, Y. O., Fedorenko, A. K., Kryuchkov, E. I. et al.: Identification of Acoustic-Gravity Waves According to the Satellite Measurement

Data, Kinemat. Phys. Celest. Bodies, 37, 273–283, https://doi.org/10.3103/S0884591321060052, 2021.

Vadas, S. L. and Crowley, G.: Sources of the traveling ionospheric disturbances observed by the ionospheric TIDDBIT sounder near Wallops

Island on 30 October 2007, J. Geophys. Res. 115, A07324. doi:10.1029/2009JA015053, 2010.

Gavrilov, N. M. and Kshevetskii, S. P.: Three-dimensional numerical simulation of nonlinear acoustic-gravity wave propagation from the

troposphere to the thermosphere, Earth, Planets and Space, 66, 88, doi.org/10.1186/1880-5981-66-88, 2014.

Gavrilov, N. M., Kshevetskii, S. P., and Koval, A. V.: Propagation of non-stationary acoustic-gravity waves at thermospheric

temperatures corresponding to different solar activity, Journal of Atmospheric and Solar-Terrestrial Physics, 172, 100-106,

https://doi.org/10.1016/j.jastp.2018.03.021, 2018.

Gavrilov, N. M. and Kshevetskii, S. P.: Identification of spectrum of secondary acoustic-gravity waves in the middle and upper atmosphere

in a high-resolution numerical model, Solar-Terrestrial Physics, 9, 3, 86-92, doi: 10.12737/stp-93202310, 2023.

Dong, W., Fritts, D. C., Hickey, M. P., Liu, A. Z., Lund, T. S., Zhang, S., et al.: Modeling studies of gravity wave dynamics in highly

structured environments: Reflection, trapping, instability, momentum transport, secondary gravity waves, and induced flow responses,

Journal of Geophysical Research: Atmospheres, 127, e2021JD035894. https://doi.org/10.1029/2021JD035894, 2022.

5. Some of the references are quite dated. Please include more recent studies to support your findings and discussions.

Answer: I excluded the reference Mitra, 1974 and included aforementioned references.

6. (Minor) The term "Brunt–Väisälää frequency" is uncommon. Typically, it is expressed as "Brunt–Väisälä frequency." Clarify why you choose to use "Brunt–Väisälää".

Answer: I find some articles with the term  "Brunt–Väisälää frequency", e.g. in article Near-Earth breakup triggered by the earthward traveling burst flow, GEOPHYSICAL RESEARCH LETTERS, VOL. 32, L13107, doi:10.1029/2005GL022983, 2005 by I.O. Voronkov, page 3, below Eq. 2; also in thesis "Ionospheric gravity wave interactions and their representation in terms of stochastic partial

differential equations" by Victor Nijimbere, Carleton University Ottawa, Ontario, Canada April, 2014  on page 35 after Eq. (3.14); also in "Stratospheric warmings-the quasi-biennial oscillation Ozone hole in the Antarctic but not Arctic -Correlations between the Solar Cycle, Polar temperatures, and an Equatorial oscillation

HOPPE Ulf-Peter

FFI/RAPPORT-2001/02263, Norwegian Defense Research Establishment, P O Box 25, NO-2027 Kjeller, Norway, on page 14, after Eq. (2.2). After your suggestion I checked this term and find that much more articles use "Brunt–Väisälä frequency" than "Brunt–Väisälää frequency" so I changed it in my article.

Comments:

1. In my opinion, the title of the work does not fully correspond to its content. The term "ionosphere-lower thermosphere coupling" is usually used in the sense of the interaction of the neutral component of the atmosphere with the ionospheric plasma, which is realized through the collision of particles. In this sense, the analysis of the dispersion equation of AGW and reflection coefficients is not enough for the study of "ionosphere-lower thermosphere coupling". In this work, the D layer is considered formally, simply as a certain altitude level of the atmosphere (60-90 km), and the ionospheric plasma is not involved in this consideration in any way. In fact, only the neutral atmospheric environment is analyzed, which is described by the HD system of equations (1)-(3).

Answer: In article Obscure waves in planetary atmospheres, page 43, Erdal Yiğit and Alexander S. Medvedev say: "Interestingly, in the mid 20th century, aeronomists—scientists who study the upper atmosphere—were largely unaware of gravity waves, and meteorologists had written them off as insignificant noise. At the end of the 1950s, Colin Hines, an aeronomist, was among the first to link numerous observed atmospheric features to gravity waves propagating from below. He was struck by the coupling and later described it in everyday terms: “The ionospheric regions would be like a light-weight tail wagged by a very massive dog, and they must respond to almost any disturbance created below.”

In article Statistical investigation of gravity wave characteristics in the ionosphere by Chum et al., in Introduction, firs sentence is: "Atmospheric gravity waves, often called travelling ionospheric disturbances (TIDs) if observed in the ionosphere, have been extensively studied in recent years for the following reasons. First, they couple different atmospheric layers and deposit momentum and energy in the layer in which they dissipate, and thus influence temperature and winds in the middle and upper atmosphere..."

In article Response of the mesosphere-thermosphere-ionosphere system to global change - CAWSES-II contribution, page 5: "It is very likely that trends in the lower atmosphere have an influence on the upper atmosphere via dynamical coupling through upward propagating waves. There is modeling evidence that fluxes of waves entering the mesosphere, lower thermosphere, and ionosphere (MLTI) have changed and will continue to change."

Also in articles: Yigit, E., Medvedev, A.S., 2015. Internal wave coupling processes in Earth's atmosphere. Adv. Space Res. 55 (4) https://doi.org/10.1016/j.asr.2014.11.020.

Yigit, E., Medvedev, A.S., 2016. Role of gravity waves in vertical coupling during sudden stratospheric warmings. Geosci. Letts. 3, 27. https://doi.org/10.1186/s40562-016-0056-1.

In article Analysis of in situ measurements of electron, ion and neutral temperatures in the lower thermosphere–ionosphere by Panagiotis Pirnaris and Theodoros Sarris , Ann. Geophys., 41, 339–354, 2023 https://doi.org/10.5194/angeo-41-339-2023, page 346: "The heating and cooling effects of GWs in the thermosphere have been extensively investigated by many simulation studies. For example, Yiğit and Medvedev (2009) used a GW parameterization that was specifically designed for thermospheric heights, which was implemented in the Coupled Middle Atmosphere and Thermosphere (CMAT2) global circulation model (Harris, 2001; Dobbin, 2005)..."

In article Dynamical coupling processes between the middle atmosphere and lower ionosphere by W.K. Hocking, J. Atmos. Terr. Phys. 1996, 58, 735–752, in Abstract: "A variety of dynamical processes are important in coupling motions within the middle atmosphere and lower ionosphere. These processes can generally be classified as either advective, wave-like or diffusive. Within the middle atmosphere, wave-like processes, and especially gravity waves, are of crucial importance. Turbulent diffusion and advection probably play lesser, but not insignificant, roles. However, whether these same concepts apply to coupling for regions outside the middle atmosphere—and especially between the upper middle atmosphere and the lower ionosphere—is not clear. In this paper the current knowledge about coupling processes between these important regions is reviewed. "

In these articles the term ionosphere-lower thermosphere coupling refers to coupling between atmospheric layers by AGW/GWs waves. Therefore I used the term coupling. If you insist it could be changed in interaction.

Besides, in my article, in the section 2. Basic equations is said:" The D layer is part of the ionosphere, where typical atmosphere models give n _n ∼ 10 ²¹ m ⁻³ for the neutral particle density and

n _p ∼ 10 ⁸ m ⁻³ for charged plasma particles, and where electric and magnetic effects play a minor role in the local atmosphere dynamics. This is why hydrodynamic (HD) equations, rather than magneto–hydrodynamic (MHD) equations, can be used to analyze wave propagation. The same is in Nina, A. and Čadež, V.: Detection of acoustic-gravity waves in lower ionosphere by VLF radio waves, Goephysical Research Letters, 40, 18, 4803-4807, https://doi.org/10.1002/grl.50931, 2013.

2. The considered model is questionable. In fact, the work examines two isothermal altitudinal layers with different temperatures, separated by a conventional boundary. The lower layer denotes the D layer, and the upper one denotes lower thermosphere. In the lower considered altitude interval of 60-90 km, the atmosphere is significantly non-isothermal. In particular, these heights include the temperature minimum in the mesopause. In the lower thermosphere (90-140 km) the largest altitudinal temperature gradient in the atmosphere is observed. That is, both the upper and lower layers considered in the work can hardly be considered isothermal. At the same time, the author uses the theory for freely propagating AGWs, developed for an isothermal atmosphere. Based on this theory, the dispersion equation and reflection coefficients were obtained. It is advisable to apply the isothermal theory of AGW in the thermosphere

above 200 km, where the temperature almost does not change with height.

Answer: I used the same approach as in the article by Nina, A. and Čadež, V.: Detection of acoustic-gravity waves in lower ionosphere by VLF radio waves, Goephysical Research Letters, 40, 18, 4803-4807, https://doi.org/10.1002/grl.50931, 2013, page 4805: "This is shown in Figure 3 taking 250 K as typical temperature of the ionosphere below 90 km. Also, in article Horizontal and vertical propagation and dissipation of gravity waves in the thermosphere from lower atmospheric and thermospheric sources by Sharon L. Vadas, JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 112, A06305, doi:10.1029/2006JA011845, 2007, on page 2: "Although the average temperature in the lower atmosphere is T ~ 250 K, the temperature increases rapidly in the lower thermosphere. During extreme solar minimum, the thermosphere is relatively cold, T~600 K. During active solar conditions however, the temperature in the thermosphere can be T~2000 K".

3. P. 4. Lines 46-47. “The standard set of HD equations describes the dynamics of adiabatic processes in a fully ionized hydrogen plasma (?) in the presence of gravity…”. The statement is incorrect. In this work, the analysis was carried out on the basis of the linear theory of AGW for a neutral isothermal atmosphere. In fact, the system of equations (1)-(3) for a neutral atmosphere is given below.

Answer: You are absolutely right. I prepared the article for plasma and did not have enough concentration to adapt the text to the neutral atmosphere case. I am sorry for this mistake.

4. P. 4. The analysis of the dispersion equation of AGW (10) given in lines 90-94 is incomplete. In the sense that inequalities (11) and (12) are valid only for the acoustic branch when Ω²-Ω²_co>0 and Ω²-Ω²_BV>0. However, for the gravitational branch, Ω²-Ω²_co<0 and Ω²-Ω²_BV<0, (See Fig.1) are always performed. Then, for the condition of free propagation of gravitational waves (K²_z>0 ), the   dispersion equation (10) follows K²_p>Ω²(Ω²-Ω²_co)/Ω²-Ω²_BV and V²_h<Ω²-Ω²_BV/Ω²-Ω²_co. That is, for the gravity branch of AGW, inequalities (11) and (12) have the opposite sign. Accordingly, the reflection conditions of the AGW should be recorded separately for the acoustic and gravitational branches. It is necessary to analyze this in the text and how it will affect the results.

Answer: In general, from the Eq.(10) and the condition K²_z>0 for the AGWs propagation, we can derive the relations (11) and (12), i.e.,  K²_p<Ω²(Ω²-Ω²_co)/Ω²-Ω²_BV and V²_h>Ω²-Ω²_BV/Ω²-Ω²_co.  From these relations the equations for propagating acoustic and gravity wave branches can be derived separately. I didn't do that in the article because I thought it is not necessary. 

a) Acoustic branch, i.e., acoustic waves which propagate when K²_z>0 for K²_p<Ω²(Ω²-Ω²_co)/Ω²-Ω²_BV and V²_h>Ω²-Ω²_BV/Ω²-Ω²_co. Here, Ω^>Ω_co and, of course, Ω>Ω_BV.

b) Gravity branch, i.e., gravity waves which propagate when K²_z>0 for K²_p>Ω²(Ω²-Ω²_co)/Ω²-Ω²_BV and V²_h<Ω²-Ω²_BV/Ω²-Ω²_co. Here, Ω<Ω_BV.

From the Eq.(10) for K²_z<0, we can,  in general, derive conditions for evanescent AGWs: K²_p>Ω²(Ω²-Ω²_co)/Ω²-Ω²_BV and V²_h<Ω²-Ω²_BV/Ω²-Ω²_co. Separately:

c) Acoustic waves are evanescent when K²_p>Ω²(Ω²-Ω²_co)/Ω²-Ω²_BV and V²_h<Ω²-Ω²_BV/Ω²-Ω²_co, Ω^>Ω_co, and Ω>Ω_BV.

d) Gravity waves are evanescent when K²_p<Ω²(Ω²-Ω²_co)/Ω²-Ω²_BV and V²_h>Ω²-Ω²_BV/Ω²-Ω²_co,  Ω<Ω_BV.

As expected, the conditions for propagating/evanescent acoustic and gravity waves are with opposite signs. This was taken into account in the analysis that follows in the article.

5. P. 5. Line 109. The two regions are characterized by the corresponding plasma densities ρ01 and ρ02. These are the densities of the neutral atmosphere, not the plasma.

Answer: Of course. The same answer as for the comment 3.

6. Р.5. Line 115. In the Earth's atmosphere, the background density decreases with height, and the temperature can have both a positive and a negative gradient. How can the boundary condition for the background parameters (16) be fulfilled with a negative temperature gradient?

Answer: Eq. (16) describes two different isothermal atmospheric regions-D layer with the temperature T1=250K and a lower thermosphere with temperature T2=500K with the plane boundary  between them. Indeed, in the real atmosphere there is a negative temperature gradient directed from the mesopause to the mesosphere. In the mesopause, or lets say boundary between D layer and lower thermosphere, with a thickness of about 5km, according Eq.(16) we have the situation where ρ₀₂>ρ₀₁. This is indication for the instability of the Rayleigh-Taylor type which could be described with nonlinear theory and numerical tools. Also, Schulthess in his article Acoustic Waves in the Upper Atmosphere, on the page 6 says: "Acoustic waves with periods greater than 4 minutes get trapped in a thermal duct in the mesopause". Similar like Dong, W., Fritts, D. C., Hickey, M. P., Liu, A. Z., Lund, T. S., Zhang, S., et al. (2022) in Modeling studies of gravity wave dynamics in highly structured environments: Reflection, trapping, instability, momentum transport, secondary gravity waves, and induced flow responses. Journal of Geophysical Research: Atmospheres, 127, e2021JD035894. https://doi. org/10.1029/2021JD035894,  Schulthess uses the numerical methods to resolve the AW/GWs behavior.

In our linear approximation we choose the boundary whose thickness tends to zero to avoid nonlinear effects. The justification can be that this 5 km thick boundary is much smaller than the wavelength of the considered waves that pass upward through that obstacle. Similar is in cited articles Marmolino et al. 1993, Jovanovic 2014 (you can find in the attachment as a supplement material) and Fleck et al 2021. The linear approximation can be understood as a limitation of this work.

7. Р.5. Line 125. The work (Jovanovic, 2014) is not available to me. Based on the inequalities (11) and (12) given by the author, I can assume that the reflection coefficient (17) and expressions (18) and (19) correctly take into account only the acoustic branch. In this regard, it is not clear how the reflection coefficients for the gravitational branch presented in Fig. 6 and 8 were calculated.

Answer: Eq.(17) describes reflection coefficient of AGWs. From this Eq. the reflection coefficient for acoustic waves is given for the frequency range Ω>Ω_co and for horizontal phase velocities V_h>1/√s=1.41 (Section 4.2.1). The reflection coefficient for gravity waves is given from the Eq.(17) for Ω<Ω_BV and V_h<Ω_BV/Ω_co=0.9 (Section 4.2.2). This could be seen in the frequency range and in the V_h values in Figs. 6 and 8. You will find the article Jovanovic, 2014 attached in the supplement.

8. I think that the dependence of the vertical wavelength on the horizontal wavelength at different frequencies (see Fig. 2, 3 and 4) is not necessarily shown in the work.

Answer: I presented AGWs in Figs. 2, 3 and 4 because these waves were detected by Nina and Čadež , 2013, see cited article.