the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 12 Aug 2025
A numerical model for solving the linearized gravity-wave equations by a multilayer method
Abstract. We developed a numerical model to solve the linearized gravity-wave equations by a multilayer approach. Specifically, the model handles the linearized equations including viscosity, thermal conduction, and ion drag. The solution methods are based on the matrix exponential formalism and encompass two main approaches: (i) global matrix methods and (ii) scattering matrix methods. Both methods are focused on determining either (i) the amplitudes of the characteristic solutions or (ii) the discrete values of the solution vector. Ascending and descending wave modes are distinguished based on the criterion that the real parts of the eigenvalues of the characteristic equation for ascending modes are smaller than those for descending modes. In global matrix methods, ascending and descending modes can be defined (i) at the upper and lower boundaries or (ii) in each layer. In contrast, scattering matrix methods necessitate explicitly determining the mode type within each layer. The model accommodates two types of lower boundary conditions and can handle both single-frequency waves and time wavepackets. Our simulations demonstrate that the solution methods are numerically stable and achieve comparable accuracies. Among them, the global matrix method for computing the amplitudes of the characteristic solutions is the most efficient.
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
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Preprint
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The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
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- Final revised paper
Journal article(s) based on this preprint
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2025-3406', Harold Knight, 05 Sep 2025
- AC1: 'Reply on RC1', Dmitry Efremenko, 22 Jan 2026
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RC2: 'Comment on egusphere-2025-3406', Stephan C. Buchert, 12 Dec 2025
Review of ANGEO manuscript egusphere-2025-340
Title: A numerical model for solving the linearized gravity-wave equations by a multi-layer method
Author(s): Alexandru Doicu et al.
MS No.: egusphere-2025-3406
MS type: Regular paperThe authors present an extensive work on atmospheric gravity waves (GWs). Among other the effect of ion drag on GW propagation and dissipation in the thermosphere is included. Reviewer 1, Harold Knight, has already provided a very thorough and detailed review of the core part of the manuscript. My review then focuses on the issue of the ion drag, how it is introduced and discussed.
Comment 1)
Lines 144-165: This discussion is about taking into account ion drag in the neutral dynamics. It is based on equation (13) which seems to have appeared 1st in Klostermeyer (1972). The implications of (13) are not entirely clear to me, so I try to rephrase my understanding of the issue: In the direction parallel to the magnetic field B the ions are dragged with the neutrals, and the unperturbed velocities u|| and ui,|| are approximately the same (in lines 172-177 an approach by Shibata (1983) which includes also diffusion and gravity is mentioned, but then not applied). But the interesting yet in the manuscript not elaborated aspect is what happens in the directions perpendicular to B?
Equations (14) and (15) suggest that the background ion velocity ui,⟂ is assumed to be zero in the reference frame where the neutral wind u is given (presumably a co-rotating frame). The drag force fID and frictional heat PID can only depend on the difference between ion and neutral velocity. So obviously ui,⟂ = 0 is assumed. At high latitudes this assumption is quite unrealistic. The aurora zone ion convection is dominated by coupling to the magnetosphere as described by Dungey (1961). The Weimer (2005) model, among others, could be used in combination with the HWM for the background difference u⟂ - uui,⟂. At mid-latitudes there is inter-hemispheric coupling which determines the ion velocity depending on inter-hemispheric wind differences, for example according to the HWM (Laundal et al., 2025; Buchert, 2020). The SAMI2 model, as far as I understand, includes inter-hemispheric coupling by particle transport, but not electrodynamically, i.e. without a mid-latitude Sq current system.
This study is about gravity waves and linearized perturbations. According to equations (16) and (17) the background state has an effect also on the perturbed fID' and PID'. My guess is that quantitatively it is quite negligible. Also reviewer 1 remarked on the very small difference that seems to arise from the ion drag. An update to a more realistic background ion velocity model could be done, and at high latitudes the effect should then become larger than observed in the present draft. Finally I would like to remark that the perturbations by gravity waves themselves should also affect the ion velocities, which would then affect the perturbed ion drag. Equation (16) and (17) are incomplete, as it is assumed that ui,⟂ faithfully remains unaffected by the perturbed u⟂', or that ui,⟂' = 0 as well. To abandon this assumption and solve the ion momentum equation (7) would be complicated and is understandably avoided in this work. However, ion drag forcing and dissipation by gravity wave perturbations might well be important and have comparable or even larger effect than a realistic background state via the perturbed densities in the 2nd term of equations (16) and (17). Even if a complete, linearized solution were available for both neutrals and ions, the very high mobility of electrons along magnetic field lines should have the effect that these are electric equipotential also within gravity waves. This seems not guaranteed with a complete neutral and ion solution unless additional constraints are introduced.
Alternatively, a possible assumption is that the ions are completely dragged by the neutrals, ui = u also in the perpendicular directions. This is equivalent to ignoring the ions for neutral dynamics. Compared to the effective assumption ui,⟂ = 0, which is chosen in the draft, this seems to me slightly preferable. The above mentioned condition that magnetic field-lines are electric equipotential also under perturbations by gravity waves would then generally be violated.
In summary, the authors have followed the approach in previous works when taking into account the ion-neutral coupling in atmospheric dynamics which is fine. My complaint is that the implications of the chosen approach had not been made clear up to now. Therefore I recommend to explicitly state that the Klostermeyer (1972) equation (13) implies that the ion velocities perpendicular to B, both background and perturbed ones are assumed to be zero, and this might be unrealistic in some situations. If possible, give quantitative estimates about an expected inaccuracy related to this assumption.
Comment 2)
Lines 181-182: "This topic will be discussed in more detail in the Conclusions." I cannot find a more detailed discussion of the topic in the Conclusions?
Comment 3)
Lines 129-130: "... neglected the Coriolis force ..." The statement is not very specific. According to Klostermeyer (1972) the Earth rotation should be taken into account for wave periods τ>1 hour. So I guess for long period/large scale gravity waves there could be effects from the Coriolis force. How is a limit "... gravity waves with an angular frequency ω > 2Ω, where Ω = 7.3 × 10−5 s−1 is the Earth’s angular velocity" justified, what does it mean in terms of small, medium, and large scales?
Comment 4)
Lines 620-624: The references to the SAMI2, MSIS and HWM models are a bit old, updates and newer versions of these models exist. I think that using not the newest of these models is fine and does not significantly affect the results and conclusions of this work. Updating the models is not necessary. But a brief statement about which models exactly were used is recommended. According to my research the latest models would be:
Huba, J. D. (2023). On the development of the SAMI2 ionosphere model. Perspectives of Earth and Space Scientists, 4, e2022CN000195. https://doi.org/10.1029/2022CN000195
with a link to the SAMI2 code on Github, https://github.com/NRL-Plasma-Physics-Division/SAMI2
Emmert, J. T., Drob, D. P., Picone, J. M., Siskind, D. E., Jones, M. Jr., Mlynczak, M. G., et al. (2021). NRLMSIS 2.0: A whole-atmosphere empirical model of temperature and neutral species densities. Earth and Space Science, 8, e2020EA001321. https://doi.org/10.1029/2020EA001321 with link to the public code.
Drob, D. P., J. T. Emmert, J. W. Meriwether, J. J. Makela, E. Doornbos, M. Conde, G. Hernandez, J. Noto, K. A. Zawdie, S. E. McDonald, et al. (2015), An update to the Horizontal Wind Model (HWM): The quiet time thermosphere, Earth and Space Science, 2, 301–319, doi:10.1002/2014EA000089.
References
J. W. Dungey (1961), Interplanetary Magnetic Field and the Auroral Zones, Phys. Rev. Lett. 6, 47, DOI:https://doi.org/10.1103/PhysRevLett.6.47
Weimer, D. R. (2005), Improved ionospheric electrodynamic models and application to calculating Joule heating rates, J. Geophys. Res., 110, A05306, doi:10.1029/2004JA010884
Laundal, K. M., Skeidsvoll, A. S., Popescu Braileanu, B., Hatch, S. M., Olsen, N., and Vanhamäki, H.: Global inductive magnetosphere-ionosphere- thermosphere coupling, Ann. Geophys., 43, 803–833, https://doi.org/10.5194/angeo-43-803-2025, 2025.
Buchert, S. C.: Entangled dynamos and Joule heating in the Earth's ionosphere, Ann. Geophys., 38, 1019–1030, https://doi.org/10.5194/angeo-38-1019-2020, 2020.
Citation: https://doi.org/10.5194/egusphere-2025-3406-RC2 - AC2: 'Reply on RC2', Dmitry Efremenko, 22 Jan 2026
Interactive discussion
Status: closed
-
RC1: 'Comment on egusphere-2025-3406', Harold Knight, 05 Sep 2025
Please see the supplement PDF file with my comments. It is ten pages.
- AC1: 'Reply on RC1', Dmitry Efremenko, 22 Jan 2026
-
RC2: 'Comment on egusphere-2025-3406', Stephan C. Buchert, 12 Dec 2025
Review of ANGEO manuscript egusphere-2025-340
Title: A numerical model for solving the linearized gravity-wave equations by a multi-layer method
Author(s): Alexandru Doicu et al.
MS No.: egusphere-2025-3406
MS type: Regular paperThe authors present an extensive work on atmospheric gravity waves (GWs). Among other the effect of ion drag on GW propagation and dissipation in the thermosphere is included. Reviewer 1, Harold Knight, has already provided a very thorough and detailed review of the core part of the manuscript. My review then focuses on the issue of the ion drag, how it is introduced and discussed.
Comment 1)
Lines 144-165: This discussion is about taking into account ion drag in the neutral dynamics. It is based on equation (13) which seems to have appeared 1st in Klostermeyer (1972). The implications of (13) are not entirely clear to me, so I try to rephrase my understanding of the issue: In the direction parallel to the magnetic field B the ions are dragged with the neutrals, and the unperturbed velocities u|| and ui,|| are approximately the same (in lines 172-177 an approach by Shibata (1983) which includes also diffusion and gravity is mentioned, but then not applied). But the interesting yet in the manuscript not elaborated aspect is what happens in the directions perpendicular to B?
Equations (14) and (15) suggest that the background ion velocity ui,⟂ is assumed to be zero in the reference frame where the neutral wind u is given (presumably a co-rotating frame). The drag force fID and frictional heat PID can only depend on the difference between ion and neutral velocity. So obviously ui,⟂ = 0 is assumed. At high latitudes this assumption is quite unrealistic. The aurora zone ion convection is dominated by coupling to the magnetosphere as described by Dungey (1961). The Weimer (2005) model, among others, could be used in combination with the HWM for the background difference u⟂ - uui,⟂. At mid-latitudes there is inter-hemispheric coupling which determines the ion velocity depending on inter-hemispheric wind differences, for example according to the HWM (Laundal et al., 2025; Buchert, 2020). The SAMI2 model, as far as I understand, includes inter-hemispheric coupling by particle transport, but not electrodynamically, i.e. without a mid-latitude Sq current system.
This study is about gravity waves and linearized perturbations. According to equations (16) and (17) the background state has an effect also on the perturbed fID' and PID'. My guess is that quantitatively it is quite negligible. Also reviewer 1 remarked on the very small difference that seems to arise from the ion drag. An update to a more realistic background ion velocity model could be done, and at high latitudes the effect should then become larger than observed in the present draft. Finally I would like to remark that the perturbations by gravity waves themselves should also affect the ion velocities, which would then affect the perturbed ion drag. Equation (16) and (17) are incomplete, as it is assumed that ui,⟂ faithfully remains unaffected by the perturbed u⟂', or that ui,⟂' = 0 as well. To abandon this assumption and solve the ion momentum equation (7) would be complicated and is understandably avoided in this work. However, ion drag forcing and dissipation by gravity wave perturbations might well be important and have comparable or even larger effect than a realistic background state via the perturbed densities in the 2nd term of equations (16) and (17). Even if a complete, linearized solution were available for both neutrals and ions, the very high mobility of electrons along magnetic field lines should have the effect that these are electric equipotential also within gravity waves. This seems not guaranteed with a complete neutral and ion solution unless additional constraints are introduced.
Alternatively, a possible assumption is that the ions are completely dragged by the neutrals, ui = u also in the perpendicular directions. This is equivalent to ignoring the ions for neutral dynamics. Compared to the effective assumption ui,⟂ = 0, which is chosen in the draft, this seems to me slightly preferable. The above mentioned condition that magnetic field-lines are electric equipotential also under perturbations by gravity waves would then generally be violated.
In summary, the authors have followed the approach in previous works when taking into account the ion-neutral coupling in atmospheric dynamics which is fine. My complaint is that the implications of the chosen approach had not been made clear up to now. Therefore I recommend to explicitly state that the Klostermeyer (1972) equation (13) implies that the ion velocities perpendicular to B, both background and perturbed ones are assumed to be zero, and this might be unrealistic in some situations. If possible, give quantitative estimates about an expected inaccuracy related to this assumption.
Comment 2)
Lines 181-182: "This topic will be discussed in more detail in the Conclusions." I cannot find a more detailed discussion of the topic in the Conclusions?
Comment 3)
Lines 129-130: "... neglected the Coriolis force ..." The statement is not very specific. According to Klostermeyer (1972) the Earth rotation should be taken into account for wave periods τ>1 hour. So I guess for long period/large scale gravity waves there could be effects from the Coriolis force. How is a limit "... gravity waves with an angular frequency ω > 2Ω, where Ω = 7.3 × 10−5 s−1 is the Earth’s angular velocity" justified, what does it mean in terms of small, medium, and large scales?
Comment 4)
Lines 620-624: The references to the SAMI2, MSIS and HWM models are a bit old, updates and newer versions of these models exist. I think that using not the newest of these models is fine and does not significantly affect the results and conclusions of this work. Updating the models is not necessary. But a brief statement about which models exactly were used is recommended. According to my research the latest models would be:
Huba, J. D. (2023). On the development of the SAMI2 ionosphere model. Perspectives of Earth and Space Scientists, 4, e2022CN000195. https://doi.org/10.1029/2022CN000195
with a link to the SAMI2 code on Github, https://github.com/NRL-Plasma-Physics-Division/SAMI2
Emmert, J. T., Drob, D. P., Picone, J. M., Siskind, D. E., Jones, M. Jr., Mlynczak, M. G., et al. (2021). NRLMSIS 2.0: A whole-atmosphere empirical model of temperature and neutral species densities. Earth and Space Science, 8, e2020EA001321. https://doi.org/10.1029/2020EA001321 with link to the public code.
Drob, D. P., J. T. Emmert, J. W. Meriwether, J. J. Makela, E. Doornbos, M. Conde, G. Hernandez, J. Noto, K. A. Zawdie, S. E. McDonald, et al. (2015), An update to the Horizontal Wind Model (HWM): The quiet time thermosphere, Earth and Space Science, 2, 301–319, doi:10.1002/2014EA000089.
References
J. W. Dungey (1961), Interplanetary Magnetic Field and the Auroral Zones, Phys. Rev. Lett. 6, 47, DOI:https://doi.org/10.1103/PhysRevLett.6.47
Weimer, D. R. (2005), Improved ionospheric electrodynamic models and application to calculating Joule heating rates, J. Geophys. Res., 110, A05306, doi:10.1029/2004JA010884
Laundal, K. M., Skeidsvoll, A. S., Popescu Braileanu, B., Hatch, S. M., Olsen, N., and Vanhamäki, H.: Global inductive magnetosphere-ionosphere- thermosphere coupling, Ann. Geophys., 43, 803–833, https://doi.org/10.5194/angeo-43-803-2025, 2025.
Buchert, S. C.: Entangled dynamos and Joule heating in the Earth's ionosphere, Ann. Geophys., 38, 1019–1030, https://doi.org/10.5194/angeo-38-1019-2020, 2020.
Citation: https://doi.org/10.5194/egusphere-2025-3406-RC2 - AC2: 'Reply on RC2', Dmitry Efremenko, 22 Jan 2026
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Alexandru Doicu
Dmitry S. Efremenko
Thomas Trautmann
Adrian Doicu
The requested preprint has a corresponding peer-reviewed final revised paper. You are encouraged to refer to the final revised version.
- Preprint
(1299 KB) - Metadata XML
Please see the supplement PDF file with my comments. It is ten pages.