the Creative Commons Attribution 4.0 License.
the Creative Commons Attribution 4.0 License.
| 23 Feb 2026
Variability of local gravity wave spectra from data of a high-resolution icosahedral-grid global model
Abstract. Atmospheric gravity waves influence the general circulation through transport of energy and momentum. Even with increasing computing capacities, parametrisation of their effects is still needed. Here, we diagnose gravity wave spectra from the data of a high-resolution ICON simulation on subdomains defined by a low-resolution ICON grid. A unique methodology is applied that avoids unnecessary interpolations and filters the data by projection on the linearised gravity wave modes, providing precise and detailed information about the gravity wave spectra. The dependence of these spectra on latitude is then studied, highlighting the importance of the zonal wind direction in the shape of the spectra. Finally, we see that the spectra can be highly simplified by using tens to hundreds of principal components, which is a key property allowing for an increase in efficiency of current gravity wave parametrisations.
Competing interests: Supervisor of ZP is an editor at this journal.
Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims made in the text, published maps, institutional affiliations, or any other geographical representation in this paper. While Copernicus Publications makes every effort to include appropriate place names, the final responsibility lies with the authors. Views expressed in the text are those of the authors and do not necessarily reflect the views of the publisher.- Preprint
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Status: final response (author comments only)
- RC1: 'Comment on egusphere-2026-807', Anonymous Referee #1, 19 Mar 2026
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RC2: 'Comment on egusphere-2026-807', Anonymous Referee #2, 06 Apr 2026
This article proposes a method to study the variability of gravity wave spectra from data of a high-resolution global model.
The method is introduced and some spectra are calculated, and the potential use of the method in gravity wave parametrizations is pointed out.
The topic is certainly relevant, and the treatment seems to be correct, but what I think is missing a bit in this paper is a clear scientific problem to address. For this reason, it is my opinion that the paper needs substantial revisions to become a stronger piece of research.General comment
The emphasis of the paper is on use of a particular method for obtaining spectra of gravity waves in the atmosphere, but there is no well-defined scientific problem to address, apart from finding the characteristics of those spectra. The authors make some vague allusions to the use of the results in gravity wave parametrizations for lower-resolution meteorological models, but this is not really pursued to any significant extent, so the reader gets the impression that the methodology is being presented for the sake of it. It would be good if the method had a more direct immediate scientific objective, linked with understanding some relevant phenomenon, even if its use in improving gravity wave parametrization is relegated to a follow-up publication.
Specific comments
Line 19: "enhance their resolution". Strange phrasing. I suggest replacing this by "have had enhancements to their resolution", or something similar.
Line 27: "One dimensional spectra of atmospheric quantities...". It is important to note that the focus in this study is on waves, not turbulence, as spectra are often referred to in the context of turbulence. I suggest adding an allusion to waves in this sentence, to clarify this.
Line 33: "localized". It is unclear what this means exactly. Please briefly explain.
Line 37: "the triangular subdomains". This may be familiar to users of ICON, but not to readers in general. Please explain why this is the natural shape of domains in ICON.
Line 42: "results are discussed and concluded". Again, strange phrasing. I suggest replacing by "results are discussed and conclusions presented".
Line 57: "local spectra". Explain in more detail what "local" means here.
Line 67: "deplaned and tapered". Whereas it is explained in adequate detail what "tapered" means, this needs to be done in more detail for "deplaned".
Lines 73-74: "defined as the zonal and meridional sides of the smallest rectangle in which the subdomain can be inscribed". As a non-expert in this area, since this procedure (inscribing the triangle in the rectangle where the spectrum is calculated requires a substantial amount of padding, I wonder if it would not make sense to do the opposite, i.e., inscribe the rectangle where the spectra are calculated in the triangle. In that case, there would be non-zero data everywhere. Would this place undue limitations on the maximum length scales that could be included in the spectrum? Please discuss this.
Lines 85-86: "F is defined by Eq. (3) as a matrix containing terms like …". This is not very clear from Eq. (3). I suggest mathematically defining and presenting both F and hat(f) separately.
Line 87: "lambda || \hat(f) ||". Briefly explain the importance and meaning of the term involving lambda.
Line 88: "this corresponds to solving the equation...". It is not clear how Eqs. (4) and (5) are linked. Please provide some intermediate steps that show this.
Lines 90-91: "Eq. (3) describes the real Fourier transform and its equivalence to the formula for the standard Fourier transform". It this really necessary? I would think that it is clear enough to present either Eq. (3) or Eq. (6).
Lines 98-99: "Note the negative sign in the definition of omega that allows us to write...". Is it really important that omega is negative, or just a convention, which relies on how the Fourier series and the Fourier coefficients are defined?
Line 105: Eq. (10). The notation f_xyt and f_klw differs from that introduced previously, where x,y,t and k,l,omega appear inside brackets, instead of as subscript indices. Please make this notation uniform throughout the paper.
Line 111: "the scaling factor alpha_u". Is this scaling factor uniformly applied across scales? If yes, please mention it explicitly.
Line 114: "The scaling factors are computed using a single time step only". Eq. (11) suggests that the scaling factor is only computed based on the spatial Fourier transform, and not the temporal one. Is this so? If yes, explain why.
Line 121: "for both gravity waves and sound waves". Actually, waves which are affected by the Coriolis parameter are usually called inertia-gravity waves. Eq. (12) expresses the dispersion relation for waves which are affected by gravity, Earth's rotation and the compressibility of air. Should they be called "gravity waves and sound waves", or rather "inertia-gravity-acoustic waves" or something similar?
Line 139: "the vector describing the polarisation relation above". It is not totally clear what this vector is. I suggest presenting the definition of the mentioned vector explicitly.
Line 159: Eq. (21). It is worth noting here that the last term in this equation corresponds to potential energy (unlike the preceding terms, which are kinetic energy).
Line 162: "unless the wave dissipates or generates". Strange phrasing. I suggest replacing it by "unless the wave dissipates or is being generated".
Line 170: "without a clear orography-based GW activity". I am not sure Fig. 1 shows this. There seems to be a certain correlation between orography elevation and wind speed.
Lines 174-175: "When averaged over the bands of horizontal wavelength size". Is this averaging also performed over the different directions? If so, this should be explicitly mentioned.
Line 185: "the symmetry of Fig. 1e is no longer present". The authors may mean "the exact symmetry of Fig. 1e", because Fig. 1f is also to a large extent symmetric, but presumably not as much as Fig. 1e.
Line 207: "not as strongly circular". Do the authors mean "not as isotropic" or "more anisotropic"? If so, I think these latter forms are more appropriate.
Line 210: "The more noticeable difference" should be instead "The most noticeable difference".
Line 216: "Global distribution gravity waves". There should be the word "of" between "distribution" and "gravity".
Lines 239-240: "The gravity wave energy is higher at the Northern Hemisphere since the wind is stronger in the winter hemisphere". Is this the reason, or is it because there is much more orography in the northern hemisphere, or possibly both reasons? If so, please correct appropriately.
Caption of Fig. 3, line 1: "Global distribution of the GW". Since GW projection was not done yet at this stage, it would better to call this "total energy", avoiding use of "GW", if I am not mistaken.
Line 276: "the most comprehensive PCs". It is unclear what "comprehensive" means here: "numerous"? If so, the latter formulation might be more specific and thus preferable.
Line 299-300: "Although the reconstructed spectra for 200 PCs for both groups do not look exactly the same as the original spectra...". 200 PCs sound quite a lot. Is using such a large value of PCs really useful for parametrization? Why do the spectra converge so slowly?
Line 303: "unique methodology". Does this mean "original"? Could the word "unique" be replaced by something more descriptive?
Figs. 5, 6 and 8: What are the units of the colour scales on these figures, if any?
Caption of Fig. 6, line 1: "Principal components 0, 1, 2 (columns)". Does this mean each of the PCs in isolation, or 0, 0+1 and 0+1+2 ? Please clarify.
Caption of Fig. 6, line 2: "which is notable". Do you mean "noticeable" instead, or "remarkable"? If the latter, in what sense?
Lines 320-321: "since we were analysing data at a single altitude only, we do not have the spectra in the wavenumber klm space". This seems to be contradicted by Figs. 5-6. Probably the authors mean something different. If so, please phrase this passage more clearly.
Lines 325-326: "the approach is relatively reliable". What term of comparison (truth) or measure of reliability do you have? If none, then this claim seems unjustified.
Caption of Fig. 8, line 1: "first column". Since earlier in the same line you mention "right column", it would be better to call this "left column" instead.
Line 336: "with a novel methodology". If this methodology is indeed novel, that should be emphasized earlier in the paper, say in the introduction or so.
Line 344: "gravity and sound waves". As previously, would not these waves be better described as "inertia-gravity-acoustic waves"?
Line 346: Eq. (A1). The appearance of the speed of sound c_s in this equation suggests that this equation set is subject to some kind of scaling, which is omitted. Either this should be described here (an appendix) or referred to a publication where it is explained.
Line 349: Eq. (A4). This equation expresses adiabatic flow. Yet, I do not think this assumption is mentioned anywhere. Please mention it explicitly.
Line 360: "it holds". Remove these words, as they are unnecessary.
Line 365: Eq. (A12). I think k_h needs to be defined here, for clarity.
Line 379: "For sufficiently large ..., this translates to ...". Does this approximation automatically exclude sound waves? If so, explain why/how.
Line 383: "minimal intrinsic frequencies can be obtained by maximizing the vertical wavenumber". It is not obvious to me why. Please explain how it works.
Line 386: "the maximum can be reached by taking the limit m^2 -> 0". Again, it is not obvious why. More details are necessary.
Line 433: The page number range of Fritts and Alexander (2003) seems to be missing.
Line 477: The page number range of Stephan et al. (2022) seems to be missing.
Citation: https://doi.org/10.5194/egusphere-2026-807-RC2
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- 1
The paper is a very thorough introduction into the application of spectral analysis to the icosahedric grid of the ICON model avoiding spurious effects necessarily generated by interpolating the data on a rectangular grid first. The data gives some convincing examples of the application. I feel, however, that the paper should be strengthened in the intuitive physical interpretation. Also, when it comes to discussion of global distributions there is a wealth of previous work from both models and satellite observations which you can compare against. This shouldn't be lengthy, but also not completely ignorant. I am wondering that you are not considering phase speed spectra and would suggest this. The paper is generally well written and of high presentation quality, thus strongly recommended for publication as soon as these points raised have been taken into account.
General comment:
The aim of the paper is, I think, to motivate other scientists working with ICON to use your tool to analyze ICON data. This means, you are aiming to a large fraction to GW physicists and people interpreting models. I think a few minor modifications (as indicated below) would help them to better understand what you propose. The paper contains no introduction to the code. I hope the tool is reasonably self explaining, otherwise I am voting for an appendix C with ashort technical introduction.
Specific comments:
L51 With a horizontal resolution of 2.5km you are expected to reasonably represent wavelengths of 25km and longer, which would correspond to GWs emitted from a single convective tower (though that would of course not be really resolved). You would then expect 20min the dominant period (works of Fovell and Lane), i.e. you are just at the Nyquist limit -> direction flips to be expected. For introducing a new method, that's all fine, but from a physical point of view I think one should do 5min time step.
L55 Only a comment, no action needed: Just to remember that this is only an estimate, since the interaction of scales would be missing in a 160km run.
L67 A picture illustrating this would be really nice. If you really care it takes minutes to understand these three senetences.
L90 Perhaps one sentence why, in contrast to a FFT on a regular rectangular grid you need a regularization?
L104 Why do you want a symmetric range? At some point you can resolve ambiguities by GW polarization relations. If that is the motivation, include here? For instance: The symmetric range is later used in Section ... to ...
L107 equidistant in a regular x-y grid I think you need to say this, because the sides of the triangles are of equal length -> equidistant
L116 For comparison with other methods and observations it would be interesting to do temp as well, in particular as b includes a vertical derivative which introduces new sources of uncertainty
L121 Do you see infrasound waves in ICON? If yes then the 10min sampling become really questionable
L139 I think at this point you need to explain a bit more. In which way are the polarization relations a vector? And what are you gaining by the projection in a physical sense. A few additional senetnces should help. Please keep in mind that you need to keep the average atmospheric scientist in the boat, if you want your method to be applied.
L146 O.k., here is the why. Still I don't see how the projection does it.
L151 in Appendix B from the extremal vertical wavenumber values m->inf and m=0, respectively.
L154 I don't think the model can really simulate waves at its vertical Nyquist. Should be a safeguard only?
L163 the wave ... generates -> is generated also not fully satisfactory
L168 arbitrary, exemplarily ? Or did you really use a (computer) dice? And why not use something where other studies exist?
F1 Lucid writing rules: same colors for same things, different colors for different things. There are more color scales than viridis. Confirming is a bit strong.
In the explanation you discuss the limits by f, N creeps in via changes in color between e and f (correct?), the extent of the k scale is Nyquist. In principle k=0 is part of the Fourier grid, but you cut to the domain size, so the white stripe in the middle (?). That this is naturally not very populated (e) is from the tapering? An additional background removal? Because b is at a mean value of 11m/s.
L187 This irritates me slightly. Physically, time is the dimension which runs only in one direction, so you should break symmetry here (same as nature). m>0 should be downward propagating, m<0 upward propagating waves and it should make sense to compare both of them in order to search for reflections and middle atmosphere sources. Mathematically, you can break of course as you like, but I am nota able to send my sensor backward in time. At least mention that you have both possibilities and do a choice here which makes it easier to discuss some features.
L191 This is a bit simpler than the limit you have stated above (Equ 20). At one point you should include, how the two relate to each other. I would suggest after equ 20 something for GWs with such and such properties these become the well known limits ...
L197 58% is surprisingly symmetric.
L203 For me I would again discuss along the physical case. Scandinavia is known to excite mountain waves. (I find the suitable... a bit awkward). Accordingly, we find a predominance of waves with ground-based frequencies close to zero. There obviously is a stronger dominance of upward propagating waves with a well defined source below. That the ridge is N-S and thus most power is in k you don't discuss in the spectra, maybe omit? What you mean by circular I don't understand. Explain or omit.
L217 Why "Although"? Mission A accomplished now comes mission B. We now extent our analysis to the global scale ... or something like that
L220 e.g. Andes or Himalayas -> These are the two you would not necessarily expect and which will disappear when you move a little higher in analysis altitude. (Andes -> summer, Himalaya a general wind shear). Not against quoting them, but I would do a few more e.g. Delany, Rocky Mountains, Greenland, Iceland, Mongolia, Catabatic winds at Antarctica. Interestingly, at this particular day, not so much Scandinavia. In addition to this you see the subtropical convection on the SH, without pointing at special locations, should be mentioned.
F3 Please do not use judgemental language: simple -> only How good or bad the removal works will depend a lot on the details of the removal.
L235 What is a non-linear GW? Do they really exist as a pure form or are GWs rather usually well described but may deviate somewhat from linearity? In particular, you would then expect deviations from the quadrature phase relations between temperatures and winds. which indead would reduce the projected energy in the cospectra. Please rethink the formulation.
L238 Why Finally ?
L242 ITCZ -> this would be the small peak around 10S on top of a wider GW maimum form 10N to 25S. Convection about subtropical sources is also known.
L248 ... are chosen narrower ... Or did you really make them conatining equla amounts of cells?
3.3 in general:
For me it would now be logical to switch to spectra in terms of phase speed and direction. This helps you very intuitively to see different wave sources and wind filtering - quite a bit of work out in the literature. The vertical wavelength then mixes the wind and the source effect and the zonal means and the variations of winds with longitude is not really helping.
The PCA on the other hand is not motivated. Why do you want to do this? What does the modes tell us? Wouldn't it be better to perform PCA on phase-speed spectra? And how would results change if you go to a different day, e.g. in March?
L271 retrograde -> generally opposite ?
L303 GW filtering -> this is for me associated with (critical level) wind filtering. You mean the projection/identification/extraction ... still not the right word, but come up with something more positive saying afterwards we have the GW only
L305 flow -> dynamics ? Its not only u and v
L306 This is here not connected and would need to be introduced in the discussion of the model setup. You may also overestimate some scales.
L311 You cannot expect the readers to know the Chew paper. Should be introduced in the intro or really discussed in the math part, if you want to delineate your method from others.
L323 I think if you have the input data, i.e. model output at high frequency its better to use frequency rather than m as it is less subject to changes by e.g. refraction. However, of course, when you have it is a strong constraint as you need to foresee the application when you do the model run and you can apply it to short runs only. A fair one sentence statement would be good to make.
For dedicated experiments which are saved at short time steps a k,l,frequency analysis is the better choice ...
L324 Please have a look at Strube et al as well. This paper discusses an altitude dependency of how well this may work. (DOI: {10.5194/amt-13-4927-2020})