The paper investigates characteristics of Rossby waveguides in the Southern and Northern hemispheres (SH/NH) during summer and winter and relationships between waveguides and quasi-stationary wave amplitudes. Several recent studies emphasized that waveguides are related to high-amplitude waves but the causality between the two is still a matter of debate. The originality of the paper is to develop a method to detect systematically at each longitude and each time the presence of waveguide based on the detection of turning latitudes where the stationary wavenumber Ks of the barotropic Rossby wave dispersion relation is equal to the zonal wavenumber k. The stationary wavenumber has been introduced in Hoskins and co-authors papers but only applied to a time-mean flow. In the present paper, it is adapted to get waveguides as function of time. The stationary wavenumber is computed by defining a background zonal flow using a spatial and temporal filter of the zonal wind. The paper also introduces the notion of waveguide depth which is somehow close to the number of zonal wavenumbers for which a waveguide exists. Quasi-stationary wave amplitudes are computed by considering the amplitude of the wave envelope where the quasi-stationary wave is here defined as the deviation of the 15-day running mean meridional wind from its climatology for each wavenumber. The definitions of the waveguides and quasi-stationary waves amplitude are appropriate and well thought. But since the 15-day running mean is used for the computation of both the background flow and the quasi-stationary waves, it is the separation in spatial scale that makes the difference between the two parts of the flow (k<2 for the background flow and k between 4 and 15 for the quasi-stationary waves). Results show that waveguide frequencies are higher in summer than winter in the NH and more frequent in the NH than in the SH. It would be good to detail more the physical interpetation for those statistics (see main comments below). Composites of high and low waveguide depths indicate that high waveguide depths are associated with a double jet structure and in that sense confirm previous studies. However, I am not entirely convinced because they rely on Figures 6 and 7 where anomalous composites are shown so a double jet structure in zonal wind anomalies does not necessarily mean a more pronounced double jet structure for the high waveguide depth than for the low waveguide depth (see main comment below where I suggest to show the composites of the low and high waveguide depths separately). Then correlations between waveguide depths and wave amplitudes are investigated. Even though the correlations are weak, there are several regions and seasons for which the positive correlations are significant. It concerns the SH and the NH summer over the Atlantic and Europe. The results are interesting in themselves but it would be good to add some comments related to previous studies and also related to the maps of waveguide depths (see below for more details). Finally sensitivity tests made by changing some values of the parameters used for the waveguide detection show that the results are quite robust. I think changing the pressure level to detect waveguide could be also another sensitivity test worth making. To conclude, the paper is well written and well organized, the results are new and interesting. However I have several major comments that need to be addressed before I recommend publication of the paper. 

Major comments:

a) Abstract: the notion of waveguide depth is obscure there. When I read the abstract I thought it was considering the depth in altitude. It would be good to add words to qualitatively define the waveguide depth. For me, the waveguide depth is closely related to the number of zonal wavenumbers for which the waveguide exists. 

b) Introduction (lines 25 to 29). It would be good to add some physical interpretation of why people think there is a link between waves amplitude and waveguides. In my opinion, this is potentially because waves are not dissipated as there are no critical latitudes in a waveguide. But I am not sure this is what the papers cited in line 28 have argued.

c) Method: it would be good to highlight that the background flow and the waves are separated by the spatial scale (k<2 for the background flow and k between 4 and 15 for the quasi-stationary waves). Since the 15-day running mean is used for the detection of both the background flow and the waves, the reader might be confused by the separation between these two parts of the flow.

d) Section 4.1 and waveguide frequencies: 



- before starting that section it would have been nice to show Ks for the time mean flow of JJA and DJF, i.e repeat Figures 3c of Hoskins and Ambrizzi (1993) and 11c of Ambrizzi et al (1995). Maybe it would be good to do it by considering the climatological flow for k<2 to be close to what is done in the present paper for the time-evolving waveguides. Such additional figures could help to better visualize the difference between summer and winter and between SH and NH. The argument made lines 160-165 to qualitatively explain why the summer NH has more frequent waveguides than the winter NH could be better understood by showing the time mean U and Ks for both seasons. Furthermore, maybe the additional argument is the fact that the jet is probably narrower in summer than winter and both the planetary vorticity gradient and relative vorticity gradient play a role in the difference between summer and winter.



- I am surprised that the SH has less waveguides than the NH as the double jet structure (separation between subtropical and eddy-drivent jet) is more marked there, at least in the climatologies. 

e) Section 4.2: this is the part of the results where I am less convinced by the conclusions. For instance, Figure 6a shows an anomalous tripolar pattern in zonal wind when computing the difference between high and low waveguide strengths. This anomaly could be the result of different changes: a more pronounced double jet structure is one possibility but it could result from a widening of the jet or some latitudinal shifts. So it would be very nice to compare composites of high waveguide strengths and low waveguide strengths separately before (or rather than) showing the difference.

f) Section 5: It is suprising that the correlations are strong in the Atlantic and over Asia and not in the Pacific while the waveguide depths are similar in the North Atlantic and North Pacific. What would be a possible explanation for that ? Or if you do not have hypothesis it would be nice to comment these results by referring to other studies. Were the studies on the relationship waveguide-wave amplitude focused in the North Atlantic and Asian regions. Do you know studies that also considered that relationship in the North Pacific ?

g) Sensitivity tests: I think it would be good to have a sensitivity test by changing the mean pressure level (e.g. 500 hPa?). Held et al. (1985) computed a barotropic equivalent level near 425 hPa and Charney (1949, see section 6) found a barotropic equivalent level closer to 550-600 hPa.

Minor comments:

1) Line 125: maybe add "temporally and zonally filtered U following the method described in section 2.1"

2) Line 130: it would be good to have a qualitative description of what the waveguide depth means. We understand mathematically in the main text but this would be useful for the abstract.

3) Line 134 and thereafter: why is cut-off latitude used rather than turning latitude ? I think turning latitude is the classical term

4) Figure 2: what are the black contours. Zonal wind at 300 mb ? Same question for Figure 4 but they disappear in Figure 5.

5) Line 178: I do not understand the end of the sentence "latitudinal cut-off ... latitude of the jet".

6) Lines 187-188: I do not understand what is meant by "Latitudes are weighted equally". Do you mean that multiplication by the cosine of latitude is applied to do the regional averages?

7) Line 196: I would add "a 'double jet structure' in anomalies is present

8) Lines 204-206: here again it would be better to see both composites rather the difference between composites

9) Figure 7: the magenta contour is difficult to see in the red-brown areas.

10) Lines 260-265: Here the importance of narrow jet is highlighted but this has not been the main argument mentioned in the main text when describing the difference between summer and winter in the NH (lines 160-165). The story of the equatorward displacement of the jet was emphasized. So please be more precise why there is a difference between summer and winter. How important are the jet width and latitude in that story ?

11) Caption Figure 9: please provide units for dimensional parameters.

see attachment

The atmospheric waveguide has a profound influence on the propagation path of stationary Rossby waves, thereby affecting when and where these waves impact surface weather and climate. In recent years, studies on the atmospheric waveguide have gained popularity among the climate community due to its significant connection to extreme events. The investigation of waveguides can be traced back to early works, notably Hoskins and Karoly (1981), followed by Hoskins and Ambrizzi (1993) and Ambrizzi et al. (1995). This current study aims to extend previous research by examining the waveguide in the context of spatially and temporally varying mean flow. Most of the analysis is focusd on this issue.

1. The separation of mean flow and perturbations is always a controversial issue when studying wave-mean flow interactions. This issue becomes even more critical when large-amplitude eddies appear in the mean flow (e.g., Wirth and Polster, 2021). However, the present study heavily relies on the separation method, and most of the findings are based on the assumption that the waveguide and Rossby waves are well-separated and independent. Therefore, I question the significance of the results, as many intraseasonal waveguide behaviors are actually reflected by long-lasting waves.

2. Regarding the methodology, using the traditional turning point perspective to identify the waveguide could be misleading, despite its extensive use in recent studies such as Petoukhov et al. (2013) and many subsequent papers. The limitations of this method have been thoroughly discussed by Wirth (2020). Therefore, the authors need to demonstrate the limitation of the method used here is nontrivial and confirm the appropriateness of the method.

3. I doubt about the characterization of the atmospheric circulation associated with the waveguide strength as "double jet streams" (Figure 6), as the zonal wind anomalies are only confined to a local scale. Additionally, as related to my major comment 1, long-lasting waves might play a role in this structure. Therefore, it is possible that the pattern seen in Figure 6 is not "double jet streams", but prominant Rossby wave activity itself. 

4. The current study primarily focuses on the waveguide effect along the subtropical jet, based on the refractive index, which essentially represents the gradient of absolute vorticity. However, recent studies (e.g., Xu et al. 2019, doi: 10.1175/JCLI-D-18-0343.1; Xu et al., 2020, doi: 10.1175/JCLI-D-19-0458.1) have presented compelling evidence of the existence of stationary Rossby waves along the eddy-driven jet, where the waveguide effect arises due to the gradient of potential vorticity. As mentioned by the author herself, this important aspect has been neglected due to the limitations of the methodology used in this manuscript. The authors briefly touch upon this issue in the manuscript, but in my opinion, more in-depth discussion is required.