the Creative Commons Attribution 4.0 License.
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| 30 Jun 2026
How to diagnose barotropic Rossby wave resonance along a circumglobal jetstream?
Abstract. Resonant amplification of Rossby waves along a circumglobal jetstream was recently hypothesized as the underlying mechanism for the occurrence of extreme weather in observed episodes with large wave amplitudes. An important part of the argument is based on refractive index theory in the framework of the linear barotropic model. The approach makes a number of assumptions and approximations with the goal to diagnose the existence of a zonal waveguide and, hence, the potential for Rossby wave resonance. The current paper compares this approach with a recently developed direct numerical method that makes no further assumptions given the chosen framework and is, hence, considered as more trustworthy. The comparison indicates that the occurrence of waveguides as diagnosed from the refractive index method is both qualitatively and quantitatively inconsistent with the occurrence of resonance in the direct numerical method. It is concluded that the previously-used waveguide diagnostic is not a reliable basis for detecting Rossby wave resonance. The code for the direct numerical method is publicly available to encourage its use in future applications.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Weather and Climate Dynamics.
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- CC1: 'Comment on egusphere-2026-3648', Isaac Held, 27 Jul 2026
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RC1: 'Comment on egusphere-2026-3648', Anonymous Referee #1, 29 Jul 2026
This paper is concerned with the arguments used by a sequence of papers over the last decade or so on mechanisms for potential increases in mid-latitude planetary wave amplitudes under climate change. The specific method included in these paper has been to consider Rossby-wave refractive index and use this as a criterion for the existence of a free stationary wave mode and hence the possibility of resonance.
The authors of this paper have previously published papers expressed skepticism about the refractive-index approach and the current paper refines and extends some of their arguments. The whole scope of this discussion is based on the premise -- implicit in the arguments of the Rossby-wave resonance proponents -- that linear two-dimensional models of Rossby wave dynamics are relevant. What the current paper shows is that the criteria on refractive index used in these arguments are simply not a useful indicator of the possibility of resonance (meaning that wave amplitudes for a particular zonal wavenumber are enhanced relative to those for other wavenumbers). The reasons why this is the case are fairly simple but are generally clearly illustrated and explained in the paper. I thought that it was a nice point that the refractive index method misses the need for modal structure in latitude and that while WKB theory can address it does not often give the correct answer. I also thought that the 'main results' summary in 488-521 was fair and clear.Â
The general ideas being used in this paper are simple, and not particularly new, but a well-defined problem has been selected and clear arguments and conclusions are presented. So to me the paper in some form seems suitable for publication in WCD. I do think that there is scope for shortening (i.e. focusing) the paper -- that might make it more effective in conveying a clear message. The authors could be more ambitious -- they mention an extension to consider a wider class of flow profiles in future work and I am a bit surprised that that material is not included here -- but that is not something I would insist on. I have made other recommendations below for changes to the paper prior to publication.Â
Detailed comments:
14: 'Like in theoretical physics' -- I'd have written something different -- 'The phenomenon of resonance is universally recognised as occurring if the external forcing projects effectively onto the spatial and temporal structure of a free mode of the system'.
20: why is the term 'saturated' included here?
34: 'It follows that' 'According to ray tracing theory, it follows that ...'?
54: 'suspicious' > 'questionable'?
55: 'underlying assumptions are barely satisfied' > 'underlying assumptions concerning spatial scales are barely satisfied'?
57: I find 'completely discounts for the wave nature of the solution' an odd description of ray tracing theory. In the following text you give examples of what ray theory neglects -- perhaps the first part of this sentence isn't needed.
65: 'Given these caveat ... approximations.' The reader may wonder, given these examples, what further there is to add on the topic. I say this to encourage the authors to make it clearer what the current paper contributes over and above what has been presented before.
70: 'performs poorly in model tests (Mooring and Linz 2026).' Not very specific -- a little more information would be helpful -- again to be clear about what is contributed by the current paper relative to previous work.
71: 'Reference for our analysis ...' and following paragraph -- again it would be better if this was written in a way that made it clear what the current paper is offering that goes beyond Wirth and Harnik (2026).Â
96-133: This is a nicely written introduction to the topic of resonance but it really is textbook material. Are the authors including it because they believe that the promoters of the 'Rossby wave resonance' arguments are fundamentally confused about what resonance is? The only substantial less-familiar point here is that whereas in an early-stage undergraduate explanation of resonance via a forced oscillator one might describe how the response varies with forcing frequency, here it is more natural (as explained by Haurwitz) to consider the zero frequency case and ask how the response varies with zonal wavenumber. That might more effectively be explained in the context of (10).
183: 'represents the square of the nondimensional stationary wavenumber' -- explain briefly why this term has been chosen.Â
186: '(10) turns singular at the critical latitudes' -- surely only if alpha=0. If alpha > 0 then, divide the equation by the (1-i epsilon) factor and note that epsilon tends to infinity in this limit, so the second derivative term and the term including the fact s^2 balance. (I see later than in fact you consider cases where alpha=0 in some parts of the flow -- but that hasn't been stated yet.)
Figure 2 caption: 'minus the imaginary part of K_s' -- I think that it needs to be noted explicitly in the text that K_s may be imaginary. That having been said, the fact that K_s^2 becomes negative in some regions, although u_0 > 0, implies that the absolute vorticity gradient is negative in some regions. Was that intended? Does that worry you? Would any of your conclusions change if K_s^2 was everywhere positive?
215: I think that 'crude' is an ill-advised term to use -- unless you feel that the only way to win your arguments is to be explicitly dismissive. 'Heuristic' -- followed by a later comment that these arguments seem difficult to justify would be more diplomatic.Â
215-227: A problem here is that you are straying from your primary message -- that arguments based on ray-tracing are unreliable, to a broader critique of details of these previous papers. So this is tending towards a 'Comment on' rather than a stand-alone paper. I don't know whether this critique has been presented in any of your previous papers. But, however justifiable the critique might be, I suggest that it would be more effective to confine discussion of these issues to your final section.Â
291: 'The refractive index method allows ... from the knowledge of the basic state alone. By contrast the direct numerical method ...' -- this might be misleading -- the information required for both is (essentially) knowledge of the basic state -- there is no extra information being supplied for the direct method (except insofar as you are specifying the structure of the flow outside of the jet and the structure of the sponge damping, and your line of argument is surely that your conclusions are somewhat insensitive to these extra pieces of information).
297-299: There's a danger that this comment is seen as rather self-serving and perhaps it is not needed. The key point is that both approaches are considering (10). The refractive index approach is an approximate one that may or may not give good insight into the solution. The 'direct' approach provides a full numerical solution to (10). Assuming that the numerical approach is correct how can the direct approach not be more trustworthy?
354: 'strong and narrow jets' -- there are two cases, one strong and one narrow.
450: I suggest briefly explaining that, since it is based on matching to Airy functions, this criterion (22) takes no account of any detailed behaviour outside the range of the turning latitudes.
525: Again there is the question of using terms like 'crude assumptions' is going to help or hinder your aim, which is to highlight strong deficiencies in the turning point criterion used by those in the 'Rossby wave resonance' community. I suggested previously that your comments on the additional conditions used by that community might be more better confined to the final section of the paper -- i.e. here -- I still think that there is a question on whether your message is more powerful if you confine it to 'the turning point criterion does not work' -- or whether you launch into a more general critique of the 'Rossby wave resonance' approach.
Figure 14: I'm not sure how helpful this will be to the reader. At one level it is a simply a Venn diagram displaying where in parameter space the two methods ('refractive index' vs 'direct numerical') predict what is called resonance (A vs B) and emphasising that there are regions with A and B, regions with B and not A, etc ..
But then there is the ingredient of the axis corresponding to wave amplitude -- this surely must mean 'wave amplitude for given forcing' and it must correspond to the prediction of the direct numerical method since the refractive index method does not make any such prediction? But then I read in 537-538 that region (v) corresponds to '... leakage coincides with weak forcing' -- suggesting that the displayed amplitude is not intended to be for given forcing, somehow the amplitude of the forcing is being taken into account as well. That seems confusing to me.551: 'For one thing ...'
570: ' ... choice of wind profile ... . However, neither of these issues affects the core of the present work, which is assessing the utility of the refractive index method ...' -- in a strict logical sense yes -- you have generated a range of examples where the refractive index method does not provide any useful information. But couldn't a proponent of the refractive index method come back and suggest that if you confined attention to a different set of flow profiles then it might have skill? An obvious question arises from the leakage of wave activity to low latitudes and to the opposite hemisphere illustrated by Figure 7. Different profiles might inhibit this. There seems to be an acceptance of this in your text in 564-568 -- but, given the fairly routine and straightforward nature of the required calculation, have you considered completing this 'future work' and including it in current paper?
Citation: https://doi.org/10.5194/egusphere-2026-3648-RC1 -
RC2: 'Comment on egusphere-2026-3648', Isaac Held, 13 Aug 2026
The comment was uploaded in the form of a supplement: https://egusphere.copernicus.org/preprints/2026/egusphere-2026-3648/egusphere-2026-3648-RC2-supplement.pdf
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Publisher’s note: the content of this comment was removed on 13 August 2026 since the comment was posted by mistake.