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the Creative Commons Attribution 4.0 License.
| 29 Jun 2026
Does the envelope of Rossby wave packets exhibit higher predictability than the underlying wave pattern?
Abstract. This study tests the long-standing hypothesis that Rossby wave packet (RWP) envelopes exhibit enhanced predictability compared to embedded weather systems, here interpreted as the individual troughs and ridges comprising the underlying wave pattern. Using a potential vorticity framework and the saturation fraction of the mean square error as a predictability metric, we derive a tendency equation for envelope errors to diagnose error-growth mechanisms. Contrary to the hypothesis, our analysis does not reveal any evidence of enhanced envelope predictability. The error dynamics of the envelope and the underlying wave pattern are strikingly similar. Crucially, while the envelope discards information about the location of troughs and ridges (phase information), it remains sensitive to the relative phase relationships among constituent wavenumbers. Phase errors in the individual wavenumbers may directly propagate into envelope errors, demonstrating that "the envelope is the wave", i.e, the dynamics of the envelope are intrinsically governed by its underlying wavenumbers, not by larger-scale dynamics. At least up to lead times of 10 days, the longest lead times considered herein, envelope predictability is thus governed by the same dynamical constraints as the synoptic-scale Rossby wave field. The misconception of RWPs as a larger-scale, dynamical phenomenon may arise by noting the continent- or ocean-basin-wide scale of RWP ''objects'' on spatial maps. Confirming previous work, we find enhanced predictability in the presence of RWPs compared to situations without RWPs. This enhanced predictability arises because RWPs organize high-amplitude anomalies coherently over large scales – not from the envelope itself. The spatial coherence of high-amplitude anomalies within RWPs is arguably facilitated by a strong midlatitude waveguide, suggesting waveguide characteristics as a promising target for future predictability research.
Competing interests: At least one of the (co-)authors is a member of the editorial board of Weather and Climate Dynamics.
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: 'Review of "Does the envelope of Rossby wave packets exhibit higher predictability than the underlying wave pattern?" by Riemer and Goelz', Maarten Ambaum, 17 Jul 2026
- RC2: 'Comment on egusphere-2026-3447', Anonymous Referee #2, 05 Aug 2026
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RC3: 'Comment on egusphere-2026-3447', Anonymous Referee #3, 06 Aug 2026
This study examines the hypothesis posed by Lee and Held (1993) that “Because the packet can remain coherent despite chaotic internal dynamics, the packet envelope should be more predictable than the individual weather systems”. As the authors suggested, as far as I can tell, this hypothesis has not been quantitatively assessed up to now. The authors compared normalized error growth of the envelope and the underlying wave fields, and their results suggested that the envelope errors might reach saturation slightly faster than errors of the wave field, suggesting that the hypothesis posed above should be rejected. They also examined the mechanisms responsible for error growth, and found that the same mechanisms (divergent flow and nonlinearity) dominate the error growth of both the envelope and the waves. They also suggested that higher amplitude waves may be more predictable even outside of the RWP envelopes. Overall, I think the topic is of interest to the forecasting and dynamics community, and this paper should eventually be published, but I do have some major issues that need to be resolved before this paper can be accepted for publication.
Major comments:
1) The authors made use of an approximation to the RWP envelope, E (defined in equation 1), which was computed based on along-latitude circle Hilbert transform of PV. Comparing this with the more commonly used RWP envelope function computed based on along-stream velocity (Fig. 3 vs. Fig. 1), E is clearly noisier, since it is computed from a noisier variable (PV), and without the along stream calculations to smooth out kinks due to the undulating properties of RWPs. It is not clear to me that the error characteristics of E are the same as those of the RWP envelope. In my opinion, the authors need to show that the error growth in the smoother RWP envelope is similar to that of E to make the results of this paper more convincing. This can probably be done by creating normalized error growth curves similar to Fig. 6b for the envelope function in place of E.
- The authors argued (line 130) the refinements would impact on both the envelope and the underlying wave pattern in a consistent manner”, which I do not agree with, since the along stream refinement by Zimin et al (2006) does not really affect the waves, but does serve to produce much smooth and more coherent envelope signals. Also, even if we apply the Wolfe and Wirth (2015) transform to compute the envelope, it is not clear that to quantify error growth of the underlying wave field, we should consider the error growth of the transformed wave field rather than the error growth of the original wave field.2) The authors justified using MSE-type error metrics instead of using object-based methods for verifying coherent RWP features stating that “It is not clear to us, however, how a comparable object-based method could be designed for the underlying wave pattern” (line 50). I’m not sure I understand this argument, since many previous studies have examined troughs and ridges using feature-based methods. In fact, the hypothesis posed by Lee and Held (1993) relates to “individual weather systems”, which, at least to me, suggested that the comparison should be between RWPs and troughs and ridges as objects, rather than the “wave” field. Hence, to me, this study does not really directly assess the hypothesis posed by Lee and Held (1993). My interpretation of their hypothesis would be something like “Can RWPs be predicted further ahead than individual troughs and ridges”?
3) In section 4.3, the authors claimed that higher amplitude PV anomalies (in terms of E) exhibit higher predictability even outside of RWPs. The following may be due to my misunderstanding. In comparing Fig. 1 and Fig. 3, it seems to me that areas outside of RWPs (defined in Fig. 1) with very high values of E (shown in Fig. 3) may frequently be located just outside of the RWP objects and are likely related to PV anomalies that are associated with the RWP, so it is not clear to me that these should be considered as being “outside” of the RWPs. Did the authors make sure that what they considered “outside” are really not associated with RWPs, and how?
4) To me, the example shown in Fig. 9 is potentially misleading. Let us consider the simplest wave packet, which is formed by the superposition of 2 waves: cos(phi1)+cos(phi2). Mathematically, this is identical to 2*cos(phi1-phi2)+cos(phi1+phi2), , where phi1 = k1x-w1t+C1, and phi2 = k2x-w2t+C2, where the k’s are the wavenumbers, and the w’s are the angular frequency. This is in the form of a wave packet, with the first factor being the envelope (lower frequency and smaller wavenumber), and the second factor the carrier wave (higher frequency and higher wavenumber). The authors’ example is similar to the case in which the errors of phi1 and phi2 are of opposite signs. This makes the error in phi1+phi2 small, while the error in phi1-ph2 will be large. The result is that the envelope will shift, but the phase of the carrier wave is not shifted, which is what is shown in Fig. 9. In this case, the error in the underlying wave pattern will largely be amplitude errors instead of phase errors, while clearly the envelope error is a phase shift. This example is in fact inconsistent with the authors’ contention that for the underlying wave, phase errors dominate over amplitude errors. Now in the case in which the errors in phi1 and phi2 are of the same sign, then the phase error in the carrier wave will be amplified, while the phase error of the envelope will be small. So, while this example illustrates that phase errors in the wave field could lead to envelope errors, to me, this is not necessarily a typical case, and does not really “explain” the results found in this paper, unless the authors could demonstrate that the phase errors in the different wavenumbers are correlated with each other in a manner that is consistent with their contrived example.
5) Regarding mechanisms responsible for error growth (Section 4.2), before I read the paper, I expected that since RWP propagation is linked to wave dispersion, the largest errors in RWP envelopes would likely be due to mechanisms that act to disrupt wave dispersion, as well as mechanisms that are generally not well handled by models, which, to me, would suggest that the error growth would be dominated by nonlinearity as well as the impact of diabatic heating (or DIV). I expect linear dynamics to be better handled by model forecasts, so is it really surprising that the dominant mechanisms for error growth in both the envelope and wave fields are nonlinearity and divergent flow?
6) In many of the comparisons (e.g. Fig. 6) there were no estimations of uncertainties, so it is not clear to me whether the differences between the different curves are significant or not. The authors should conduct an uncertainty analysis to examine how significant the results are.
Other comments:
a) Lines 15-16: After reading this paper, it seems to me that this is merely a hypothesis posed near the end of the paper (lines 373-375) without much support from the results presented. As such, this should probably not belong to the abstract.
b) Line 155: “where the amplitude of E exceeds” – is that the instantaneous value of E? In some discussions I thought it might be the climatological value of E. Can the authors please clarify?
c) Line 170-171: “whereas the climatological mean of E is comparable to that of the variance”. Not sure what this means. E and its variance would have different dimensions. How can they be comparable?
d) Equation 12: Divergence instead of gradient?
e) Line 230: “tropospheric” – should be tropopause?Citation: https://doi.org/10.5194/egusphere-2026-3447-RC3
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In my opinion this paper addresses the interesting and intriguing question whether Rossby wave envelopes exhibit higher predictability than the underlying waves. There are good arguments why this may be the case although there are equally good arguments for the opposite. The authors argue that the predictability of the wave envelope is no better than that of the underlying waves.
I found the paper interesting and well written, although a bit dense and not so easy to follow. I also have some reservations about some of the descriptions, but I would not consider those major issues, in the sense of requiring a major revision.
There is one more major point which I found hard to think through its implications, but which seems central to your argument: the normalisation of the errors with their individual saturation values. Perhaps it is worth emphasising again what you mean to do here. May I suggest you spend a bit more time at the start of your section 4, or perhaps in section 3.2 to flesh out the implications and interpretation of this methodological choice of normalisation?
If I understand this well then the normalisation allows us to see how the MSE grows as compared to what the maximum MSE could ever be for that individual component. However, Fig 6a clearly shows that in terms of the PV error, the underlying wave still grows larger errors, presumably indicating the extent to which phase errors contribute to the underlying PV anomalies. In other words, it is still the case that the phase errors introduce a further mechanism of error growth, which is by definition not seen by the envelope. I think that what you actually show is that this phase error grows at the same rate as the amplitude error. If the phase error grew faster, then the underlying wave error growth would contain a faster initial timescale compared to the growth of the envelope error.
I think this distinction between phase error and amplitude error can be easily extracted using Hilbert transforms. Perhaps the authors would find this of interest to pursue, but as reviewer I feel I am not in a position to insist on this.
In my opinion that is a much clearer way to describe the main conclusion (if you agree with my interpretation): phase errors grow at the same rate as amplitude errors. (I do realize that you suggest that the Baumgart and Jankov papers, which I have not studied, show that the phase error grows quicker at the start; this may be contradicted by your results in this paper.)
Minor comments:
L1: "... the long standing hypothesis ...":It seems that the Lee & Held paper is the only reference to this hypothesis. To call this a "long standing hypothesis" seems a rather grand description for a single somewhat vague idea published some 30 years ago, but has not had much specific follow up since. Perhaps just state the straightforward question: is the envelope of a Rossby wave better predictable than the Rossby wave itself?
L8 (and several other times in the ms): "the envelope is the wave": I think this is not at all an accurate description of what you actually found, and I strongly suggest you do not use this phrase, which just seems misleading to me. I think you found that the wave amplitude has the same predictability as its phase. I am sure you agree that in a literal sense it is not at all true that the envelope is the wave: a wave includes phase information, the envelope only contains the amplitude information. They are different things by definition.
L34-43: A nice description. I think it would be worth to follow this up with a description why it also could be argued that the envelope has the same predictability as the underlying wave. I disagree with the "larger scale" argument: the larger scale envelope would just contain fewer wavenumbers (uncertainty relation from QM!): it is just an expression of what waves make up the packet, not whether the packet has some intrinsic larger scale with its own physics.
Fig 2: I would strongly advise to use the same colourbars for the winter and summer pictures; that way we can properly compare. It looks like there are few days with RWPs in winter, compared to summer (although it is not so easy to see with the different colourbars) which comes as a big surprise to me, given that winter is baroclinically much more active. Are there fewer but longer lived events in summer perhaps?
L123: I do not recognize the Hilbert transform in this equation; perhaps worth adding in the discussion and in the equation? I am also surprised by the notation for the Fourier components of q. Why not use something like $\hat{q}$?
L134: Are the differences between RWPs and maxima of E just due to the different basic states used to find anomalies? Perhaps make explicit in the paper?
L168: Perhaps worth noting that variances may be about twice as big, but the RMS difference is of course rather less.
L177: the neglect of N in the error growth. At first glance, this seems like a bad approximation for timescales beyond a week, also in the context of much recent interest in the contribution of diabetic effects on Rossby waves. Can you justify this better? The term N being smaller than the advection terms in a PV equation does not mean that this is also true for the anomaly equation (i.e., the advecting winds could be very well predicted, while the N might be rather uncertain).
L198: "residuum": I think that is German. Change to "residual"
L209: "ir-rotational" -> "irrotational"
L272: "orange" -> "brown"
L300: perhaps worth explicitly explaining at this point that you from now on will call the sum of BC and ADV "moist-baroclinic". I think it is a rather informal way of interpreting moist baroclinic processes how we normally understand them.
L305/6: I was confused why they would not add up to 1 in the figure. Explain more clearly what is used to normalise the curves.
L336/7: Sorry - I did not understand this sentence at all. Rephrase?
Section 4.3: I had (and still have) great difficulty in following this section. I think there is so much confusing methodology that the reader simply loses track. Please consider rewriting some of this and aim a bit more directly at the question rather than let the reader drown in the various percentiles inside and outside RWPs. Perhaps I am not clever enough to understand what is happening, but in general, I think you need to make things easy for the reader.
L405: I did not understand this conclusion at all: if you control for amplitude you find that predictability in and outside RWPs is the same, how can you conclude that RWPs "organize" high amplitude waves? Again, it probably hinges on the fact that I struggled to get much insight out of section 4.3.
Maarten Ambaum