| 18 Dec 2025
Revisiting barotropic instability from the perspective of wave evolution theory
Abstract. The instability of Rossby waves has been a long-standing topic in dynamical meteorology. The classic theoretical analysis had provided in–depth physical understanding of the problem. However, developing a systematic and quantitative comprehension of wave energy and amplitude evolution remains challenging. With an eye to such issues, this investigation provides a novel and practicable algorithm to solve the wave action conservation equation. Theoretical analysis establishes that wave packet energy evolves through two competing factors: direct proportionality to intrinsic frequency and inverse proportionality to group velocity magnitude. Energy density attains extremal values at turning points where group velocity magnitudes become extremized. To ensure a ray can be reflected by a turning point, zonal phase speed must be smaller than an upper limit determined by dispersion relation at the turning point. Crucially, a specific zonal phase speed range emerges below this maximum threshold where concurrent transient growth of both wave energy and amplitude occurs when a ray is moving toward the turning point, with the upper limit corresponding to optimal wave development conditions. Numerical experiments on a prototype westerly jet reveal distinct instability mechanisms: substantial transient growth–capable of triggering nonmodal instability–arises when rays moves toward turning points, while exponential amplification characteristic of modal instability develops at inflection points with positive energy growth rate. The derived zonal phase speed thresholds and transient growth metrics form a diagnostic framework applicable to observed atmospheric flows, enabling quantitative evaluation of both modal and nonmodal instability potentials. By unifying wave evolution dynamics with classical instability criteria, this work provides an operational bridge between theoretical predictions and real–world flow diagnostics.
The Abstract and main body are interesting and generally well written. The references are fairly complete, with one major exception and some minor ones noted below. The major omission is awareness of the Hurdle Theorem (see below), which establishes a sufficient condition for shear instability and is complementary to the necessary conditions for instability that the paper cites. The paper should be publishable after taking care of the following comments and minor issues.
Comments
Line 34-49: The emphasis on classical sufficient conditions for stability is outdated and needs to be rewritten to bring it up to date. In particular, a major sufficient condition for inviscid shear instability has been established, called the Hurdle Theorem. See Deguchi et al. (2024, J. Fluid Mech., 997, A25, doi:10.1017/jfm.2024.728) and Read and Dowling (2026, Encycl. Atmos. Sci. (3rd Ed.) 4, 263—283, Academic Press, doi:10.1016/B978-0-323-96026-7.00211-3).
Ray tracing is heavily used, but without first establishing the context for which it is accurate and discussing general conditions when it is inaccurate, such as inhomogeneous environments and proximity to focusing points (caustics). This can be fixed by adding a short paragraph discussing the issues early in the introduction. There is a brief mention of such an issue on Line 132, another on Line 144-152, and a workaround on Line 191+ (Section 3). These could usefully be tied into an introductory short paragraph on the general strengths and shortcomings of ray tracing.
On a related note, it is not clear while reading this paper at what point or points in the evolution of an unstable shear flow the theory applies. When a shear flow is truly barotropically unstable, the end result often bears little resemblance to the initial conditions, a point made in e.g., the review by Read and Dowling (2026). It would help to add a few guideposts throughout the paper that indicate whether we are always teetering on marginal stability or not, and at what point does the evolution of unstable flow render the discussion moot.
It should be made clear early in the paper that the system under study has a flat bottom topography and thus misses out on an entire class of marginally stable cases that are especially relevant to Jupiter and Saturn. See Deguchi et al. (2024) for an extended discussion of this point. This limitation simply needs to be made explicit somewhere in the paper’s introduction, and ties into the conclusions and future work, for example the comment on Line 472 about “real-world atmospheric flows”.
The Hurdle Theorem (Deguchi et al. 2024) can be readily applied to the u = sech2(y) prototype (29) analysed in Section 4, which will significantly enhance the discussion, including the material shown in Fig. 1 c), which currently only illustrates sufficient-for-stability criteria and is lacking sufficient-for-instability criteria. Some of the main figures later in the paper are also ripe for addition of Hurdle Theorem regions.
Minor
Line 14: “by dispersion relation” -> “by the dispersion relation”
Line 17-18: the clause-isolating hyphens should be longer em-dashes, as in growth—capable
Line 26: “they may play” -> “they play”
Line 59: “packet, w to” needs to be fixed
Line 70: “Li et al.(2021a) -> “Li et al. (2021a)”
Line 91: The epsilon and Psi symbols are running into each other in (3). There are several spacing issues in the equations throughout the manuscript. Presumably these will at least get fixed in the typesetting.
Line 194 and (15): This transform needs to be better motivated and referenced in terms of how and why it is helpful and how old this strategy is.
Line 215-220: There are remarkably few citations in the beginning of Section 3. Please add some citations to similar work, or add emphasis that none exists.
Page 11: To this point there has been a noticeable lack of any figures. The effect is to make reading the paper more tedious than this important subject deserves. Please add a couple of examples to illustrate some of the key points being made, which will prop up the reader’s morale, particularly new students.
Line 438: It is arguable that the explicit physical understanding of Rossby wave instability in the context of the reciprocal Rossby-Mach number (e.g., Deguchi et al. 2024) has not been well known for several decades, which undermines the point of this sentence. This needs to be updated.