{"id":"cd447ba9-60b1-40f9-8dd0-e832b6feedb5","arxiv_id":"2603.15287","paper_version":2,"verdict":"CONDITIONAL","confidence":"HIGH","novelty_score":6.5,"correctness_risk":"medium","formal_verification":"none","parameter_count":3,"one_line_summary":"Without shear, sharp-stratification convergence to bilayer Euler is proved; with shear, numerical KH growth ~δ^{-1} prevents full Sobolev justification of bilayer Euler and shallow-water models.","lead":"Linear continuously stratified Euler equations converge to bilayer Euler when the pycnocline thins and there is no shear. With shear, numerical dispersion relations show Kelvin-Helmholtz growth rates that blow up as the interface sharpens, blocking full Sobolev justification of both bilayer Euler and bilayer shallow-water models.","discovery_kind":"extension","skeptic_critique":{"model":"grok-4.5","headline":"The δ^{-1} growth scaling (and thus the absolute obstruction to Sobolev justification of bilayer SW) rests on unproven fidelity of truncated modal eigenvalues to the true continuous spectrum.","rationale":"The reader correctly isolates the only load-bearing soft spot: the V=0 half (Prop. 6.1) is a clean energy-estimate justification with an explicit rate, while the V\neq0 half is careful but purely numerical spectral analysis whose key asymptotic (Im(ω*)~δ^{-1}) is not rigorously controlled. Appendix C mitigates but does not close the gap, exactly as the reader notes. No stronger internal inconsistency or derivation error appears; the conditional verdict already reflects the right degree of caution. The proposed test directly probes the numerical fidelity that underpins Conjecture 6.5 and the SW-obstruction claim without requiring new theory.","tokens_in":44100,"tokens_out":707,"duration_ms":14148,"concrete_test":"For a fixed small δ (e.g. 10^{-3}), recompute the full set of eigenvalues of B_k with successively doubled vertical resolution (ℓ_r) and number of retained modes ℓ (up to at least 200), using both the original trapezoidal quadrature and a higher-order spectral collocation for the Sturm-Liouville eigenfunctions; if the extracted max_k Im(ω) fails to stabilize to within 5 % of the reported δ^{-1} value, or if the unstable band collapses, the scaling claim (and the SW-obstruction argument of §6.3.2) is unreliable.","verdict_should_be":"UNCHANGED","load_bearing_attack":"The central claim that growing modes with Im(ω*_δ)≈δ^{-1} (k*_δ≈δ^{-1}) block uniform-in-δ Sobolev well-posedness of the continuous system—and therefore full justification of both bilayer Euler and bilayer shallow-water—depends entirely on the eigenvalues of the finite-mode matrices B_k (Def. 5.1, eqs. (5.8)–(5.9), (5.13)) faithfully capturing the unstable band of the continuous linearized operator (3.32)/(3.33). Appendix C only checks residual consistency of the reconstructed (c,w) pairs against the Taylor-Goldstein equation (C.2)–(C.4); because the spatial operator T is neither self-adjoint nor skew-adjoint, a small residual does not imply spectral proximity. The ad-hoc Im(c)>3·10^{-3} threshold used to extract k_max,δ (Fig. 17) and the continuous-spectrum pollution visible for Re(c)∈ range(V^δ) further leave open the possibility that the observed δ^{-1} scalings (Figs. 18, 21, 22 and Conjecture 6.5) are truncation artifacts rather than genuine features of the continuous problem.","agreement_with_reader":"agree"},"referee_report":{"model":"grok-4.5","summary":"The paper studies the sharp-stratification limit of the linearized stratified Euler equations in a strip as the pycnocline thickness δ\to0. In the absence of shear, it proves quantitative convergence in Sobolev spaces of solutions toward the linear bilayer Euler equations (Proposition 6.1), using isopycnal coordinates and energy estimates on the difference. In the presence of a sharp shear profile V^δ, it develops a normal-mode discretization, computes the dispersion relation numerically, and reports Kelvin–Helmholtz-type unstable bands whose maximal growth rates appear to scale as Im(ω*_δ)≈δ^{-1} (and k*_δ≈δ^{-1}). From this numerical evidence the authors formulate Conjecture 6.5 and argue that such growth rates obstruct full justification, in finite-regularity Sobolev spaces on a δ-independent time interval, of both the bilayer Euler equations and the bilayer shallow-water equations as reduced models of the continuous system.","tokens_in":44409,"tokens_out":1640,"duration_ms":24081,"significance":"The stable-case result (Proposition 6.1 and Appendix B) is a genuine full justification with an explicit rate O(δ^{1/2}|log δ|(1+t)), carefully written energy estimates, and a clear use of isopycnal coordinates to compare continuous and bilayer unknowns. The modal numerical scheme (Section 5), its semi-discrete convergence proposition (Proposition 5.2), and the open scripts are strengths that make the unstable-case study reproducible. The claim that KH growth may also block justification of the well-posed bilayer shallow-water system is interesting and, if the scalings hold, would be a useful caution for geophysical modeling. The paper correctly separates proved statements from numerical conjectures.","major_comments":[{"comment":"The obstruction argument in §6.3 (and the bilayer-SW conclusion in particular) rests on Conjecture 6.5, whose δ^{-1} growth and k_max,δ≈δ^{-1} scalings are read from eigenvalues of the truncated matrices B_k (Definition 5.1, (5.8)–(5.9), (5.13)). Appendix C only checks residual consistency of reconstructed (c,w) pairs against the Taylor–Goldstein residual (C.2)–(C.4). As the authors note, the spatial operator T is neither self-adjoint nor skew-adjoint, so a small residual does not imply spectral proximity. Continuous-spectrum pollution for Re(c) in the range of V^δ is visible, and k_max,δ is extracted with an ad-hoc Im(c)>3·10^{-3} threshold (Fig. 17). Without a mode-refinement study that tracks Im(ω*_δ) and k*_δ under simultaneous increase of vertical modes ℓ and Fourier cutoff, or an independent diagnostic (e.g. direct discretization of the TG equation), the claim that growing modes wi","section":null},{"comment":"Proposition 5.2 proves convergence of the modal truncation for the evolution problem under V∈W^{2,∞}, but the dispersion computation in §6.2 uses the same truncation for a family of profiles with ||V^δ'||_∞∼1/δ, which is not uniform in δ. The energy estimate of Proposition 3.7 likewise requires V∈W^{1,∞} with constants that blow up as δ\to0. The paper should clarify whether the observed unstable band and its δ-scalings remain stable under this non-uniformity, or whether the semi-discrete spectrum could be polluted by the increasingly steep shear layer for the values of δ and ℓ used in Figs. 18–22.","section":null},{"comment":"In the stable case, the measured numerical rate for Err versus δ (Fig. 11, §6.1) is roughly δ^{0.56} and the points are not well aligned; the authors note that this does not conclusively confirm the theoretical δ^{1/2}|log δ| rate of Proposition 6.1. Given that the theoretical rate is the main proved contribution, a short discussion of why the numerical rate is inconclusive (resolution of the pycnocline, number of modes, choice of initial data (6.4), or the L^∞-in-time error definition (6.5)) would strengthen confidence that the numerics and analysis are consistent.","section":null}],"minor_comments":[{"comment":"Several figures (e.g. Figs. 14–16, 23) are hard to read in grayscale; consider distinct markers or line styles in addition to color.","section":null},{"comment":"Notation for the number of vertical modes switches between ℓ, N, and ℓ in captions and text (e.g. §6.2.2); unify.","section":null},{"comment":"The date line reads “March 17, 2026”; confirm this is intentional.","section":null},{"comment":"In (3.14) and Lemma 3.1 the asymptotic n c_n \to c is stated without an explicit reference for the constant; a pointer to [AM87] is given later but could be placed at first use.","section":null},{"comment":"Typographical: “Saint-Andrew cross” / “Saint Andrew’s cross” appear in both forms; standardize. Occasional missing spaces before parentheses and “Grönwall’s” spelling vary.","section":null},{"comment":"Remark 6.6 on analytic spaces is useful; a short pointer to the vortex-sheet literature already cited ([SSBF81], [CO86]) in the introduction would help non-specialist readers.","section":null}],"recommendation":"major_revision","confidential_remarks":"The stable-case analysis is solid and publishable on its own. The unstable-case numerical study is interesting but currently over-sold relative to the spectral-approximation gap that the authors themselves flag in Appendix C. I would accept after the authors either (i) add a convincing mode-convergence study for Im(ω*) and k_max,δ or (ii) rewrite §6.3 so that the bilayer-SW obstruction is clearly labeled as a numerical indication rather than a firm consequence. Scope is appropriate for a fluids / applied-PDE journal."},"author_rebuttal":null,"desk_editor":{"model":"grok-4.5","letter":"The paper splits cleanly. Without shear it gives a genuine full-justification theorem (Prop. 6.1): linear stratified Euler with a thin pycnocline converges to linear bilayer Euler in isopycnal coordinates at rate δ^{1/2}|log δ|. The energy estimates, pressure theory and difference estimates in Appendix B are standard and carefully written; the rate is proved, not fitted. That part is new relative to Jam01 and ABD24 and is ready for use.\n\nWith a sharp shear the story is numerical. The modal truncation (finite vertical Sturm-Liouville modes + Fourier + RK4) is well-specified, comes with a convergence proposition for the semi-discrete system, and produces dispersion plots that look like the bilayer KH picture plus a high-frequency cut-off k_max,δ ~ δ^{-1}. The growth-rate scaling Im(ω*) ~ δ^{-1} is then used to argue that even the well-posed bilayer shallow-water system cannot be fully justified in Sobolev spaces on a δ-independent interval. That obstruction argument is cleanly written and, if the scaling holds, important for modelers.\n\nThe soft spot is exactly the one the stress-test flags: Appendix C only checks residual consistency of reconstructed (c,w) pairs against Taylor-Goldstein. Because the spatial operator is neither self-adjoint nor skew-adjoint, a small residual does not guarantee spectral proximity. The ad-hoc Im-threshold used to extract k_max,δ and the continuous-spectrum pollution visible inside the range of V^δ leave open the possibility that the precise δ^{-1} scalings are truncation artifacts. The qualitative presence of KH in the continuous model looks robust; the absolute prohibition of bilayer-SW justification is still conditional on better spectral analysis.\n\nWho it is for: people who work on free-boundary Euler, stratified GFD, or the justification of layered models. The V=0 theorem is citable now; the shear numerics are a useful warning and a clear target for analysis. I would send it to referees. The math is honest, the code is available, and the central claims are stated with the right degree of caution.","headline":"Solid V=0 justification with a proved rate; the shear-case numerics make a credible case that KH also blocks bilayer SW, but the δ^{-1} scaling remains unproven spectral fidelity.","tokens_in":45031,"tokens_out":580,"would_cite":true,"duration_ms":6478,"reading_group":"yes","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","76B70","76M22","86A05"],"pacs":[],"model":"grok-4.5","headline":"With shear, Kelvin-Helmholtz growth rates blow up as the pycnocline thins, so continuous stratified Euler cannot fully justify bilayer Euler or bilayer shallow-water models in Sobolev spaces.","keywords":["stratified Euler equations","sharp stratification limit","bilayer models","Kelvin-Helmholtz instability","dispersion relation","normal modes","pycnocline","shallow-water limit"],"falsifier":"A rigorous spectral analysis (or a high-resolution independent computation) of the Taylor-Goldstein operator for the family of sharp density-and-shear profiles that either confirms or refutes the conjectured scalings k_max,δ≈δ^{-1} and Im(ω*)≈δ^{-1}.","tokens_in":44953,"feed_emoji":"〜️","tokens_out":784,"duration_ms":8748,"temperature":0.7,"pith_summary":"Ocean models often replace a thin continuous density jump (a pycnocline of thickness δ) by two layers of constant density. Without background shear the paper proves that solutions of the linearized stratified Euler equations converge to the bilayer Euler equations as δ\to0, with an explicit error of order δ^{1/2}|log δ|. With a sharp shear profile the same limit is obstructed by Kelvin-Helmholtz instabilities. A modal numerical scheme for the dispersion relation shows that the continuous system itself develops unstable modes whose growth rates scale like 1/δ (and whose wave numbers also scale like 1/δ). Those rates prevent uniform-in-δ energy estimates on any fixed time interval for generic Sobolev data. The same obstruction survives the shallow-water limit, so the bilayer shallow-water equations likewise cannot be fully justified from continuous stratification in Sobolev spaces, even though the bilayer model itself is well-posed.","feed_headline":"Shear makes bilayer ocean models unjustifiable as pycnocline thins","feed_subtitle":"Growth rates blow up like 1/δ, blocking Sobolev convergence to both Euler and shallow-water bilayers","key_machinery":"Normal-mode decomposition of the linearized stratified Euler equations (Sturm-Liouville eigenfunctions of the Taylor-Goldstein operator together with the resulting finite-dimensional matrices B_k) that converts the continuous dispersion relation into a computable eigenvalue problem whose imaginary parts track the Kelvin-Helmholtz growth rates.","core_discovery":"In the presence of a shear profile that becomes discontinuous as δ\to0, numerical computation of the dispersion relation of the linearized stratified Euler equations reveals Kelvin-Helmholtz modes whose maximal growth rates satisfy Im(ω*_δ)≈δ^{-1}. Consequently neither the bilayer Euler equations nor the bilayer shallow-water equations can be fully justified, in finite-regularity Sobolev spaces on a time interval independent of δ, as reduced models of the continuous system.","pith_inferences":[],"forward_implications":[],"fun_headline_variants":["Shear drives 1/δ KH growth, blocking Sobolev bilayer limits","Discontinuous shear voids bilayer Euler and shallow-water models","Pycnocline thinning with shear yields unbounded KH modes","Growth rates ~δ^{-1} halt finite-time bilayer justification","Kelvin-Helmholtz prevents sharp-stratification bilayer reduction"],"cache_read_input_tokens":32896,"weakest_assumption_plain":"The eigenvalues of the truncated modal matrices are assumed to approximate the true continuous spectrum closely enough that the observed 1/δ growth-rate scaling is not an artifact of truncation or continuous-spectrum numerical noise.","fun_headline_variants_meta":{"raw":{"variants":["Shear drives 1/δ KH growth, blocking Sobolev bilayer limits","Discontinuous shear voids bilayer Euler and shallow-water models","Pycnocline thinning with shear yields unbounded KH modes","Growth rates ~δ^{-1} halt finite-time bilayer justification","Kelvin-Helmholtz prevents sharp-stratification bilayer reduction"]},"model":"grok-4.5","effort":"low","cost_usd":0.004396,"raw_usage":{"total_tokens":1304,"prompt_tokens":809,"num_sources_used":0,"completion_tokens":89,"cost_in_usd_ticks":43960000,"prompt_tokens_details":{"text_tokens":809,"audio_tokens":0,"image_tokens":0,"cached_tokens":128},"completion_tokens_details":{"audio_tokens":0,"reasoning_tokens":406,"accepted_prediction_tokens":0,"rejected_prediction_tokens":0}},"tokens_in":809,"tokens_out":89,"duration_ms":4185,"temperature":1.0,"reasoning_tokens":406,"cache_read_input_tokens":128,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-07-14T20:30:18.806293+00:00","model_set":{"reader":"grok-4.5"},"falsifier":"A rigorous spectral analysis (or a high-resolution independent computation) of the Taylor-Goldstein operator for the family of sharp density-and-shear profiles that either confirms or refutes the conjectured scalings k_max,δ≈δ^{-1} and Im(ω*)≈δ^{-1}.","supporting_citations":[],"review_version":1}