{"id":"3b187c88-9ffd-457b-82c6-31414bf6e383","arxiv_id":"2508.00444","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":7.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"For circular two-phase flows, smooth wind profiles destabilize water waves only through critical layers, and under sign conditions these layers do trigger instability, subject to a semicircle bound.","lead":"This paper studies how wind destabilizes water waves when two fluids meet along a circular interface, rather than a flat one. It derives the linearized equations, proves a semicircle bound on unstable wave speeds, and identifies critical layers as the source of instability.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"Theorem 3.3's sign condition is internally inconsistent: when c_+^(k)<0, the hypotheses imply ˙ϖ_-(s_j)>0, making c♯ in (3.38) negative, so the proof cannot yield Im c>0.","rationale":"The reader's weak point (unproved Lemmas 3.4-3.6) concerns external support; my concern is internal and affects the statement of the central theorem itself. Even if one accepts all three quoted lemmas, the sign of c♯ in (3.38) contradicts the theorem's stated hypotheses when c_+^(k)<0. Since (3.17) fixes h_+^I(0)>0 independently of the sign of c_+^(k), and c_+^(k) ˙ϖ_-(s_j)≤0 then forces the sum in (3.38) to be positive, the proof cannot produce an unstable c near c_+^(k). This is not a matter of consensus or a minor typo: the stated sufficient condition is wrong in a parameter regime explicitly allowed by the theorem. The paper can likely be repaired by adding the natural assumption c_+^(k)>0>c_-(k) (or replacing the product sign by a branch-appropriate sign), which is consistent with the quiescent-water examples; hence the disposition remains conditional rather than an outright rejection. My test with A+B=-10 and c_+<0 is a concrete within-theory check that would settle the issue; the same computation also exposes the (3.35a) denominator typo that should be corrected in any revision.","tokens_in":21766,"tokens_out":22882,"duration_ms":210692,"concrete_test":"Take explicit admissible data: inner Taylor-Couette A+B=-10, B=0, α/ϱ_+=40, k=2, R_in=1/2 (so c_+^(k)≈-7.03<0), wind w_-(s)=c_+^(k)+a(s-1) with a>0 on (0,logRout), so s=1 is a regular critical point with ˙ϖ_-(1)=2a>0 and c_+^(k) ˙ϖ_-(1)<0. For this data compute c♯ from (3.38) using (3.17): h_+^I(0)>0 and S=2a bξ♯_1(1)/a>0, so c♯<0. Then check the fixed-point system (3.39)-(3.40) for ε=10^-3: the only zero has c_I<0 (or none with c_I>0), disproving the c_+-claim. Analytical shortcut: with c_I>0, the dispersion balance c_I=ε Im ζdot_-(0) h_+^I and Lemma 3.6's Im ζdot_-(0)≈-π ˙ϖ_-(s*) bξ_1(s*)/|˙w_-(s*)| have opposite signs when ˙ϖ_->0.","verdict_should_be":"UNCHANGED","load_bearing_attack":"Even granting Lemmas 3.4–3.6, the proof of Theorem 3.3 has a sign defect. From (3.17), h_+^I(0)>0 whenever c_+^(k)≠w_-(0), because the numerator is a square and the denominator is c_+^(k)-c_0>0 for the larger root. The constructed c♯ in (3.38) is c♯ = -π h_+^I(0) Σ_j ˙ϖ_-(s_j) bξ♯_1(s_j)/|˙w_-(s_j)|, with bξ♯_1(s_j)>0 (since ξ_1=|ζ_-|^2≥0 and Lemma 3.6 gives eξ_1(σ'_j)≠0). If c_+^(k)<0, the theorem's hypothesis c_+^(k) ˙ϖ_-(s_j)≤0 forces ˙ϖ_-(s_j)≥0 (strict for at least one j), hence the sum S>0 and c♯<0. The proof's assertion 'It follows from Lemma 3.6 that c♯>0' is therefore unjustified and false in this regime. Since c_I is set to ε(c♯+ν2) in (3.40), c_I is negative near ν2=0, and the fixed-point system (3.39) has no zero with c_I>0; the dispersion balance c_I=ε Im ζdot_-(0) h_+^I opposes the sign obtained from Lemma 3.6. Thus the c_+-statement of Theorem 3.3 fails for admissible Taylor-Couette parameters with c_+^(k)<0 (e.g., strong negative background flow), and the 'same result' for c_- requires the two roots to have opposite signs. The theorem needs either an explicit positivity assumption on c_+^(k) or a corrected sign condition, e.g., ˙ϖ_-(s_j)<0 for the positive branch and ˙ϖ_-(s_j)>0 for the negative branch. A related typo: the denominator in (3.35a) should be |˙w_-(s')|, not |w_-(s')|.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","summary":"The paper studies two-phase incompressible Euler equations with a free interface, focusing on perturbations of circular steady flows. It derives a general linearized formulation, then specializes to axisymmetric circular background flows, obtaining a Rayleigh-type ODE and a dispersive relation. The main results are a semicircle bound on unstable phase velocities, a necessary condition relating instability to critical layers of the wind profile, a sufficient condition for instability when the phase velocity is a regular value of the wind profile and a sign condition on the vorticity derivative holds, and an example showing instability for a non-smooth wind profile whose critical layer is disjoint from the vorticity-derivative support.","tokens_in":41,"tokens_out":17823,"duration_ms":294878,"significance":"If the results are correct, the paper provides a meaningful extension of the shear-flow instability theory of Bühler, Shatah, Walsh, and Zeng to circular flows, and it contributes a general linearization of two-phase free-boundary problems. The semicircle theorem is a useful addition, and the necessary/sufficient critical-layer statements are the first of their kind for wind-perturbed circular water waves. The manuscript also contains explicit computations in examples that are valuable for testing the theory. However, the sufficiency theorem contains a sign inconsistency that affects a central claim, and the main proof relies on unproved estimates imported from another paper; these issues must be resolved before the main results can be accepted.","major_comments":[{"comment":"The proof's assertion 'It follows from Lemma 3.6 that c♯>0' is not justified and is false under the stated hypotheses for a range of admissible parameters. From (3.17), h_+^I(0)>0 and h_-^I(0)<0 whenever the relevant root differs from w_-(0). Since c♯ = -π h_±^I(0) Σ_j ˙ϖ_-(s_j) bξ♯_1(s_j)/|˙w_-(s_j)|, and since bξ♯_1(s_j)>0 and |˙w_-(s_j)|>0, c♯>0 is equivalent to h_±^I(0) Σ_j ˙ϖ_-(s_j)bξ♯_1(s_j)/|˙w_-(s_j)| < 0. The hypothesis c_+^{(k)}˙ϖ_-(s_j)≤0 gives sign(Σ)= -sign(c_+^{(k)}), so the claimed positivity requires sign(c_+^{(k)})=+1. If c_+^{(k)}<0, which is attainable by taking A+B strongly negative in (3.15), the hypothesis forces ˙ϖ_-(s_j)≥0, hence Σ≥0 and c♯≤0; then the fixed-point system (3.39)-(3.40) cannot produce c_I>0 near ν2=0. Similarly, for the c_- branch the same argument requires c_-^{(k)}<0, so the final sentence of Theorem 3.3 is false when c_-^{(k)}>0. The theorem should either impose explicit sign conditions on c_±^{(k)} or replace the factor c_±^{(k)} in the sign condition by the relevant c_±^{(k)}-c_0, which is what (3.17) actually shows controls the sign of h_±^I(0).","section":"§3.4, Theorem 3.3 and Eq. (3.38)"},{"comment":"The proof of Theorem 3.3 depends entirely on Lemmas 3.4-3.6, which are imported from [3] with only the indication 'cf. [3; Proposition 4.2, 4.3, §4.4]'. These lemmas were developed for shear flows with a flat interface, and their adaptation to the annular circular-flow boundary-value problem (3.19) with ζ_-(0)=1, ζ_-(logRout)=0, and the coupling to the dispersive relation (3.16) is not demonstrated. In particular, the jump conditions (3.35), the sign of eΦ(0) in Lemma 3.6, and the uniform error estimates O(|c_I|^μ) are load-bearing for the existence argument. The paper should either provide complete proofs adapted to the circular-flow setting or state the precise hypotheses under which the quoted results apply and verify all of them, including the boundary conditions at s=0 and s=logRout and the coupling through h_±^I.","section":"§3.4, Lemmas 3.4-3.6"}],"minor_comments":[{"comment":"The denominator in the jump condition (3.35a) should be |˙w_-(s')|, not |w_-(s')|; the preceding asymptotic formula (3.34) shows that the factor |˙w_-(s')| is what arises from the change of variables near the regular point s'.","section":"Eq. (3.35a)"},{"comment":"The displayed limiting system (3.31) appears garbled in its third component: the term involving c_I ˙ϖ_-(s)/(w_-(s)-c_R)^2 should be reconciled with the distributional limit used in (3.34), where the denominator contains c_I^2 before taking the limit. Please rewrite the limiting system consistently and state precisely which terms are retained to produce the jumps in (3.35).","section":"Eq. (3.31)"},{"comment":"In the non-smooth example, the statement that 'the critical layer is away from spt(˙ϖ_-)' would be easier to verify if the actual location of the critical layer and the positive separation from s_* were written explicitly, since the paper only gives w_-(s_*) = λ_+^{(k)} + ω_*/|k|(1-e^{-2|k|s_*}) and leaves the separation implicit.","section":"§3.5, Example 3.7"}],"recommendation":"major_revision","confidential_remarks":"The main concern is the sign defect in Theorem 3.3: the theorem as stated is false for a natural range of Taylor-Couette parameters. The fix appears local, but because Theorem 3.3 is the central sufficiency result, I would not accept the manuscript without a corrected statement and a repaired proof. In addition, the heavy reliance on unproved lemmas from [3] should be disclosed more carefully; if those lemmas cannot be adapted, the main existence proof fails. The semicircle argument and the general linearization in Section 2 seem sound and should be preserved."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"Two things to know. First, Section 2's coordinate-free linearization of the two-phase free-boundary problem is a genuine contribution: it removes the graph assumption, and the jump conditions emerge cleanly from the geometry. The semicircle theorem in Section 3.1 also checks out and is a natural extension of Howard's argument. The examples, especially the Taylor-Couette water-wave cases and the Lipschitz wind profile, are instructive. Second, the central instability result, Theorem 3.3, has a sign flaw that matters. The proof defines c_sharp in (3.38) and asserts that Lemma 3.6 gives c_sharp > 0. That is only true when the branch root and the derivative of the wind vorticity have opposite signs. For the c_+ branch, h_+^I(0) > 0, so c_sharp > 0 requires the sum over critical points to be negative, i.e. dot-vartheta_-(s_j) < 0. The theorem's hypothesis c_+^(k) dot-vartheta_-(s_j) <= 0 forces dot-vartheta_-(s_j) >= 0 whenever c_+^(k) < 0, so in that regime c_sharp is negative and the fixed-point argument cannot produce Im c > 0. The statement needs either an explicit positivity assumption on c_+^(k) or a branch-dependent sign condition: dot-vartheta < 0 for the positive branch and dot-vartheta > 0 for the negative branch. As written, the proof of Theorem 3.3 does not go through for admissible Taylor-Couette parameters with strong negative background flow. The second soft spot is the import of Lemmas 3.4-3.6 from [3]. These are stated with boundary conditions specific to the circular-flow ODE, but the paper only says \"cf. [3]\" and does not show the adaptation in detail. A referee should ask for a proof or a precise transfer argument. Smaller issues: the denominator in (3.35a) should be |dot-w_-(s')|, not |w_-(s')|, and the notation c_+^(k) < w_-([0,log Rout]) in Proposition 3.2 needs unpacking. Who this is for: people working on two-phase Euler stability and on critical-layer mechanisms. Section 2.1 and the semicircle theorem are worth citing even if the instability part needs repair. The paper deserves refereeing: the derivation and semicircle result are solid, and the critical-layer mechanism is promising. But I would not accept Theorem 3.3 in its current form; it needs a corrected sign condition and a careful check of which branch can actually produce instability.","headline":"Useful linearization and semicircle theorem, but the main instability theorem has a sign error in the c_+<0 regime and relies on unproved imports from [3].","tokens_in":22747,"tokens_out":7778,"would_cite":true,"duration_ms":69993,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["35Q35","76E07"],"pacs":[],"model":"deepseek-v4-flash","headline":"A light wind shearing over a Taylor-Couette water flow destabilizes surface waves precisely when the wind's angular velocity matches the wave phase velocity at a critical layer, if the density ratio is small and the wind vorticity…","keywords":["two-phase Euler equations","vortex sheets","circular flows","critical layers","Rayleigh equation","Taylor-Couette flow","linear instability","surface waves"],"falsifier":"Numerically solve (3.19) with the dispersive relation (3.16) for a $C^4$ wind profile satisfying every hypothesis of Theorem 3.3; if no root with $\\operatorname{Im} c > 0$ exists as $\\varepsilon \\to 0$, the existence claim fails, and if a growing root appears when the sign condition is reversed at an outermost critical point, the condition is shown non-sharp.","tokens_in":21500,"feed_emoji":"🌊","tokens_out":13305,"duration_ms":116241,"temperature":0.7,"pith_summary":"This paper asks when wind can destabilize surface waves on a circular water flow. It derives the linearized two-phase Euler equations around general background flows, then specializes to axisymmetric circular flows with a free circular interface, reducing the perturbation problem to a Rayleigh-type ODE and an algebraic dispersive relation. It proves a semicircle bound on where unstable phase velocities can lie, and then shows that for small density ratio $\\varepsilon = \\varrho_-/\\varrho_+$, smooth wind profiles generate instability exactly through critical layers—locations where the wind's angular velocity equals the unperturbed water-wave phase velocity. The main theorem states that if that phase velocity is a regular value of the wind profile and the derivative of the wind vorticity at each preimage point obeys $c_+^{(k)}\\dot{\\varpi}_-(s_j) \\le 0$ with strict inequality at an outermost point, then for sufficiently small $\\varepsilon$ a growing mode exists with phase velocity $O(\\varepsilon)$ from the neutral value.","feed_headline":"Critical layers trigger wind-wave instability on circular flows","feed_subtitle":"A smooth wind over a circular Taylor-Couette flow turns unstable exactly where wind speed matches wave phase speed.","key_machinery":"The load-bearing object is the singular Rayleigh-type boundary-value problem (3.19) for the outer fluid, coupled to the dispersive relation (3.16). Solutions are tracked through the four real quadratic quantities $(\\xi_1,\\xi_2,\\xi_3,\\Phi) = (|\\zeta_-|^2, \\operatorname{Re}(\\zeta_-\\dot{\\zeta}_-^*), |\\dot{\\zeta}_-|^2, \\operatorname{Im}(\\zeta_-\\dot{\\zeta}_-^*))$, which satisfy the first-order system (3.29). As $\\operatorname{Im} c \\to 0$ and $\\operatorname{Re} c$ crosses a regular value of $w_-$, the system develops poles whose residues give the jump conditions (3.35); the sign of each jump is set by $\\dot{\\varpi}_-(s_j)/|\\dot{w}_-(s_j)|$, and the boundary value $\\Phi(0) = \\operatorname{Im}\\{\\dot{\\zeta}_-(0)\\}$ records the accumulated jumps. The dispersive relation supplies analytic maps $h_R, h_I$ whose zero set in $(\\nu_1,\\nu_2,\\varepsilon)$ is solved by the implicit function theorem, turning a nonzero $\\Phi(0)$ into a positive $\\operatorname{Im} c$.","core_discovery":"The central result is a critical-layer characterization of wind-generated instability near circular flows. For the limiting water-vacuum problem around a Taylor-Couette flow (angular velocity $A/r^2 + B$, constant vorticity inside), fix a wave number $k$ and suppose the neutral phase velocity $c_+^{(k)}$ is real and the water-wave mode is stable without wind. When a light outer wind of angular velocity $w_-(s)$ is added, Proposition 3.2 shows that if $c_+^{(k)}$ lies outside the range of $w_-$, then every nearby mode is real for small $\\varepsilon$: no critical layer, no instability. Theorem 3.3 is the converse: if $c_+^{(k)}$ is a regular value of a $C^4$ profile $w_-$ (all preimage points have $\\dot{w}_-(s_j) \\neq 0$) and $c_+^{(k)} \\dot{\\varpi}_-(s_j) \\le 0$ at every preimage point, strictly at one of the two outermost points, then the coupled ODE-dispersion problem has a solution with $\\operatorname{Im} c > 0$ and $|c - c_+^{(k)}| = O(\\varepsilon)$, i.e., a genuinely growing surface wave. The theorem thus makes critical layers both necessary and sufficient for smooth wind profiles, and it confines unstable phase velocities to a semicircular region determined only by the extreme angular velocities of the two fluids.","pith_inferences":["Editorial inference: the same critical-layer mechanism should survive for non-axisymmetric perturbations of circular interfaces, since the radial structure of the Rayleigh equation, not the angular ansatz, carries the argument.","Editorial inference: a numerical continuation in $\\varepsilon$ of the roots of (3.16)-(3.19) could reveal how small the density ratio must be for the $O(\\varepsilon)$ bound and whether the proof's implicit constants are practically relevant.","Editorial inference: the nonsmooth example suggests a square-root scaling, $\\lambda_I = O(\\varepsilon^{1/2})$, for Lipschitz wind profiles versus the linear scaling for smooth ones; measuring that exponent would distinguish the two instability mechanisms.","Editorial inference: the necessity argument plausibly extends to other axisymmetric base flows such as vortex patches, giving a route to stability criteria for droplets and other non-graph interfaces."],"forward_implications":["For smooth wind profiles, wind-generated instability of circular Taylor-Couette water waves is governed by critical layers alone: if the wind's angular velocity misses the neutral phase velocity, no growing mode appears at small density ratio.","When the sign condition holds, the unstable mode's phase velocity stays within $O(\\varepsilon)$ of the water-vacuum value, so lighter winds produce slower growth of order $\\varepsilon$.","The semicircle theorem bounds all unstable phase velocities by the minimum and maximum angular velocities of the two fluids, independent of the detailed profile shape, so numerical searches can be restricted to a known disk.","Capillary forces suppress high wave numbers; without surface tension, constant-vorticity circular flows are linearly unstable at large $|k|$, matching the classical ill-posedness of vortex sheets.","For merely Lipschitz wind profiles, instability can occur even when the critical layer is disjoint from the support of the vorticity derivative, so the smoothness hypothesis in the necessity result is essential."],"supporting_citations":[{"why":"Supplies the singular ODE estimates and jump conditions used as Lemmas 3.4-3.6, the analytic backbone of Theorem 3.3.","marker":"[3]"},{"why":"Provides the semicircle method for bounding unstable phase velocities that Theorem 3.1 adapts to circular two-phase flows.","marker":"[9]"},{"why":"Gives the spectral and semicircle framework for linearized two-phase shear flows that the paper extends to circular backgrounds.","marker":"[14]"},{"why":"Identifies the wind-profile critical-layer mechanism that the paper makes rigorous in the circular setting.","marker":"[17]"}],"fun_headline_variants":["Critical layers make wind-water waves unstable on circular flows","Wind-wave instability on circular flows pinned to critical layers","Critical layer criterion: wave growth near Taylor-Couette flows","Semicircle rule for wind-driven wave instability on circular flows","When wind speed matches wave phase, circular flow gets unstable"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"The proof imports, rather than proves, the estimates that control solutions of the singular equation across critical layers and define the jump conditions; if those imported estimates do not hold for the annular boundary-value problem with its specific boundary data, the sufficiency theorem loses its ground.","fun_headline_variants_meta":{"raw":{"variants":["Critical layers make wind-water waves unstable on circular flows","Wind-wave instability on circular flows pinned to critical layers","Critical layer criterion: wave growth near Taylor-Couette flows","Semicircle rule for wind-driven wave instability on circular flows","When wind speed matches wave phase, circular flow gets unstable"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000646,"raw_usage":{"total_tokens":2956,"prompt_tokens":924,"completion_tokens":2032,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":540,"completion_tokens_details":{"reasoning_tokens":1951}},"tokens_in":540,"tokens_out":2032,"duration_ms":14853,"temperature":1.0,"reasoning_tokens":1951,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-06T10:09:25.780384+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"Numerically solve (3.19) with the dispersive relation (3.16) for a $C^4$ wind profile satisfying every hypothesis of Theorem 3.3; if no root with $\\operatorname{Im} c > 0$ exists as $\\varepsilon \\to 0$, the existence claim fails, and if a growing root appears when the sign condition is reversed at an outermost critical point, the condition is shown non-sharp.","supporting_citations":[{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Supplies the singular ODE estimates and jump conditions used as Lemmas 3.4-3.6, the analytic backbone of Theorem 3.3."},{"cited_title":"Howard, Note on a paper of John W","cited_arxiv_id":null,"evidence_quote":"Provides the semicircle method for bounding unstable phase velocities that Theorem 3.1 adapts to circular two-phase flows."},{"cited_title":null,"cited_arxiv_id":null,"evidence_quote":"Gives the spectral and semicircle framework for linearized two-phase shear flows that the paper extends to circular backgrounds."},{"cited_title":"Miles, On the generation of surface waves by shear flows, J","cited_arxiv_id":null,"evidence_quote":"Identifies the wind-profile critical-layer mechanism that the paper makes rigorous in the circular setting."}],"review_version":1}