Interactions between Wind and Water Waves near Circular Flows

Changfeng Gui; Sicheng Liu

REVIEW 2 major objections 3 minor 22 references

Interactions between Wind and Water Waves near Circular Flows

T0 review · 2 major / 3 minor · reviewed 2026-08-06 · deepseek-v4-flash

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Pith's one-line read 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…

desk verdict 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]. read the letter →

arxiv 2508.00444 v1 pith:GXDCPH4A submitted 2025-08-01 math.AP math-phmath.MP

Changfeng Gui , Sicheng Liu This is my paper · ORCID

classification math.APmath-phmath.MP MSC 35Q3576E07

keywords two-phaseEulerequationsvortexsheetscircularflowscriticallayersRayleighequationTaylor-Couetteflowlinearinstabilitysurfacewaves

verification ladder T0 review T1 audit T2 compute T3 formal

The pith

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The reading

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.

What carries the argument

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 $\operatorname{Im} c$.

What would settle it

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.

Watch

Extended reading notes

Core claim

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.

Load-bearing premise

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.

Editorial extensions

If this is right

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.

Reading between the lines

Editorial extensions of the paper, not claims the author makes directly.

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.

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Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, and a circularity audit.

Desk Editor's Note

read the 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.

Referee Report

2 major / 3 minor

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.

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 (2)

[§3.4, Theorem 3.3 and Eq. (3.38)] 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).
[§3.4, Lemmas 3.4-3.6] 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.

minor comments (3)

[Eq. (3.35a)] 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'.
[Eq. (3.31)] 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).
[§3.5, Example 3.7] 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.

Circularity Check

0 steps flagged · score 1.0 of 10

No significant circularity: the main instability results are derived from the linearized Euler equations and dispersive relation; the only self-citation is a standard geometric identity, and the quoted technical lemmas are external prior work.

full rationale

The derivation chain is self-contained. The reduction to the Rayleigh-type ODE (2.24)/(3.19) and the dispersive relation (2.33)/(3.16) is obtained by explicit linearization of the two-phase Euler free-boundary problem in §2, not assumed. Theorem 3.1's semicircle bound follows from energy identities derived from the ODE and the dispersive relation. Proposition 3.2's necessity argument derives the estimate |Im ζdot_-(0)| ≲ ℓ^{-6}|c_I| and then uses the dispersive relation (3.16) to obtain |c_I| ≲ ε ℓ^{-6}|c_I|, which is a genuine contraction rather than a restatement of the conclusion. Theorem 3.3's sufficiency proof is a fixed-point construction around the ε=0 roots c_±^(k), with the sign condition entering only to determine the sign of the limiting imaginary part; the conclusion is not fitted into the hypotheses. The only self-citation is [13] for the standard geometric identities (2.6)–(2.7), which is not load-bearing. Lemmas 3.4–3.6 are quoted from external prior work [3] and are not re-proved here; this is a reliance on published external results and a possible completeness gap for the annular boundary setup, but not a circular reduction. A separate correctness concern, noted by the skeptic, is that the proof's assertion 'It follows from Lemma 3.6 that c♯>0' in (3.38) appears to have a sign defect when c_+^(k)<0 and the stated hypothesis forces ˙ϖ_-(s_j) ≥ 0, giving c♯ < 0; however, that is an internal sign inconsistency in the theorem's proof, not a predicted quantity that reduces to its input by construction. Overall circularity is minimal.

Assumptions & free parameters 0 free parameters · 4 assumptions · 0 invented entities

The central instability results rest on the Euler two-phase model, the normal-mode reduction, and the unproved import of critical-layer ODE lemmas from [3]. No free parameters are fitted; the constants in the examples are model inputs.

assumptions (4)

domain assumption Two-phase flow dynamics are governed by incompressible Euler equations (1.1)-(1.2) with surface tension parameter α≥0.
Physical model stated at the start; all results are derived for this model.
domain assumption Linear instability is equivalent to existence of a normal-mode solution (u,q,ψ⊥) = e^{λt+ikθ}(u(r),q(r),ψ) with Im{c}>0 solving the boundary value problems (2.32) and dispersive relation (2.33).
Standard linearized stability reduction, introduced in Section 2.2 and used throughout Section 3.
domain assumption Lemmas 3.4-3.6 from [3] (Propositions 4.2, 4.3 and Section 4.4) hold for the circular-flow ODE (3.19) with boundary conditions (2.32).
These singular-ODE estimates are invoked without proof in the proof of Theorem 3.3; the adaptation to the annular domain and the boundary at s=0 is not demonstrated.
domain assumption The water-vacuum limit (ε=0) has two distinct real phase velocities c_±^{(k)} for the fixed wave number.
Assumed in Section 1.2.3 and used to set up the perturbation problem; if the unperturbed wave is already unstable, the near-ε analysis changes.

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Cite this review

Pith. "Pith review of Interactions between Wind and Water Waves near Circular Flows." pith.science (2026). https://pith.science/paper/GXDCPH4A

@misc{pith2026250800444,
  author       = {Pith},
  title        = {Pith review of: Interactions between Wind and Water Waves near Circular Flows},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/GXDCPH4A}},
  note         = {Machine review of arXiv:2508.00444}
}

read the original abstract

This manuscript concerns the dynamical interactions between wind and water waves, which are characterized through two-phase free interface problems for the Euler equations. We provide a comprehensive derivation on the linearized problems of general two-phase flows. Then, we study the instability issues of perturbing waves around circular steady solutions, and we demonstrate a semi-circle result on the possible locations of unstable modes. We also present necessary conditions and sufficient ones for the instability of wind-perturbing water waves near Taylor-Couette flows.

Discussion (0). Continue with ORCID to comment.

Reference graph

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Pith tools

Reviewed August 6, 2026 · model on record in the stance chip above.

[1] [3]

Oliver B ¨uhler, Jalal Shatah, Samuel Walsh, and Chongchun Zeng, On the wind generation of water waves, Arch. Ration. Mech. Anal. 222 (2016), 827–878

work page 2016

[2] [1]

Thomas Beale, Thomas Y

J. Thomas Beale, Thomas Y . Hou, and John S. Lowengrub, Growth rates for the linearized motion of ﬂuid interfaces away from equilibrium , Comm. Pure Appl. Math. 46 (1993), 1269–1301

work page 1993

[3] [2]

Didier Bresch and Michael Renardy, Kelvin-Helmholtz instability with a free surface , Z. Angew. Math. Phys. 64 (2013), 905–915

work page 2013

[4] [4]

Pure Appl

Ching-Hsiao Arthur Cheng, Daniel Coutand, and Steve Shkoller, On the motion of vortex sheets with surface tension in three-dimensional Euler equations with vorticity , Comm. Pure Appl. Math. 61 (2008), 1715–1752

work page 2008

[5] [5]

P. G. Drazin and W . H. Reid, Hydrodynamic stability, second ed., Cambridge Mathematical Li- brary, Cambridge University Press, Cambridge, 2004, With a foreword by John Miles

work page 2004

[6] [6]

Ebin, The equations of motion of a perfect ﬂuid with free boundary are not well posed , Comm

David G. Ebin, The equations of motion of a perfect ﬂuid with free boundary are not well posed , Comm. Partial Diﬀerential Equations 12 (1987), 1175–1201

work page 1987

[7] [7]

Ebin, Ill-posedness of the Rayleigh-Taylor and Helmholtz problems for incompressible ﬂuids, Comm

David G. Ebin, Ill-posedness of the Rayleigh-Taylor and Helmholtz problems for incompressible ﬂuids, Comm. Partial Diﬀerential Equations 13 (1988), 1265–1295

work page 1988

[8] [8]

Thierry Gallay, Stability of vortices in ideal ﬂuids: the legacy of Kelvin and Rayleigh , arXiv preprint (2019), arXiv:1901.02815

work page Pith review arXiv 2019