Instabilities of internal gravity waves in the two-dimensional Boussinesq system

A. Maspero; R. Bianchini; S. Pasquali

REVIEW 3 major objections 3 minor 41 references

Instabilities of internal gravity waves in the two-dimensional Boussinesq system

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

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Pith's one-line read The paper proves that small-amplitude internal gravity waves in an inviscid stably stratified Boussinesq fluid are linearly unstable, with an explicit linear-in-amplitude growth rate.

desk verdict First rigorous PSI-type instability for inviscid internal waves, with a proof that mostly holds up; the main caveat is a typo in Proposition 3.3 that needs fixing. read the letter →

arxiv 2507.10390 v1 pith:WZYO52QO submitted 2025-07-14 math.AP

R. Bianchini , A. Maspero , S. Pasquali This is my paper · ORCID

classification math.AP MSC 35Q3576E2076B70

keywords BoussinesqequationsinternalgravitywavesspectralinstabilitymodulationalparametricsubharmonicFloquet-Blochdecompositioninviscidstratifiedfluid

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 proves that small-amplitude internal gravity waves in a two-dimensional inviscid, stably stratified Boussinesq fluid are linearly unstable: the linearization about such a wave has an eigenvalue with strictly positive real part. This is the first rigorous derivation of the Parametric Subharmonic Instability (PSI) in the inviscid setting, in which an initially excited primary wave transfers energy to two subharmonic waves of lower frequency. The result matters for oceanography because viscous effects are often negligible there, so the inviscid instability is the physically relevant one. The proof reduces the problem to a two-by-two matrix whose eigenvalues have an explicit first-order growth rate, linear in the wave amplitude.

What carries the argument

The central object is the resonant set $R_k$, the curve of Floquet parameters $\mu$ for which the unperturbed linearized operator $L_{\mu,0}$ has a double eigenvalue $\lambda_+ = \lambda^-_k(\mu) = \lambda^+_0(\mu)$ on the invariant subspace $H^1_k$ of functions whose Fourier coefficients are supported on wavevectors proportional to the primary wavevector $k$. Kato's similarity transformation and spectral projectors show that this double eigenvalue persists as an isolated pair for $\epsilon>0$, reducing the problem to the eigenvalues of a $2\times 2$ matrix. The product of the off-diagonal entries of that matrix yields the explicit growth-rate function $e(\mu)$ in equation (2.38), whose sign controls the instability.

What would settle it

Direct numerical diagonalization of the restricted operator $L_{\mu,\epsilon}|_{H^1_k}$ for a concrete case, say $k=(1,1)$, small $\epsilon$, and $\mu$ on the resonant branch $R_k$, should reproduce an unstable eigenvalue whose real part equals $\epsilon\sqrt{e(\mu)}$ at leading order; finding no such eigenvalue, or a leading rate that disagrees, would refute the theorem.

Watch

Extended reading notes

Core claim

The paper's central claim is that for every wavevector $k=(m,n)$ with $m,n>0$ and every sufficiently small amplitude $\epsilon$, the operator obtained by linearizing the two-dimensional inviscid Boussinesq equations in vorticity-stream form around the plane wave solution has at least one eigenvalue with strictly positive real part. The sharper statement is that, restricting to the invariant subspace of functions supported on wavevectors proportional to $k$ and choosing the Floquet parameter $\mu$ on the resonant set defined by $\Omega(k)-\Omega(k+\mu)=\Omega(\mu)$, the perturbed eigenvalues take the form $\lambda_+ + iO(\epsilon^2) \pm \epsilon\sqrt{e(\mu)} + O(\epsilon)$, where $e(\mu)$ is given explicitly and is positive in the small-$|\mu|$ and large-$|\mu|$ limits. Because $e(\mu)>0$, the real part of one of the eigenvalues is positive, an instability whose leading growth rate is linear in the amplitude $\epsilon$.

Load-bearing premise

The linear instability is a spectral point of the Floquet-Bloch operator on the torus, and the growing solution $e^{\lambda t} e^{i\mu x} v(x)$ is not square-integrable over $\mathbb{R}^2$, so the theorem does not provide an $L^2$ eigenfunction of the operator on the whole plane.

Editorial extensions

If this is right

If the theorem is right, every sufficiently small-amplitude internal plane wave in the inviscid two-dimensional Boussinesq system is modulationally unstable, with growth rate proportional to amplitude.
The explicit formula for $e(\mu)$ makes it possible to predict, for a given primary wave vector $k$, which resonant subharmonic pairs grow fastest.
The instability persists all along each resonant branch, from small to large Floquet parameters, as the asymptotic formulas (2.40) and (2.41) show positivity in both limits.
The method extends the rigorous modulational-instability toolkit to a non-Hamiltonian, reversible system with an unbounded perturbation, where classical perturbation theory requiring boundedness does not apply.
The result rigorously confirms the physical PSI picture: a primary wave of frequency $\Omega(k)$ transfers energy to two secondary waves whose frequencies are, to leading order, half of $\Omega(k)$.

Reading between the lines

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

One testable extension is to compute the spectrum of the truncated operator on $H^1_k$ numerically for a fixed $k$ and small $\epsilon$, and check that the leading growth rate matches $\epsilon\sqrt{e(\mu)}$; agreement to order $\epsilon$ would confirm the sharpness of the formula.
Because the unstable Bloch mode is not square-integrable over $\mathbb{R}^2$, the established instability is spectral; whether nonlinear effects convert it into actual norm growth for localized data remains an open question that the paper does not address.
The structure of the proof suggests that the resonance condition, rather than the specific form of $\Omega$, drives the instability, so an analogous result may hold for more general stable density profiles with a homogeneous dispersion relation.
The relation in Proposition 6.1 between $e(\mu)$ and the physical interaction coefficients $I_\pm(\mu)$ indicates that the rigorous growth rate coincides with the weakly-nonlinear triad expansion in the small-Floquet limit, which could be used to calibrate reduced models of internal-wave energy transfer.

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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

This paper is worth reading: it gives the first rigorous proof of PSI-type instability for inviscid internal waves in the 2D Boussinesq system, and the proof strategy is genuinely novel. The main theorem establishes a positive-real-part spectral point for the linearization around a small-amplitude plane wave, with an explicit first-order growth rate. The resonance-set analysis (Lemma 2.5) and the spectral-isolation argument (Prop 2.7) are careful, and the comparison with the physical literature in Section 6 reads as a sensible consistency check, not a fitted input. The paper also ships a Mathematica script for the entanglement coefficients, which helps reproducibility.

The soft spots are mostly presentation-level, but one sits in a load-bearing place. In Proposition 3.3, the displayed formulas (3.18)-(3.19) carry an extra factor of epsilon on the right-hand side. The Section 5 derivation actually computes (beta1 - gamma1 w) epsilon, which cancels to an O(1) numerator; as printed, b1 and b0 become O(epsilon), so the off-diagonal entries in (3.20) are O(epsilon^2) and the eigenvalue splitting in (2.37) does not follow. This is clearly a typo — the final expressions for b1,b0 in (2.36) are O(1) and match the corrected identities — but a referee will need the authors to fix it. Also, the introduction claims e(mu) is strictly positive in both regimes |mu|<<1 and |mu|>>1, yet (2.40) is negative on the phi- branch near zero; the claim should be restricted to the phi+ branch, which is enough for the theorem. On the functional-analytic side, Theorem 1 states an eigenvalue of L_epsilon when the proof actually produces a Floquet–Bloch eigenvalue with a non-L^2 Bloch wave; this is standard in the modulational-instability literature, but it should be stated explicitly to avoid overclaiming.

None of these affect the central result: the existence of a positive-real-part spectral point and the explicit growth rate stand. The paper is a real contribution and deserves a serious referee. I would send it to review, and I would cite it once the typos are corrected.

Referee Report

3 major / 3 minor

Summary. The paper studies the two-dimensional inviscid Boussinesq system and proves a modulational instability result for small-amplitude internal plane waves. The authors linearize around an exact plane-wave solution, use Floquet-Bloch decomposition, restrict to the invariant subspace of harmonics of the primary wavevector, and show that for resonant Floquet parameters a purely imaginary double eigenvalue of the unperturbed operator splits into a pair with nonzero real part. The growth rate is computed explicitly as ϵ√e(µ) with an explicit function e(µ), and its small- and large-Floquet-parameter asymptotics are given. The paper also compares the resulting rate with the physical literature on parametric subharmonic instability and includes a Mathematica code for the algebraic computations.

Significance. If the result is correct, this is the first rigorous treatment of inviscid parametric subharmonic instability for internal gravity waves, a mechanism that is widely used in oceanography and experimental fluid mechanics. The explicit, parameter-free formula for the first-order growth rate and the verification of both small- and large-Floquet-parameter regimes are concrete and falsifiable predictions. The analytic setup, including the characterization of the resonant set in Lemma 2.5 and the spectral-isolation argument in Proposition 2.7, is a nontrivial adaptation of recent Stokes-wave techniques. The availability of machine-checkable code for the entanglement coefficients is a further strength. However, as detailed below, two load-bearing points need correction: a displayed algebra error in Proposition 3.3 that temporarily breaks the proof of Theorem 2, and an overstatement in Theorem 1 about the nature of the instability (spectral versus L²-eigenvalue instability).

major comments (3)

[Proposition 3.3, Eqs. (3.18)-(3.19)] Equations (3.18) and (3.19) contain an extra factor ϵ on the right-hand side. The coefficients β1, β0, γ1 are ϵ-independent Taylor coefficients defined in (3.16), and the derivation in Section 5 (the displayed chain ending with 'proving formula (3.18)') actually computes (β1−γ1w)ϵ = ϵ/8·..., i.e. β1−γ1w = 1/8·... . As printed, the quantities b1 and b0 defined in the proof of Theorem 2 become O(ϵ), the off-diagonal entries in (3.20) become O(ϵ²), and the eigenvalue splitting ±ϵ√e(µ) in (2.37) does not follow. The final formulas (2.36) confirm that the intended identities are O(1). This is a local but load-bearing typo; it must be corrected and the surrounding computation reconciled.
[Theorem 1 and Eqs. (2.6)-(2.7)] Theorem 1 claims that the linearized operator Lϵ on L²(R²) has an unstable eigenvalue. What is actually constructed is a Bloch eigenvalue of L_{µ,ϵ} on L²(T²); the corresponding spacetime function h(t,x)=e^{λt}e^{iµx}v(x) is not square-integrable over R², as the text itself notes after (2.6)-(2.7). Thus the result establishes spectral instability (Re σ(L_{µ,ϵ})>0 for some µ, equivalently the spectrum of Lϵ intersects the right half-plane) rather than existence of an L² eigenfunction. This is a standard accepted notion in modulational-instability theory, but Theorem 1 and the abstract should state this explicitly so that the claim is not overread as an L²-instability result.
[Lemma 3.1] Lemma 3.1 is the main perturbative tool and is stated without proof. Because L0 is unbounded and the introduction emphasizes that the perturbation is not bounded, the analyticity of the spectral projectors P_{µ,ϵ} and the existence of the contour integral for small ϵ require justification. On the invariant subspace H¹_k this is straightforward: in view of (4.15), the jets L^{±k}_1 act on modes nk through the constants k⊥·µ, so L1 is actually bounded on H¹_k. Please add a proof or a precise Kato-type statement verifying the hypotheses, so that the reduction to the 2×2 matrix representation is fully justified.

minor comments (3)

[Lemma 4.6, Eq. (4.19)] The reversibility identities in (4.19) appear to be missing complex conjugation or an overline; as typeset, the first identity is tautological. Please check the intended relations.
[Appendix B, Step 2 near 0] For ℓ=−1 the leading coefficient displayed after (B.7) contains the factor m²−2n²; a parenthetical noting that this factor is nonzero for nonzero integers would remove an apparent edge case.
[General notation] The notation r(ϵⁿ) is defined in the introduction, but in (2.37) the expression ±ϵ√e(µ)+r(ϵ) could be confused with the earlier real-valued remainders r(ϵ²); a brief reminder or a different symbol would improve readability.

Circularity Check

0 steps flagged · score 0.0 of 10

No circularity: the PSI growth rate is derived from the linearized Boussinesq operator with no fitted input, and the comparison with the physics literature is a post-hoc consistency check.

full rationale

The derivation is self-contained. The instability growth rate e(µ) is computed from the linearized Boussinesq operator: Theorem 2's b1(µ) and b0(µ) in (2.36) are obtained by expanding the 2x2 matrix (3.7) using the jets (4.3)-(4.4) and the entanglement coefficients (4.18), with no fitted constants. The resonance set R_k in (2.28) is a mathematical definition, not an assumption of instability; Lemma 2.5 and Proposition 2.7 only locate where the unperturbed eigenvalue has algebraic multiplicity two, and Theorem 2 then proves e(µ)>0 along the relevant parametrizations. The comparison with Dauxois et al. in Section 6 occurs after the derivation and is a consistency check, not an input. Self-citations to [9,10,15] supply technical vocabulary (entanglement coefficients, Kato transformations, the plane-wave ansatz), but the paper proves the lemmas it uses rather than importing the conclusion by citation. The apparent extra factor ϵ in (3.18)-(3.19) is a typographical/correctness issue: the Section 5 algebra and the final O(1) expressions for b1,b0 in (2.36) show the ϵ belongs on the left-hand side of those identities, so a reader must repair the displayed formulas, but this does not make the derivation circular. No load-bearing step reduces to its own input, and no fitted parameter is later relabelled as a prediction.

Assumptions & free parameters 0 free parameters · 5 assumptions · 0 invented entities

The paper's conclusion depends on the standard Floquet-Bloch spectral characterization of periodic-coefficient operators, Kato's perturbation theory, and the physical Boussinesq model; no constants are fitted and no new physical entities are introduced.

assumptions (5)

standard math Floquet-Bloch decomposition characterizes the L^2 spectrum of periodic-coefficient operators as the union of spectra of Bloch operators L_{mu,epsilon} on the torus.
Invoked in (2.6) to reduce the analysis to L^2(T^2); standard but load-bearing for the spectral conclusion.
standard math Kato perturbation theory for isolated eigenvalues and spectral projectors applies to the unbounded operator family L_{mu,epsilon}: Y_k -> X_k.
Section 3, Lemma 3.1 constructs projectors via contour integrals of resolvents; requires the resolvent to exist on Gamma and the spectral gap, which the paper verifies for epsilon small.
domain assumption The 2D inviscid Boussinesq system (1.1) is an adequate model for internal gravity wave instability.
The entire result concerns this model; the physical relevance to oceanography is asserted in the introduction but not derived.
domain assumption The resonance condition (2.27) defines the set R_k of Floquet parameters studied; instability needs only one such parameter with e(mu)>0.
The choice is motivated by PSI/TRI (Remark 2.4). This restricts attention to a subfamily of perturbations but is sufficient for the proof of existence.
domain assumption A Bloch eigenvalue with positive real part constitutes linearized instability of the original PDE.
Section 2.1 (2.7) produces the growing solution h(t,x)=e^{lambda t}e^{i mu x}v(x), which is not square-integrable; the paper relies on the standard spectral notion of instability rather than L^2 instability.

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Pith. "Pith review of Instabilities of internal gravity waves in the two-dimensional Boussinesq system." pith.science (2026). https://pith.science/paper/WZYO52QO

@misc{pith2026250710390,
  author       = {Pith},
  title        = {Pith review of: Instabilities of internal gravity waves in the two-dimensional Boussinesq system},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/WZYO52QO}},
  note         = {Machine review of arXiv:2507.10390}
}

read the original abstract

We consider a two-dimensional, incompressible, inviscid fluid with variable density, subject to the action of gravity. Assuming a stable equilibrium density profile, we adopt the so-called Boussinesq approximation, which neglects density variations in all terms except those involving gravity. This model is widely used in the physical literature to describe internal gravity waves. In this work, we prove a modulational instability result for such a system: specifically, we show that the linearization around a small-amplitude travelling wave admits at least one eigenvalue with positive real part, bifurcating from double eigenvalues of the linear, unperturbed equations. This can be regarded as the first rigorous justification of the Parametric Subharmonic Instability (PSI) of inviscid internal waves, wherein energy is transferred from an initially excited primary wave to two secondary waves with different frequencies. Our approach uses Floquet-Bloch decomposition and Kato's similarity transformations to compute rigorously the perturbed eigenvalues without requiring boundedness of the perturbed operator - differing fundamentally from prior analyses involving viscosity. Notably, the inviscid setting is especially relevant in oceanographic applications, where viscous effects are often negligible.

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Reference graph

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Pith tools

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

[1] [1]

H. D. Abarbanel, D. D. Holm, J. E. Marsden, and T. S. Ratiu. Nonlinear stability analysis of stratified fluid equi- libria.Philosophical Transactions of the Royal Society of London. Series A, Mathematical and Physical Sciences, 318(1543):349–409, 1986

work page 1986

[2] [2]

Akylas and C

T. Akylas and C. Kakoutas. Stability of internal gravity wave modes: from triad resonance to broadband instability. Journal of Fluid Mechanics, 961:A22, 2023

work page 2023

[3] [3]

Bedrossian, R

J. Bedrossian, R. Bianchini, M. C. Zelati, and M. Dolce. Nonlinear inviscid damping and shear-buoyancy instability in the two-dimensional Boussinesq equations.Comm. Pure Appl. Math., 76(12):3685–3768, 2023

work page 2023

[4] [4]

Benielli and J

D. Benielli and J. Sommeria. Excitation and breaking of internal gravity waves by parametric instability.Journal of Fluid Mechanics, 374:117–144, 1998

work page 1998

[5] [5]

T. B. Benjamin. Internal waves of finite amplitude and permanent form.Journal of Fluid Mechanics, 25(2):241–270, 1966

work page 1966

[6] [6]

T. B. Benjamin. Instability of periodic wavetrains in nonlinear dispersive systems.Proceedings of the Royal Society of London. Series A. Mathematical and Physical Sciences, 299(1456):59–76, 1967

work page 1967

[7] [7]

T. B. Benjamin. On the Boussinesq model for two-dimensional wave motions in heterogeneous fluids.Journal of Fluid Mechanics, 165:445–474, 1986

work page 1986

[8] [8]

T. B. Benjamin and J. E. Feir. The disintegration of wave trains on deep water Part 1. Theory.Journal of Fluid Mechanics, 27(3):417–430, 1967. 22 R. BIANCHINI (†), A. MASPERO (⋆), AND S. PASQUALI (∗)

work page 1967