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Modelling intermediate internal waves with currents and variable bottom

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Pith's one-line read The paper derives a variable-coefficient Intermediate Long Wave Equation for thermocline waves under a shear current and over a slowly varying sea floor, recovering the integrable ILWE, Benjamin-Ono, and Korteweg-de Vries equations as…

desk verdict New variable-coefficient ILWE with currents and topography, but the WKB ansatz is inconsistent and the key algebra is hidden; the central result is plausible but unverified. read the letter →

arxiv 2506.10123 v1 pith:CV43DMRC submitted 2025-06-11 nlin.PS physics.flu-dyn

Rossen Ivanov , Lyudmila Ivanova This is my paper · ORCID
classification nlin.PSphysics.flu-dyn MSC 76B5576B1535Q5337K10
keywords internalwavesIntermediateLongWaveEquationDirichlet-NeumannoperatorshearcurrentvariablebottomBenjamin-OnoKorteweg-deVriesHamiltonianformulation
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 derives a single asymptotic equation for the motion of the thermocline—the interface between a deep, light upper layer and a thin, denser lower layer—when a depth-dependent current and a slowly varying sea floor are both present. The equation, (48), is a variable-coefficient Intermediate Long Wave Equation: it tracks how the wave amplitude changes as the lower-layer depth varies, how nonlocal dispersion acts through the operator $T = -i\coth(h_1 D)$, and how the quadratic nonlinearity steepens the wave. For a flat bottom it reduces to the known integrable ILWE, and in the deep-upper-layer and very-long-wave limits it reduces to the Benjamin-Ono and Korteweg-de Vries equations. The paper also derives a higher-order version (77) and shows that at a certain critical lower-layer depth the leading quadratic nonlinearity vanishes, leaving higher-order and nonlocal terms to govern the wave. If the model is right, it gives a single formula that interpolates between the standard long-wave descriptions as the thermocline depth changes.

What carries the argument

The load-bearing object is the lower-layer Dirichlet-Neumann operator, which maps the interface value of the velocity potential to the normal velocity at the interface and encodes the variable bottom through the expansion $G(b,\eta)=\delta^2 D b(X)D + \delta^3 D\eta D - \frac{\delta^4}{3}D^2 b^3(X)D^2 + O(\delta^5)$. Combined with the upper-layer operator, it converts the fluid equations into the quasi-Hamiltonian system (23)–(25). The reduction to one equation uses a slow variable $X=\varepsilon x$, a single-travelling-wave ansatz, the leading-order linear relation $u = \rho c_0(X)b(X)^{-1}\eta$, and the nonlocal operator $T = -i\coth(h_1 D)$. Compatibility of the two evolution equations fixes the constants (45)–(47) and yields the variable-coefficient ILWE (48).

What would settle it

Evolve a single ILWE soliton over a smooth bottom shoal in the full two-layer fluid equations, with the current profile (4), and measure the amplitude of the counter-propagating component of the linear system (36): if it is not at most $O(\delta)$ relative to the incident wave, the one-component ansatz behind (48) fails. A second check is to track the soliton amplitude across a shoaling lower layer and compare it with the adiabatic prediction $\eta_{\max}(b)\approx \mu_1 b^{1/4}\tan(\mu_2 b^{-7/4})$; systematic deviation would indicate neglected reflection or the birth of new solitons.

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Extended reading notes

Core claim

The central claim is that the full two-layer fluid equations with a piecewise-linear shear current and a slowly varying bottom reduce, at order $\delta = h/L$, to the one-component variable-coefficient ILWE (48) for the interfacial elevation $\eta(x,t)$. The reduction fixes the coefficient of the depth-growth term through $c_0'(X)$ and identifies the nonlocal dispersive term $T\eta_{xx}$ and the quadratic nonlinearity $\eta\eta_x$ as the three balancing effects of order $\delta$. When the bottom is flat, $b=h$, the equation becomes the integrable ILWE (52), whose one-soliton solution is (54); in the limits $T\to H$ and $kh_1\to 0$ it gives the Benjamin-Ono and KdV equations. The paper further derives the second-order ILWE (77), which contains cubic and nonlocal nonlinear terms and is not integrable.

Load-bearing premise

The derivation assumes the wave is a single component travelling in one direction, with no reflected wave of comparable size created as the bottom depth changes; if reflection or coupling between the two propagation directions is significant over the varying bottom, equation (48) is not the complete leading-order description.

Editorial extensions

If this is right

  • With a flat bottom, equation (48) becomes the integrable ILWE (52), whose one-soliton solution (54) has amplitude set by the parameter $k_0$ and width set by $h_1$.
  • In the very-long-wave limit $kh_1\to 0$, the model reduces to the KdV equation (57); in the deep-upper-layer limit $T\to H$, it reduces to the Benjamin-Ono equation.
  • Over a slowly varying bottom, mass and energy of the ILWE become adiabatic invariants, and the soliton amplitude follows $\eta_{\max}(b) \approx \mu_1 b^{1/4}\tan(\mu_2 b^{-7/4})$ as the lower layer shallows, assuming no new solitons are created.
  • At the critical lower-layer depth (51), the coefficient of $\eta\eta_x$ vanishes, so the leading-order balance is carried by the higher-order and nonlocal terms of the second-order ILWE (77).

Reading between the lines

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

  • A natural extension would be a two-component ansatz keeping both roots of the dispersion relation (38); the resulting equations would yield a reflection coefficient for internal solitons over a shoal, which the one-component model neglects.
  • The vanishing-nonlinearity condition (49) suggests a testable situation: with a shear current satisfying $\gamma_1^2 > (\rho/4\rho_1)\gamma^2$, a wave of the right speed should pass through a depth at which the $\eta\eta_x$ term disappears, exposing the higher-order and nonlocal terms.
  • Because the adiabatic amplitude law (67) assumes no new solitons are born, comparing it with the known fission behaviour of variable-bottom KdV solitons in the long-wave regime would show whether the nonlocal operator $T$ shifts the fission threshold.
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Editorial analysis

A structured set of objections, weighed in public.

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

Referee Report

3 major / 4 minor

Summary. This paper derives an asymptotic model for internal interfacial waves between two layers of inviscid, incompressible fluid, assuming a flat upper surface, a piecewise linear shear current, and a slowly varying bottom. Using a Hamiltonian formulation with Dirichlet–Neumann (DN) operators (with the bottom-dependent DN expansion taken from the authors' earlier work), the authors obtain a coupled system for the interface elevation η and tangential velocity u, then reduce it to a scalar variable-coefficient Intermediate Long Wave Equation (ILWE), equation (48). For flat bottom the equation is claimed to reduce to the known integrable ILWE, whose Benjamin–Ono and Korteweg–de Vries limits are presented. The paper also contains a higher-order ILWE for flat bottom (Appendix 1) and a discussion of adiabatic invariants for slowly varying depth.

Significance. If the central derivation were correct, the paper would provide a single asymptotic model interpolating among KdV, ILWE, and BO regimes for internal waves with currents and variable topography; the explicit reduction to the known flat-bottom integrable equations is a useful check. The authors clearly list all model coefficients and identify a physically interesting condition under which the quadratic nonlinearity vanishes. The strengths are the careful Hamiltonian/DN setup and the explicit limits. However, the validity of the main new result—the variable-coefficient ILWE (48)—is compromised by an inconsistent WKB ansatz, as detailed in the major comments; the central claim is therefore not established as it stands.

major comments (3)
  1. [Section 6, Eq. (37)] The WKB ansatz η = η0 e^{ik(x−c(X)t)}, u = u0 e^{ik(x−c(X)t)} is not self-consistent. Since X = εx, one has ∂_x e^{ik(x−c(X)t)} = ik(1 − ε t c'(X)) e^{ik(x−c(X)t)}. The O(ε t) correction is discarded when deriving the dispersion relation (38), but at times t = O(1/ε) it is O(1), the same size as the leading-order terms. A valid WKB treatment of a time-independent, slowly varying medium requires a conserved frequency ω and a slowly varying wavenumber k(X) = ω/c(X), with phase θ = ε^{-1} ∫ k(s) ds − ωt; the ansatz (37) instead fixes k and lets ω = k c(X) vary, so it does not satisfy the eikonal equation θ_t + c θ_x = 0. The secular term is therefore not a small correction but a consistency failure, and the derivation of (38), and hence of the compatibility calculation leading to (48), does not establish the claimed variable-coefficient ILWE.
  2. [Section 6, Eqs. (43)–(47)] The compatibility calculation that determines f(X), α1, and α2 is not shown; the text only states that 'the compatibility ... determines uniquely' the constants. This omission would be a minor issue if the reduction were standard, but because the substitution of (43) into (41)–(42) involves derivatives of the phase (37) with the secular term from the previous comment, the omitted algebra cannot be checked independently. The authors should either provide the full calculation or specify the precise assumptions under which (43)–(47) follow.
  3. [Section 6, Eq. (39)] The quadratic dispersion relation (38) admits two roots, corresponding to left- and right-travelling waves. The single-component ansatz (37) selects one branch and implicitly neglects reflected waves and coupling between the two modes, which a slowly varying bottom generically generates. No estimate of the reflected wave amplitude or a unidirectionality argument is provided. Even if the eikonal issue were repaired, such an estimate would be necessary to justify that (48) is a complete leading-order description over variable topography.
minor comments (4)
  1. [Section 3 and Section 5] There are several typographical errors: 'simmilar' should be 'similar' (Section 3), 'parametetr' should be 'parameter', 'neigbourhood' should be 'neighbourhood', and 'the 5-th assumption' should be 'the fifth assumption' (Section 5).
  2. [Section 6, Eq. (48)] The notation for the operator T = −i coth(h1D) is introduced just before (32), but its action on functions of x is not explicitly defined; a brief reminder (e.g., via Fourier multiplier) would help readers not familiar with this convention.
  3. [Appendix 1, Eqs. (70)–(75)] The coefficients of the higher-order ILWE are listed without any derivation or reference to a computer-algebra script. Since this is a claimed new result, providing at least a sketch of the matching or stating that the calculation is symbolic and available upon request would improve reproducibility.
  4. [Section 8, Eqs. (59)–(60)] The derivation of the transformed equation for E(x,t) = √c(X) η is not detailed; in particular, the treatment of the X-dependence of c in the O(δ) terms should be stated explicitly, because ∂_x(√c) is O(ε) and hence of the same order as the retained corrections.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the variable-coefficient ILWE is a genuine asymptotic reduction from stated assumptions and prior independent DN expansions.

full rationale

The paper's central result, equation (48), is obtained by a standard asymptotic reduction from the Hamiltonian system (34)-(35), which are derived from the physical setup and the Dirichlet-Neumann operator expansions. The DN expansions in (26) and (29) are quoted from earlier works, including the authors' own [29], but that prior work is an independent published mathematical derivation that does not presuppose equation (48); it is not fitted to the present result and is externally verifiable. The derivation of (48) proceeds by solving the leading-order linear system, posing the correction ansatz (43), and imposing compatibility to determine f(X), alpha1, and alpha2. No parameter is fitted to data, and no step defines the predicted quantity in terms of itself. Self-citations appear only for standard Hamiltonian formulations, DN-operator techniques, and known ILWE/BO/KdV results, none of which are load-bearing in a circular sense. The flat-bottom limit correctly reduces to the known integrable ILWE, providing an external consistency check. Thus the derivation is self-contained with respect to circularity, regardless of any questions about WKB validity or completeness of the ansatz.

Assumptions & free parameters 0 free parameters · 6 assumptions · 0 invented entities

The derivation rests on the Hamiltonian/DN operator formalism and the scaling assumptions inherited from prior work by the same group and others. These are unproved inputs for this paper; they are standard in the field, but the paper provides no independent verification.

assumptions (6)
  • domain assumption The Hamiltonian formulation (23)-(24) with the DN operators G and G1 correctly describes the dynamics of the interface η and momentum ξ for the piecewise linear current profile (4).
    This is the starting point of the derivation, taken from prior work [9,19]; the paper does not re-derive it but relies on its correctness.
  • standard math The Dirichlet-Neumann operator expansion G(b,η) = δ^2 D b D + δ^3 D η D - (δ^4/3) D^2 b^3 D^2 + O(δ^5) for a slowly varying bottom (equation 28).
    Quoted from [29]; it is used to compute the Hamiltonian to the relevant order and is central to the variable-bottom terms.
  • standard math The upper-layer DN operator expansion G1(η) = δ D tanh(h1 D) - δ^3 D η D + δ^3 D tanh(h1 D) η D tanh(h1 D) + O(δ^5) (equation 29).
    Standard expansion from [19] for a flat, deep upper layer.
  • domain assumption The scaling assumptions in Section 5: ε=O(δ), h/h1=O(δ), h1 k=O(1), ξ=O(1), |β|max << ε^{1/3}, β'(x)=O(ε).
    These delimit the intermediate long-wave regime and justify the small-amplitude, slowly-varying-bottom approximations.
  • domain assumption The energy of the bottom boundary layer of the current profile (4) is constant and does not contribute to the interfacial dynamics, as stated in [8].
    Borrowed from [8]; needed for the Hamiltonian to depend only on η and ξ.
  • domain assumption The commutator [D,b(X)] = -i b'(X) is O(ε) and can be neglected at the orders retained in the expansion.
    Follows from the slow variation of the bottom and is used repeatedly to commute b(X) with D.

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Pith. "Pith review of Modelling intermediate internal waves with currents and variable bottom." pith.science (2026). https://pith.science/paper/CV43DMRC

@misc{pith2026250610123,
  author       = {Pith},
  title        = {Pith review of: Modelling intermediate internal waves with currents and variable bottom},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/CV43DMRC}},
  note         = {Machine review of arXiv:2506.10123}
}
read the original abstract

A model for internal interfacial waves between two layers of fluid in the presence of current and variable bottom is studied in the flat-surface approximation. Fluids are assumed to be incompressible and inviscid. Another assumption is that the upper layer is considerably deeper with a lower density than the lower layer. The fluid dynamics is presented in Hamiltonian form with appropriate Dirichlet-Neumann operators for the two fluid domains, and the depth-dependent current is taken into account. The well known integrable Intermediate Long Wave Equation (ILWE) is derived as an asymptotic internal waves model in the case of flat bottom. For a non-flat bottom the ILWE is with variable coefficients. Two limits of the ILWE lead to the integrable Benjamin-Ono and Korteweg-de Vries equations. Higher-order ILWE is obtained as well.

Figures

Figures reproduced from arXiv: 2506.10123 by the authors.

Figure 1
Figure 1. Coordinates and sketch of the fluid domain. The flat surface is given by [PITH_FULL_IMAGE:figures/full_fig_p004_1.png] view at source ↗
Figure 2
Figure 2. The dependence (67) with µ1 = 1, µ2 = π/4. Perhaps a more accurate estimate for ηmax(b) can be obtained from the energy consid￾eration. In the long-wave regime this is reported, for example, in [39]. 9 Conclusions We have presented a study of the propagation of interfacial internal waves in the case when the lower layer is significantly thinner than the upper layer. The flat surface approximation 17 [PITH_FULL_IMAG… view at source ↗

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    1968 doi

Pith tools

Reviewed August 7, 2026 · model on record in the stance chip above.