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Three-layer water flows: Dirichlet-Neumann operators and approximations

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Pith's one-line read The full nonlinear three-layer water-wave equations admit a Hamiltonian form in six surface variables.

desk verdict A solid formal extension of the Craig et al. Hamiltonian machinery to three-layer free-surface flows, with a repairable gap in the operator inversion that needs a function-space fix. read the letter →

arxiv 2608.06314 v1 pith:EDIXXWZX submitted 2026-08-06 physics.flu-dyn math-phmath.MP

Rossen Ivanov , Calin-Iulian Martin This is my paper · ORCID
classification physics.flu-dynmath-phmath.MP MSC 35Q3135Q35
keywords three-layerwaterwavesHamiltonianformulationDirichlet-NeumannoperatorsinternaldispersionrelationBoussinesqapproximationrigidlidmodelstratifiedflows
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

Three-layer water flows — a free surface, two internal interfaces, a flat bottom, and constant density in each layer — obey strongly coupled nonlinear equations. The paper claims that, for irrotational flow, these equations are exactly equivalent to a Hamiltonian system in six one-dimensional variables: the three elevations $\eta_1,\eta_2,\eta_3$ and three momenta $\xi_1,\xi_2,\xi_3$ built from density-weighted traces of the velocity potentials. The Hamiltonian is written explicitly using Dirichlet-Neumann operators for each layer, so all information about the fluid interior is condensed into boundary quantities. From this formulation the paper derives the linear dispersion relation, analyzes its long-wave limit where the squared wave speed $c^2$ satisfies a bi-cubic equation, and provides bounds and approximate formulas for the six real propagation speeds. A sympathetic reader would care because this gives one energy-based framework from which surface and internal wave models — including rigid-lid, Boussinesq, and two-layer limits — follow systematically.

What carries the argument

The carrying object is the Dirichlet-Neumann operator for each layer: the boundary map that sends a harmonic function's trace on a layer's boundary to its normal derivative, with the surface geometry encoded in the elevations. The top and middle layers each get a $2\times2$ operator matrix and the bottom layer a single operator; their leading Fourier-multiplier symbols are $k\coth(kh_i)$, $k\,\mathrm{csch}(kh_i)$ and $k\tanh(kh_b)$. The algebraically central object is the operator $D=(\rho_2 G+\rho_3\Gamma_{22})\Gamma_{12}^{-1}(\rho_2G_{11}+\rho_1\Gamma_{11})-\rho_1\rho_3\Gamma_{21}$, whose invertibility lets the five potential traces be written as functions of $(\eta_i,\xi_i)$; this converts the total energy into the explicit Hamiltonian (3.33). The paper then uses systematic Taylor expansions of the DN operators in the elevations to move from the nonlinear Hamiltonian equations to the linear dispersion relation and, in the rigid-lid case, to a Boussinesq system.

What would settle it

Compute the symbol of $D$ for a monochromatic perturbation of wavenumber $k$ and search over admissible densities and depths for a zero eigenvalue; if one exists, the closed Hamiltonian (3.33) fails there. Alternatively, numerically solve the full linear system (4.11)-(4.12) across a random sample of ocean- and lake-like parameters and compare with the approximate speeds (5.23)-(5.27); a parameter region with relative error exceeding a few percent would delimit the approximations' validity.

Watch

Extended reading notes

Core claim

The central claim is a Hamiltonian reformulation of the full nonlinear three-layer irrotational water-wave problem. With elevations as coordinates and momenta defined by $\xi_1=\rho_1\Phi_1^1$, $\xi_2=\rho_2\Phi_2^2-\rho_1\Phi_1^2$, $\xi_3=\rho_3\Phi^3-\rho_2\Phi_2^3$, the authors show that the Euler equations and all boundary conditions are equivalent to $\delta H/\delta \eta_i=-\xi_{i,t}$ and $\delta H/\delta \xi_i=\eta_{i,t}$ for $i=1,2,3$. The kinetic energy is expressed through three-layer Dirichlet-Neumann operators; inverting an operator $D$ expresses the five velocity-potential traces in terms of the three momenta, yielding a closed Hamiltonian $H(\eta_i,\xi_i)$ with coefficients $A_{ij}$ built from the DN operators. Linearization about the rest state produces a $6\times 6$ system whose characteristic equation is the dispersion relation; in the long-wave limit it becomes a cubic in $c^2$, whose six real roots are the three right-moving and three left-moving wave speeds. The same DN machinery, with the upper surface fixed flat, yields the rigid-lid system, its bi-quadratic dispersion relation, a coupled Boussinesq system, and the two-layer free-surface limits.

Load-bearing premise

The derivation assumes that the operator $D$ used to solve for the velocity-potential traces can be inverted on the spaces of allowed surface elevations and densities; the paper does not prove this, and if $D$ degenerates for some configuration the explicit closed-form Hamiltonian is not valid.

Editorial extensions

If this is right

  • A single energy functional now generates the full nonlinear motion of the free surface and both interfaces, so no separate interface-by-interface derivation is needed.
  • Linearizing the Hamiltonian equations gives the dispersion relation at arbitrary wavelength, and its long-wave limit is a bi-cubic equation in $c^2$ with six real roots: three right-moving and three left-moving modes with speeds of different orders.
  • The bounds and approximations in Section 5 give practical estimates of the three positive speeds directly from densities and layer thicknesses, and they match numerical roots for ocean, lake, and laboratory parameter sets.
  • In the rigid-lid case the same DN formalism yields a bi-quadratic dispersion relation whose two positive speeds have opposite interface polarities, and a Boussinesq system follows by expanding the operators.
  • Setting $\rho_3=\rho_2$ or $\rho_1=0$ recovers the known two-layer free-surface dispersion relation and Boussinesq system, so the three-layer model contains earlier two-layer models as limits.

Reading between the lines

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

  • The six-variable reduction likely extends to $N$ layers by continuing the density-weighted momentum definition, since equations (3.38)-(3.51) already display a recursive structure; this is an extrapolation, not proven here.
  • Because the Boussinesq and rigid-lid systems are obtained by expanding DN operators inside a Hamiltonian framework, one would expect them to inherit exact energy conservation; a numerical check of conserved quantities would be a direct test.
  • The approximate formulas (5.23)-(5.27) could be used to predict the three mode speeds in a laboratory three-layer tank, and the predicted polarity relation between interface displacements is a concrete observable signature.
  • A spectral check of $D$ for the extreme-density laboratory cases in Table 1 would show whether the closed-form Hamiltonian (3.33) remains valid outside the oceanographic parameter range.
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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

2 major / 4 minor

Summary. The paper studies two-dimensional, inviscid, incompressible, irrotational water waves in a three-layer fluid with a free surface, two interfaces, and a flat bottom. The main claim is that the full nonlinear governing equations admit a Hamiltonian formulation in six variables: the three surface/interface elevations and three momenta obtained from traces of the velocity potentials, with the Hamiltonian expressed through Dirichlet-Neumann operators for each layer. From this formulation the paper derives the linear dispersion relation, analyzes the long-wave limit as a bi-cubic equation for the squared wave speeds, gives bounds and approximate formulas for the six real speeds, and then treats the rigid-lid model, including its linear dispersion relation, the long- and short-wave limits, and a Boussinesq approximation. Limits to two-layer models are recovered in several places and compared with the literature.

Significance. If the technical gaps discussed below are closed, this paper would be a useful and fairly comprehensive reference: it extends the two-layer Hamiltonian/DN-operator framework of Craig, Guyenne and Kalisch to the three-layer free-surface case, provides explicit dispersion relations and root bounds, and derives a Boussinesq system for the rigid-lid model. The algebraic derivations are largely explicit, the two-layer limits are checked against known formulas, and the approximate speed formulas are tested against numerical roots in Table 1. The paper is formal-analytical rather than numerical or rigorous in its functional-analytic aspects, but the scope and the systematic use of DN operators are genuine strengths.

major comments (2)
  1. [§3.4, Eqs. (3.28)–(3.32)] The reduction from the five boundary traces to the three momenta ξ_i is the load-bearing step of the Hamiltonian formulation, but the derivation uses the inverses G^{-1}(η3), G11^{-1}, Γ12^{-1}, Γ21^{-1}, and D^{-1} without stating hypotheses on the elevations, densities, or function spaces. The bottom-layer DN operator has leading symbol k tanh(k h_b), so G(η3) annihilates constants and is not invertible on all of S(R); the zero-mean condition on the η_i does not by itself put the argument Γ21(φ2)_{s2}+Γ22(φ2)_{s3} into the range of G(η3). Thus the closed forms (3.31)–(3.34) are formal, and the claimed equivalence between the Euler system and the six-variable Hamiltonian system is not established for configurations near k=0 or when the relevant traces fall outside the domains of the inverses. This issue appears repairable with a suitable zero-mean Sobolev setting and a verification of the range conditions, but as written it is a gap in the central claim.
  2. [§5.1, Eq. (5.1) and Proposition 5.1] The long-wave analysis treats the cubic P(X) as having three real positive roots: the bounds in (5.20) and the approximate speeds (5.23), (5.26), and (5.27) all rely on that assumption. The paper cites [2,4,70] for the reality of the roots, and Proposition 5.1 only proves that any real roots lie in (0, gH), not that three real roots exist. Since the dispersion relation is central and Table 1 compares approximate formulas with the numerical roots of (5.1), the proof should be supplied (for example via a discriminant computation) or the precise hypotheses under which the cited results apply should be stated.
minor comments (4)
  1. [§3.4, after Eq. (3.18)] The unit normal n1 is said to be attached to the surface y=-h1+η1(x,t), but the free surface is defined in (2.1) as y=h1+η1(x,t); this is a typo in the definition of n1.
  2. [Eq. (3.9)] The variation of the trace Φ3 is written without its left-hand side; it should read δΦ3 = (φ3,y)_{s3} δη3 + (δφ3)_{s3}.
  3. [§3.4 and Remark 4.1] The operator D introduced in (3.31) for the inverse of the 2×2 system is distinct from the Fourier multiplier D=-i∂x defined in (4.4); using different symbols for these two objects would avoid a confusing collision of notation.
  4. [Throughout] There are several typographical slips, including “Bousinessq” in Section 1 and some nonstandard hyphenation; these should be corrected during editorial processing.

Circularity Check

0 steps flagged · score 0.0 of 10

No significant circularity: the Hamiltonian formulation is derived from the Euler equations, and the wave speeds are analytical solutions of the derived dispersion relation rather than fitted inputs.

full rationale

The central claim—the Hamiltonian formulation of the three-layer Euler equations—is established by direct computation: the variations of the energy functional H are evaluated using the divergence theorem and the chain rule, giving (3.13)–(3.15) and (3.16), which reproduce the evolution equations (2.18)–(2.20) and the kinematic conditions. No result is assumed from the conclusion; the canonical variables (η_i, ξ_i) are introduced by the standard Benjamin–Bridges/Craig–Guyenne–Sulem Legendre transform, and the reduction of the five potential traces to three momenta in (3.31)–(3.32) is an algebraic solution of the operator system (3.30), not an input. The dispersion relation is obtained by linearizing the derived Hamiltonian equations and diagonalizing the resulting matrix M(k); the approximate speed formulas (5.23), (5.26), (5.27) are analytic solutions of the derived bi-cubic (5.1) under explicit dominance assumptions, and Table 1 compares them with numerical roots of the same equation—an internal consistency check, not a fitted prediction. The paper cites the authors' prior work ([19], [42]) for standard Dirichlet–Neumann operator expansions and for a standard linear ODE solution formula, but these citations are not load-bearing for the Hamiltonian equivalence or the dispersion relation; no uniqueness claim from self-citation is used to exclude alternatives. The unproved invertibility of the operators G(η3), Γ12, D in (3.28)–(3.31) is a well-posedness and domain gap, not a circular step, since the algebraic reduction would be valid wherever the inverses exist. No equation is defined in terms of the result it is asked to predict.

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

The paper introduces no new physical entities or free parameters. All parameters are physical inputs (densities, depths, gravity). The main assumptions are the standard modeling assumptions of irrotational, inviscid, piecewise constant density flow, plus technical assumptions about the invertibility of a key operator and the reality of the roots of the dispersion relation, which are not fully proved in the paper.

assumptions (5)
  • domain assumption The flow is irrotational in each layer (equation 2.9).
    This allows the introduction of velocity potentials phi_i in each layer, which is essential for the Hamiltonian formulation.
  • domain assumption Densities are constant in each layer and satisfy rho1 < rho2 < rho3.
    This models stable stratification and is used to determine the signs of coefficients in the dispersion relation and the bounds.
  • domain assumption The surface elevations eta_i are Schwartz functions with zero mean (equation 2.2).
    This technical condition ensures the Fourier transform and the Hamiltonian framework are well defined.
  • domain assumption The operator D in (3.31) is invertible on the relevant function spaces.
    The paper solves for the traces (phi2)_s2 and (phi2)_s3 using D^{-1} without proving invertibility.
  • domain assumption All roots of the bi-cubic (5.1) are real, as claimed by references [2,4,70].
    The reality of the six wave speeds is essential for the bounds in Proposition 5.1 and the approximate formulas; the paper does not provide a proof.

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Cite this review

Pith. "Pith review of Three-layer water flows: Dirichlet-Neumann operators and approximations." pith.science (2026). https://pith.science/paper/EDIXXWZX

@misc{pith2026260806314,
  author       = {Pith},
  title        = {Pith review of: Three-layer water flows: Dirichlet-Neumann operators and approximations},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/EDIXXWZX}},
  note         = {Machine review of arXiv:2608.06314}
}
read the original abstract

The object of investigation in this paper are the nonlinear equations of motion for two-dimensional inviscid water flows with piecewise constant density stratification in a three-layer fluid with a flat bottom, a free surface and two interfaces. We establish a Hamiltonian formulation for the nonlinear governing equations in this setup. The Hamiltonian of the system and the equations of motion of the surface and of the interfaces are expressed with the help of the Dirichlet-Neumann (DN) operators, which are introduced for each of the layers. Then, the linear equations for small amplitudes of the elevation of the surface and of the interfaces in the leading order are derived from which a bi-cubic equation for the dispersion relation is obtained, whose solutions are analysed. The six real solutions for the possible propagation speeds (three positive, related to right-moving waves and three negative, related to left-moving waves) have magnitudes of different order. Upper and lower bounds for the previously mentioned roots are also given in terms of the coefficients of the equation. Subsequently, approximate formulae for the propagation speeds are derived. The importance of the DN operators is further illustrated in a separate analysis of the three-layer model with flat surface (rigid lid). The full nonlinear evolution equations are expressed again in terms of the DN operators, and the equations in the linear regime and the weakly nonlinear propagation regime (the Boussinesq approximation) are derived by a proper expansion of the DN operators. Limits to the two-layer free surface model are obtained as well. The obtained results are applicable to internal waves in lakes and in the ocean as well as to laboratory experiments with three superimposed fluid layers.

Figures

Figures reproduced from arXiv: 2608.06314 by the authors.

Figure 1
Figure 1. The system under study. Ω1 := {(x, y, t) : x ∈ R, t ∈ R, η2(x, t) < y < h1 + η1(x, t)}, Ω2 := {(x, y, t) : x ∈ R, t ∈ R, −h3 + η3(x, t) < y < η2(x, t)}, Ω3 := {(x, y, t) : x ∈ R, t ∈ R, −h < y < −h3 + η3(x, t)}, (2.1) where h1, h3, h are positive constants such that h > h3 and ηi(x, t) ∈ S(R), for i = 1, 2, 3, are Schwartz functions such that for all i = 1, 2, 3 the relations Z R ηi(x, t) dx = 0 for all t, (2.2) hol… view at source ↗
Figure 2
Figure 2. Relations between the derived models: limits and approx￾imations. nonlinear way, however, as we have mentioned, there is a systematic approach for their derivation. In the Boussinesq limit the above system produces the system (7.72), see the derivation in [26], although we have obtained it as a limit from (7.71) [PITH_FULL_IMAGE:figures/full_fig_p039_2.png] view at source ↗

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