{"id":"d5e796a7-ac6b-4d9d-ba47-5affd4f91385","arxiv_id":"2506.10123","paper_version":1,"verdict":"CONDITIONAL","confidence":"MODERATE","novelty_score":6.0,"correctness_risk":"medium","formal_verification":"none","parameter_count":0,"one_line_summary":"A new asymptotic model, the variable-coefficient Intermediate Long Wave Equation, is derived for interfacial waves with shear currents and a slowly varying bottom, with higher-order corrections and a critical-depth condition.","lead":"The paper derives a variable-coefficient Intermediate Long Wave Equation that models internal waves on the thermocline when a shear current and a slowly varying seafloor are present. This gives oceanographers a unified model for intermediate-length internal waves, connecting the previously separate flat-bottom and variable-bottom cases.","discovery_kind":"new_application","skeptic_critique":{"model":"deepseek-v4-flash","headline":"The WKB ansatz (37) is not self-consistent: phase derivatives generate secular terms, so the derivation of (48) is not a valid asymptotic reduction.","rationale":"The paper has genuine strengths: the Hamiltonian setup with Dirichlet–Neumann operators is standard, the flat-bottom reduction (52) to the known ILWE appears correct for the no-current case, and the adiabatic-invariant analysis in Section 8 is a reasonable application of the derived model. The reader's CONDITIONAL verdict focused on the unverified algebra and the neglect of reflected waves. My stress-test identifies a more fundamental and concrete flaw: the WKB ansatz (37) is not self-consistent because it treats c(X) as slowly varying in the phase but does not include the phase corrections required by a valid WKB expansion. The resulting secular term is of the same order as the corrections being calculated, so the derivation of (48) is not a rigorous asymptotic reduction. This is a load-bearing concern because it affects the linear leading-order propagation itself, not just the O(δ) corrections. The concrete test—evaluating the residual of (37) in (36) and then redoing the analysis with the proper eikonal—would settle whether the final equation (48) is correct or whether the bottom-variation coefficient must be modified. Until that check is done, the central claim is unverified, hence UNVERDICTED rather than ACCEPT or CONDITIONAL.","tokens_in":15437,"tokens_out":25930,"duration_ms":280103,"concrete_test":"Substitute the ansatz (37) into the linear part of the leading-order system (36), retaining the full x-derivative of the phase: for η = η0 e^{ik(x−c(X)t)}, compute η_t = −ikc η and η_x = ik(1 − ε t c'(X)) η. Evaluate the residual R = η_t + c η_x + (b/ρ) u_x (and the analogous u-equation). Show that R contains a term −iε t k c'(X) c η that is O(ε t) and does not vanish for t = O(1/ε), indicating the ansatz is not a leading-order solution. Then re-derive the leading-order dispersion and transport using the proper WKB phase θ = ε^{-1}∫ k(s) ds − ωt with ω constant and k(X) = ω/c(X), together with a slowly varying amplitude A(X). Compare the resulting variable-coefficient term with (48); if the coefficient of the bottom term changes, the central claim fails.","verdict_should_be":"UNVERDICTED","load_bearing_attack":"The central result (48) is obtained from the WKB ansatz (37), η = η0 e^{ik(x−c(X)t)}, u = u0 e^{ik(x−c(X)t)}, with X = εx and slowly varying c(X). In the leading-order system (36), the x-derivative of the phase must be evaluated consistently: ∂_x e^{ik(x−c(X)t)} = ik(1 − ε t c'(X)) e^{...}. The extra term −iε t k c'(X) is O(ε) and grows linearly in t, becoming O(1) at t ~ 1/ε. This term is not balanced by any phase or amplitude correction in (37). Consequently, the phase does not satisfy the eikonal equation θ_t + c θ_x = 0 beyond t = 0, and the leading-order dispersion relation (38) is derived by discarding a term of the same order as the corrections being sought. A valid WKB for a time-independent 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 keeps k constant and lets ω = k c(X) vary, which is inconsistent with frequency conservation. This is not a mere bookkeeping issue: it undermines the definition of the propagation speed in (38) and the compatibility calculation that yields (48). Therefore, the derivation presented does not establish the variable-coefficient ILWE.","agreement_with_reader":"partial"},"referee_report":{"model":"deepseek-v4-flash","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.","tokens_in":15732,"tokens_out":8301,"duration_ms":95109,"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":[{"comment":"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.","section":"Section 6, Eq. (37)"},{"comment":"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.","section":"Section 6, Eqs. (43)–(47)"},{"comment":"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.","section":"Section 6, Eq. (39)"}],"minor_comments":[{"comment":"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).","section":"Section 3 and Section 5"},{"comment":"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.","section":"Section 6, Eq. (48)"},{"comment":"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.","section":"Appendix 1, Eqs. (70)–(75)"},{"comment":"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.","section":"Section 8, Eqs. (59)–(60)"}],"recommendation":"major_revision","confidential_remarks":"The paper relies heavily on the authors' previous work [29] for the DN expansion, and the referee found no evidence of missing attribution. The stress-test concern about the WKB ansatz is, in my assessment, correct and load-bearing; the authors should be asked to repair the derivation before the paper can be considered for publication."},"author_rebuttal":null,"desk_editor":{"model":"deepseek-v4-flash","letter":"First, the important thing: the variable-coefficient ILWE, equation (48), is a genuinely new combination of the intermediate long-wave regime with a shear current and a slowly varying bottom. The paper also gets the flat-bottom limit right and shows how the BO and KdV equations emerge in the short-wave and long-wave limits. The critical-depth condition (49)-(51) is a nice touch. Worth paying attention to.\n\nBut the derivation has a real problem. The ansatz (37) uses η = η0 e^{ik(x−c(X)t)} with X = εx. The x-derivative of this phase produces a term proportional to ε t c′(X), so the phase does not satisfy the eikonal equation θ_t + c(X) θ_x = 0 away from t=0. In a time-independent medium, the frequency should be conserved and the wavenumber should vary; fixing k while letting c(X) vary is not a consistent WKB. This is not a bookkeeping issue. The unwanted term is the same order as the corrections the authors compute, and it grows linearly in time. The compatibility calculation that gives (48) is therefore built on an inconsistent leading-order solution.\n\nOn top of that, the key algebra is mostly hidden. The coefficients f, α1, α2 in (43) and the whole of Appendix 1 are stated without derivation. A reader cannot check the central equation without reproducing a substantial computation by hand.\n\nI should say the result might still be right. A proper multiple-scales derivation with a phase integral S(X)/ε − ωt, or a slow-time variable, could very well produce something equivalent to (48). The paper does not supply that, and the gap is structural rather than cosmetic.\n\nThe audience is people working on internal waves over topography who want a single model spanning the ILWE–BO–KdV regimes. They will find the paper useful for orientation, but they should not yet rely on (48). I would not cite it as a proven equation in my own work.\n\nFor peer review: yes, send it out, but with a request for a major revision that (a) replaces the inconsistent WKB ansatz with a well-posed multiple-scales derivation or provides a rigorous justification, and (b) includes the missing algebra, ideally supplemented by a symbolic check. As it stands, it is a promising idea with a shaky proof.","headline":"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.","tokens_in":16229,"tokens_out":5730,"would_cite":false,"duration_ms":66997,"reading_group":"maybe","serious_thinker":"yes","would_accept_peer_review":true},"rs_alignment":null,"lean_confirmation":null,"pith_extraction":{"msc":["76B55","76B15","35Q53","37K10"],"pacs":[],"model":"deepseek-v4-flash","headline":"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…","keywords":["internal waves","Intermediate Long Wave Equation","Dirichlet-Neumann operator","shear current","variable bottom","Benjamin-Ono equation","Korteweg-de Vries equation","Hamiltonian formulation"],"falsifier":"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.","tokens_in":15251,"feed_emoji":"🌊","tokens_out":12383,"duration_ms":121744,"temperature":0.7,"pith_summary":"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.","feed_headline":"Variable-bottom internal waves collapse into one long-wave equation","feed_subtitle":"Asymptotic model reduces to the integrable ILWE and to the Benjamin-Ono and KdV limits.","key_machinery":"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).","core_discovery":"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.","pith_inferences":["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."],"forward_implications":["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)."],"supporting_citations":[{"why":"Provides the expansion of the lower-layer Dirichlet-Neumann operator for a variable bottom, which is the starting point for the Hamiltonian (24) and for equation (48).","marker":"[29]"},{"why":"Establishes the Dirichlet-Neumann operator and Hamiltonian formulation for interfacial waves that the derivation uses.","marker":"[19]"},{"why":"Gives the flat-surface quasi-Hamiltonian system with piecewise-linear shear current that the variable-bottom case extends.","marker":"[9]"},{"why":"Derives the constant-depth ILWE with currents and its soliton, Benjamin-Ono, and KdV limits, which equation (48) generalises to a variable bottom.","marker":"[22]"},{"why":"Supplies the Hamiltonian model for wave-current interactions in the long-wave regime whose KdV speed is compared with the present model's leading-order speed.","marker":"[28]"},{"why":"Motivates the background current profile $U(z)$ used for the Equatorial Undercurrent, connecting the setup to oceanographic conditions.","marker":"[8]"}],"fun_headline_variants":["Internal waves with current and bottom unify into one equation","From two-layer flow to integrable ILWE and its limits","Variable-bottom reduction yields single long-wave equation","All internal wave regimes collapse to one ILWE family","One model for internal waves: ILWE, BO, and KdV in limits"],"cache_read_input_tokens":3200,"weakest_assumption_plain":"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.","fun_headline_variants_meta":{"raw":{"variants":["Internal waves with current and bottom unify into one equation","From two-layer flow to integrable ILWE and its limits","Variable-bottom reduction yields single long-wave equation","All internal wave regimes collapse to one ILWE family","One model for internal waves: ILWE, BO, and KdV in limits"]},"model":"deepseek-v4-flash","effort":"low","cost_usd":0.000994,"raw_usage":{"total_tokens":4172,"prompt_tokens":865,"completion_tokens":3307,"prompt_tokens_details":{"cached_tokens":384},"prompt_cache_hit_tokens":384,"prompt_cache_miss_tokens":481,"completion_tokens_details":{"reasoning_tokens":3224}},"tokens_in":481,"tokens_out":3307,"duration_ms":26034,"temperature":1.0,"reasoning_tokens":3224,"cache_read_input_tokens":384,"cache_creation_input_tokens":0},"cache_creation_input_tokens":0},"created_at":"2026-08-07T04:34:02.096388+00:00","model_set":{"reader":"deepseek-v4-flash"},"falsifier":"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.","supporting_citations":[{"cited_title":"Hamiltonian approach to modelling interfacial internal waves over variable bottom","cited_arxiv_id":"2203.02590","evidence_quote":"Provides the expansion of the lower-layer Dirichlet-Neumann operator for a variable bottom, which is the starting point for the Hamiltonian (24) and for equation (48)."},{"cited_title":"Craig, P","cited_arxiv_id":null,"evidence_quote":"Establishes the Dirichlet-Neumann operator and Hamiltonian formulation for interfacial waves that the derivation uses."},{"cited_title":"The Dynamics of Flat Surface Internal Geophysical Waves with Currents","cited_arxiv_id":"1611.06581","evidence_quote":"Gives the flat-surface quasi-Hamiltonian system with piecewise-linear shear current that the variable-bottom case extends."},{"cited_title":"On the intermediate long wave propagation for internal waves in the presence of currents","cited_arxiv_id":"2007.04375","evidence_quote":"Derives the constant-depth ILWE with currents and its soliton, Benjamin-Ono, and KdV limits, which equation (48) generalises to a variable bottom."},{"cited_title":"Hamiltonian model for coupled surface and internal waves in the presence of currents","cited_arxiv_id":"1702.01441","evidence_quote":"Supplies the Hamiltonian model for wave-current interactions in the long-wave regime whose KdV speed is compared with the present model's leading-order speed."},{"cited_title":"On the dynamics of internal waves interacting with the equatorial undercurrent","cited_arxiv_id":"1510.04096","evidence_quote":"Motivates the background current profile $U(z)$ used for the Equatorial Undercurrent, connecting the setup to oceanographic conditions."}],"review_version":1}