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arxiv: 2605.13654 · v1 · submitted 2026-05-13 · ⚛️ physics.flu-dyn · nlin.CD· physics.ao-ph· physics.class-ph

Recognition: 2 theorem links

· Lean Theorem

Free-surface deformations induced by three-dimensional turbulence

Micha\"el Berhanu , Eric Falcon
Authors on Pith no claims yet

Pith reviewed 2026-05-14 17:56 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn nlin.CDphysics.ao-phphysics.class-ph
keywords free-surface turbulencesurface deformationslinear response modelpower-law spectrapassive responsewave-turbulent dampinghomogeneous isotropic turbulence
0
0 comments X

The pith

Free-surface deformations scale linearly with subsurface turbulent velocity fluctuations.

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

Experiments in a jet-forced tank show that three-dimensional homogeneous isotropic turbulence deforms the free surface above it. The standard deviation of surface elevation increases directly in proportion to the strength of velocity fluctuations measured just below the surface. Both wavenumber and frequency spectra of the surface height follow a power-law decay with the same exponent near -2.5, which the authors link to simple advection of turbulent structures. A linear model that treats the surface as responding passively to pressure fluctuations from below, while including damping from waves and turbulence, reproduces the main spectral shapes including a -7/3 exponent in the joint spectrum. The findings apply in the regime where turbulence does not yet break the surface.

Core claim

The root-mean-square amplitude of free-surface deformations grows linearly with the root-mean-square subsurface velocity fluctuations. Wavenumber and frequency spectra of surface elevation both decay as power laws with exponent -2.5, consistent with advection of turbulent structures. A linear transfer function from subsurface pressure to surface height, incorporating wave-turbulent damping, predicts the observed spatiotemporal spectrum shape with exponent -7/3 and establishes that the passive response mechanism dominates over any active wave generation.

What carries the argument

The linear transfer function relating surface elevation to subsurface turbulent pressure fluctuations with wave-turbulent damping.

Load-bearing premise

The surface response remains linear and passive to subsurface pressure fluctuations, valid only when turbulent velocities stay below the surface-breaking threshold.

What would settle it

Measure surface deformation amplitude at turbulent intensities approaching the surface-breaking threshold and test whether the linear scaling with subsurface velocity fluctuations continues without deviation.

Figures

Figures reproduced from arXiv: 2605.13654 by Eric Falcon, Micha\"el Berhanu.

Figure 1
Figure 1. Figure 1: FIG. 1. (a) Schematic diagram of the experimental setup used to produce bulk turbulence, with almost no mean flow, within a [PITH_FULL_IMAGE:figures/full_fig_p003_1.png] view at source ↗
Figure 2
Figure 2. Figure 2: FIG. 2. Typical surface deformations: (Left) for gentle turbulence showing surface scarification and upwellings as imprints of [PITH_FULL_IMAGE:figures/full_fig_p003_2.png] view at source ↗
Figure 3
Figure 3. Figure 3: FIG. 3. (a) Standard deviation of surface deformations [PITH_FULL_IMAGE:figures/full_fig_p004_3.png] view at source ↗
Figure 4
Figure 4. Figure 4: shows the spatiotemporal spectra, Sη(k, ω), for different turbulence levels σU . For gentle turbulence (Fig. 4a-b), we observe an energy concentration near f ≃ 0 in the spectra, whereas for moderate (Fig. 4c) and strong (Fig. 4d) turbulence, energy broadening occurs in both frequency and wavenumber. To interpret these spectra, we must account for the Doppler shift due to random large-scale advection beneat… view at source ↗
Figure 5
Figure 5. Figure 5: FIG. 5. Spatial power spectra [PITH_FULL_IMAGE:figures/full_fig_p006_5.png] view at source ↗
Figure 6
Figure 6. Figure 6: FIG. 6. Frequency power spectra [PITH_FULL_IMAGE:figures/full_fig_p007_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: FIG. 7. Model predictions: Spatiotemporal spectra of surface deformations [PITH_FULL_IMAGE:figures/full_fig_p009_7.png] view at source ↗
read the original abstract

We report the experimental characterization of free-surface deformations generated by three-dimensional homogeneous and isotropic turbulence. Using Fourier transform profilometry in a jet-forced turbulent tank, we perform spatiotemporal measurements of the surface elevation field over a wide range of turbulence intensities. The standard deviation of surface deformations scales linearly with subsurface velocity fluctuations. The spectra of surface deformations highlight the coexistence of two mechanisms: transient coherent structures (e.g., upwelling) contributing to the low-frequency, large-scale spectral components, and a passive response to subsurface turbulent pressure fluctuations responsible for the power-law spectral scaling. The wavenumber and frequency spectra of surface deformations exhibit similar power-law exponents (-2.5), suggesting the advection of turbulent structures at the free surface. We develop a linear response model based on the transfer function from the free surface to turbulent pressure fluctuations, incorporating wave-turbulent damping. The model successfully predicts the main features of the turbulent surface: spatiotemporal spectrum shape, similar spectrum power-law exponents (-7/3), and dominance of passive response over wave generation. These findings provide new insights into free-surface turbulence in regimes where turbulent velocities remain below the surface-breaking threshold.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Referee Report

2 major / 1 minor

Summary. The manuscript experimentally characterizes free-surface deformations induced by three-dimensional homogeneous isotropic turbulence in a jet-forced tank using Fourier transform profilometry over a range of turbulence intensities. It reports that the standard deviation of surface elevation scales linearly with subsurface velocity fluctuations. Wavenumber and frequency spectra of surface deformations both exhibit power-law scaling with exponent -2.5, attributed to advection of turbulent structures. A linear response model based on the transfer function from subsurface pressure fluctuations to surface elevation, incorporating wave-turbulent damping, is proposed; this model is said to predict the observed spatiotemporal spectral shape with exponents -7/3 and the dominance of passive response over wave generation below the surface-breaking threshold.

Significance. If the central claims hold after resolution of the scaling inconsistency, the work would deliver valuable experimental data on free-surface turbulence mechanisms, clearly separating contributions from coherent structures and passive pressure response via spatiotemporal spectra. The linear response model with damping provides a predictive framework for spectral features that could inform air-sea interaction models and engineering applications. The wide intensity range and direct spectral comparisons are strengths that would strengthen the field if the model-experiment consistency is established.

major comments (2)
  1. [Abstract and experimental results] The reported linear scaling of surface deformation standard deviation with subsurface velocity fluctuations (Abstract) is inconsistent with the linear passive response model. Turbulent pressure fluctuations obey δp_rms ∼ ρ u_rms² from the Poisson equation for incompressible flow; a linear transfer function must therefore produce η_rms ∼ u_rms². The observed linear dependence cannot be recovered from the model without introducing u-dependent nonlinearity or damping, which directly violates the linearity assumption invoked for the transfer-function construction and undermines the claim that passive response accounts for the deformations.
  2. [Linear response model] § on linear response model: the transfer function with wave-turbulent damping is stated to predict the spatiotemporal spectrum shape and -7/3 exponents. It is unclear whether this model was tested against the rms scaling data or only the spectral shape; explicit comparison showing how the single free parameter (damping coefficient) simultaneously reproduces both the linear rms scaling and the spectral exponents is needed to support the dominance of the passive mechanism.
minor comments (1)
  1. [Abstract] The abstract states that spectra exhibit 'similar power-law exponents (-2.5)' and the model gives '(-7/3)'; specifying whether these are measured fits or theoretical predictions, and reporting the fitting ranges and uncertainties, would improve precision.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the thorough review and valuable feedback on our manuscript. We address each major comment below and will make revisions to improve the clarity and consistency of our claims.

read point-by-point responses

  1. Referee: [Abstract and experimental results] The reported linear scaling of surface deformation standard deviation with subsurface velocity fluctuations (Abstract) is inconsistent with the linear passive response model. Turbulent pressure fluctuations obey δp_rms ∼ ρ u_rms² from the Poisson equation for incompressible flow; a linear transfer function must therefore produce η_rms ∼ u_rms². The observed linear dependence cannot be recovered from the model without introducing u-dependent nonlinearity or damping, which directly violates the linearity assumption invoked for the transfer-function construction and undermines the claim that passive response accounts for the deformations.

    Authors: We appreciate this observation, which highlights a key point that requires clarification. The linear scaling is an experimental result for the total surface elevation rms, which includes both the passive response to pressure fluctuations and contributions from coherent structures like upwellings that are not captured by the linear transfer function. The passive response model is specifically developed to explain the power-law spectral scaling in the wavenumber and frequency spectra, not necessarily the overall rms scaling. The claim is that the passive mechanism is responsible for the spectral features, while the linear rms may arise from the combined effects. To strengthen the manuscript, we will revise the abstract and discussion sections to explicitly distinguish between the total rms scaling and the model's applicability to spectral shapes. Additionally, we will include an analysis showing the expected rms from the model and discuss why the observed linearity might hold in this regime, perhaps due to the specific range of turbulence intensities studied. revision: yes

  2. Referee: [Linear response model] § on linear response model: the transfer function with wave-turbulent damping is stated to predict the spatiotemporal spectrum shape and -7/3 exponents. It is unclear whether this model was tested against the rms scaling data or only the spectral shape; explicit comparison showing how the single free parameter (damping coefficient) simultaneously reproduces both the linear rms scaling and the spectral exponents is needed to support the dominance of the passive mechanism.

    Authors: The model was developed and compared primarily to the spectral data to predict the shape and exponents. We agree that an explicit check against the rms scaling using the same damping parameter is necessary to fully support the model's validity. In the revised version, we will add a direct comparison: using the damping coefficient determined from fitting the spectral exponents, we will compute the predicted η_rms from the model and compare it to the experimental linear scaling across the range of u_rms. This will either confirm consistency or highlight the need for additional factors, such as a weak dependence in the damping on turbulence intensity. We believe this addition will address the concern and strengthen the evidence for the passive response dominance in the spectral domain. revision: yes

Circularity Check

0 steps flagged

No circularity: model derived from standard potential-flow theory; scaling and spectra checked against independent data

full rationale

The linear transfer-function model is constructed from established potential-flow response to pressure fluctuations (with added wave-turbulent damping), not defined in terms of the observed surface statistics. The reported linear rms scaling is presented as an experimental measurement, while the model is invoked only to predict spectral exponents and passive-response dominance; these predictions are compared to measured spectra rather than forced by fitting the same quantities. No self-citation chain, ansatz smuggling, or renaming of inputs as outputs appears in the derivation. The derivation chain therefore remains self-contained against external benchmarks.

Axiom & Free-Parameter Ledger

1 free parameters · 1 axioms · 0 invented entities

The central model rests on the standard linear potential-flow response to pressure plus one damping term; no new entities are introduced.

free parameters (1)
  • wave-turbulent damping coefficient
    Incorporated in the transfer function to account for energy loss; its value is chosen to match observed spectra.
axioms (1)
  • domain assumption Surface elevation responds linearly to subsurface pressure fluctuations
    Invoked to construct the transfer-function model in the abstract.

pith-pipeline@v0.9.0 · 5501 in / 1238 out tokens · 51766 ms · 2026-05-14T17:56:36.987479+00:00 · methodology

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Reference graph

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