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arxiv: 2605.28907 · v1 · pith:FWK6R4ATnew · submitted 2026-05-27 · ⚛️ physics.flu-dyn · physics.ao-ph

On the limiting geometry of unsteady breaking waves subject to co-flowing wind: spectrally-informed versus locally-measured steepness

Rui Cao (1 , 2) , Enrique M. Padilla (3) , Xu Chen (1) , Adrian H. Callaghan (2) ((1) Ocean University of China , (2) Imperial College London , (3) Universitat Polit\`ecnica de Catalunya) This is my paper

Pith reviewed 2026-06-29 09:36 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.ao-ph
keywords wave breakingcrest steepnesswind forcingincipient breakingsurface profileslaboratory experimentsspectral steepnessfront-face slope
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The pith

Crest-front steepness at incipient breaking holds a lower threshold near 0.2 that decreases with wind speed as crests lean less forward.

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

The paper contrasts two steepness measures for unsteady breaking waves under co-flowing wind: spectrally-informed group steepness from fixed-point records, which rises near breaking due to high-frequency content, and locally measured crest steepness from spatial profiles. It concludes that crest-front steepness at the exact onset of breaking is the most meaningful of the metrics tested. This quantity shows a consistent lower threshold of about 0.2 for breaking to begin, with values above the threshold falling as wind speed rises because crests become less forward-leaning. The result points to wind-modified dispersion and aerodynamic effects competing with wave-wave interactions, so geometry alone does not fully determine breaking. A reader would care because the finding offers a potential diagnostic variable for predicting when waves break and how much energy they dissipate under wind forcing.

Core claim

The crest-front steepness S_front(tb), which measures the front-face slope at the instant of incipient breaking, exhibits a consistent breaking-onset lower-bound threshold of S_front(tb)≈0.2. Values above this threshold decrease with wind speed as crests become less forward leaning. This behavior is attributed to wind-modified dispersion, enhanced high-frequency spectral content, and aerodynamic sheltering, indicating that wind-wave and wave-wave interactions act as competing mechanisms in triggering breaking through processes beyond what geometry alone can explain. Even so, S_front(tb) retains strong potential as a controlling variable for studies of breaking energetics and crest-scale dyna

What carries the argument

Crest-front steepness S_front(tb) at incipient breaking time tb, extracted from local spatial surface profiles to give the slope of the front face of the crest.

Load-bearing premise

The image-processing method used to extract local surface profiles captures unbiased geometry exactly at the instant of incipient breaking, and the laboratory campaigns cover the relevant spectral and wind conditions without selection bias.

What would settle it

Observing breaking events with crest-front steepness below 0.2 or sustained non-breaking waves with crest-front steepness well above 0.2 under comparable wind and spectral conditions would falsify the reported threshold.

Figures

Figures reproduced from arXiv: 2605.28907 by 2), (2) Imperial College London, (3) Universitat Polit\`ecnica de Catalunya), Adrian H. Callaghan (2) ((1) Ocean University of China, Enrique M. Padilla (3), Rui Cao (1, Xu Chen (1).

Figure 1
Figure 1. Figure 1: Illustrations of the parameters defining local crest geometry at (a) incipient breaking and (b) overturning. The wave propagates in the +x direction. The profiles shown are from a real breaking event where its vertical dimension is exaggerated by a factor of 6.5 for clarity. In both panels, xb marks the crest location at the brink of breaking, L the half-wavelength between zero crossings, and a ′ and L ′ t… view at source ↗
Figure 2
Figure 2. Figure 2: (a) Schematic of the experimental set-up for SIREN. An earlier version of this panel was published in Cao et al. (2026). (b, c) Target spectra (Sηη(f) = a(f) 2/[2 δf]) and simulated surface elevations at the focal point for wave groups with equal A = 20 mm, γ = 2, and varying Tp. The inset in (b) shows the normalised spectra across different Tp values, with frequency f scaled by fp and variance density nor… view at source ↗
Figure 3
Figure 3. Figure 3: (a) Schematic of the experimental set-up for BUBER and EURUS. As in figure 2(a), all instruments are shown, although the hydrophone, top-view camera, and bubble-camera data are not used here. The top-view camera was used only in BUBER, while the two wind probes, located 5.7 m and 9.4 m from paddle 1, were used only in EURUS. (b) Time history of wind speed measured 0.11 m above the still water level for an … view at source ↗
Figure 4
Figure 4. Figure 4: Mechanics and performance of SDBW-I. The first and second rows correspond to incipient breaking (vertical front face) and overturning, respectively: (a, e) snapshots of case S-G2Tp12A095 with partial CMG detection; (b, f) zoomed-in regions; (c, g) binary images from Otsu threshold; and (d, h) extracted air–water interface along the inner face of the crest. (i, j) SDBW-I detection compared with Cam2 imagery… view at source ↗
Figure 5
Figure 5. Figure 5: Spatial evolution of the three spectrally-informed wave-group steepness measures upstream and downstream of the focal/breaking locations. In each panel, hollow and filled symbols denote non￾breaking and breaking wave groups, respectively, with colour indicating the linear amplitude sum A. Panels (a)–(c) show SIREN cases with γ = 2 and varying wave scale; panels (d)–(f) show SIREN cases with fixed Tp = 1.1 … view at source ↗
Figure 6
Figure 6. Figure 6: (a)–(c) Comparisons of Sp, Ss, and Sn between the most upstream location (χ1) and a location just upstream of xb (χ5) for all non-breaking and breaking waves from SIREN. (d)–(f) Equivalent comparisons for breaking cases from BUBER and EURUS, using x3 and x7, whose fetch distances are comparable to χ1 = 4.5 m and χ5 = 9.0 m in SIREN (see figures 2a and 3a). 15 [PITH_FULL_IMAGE:figures/full_fig_p015_6.png] view at source ↗
Figure 7
Figure 7. Figure 7: (a, b) Spectrally resolved distributions of Sp and Sn at x7 = 9.1 m, just upstream of breaking. Values are normalised by their respective peaks, and shaded green regions mark the prescribed paddle-wave generation range. (c, d) Cumulative spectrally resolved wave-group steepness as the cut-off frequency is progressively extended, normalised by the reference values at fi/fp = 3 (horizontal dashed lines). Res… view at source ↗
Figure 8
Figure 8. Figure 8: Time-evolving histories of the locally-measured zero-crossing steepness, Szc(t), and crest-front steepness, Sfront(t), before and after incipient breaking at tb. Results are ensemble averages over five BUBER breaking waves with similar Sn. Shaded regions indicate ±1 standard deviation, and solid points at (t − tb)/Tp = 0 mark values of local steepness at incipient breaking. wind conditions. 4 Relationships… view at source ↗
Figure 9
Figure 9. Figure 9: Comparisons between local crest steepness at incipient breaking (Sb, left column, and Sfront(tb), right column) and upstream wave-group steepness (Sp(χ1), Ss(χ1) and Sn(χ1)) for SIREN breaking cases across different wave scales and spectral bandwidths. Dash-dotted lines in the left panels show linear least-squares fits. Grey dashed lines in the right panels indicate the average minimum value of Sfront(tb) … view at source ↗
Figure 10
Figure 10. Figure 10: Measured spectrally-resolved steepness distributions of Fourier-decomposed components for the SIREN γ = 2 (non-breaking) wave groups with varying Tp but comparable wave-group steepness Sn(χ1) ∈ [0.30, 0.35]. Panel (a) shows the absolute steepness density, aiki/δf, as a function of frequency, and panel (b) shows the corresponding distributions normalised by the spectral peak, aiki/[a(fp)kp], as a function … view at source ↗
Figure 11
Figure 11. Figure 11: Comparisons of locally-measured crest steepness (Sb and Sfront(tb)) with spectrally-informed wave-group steepness (Sp(x3), Ss(x3), and Sn(x3)) for breaking cases from BUBER and EURUS under different wind speeds. Panels (a, d, g) show Sb, and panels (b, e, h) show Sfront(tb). Grey dashed lines mark the average breaking-onset threshold identified in figure 9. Red solid lines are linear fits to the strongest… view at source ↗
Figure 12
Figure 12. Figure 12: (a) Crest-front distance L ′ (tb) and (b) crest asymmetry parameter ζ (defined in equa￾tion (13)), plotted against Sn(x3) for BUBER and EURUS breaking waves under varying co-flowing wind conditions. In (b), the dashed line indicates ζ = 1 (corresponding to a symmetric crest) and the region shaded indicates the forward-leaning regime. breaking.This wind-induced change in crest asymmetry appears strongest a… view at source ↗
Figure 13
Figure 13. Figure 13: (a) Surface elevation time-series of wind-induced wave fields at the two wind speeds used in EURUS, upon reaching a statistically-stationary state. Measurements were taken at gauge 7 (x = x7), the gauge closest to the focal/breaking location in EURUS (see also figure 3a). (b) Corresponding FFT variance density spectra calculated from 256 s surface elevation records. The region enclosed by the vertical das… view at source ↗
Figure 14
Figure 14. Figure 14: (a) Surface elevation time-series of a wind-induced, statistically stationary sea state at U10 = 6.0 m s−1 , recorded at different fetches: x4 = 6 m, x7 = 9.1 m, x9 = 13 m, and x11 = 15.7 m (see also figure 3a). Also shown are the FFT variance density spectra for (b) U10 = 3.0 m s−1 and (c) U10 = 6.0 m s−1 , at these fetches. Again, the two black vertical dashed lines mark the lower and upper frequency li… view at source ↗
Figure 15
Figure 15. Figure 15: Surface elevation time-series η(x, t) (left panels) and normalised FFT phase angles ϕ(f) π −1 (right panels) for G2Tp13A040 from BUBER (no wind) and EURUS (with wind). Records were taken at (a, b) x1 = 2.4 m, (c, d) x3 = 4.5 m, (e, f) x5 = 7.0 m, (g, h) x7 = 9.1 m, closest to the focal/breaking location, and (i, j) x9 = 13 m (see also figure 3a). All cases had the same prescribed focal location xf and tim… view at source ↗
Figure 16
Figure 16. Figure 16: Wind-induced modulation of the dispersion relation under different wind speeds. (a) Wind￾modulated frequency fi, mod versus the intrinsic frequency fi . (b) Ratio of wind-modulated phase speed ci, mod to the intrinsic phase velocity ci , plotted as a function of fi . (a) (b) [PITH_FULL_IMAGE:figures/full_fig_p031_16.png] view at source ↗
Figure 17
Figure 17. Figure 17: FFT variance density spectra for G2Tp13A040 from BUBER (no wind) and EURUS (with wind), recorded at (a) x4 = 6.0 m and (b) x7 = 9.1 m. Orange lines show spectra of purely wind-induced waves under the strongest wind speed, U10 = 6.0 m s−1 . Vertical dashed lines mark the active frequency range of wave-generation paddle, corresponding to the target spectrum. The x-axis is plotted on a logarithmic scale to h… view at source ↗
read the original abstract

Wave steepness is a key geometric variable for describing breaking occurrence and its consequences, including energy dissipation and air entrainment. Using three laboratory campaigns under varying spectral conditions and co-flowing wind forcing, we contrast two types of steepness commonly used for unsteady breaking waves: spectrally-informed wave-group steepness (prognostic), obtained from fixed-point surface-elevation records, and locally-measured crest steepness (diagnostic), obtained from spatial surface profiles extracted using the SDBW-I image-processing method developed herein. For the former, the long-adopted $\mathcal{S}_n$ (linear sum of Fourier-component steepness) increases appreciably within about two dominant wavelengths upstream of breaking because of its sensitivity to evolving high-frequency content. When measured sufficiently far upstream, however, wave-group steepness remains approximately linearly related to the local zero-crossing steepness $\mathcal{S}_b$ across bulk unforced conditions. Notwithstanding this, we argue that the crest-front steepness, $\mathcal{S}_{\mathrm{front}}(t_b)$, which delineates the front-face slope at incipient breaking, is the most physically meaningful metric examined here. It exhibits a consistent breaking-onset lower-bound threshold of $\mathcal{S}_{\mathrm{front}}(t_b)\approx0.2$, while values above this threshold decrease with wind speed as crests become less forward leaning. This may be attributed to wind-modified dispersion, enhanced high-frequency spectral content and aerodynamic sheltering, suggesting that wind--wave and wave--wave interactions act as competing mechanisms in triggering breaking through kinematic and energetic processes beyond what geometry alone can explain. Even so, $\mathcal{S}_{\mathrm{front}}(t_b)$ has strong potential as a controlling variable for future studies of breaking energetics and crest-scale dynamics.

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 / 0 minor

Summary. The paper reports laboratory results from three campaigns on unsteady breaking waves with co-flowing wind. It contrasts spectrally-informed steepness (e.g., S_n from Fourier components, which grows near breaking due to high-frequency content) with locally measured crest steepness extracted via the new SDBW-I image-processing method. The central claim is that crest-front steepness S_front(t_b) at incipient breaking provides the most physically meaningful metric, exhibiting a consistent lower-bound threshold of approximately 0.2 that decreases with wind speed as crests become less forward-leaning, attributed to competing wind-wave and wave-wave interactions.

Significance. If substantiated, the result supplies a diagnostic geometric criterion for breaking onset that incorporates wind effects, offering a potential improvement over purely spectral or upstream steepness measures for modeling energy dissipation and crest dynamics. The multi-campaign design and explicit contrast between prognostic and diagnostic steepness metrics constitute a clear methodological strength.

major comments (2)
  1. [Abstract] Abstract: the headline threshold S_front(t_b)≈0.2 is stated without reported sample sizes, error bars, or quantitative validation of the SDBW-I pipeline (e.g., cross-checks against wave probes or sensitivity to thresholding/temporal interpolation), which directly bears on whether the lower-bound claim and its wind-speed trend can be considered robust.
  2. [Abstract] Abstract and methods description: the assertion that SDBW-I returns unbiased local surface profiles exactly at incipient breaking t_b lacks explicit criteria for breaking detection or tests for systematic offsets in t_b identification or perspective-corrected slope reconstruction; any such offset would shift the reported 0.2 threshold and undermine the claim that it is the most physically meaningful metric.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their constructive comments, which help clarify the presentation of our results on crest-front steepness as a breaking-onset metric. We respond point-by-point below and indicate revisions to the manuscript.

read point-by-point responses

  1. Referee: [Abstract] Abstract: the headline threshold S_front(t_b)≈0.2 is stated without reported sample sizes, error bars, or quantitative validation of the SDBW-I pipeline (e.g., cross-checks against wave probes or sensitivity to thresholding/temporal interpolation), which directly bears on whether the lower-bound claim and its wind-speed trend can be considered robust.

    Authors: The abstract is space-limited, but the full manuscript reports data from three laboratory campaigns. We will revise the abstract to state the total number of breaking events analyzed and to reference the validation of SDBW-I (cross-checks with wave probes and sensitivity tests) that appears in the methods and results sections. These additions will make the robustness of the ≈0.2 threshold and its wind-speed dependence more immediately apparent without altering the central claim. revision: yes

  2. Referee: [Abstract] Abstract and methods description: the assertion that SDBW-I returns unbiased local surface profiles exactly at incipient breaking t_b lacks explicit criteria for breaking detection or tests for systematic offsets in t_b identification or perspective-corrected slope reconstruction; any such offset would shift the reported 0.2 threshold and undermine the claim that it is the most physically meaningful metric.

    Authors: We agree that the criteria used to identify t_b and any associated sensitivity tests should be stated explicitly. The manuscript already describes the SDBW-I pipeline and its application at incipient breaking, but we will expand the methods section to include the precise operational definition of t_b, the procedure for perspective correction, and the results of sensitivity analyses to thresholding and temporal interpolation. These additions will directly address concerns about possible systematic offsets while preserving the argument that S_front(t_b) is the most physically meaningful metric among those examined. revision: yes

Circularity Check

0 steps flagged

No circularity; threshold is direct empirical observation from lab data

full rationale

The paper reports laboratory measurements of crest-front steepness S_front(tb) at incipient breaking across three campaigns, identifying an observed lower-bound threshold ≈0.2 that decreases with wind speed. No derivation chain, equations, or predictions are presented that reduce this threshold to a fitted parameter, self-referential definition, or self-citation load-bearing premise. The SDBW-I method is described as the extraction tool, but the threshold itself emerges from the data rather than any construction that equates output to input by definition. This matches the default expectation of no significant circularity for measurement-based results.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

The central claim rests on empirical observations from laboratory wave tanks and a custom image-processing pipeline. No free parameters are introduced or fitted in the abstract. Standard fluid-dynamics assumptions underpin the interpretation of surface profiles and dispersion.

axioms (1)
  • standard math Incompressible, irrotational flow assumptions standard in water-wave theory
    Implicit in the use of surface-elevation records and steepness definitions derived from potential-flow concepts.

pith-pipeline@v0.9.1-grok · 5906 in / 1295 out tokens · 40166 ms · 2026-06-29T09:36:58.405477+00:00 · methodology

discussion (0)

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Reference graph

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