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arxiv: 2606.01705 · v1 · pith:WERDQYQEnew · submitted 2026-06-01 · ⚛️ physics.flu-dyn · physics.ao-ph

Breaking-induced energy dissipation of surface gravity waves at varying scales and co-flowing wind stresses

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 2026-06-28 13:09 UTC · model grok-4.3

classification ⚛️ physics.flu-dyn physics.ao-ph
keywords wave breakingenergy dissipationsurface gravity waveswind stresscrest steepnessbreaking onsetlaboratory wave groups
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The pith

Variations in wave scale and wind primarily change breaking dissipation by shifting the onset threshold, leading to a scaling ΔE_br/E0 proportional to crest asymmetry, steepness and normalized breaking duration.

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

The paper measures energy lost to breaking in laboratory wave groups of different sizes and with added wind stress. It isolates breaking dissipation from other losses using a refined accounting method. Scale and wind changes affect how much energy dissipates mainly by altering when breaking begins. The authors track this onset with the front steepness of the wave crest at the start of breaking. This produces a scaling relation that incorporates how much the crest leans forward, its local steepness, and the fraction of the wave period spent breaking.

Core claim

Breaking-induced energy dissipation varies with wave scale and co-flowing wind mainly through shifts in the breaking onset threshold. Characterizing onset and crest geometry by the crest-front steepness at incipient breaking yields the scaling ΔE_br/E0 ∝ β* S_b (τ_b/T_b), with β* as crest forward leaning, S_b as local steepness, and τ_b/T_b as non-dimensional breaking duration. The breaking strength parameter b relates approximately linearly to S_front(tb) once the threshold is accounted for.

What carries the argument

Crest-front steepness at incipient breaking S_front(tb), used to characterize breaking onset and local crest geometry across scales and wind conditions.

If this is right

  • Wave scale variations affect fractional dissipation and rate chiefly by modifying the breaking onset threshold.
  • Co-flowing wind reduces both ΔE_br/E0 and dissipation rate by causing earlier breaking with reduced crest forward lean.
  • Crest asymmetry and breaking duration play key roles in setting the amount of energy dissipated.
  • The breaking strength parameter b follows an approximately linear dependence on S_front(tb) after the onset threshold is considered.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

  • The scaling could be tested for improving energy-loss terms in numerical wave models that include wind forcing.
  • The framework for isolating breaking dissipation might extend to groups with opposing winds or broader frequency spreads.
  • Field observations of crest geometry at breaking start could check whether the same threshold dependence holds outside the laboratory.

Load-bearing premise

The crest-front steepness at incipient breaking accurately characterizes breaking onset and local crest geometry across scales and wind conditions.

What would settle it

A set of measurements at a new wave scale or wind stress where the proposed scaling ΔE_br/E0 ∝ β* S_b (τ_b/T_b) fails to collapse the data once S_front(tb) is used to set the threshold.

Figures

Figures reproduced from arXiv: 2606.01705 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: Illustration of the definition of the integration time window, ∆T = 30 s, used in equation (6). The example shown corresponds to a wave group from the SIREN campaign with γ = 2, Tp = 1.2 s, and A = 80 mm. The time domain plotted covers the full 64 s repeat period, and the start and end times of the integration window are indicated, with squares and circles marking the streamwise locations of the 14 wave ga… view at source ↗
Figure 2
Figure 2. Figure 2: Illustration of the T10 framework applied to two sets of SIREN wave groups with γ = 2 and Tp = 1.3 s, including two non-breaking and two breaking cases. The wave gauge locations are shown as relative distances from wave gauge 1. Exponential fits following equation (11) are applied to the spatial evolution of E(x). For breaking cases, separate fits are performed upstream and downstream of the identified bre… view at source ↗
Figure 3
Figure 3. Figure 3: Comparison of the background energy dissipation (∆Ef r) and breaking-induced energy dissipation (∆Ebr) quantified using the D08 and C23 methods for selected SIREN wave groups with Tp = 1.3 s and γ = 2. Panels (a) and (b) show the absolute and fractional energy dissipation, respectively, as functions of the upstream wave-group energy, E(xI ). A range of E(xI ) values between approximately 145 and 180 J m−1 … view at source ↗
Figure 4
Figure 4. Figure 4: Total energy lost over the control volume quantified for all the SIREN wave groups plotted as a function of E(xI ). Data points of different colours correspond to breaking wave groups with varying Tp. Non-breaking wave groups are shown as grey diamonds, while coloured symbols denote breaking cases. Fitting equation (12) to the non-breaking cases yields ζf = 0.26 ± 0.02 and υf = 0.94 ± 0.015, with a coeffic… view at source ↗
Figure 5
Figure 5. Figure 5: (a) Schematic illustration of the evolution of the wave energy E(x) within the control volume in the presence of wave breaking. E(xb1) and E(xb2) denote the wave-group energy at the locations of incipient breaking (xb1) and at the end of the breaking region (xb2), respectively. The control volume is divided into three streamwise segments: an upstream region of length ∆xu, the breaking region between xb1 an… view at source ↗
Figure 6
Figure 6. Figure 6: Demonstration of the spatial parameters characterising the crest profile at incipient breaking (tb). weak. The second type of steepness directly quantifies the local crest geometry at incipient breaking and therefore reflects the cumulative effects of non-linear wave evolution, energy focusing and wind forcing acting up to the breaking point. This type is here referred to as local steepness. Following prev… view at source ↗
Figure 7
Figure 7. Figure 7: Breaking-induced fractional energy dissipation ∆Ebr/E(xI ) plotted as functions of various measures of wave (group) steepness for SIREN breaking waves: panels (a, b) use spectral measures representing wave group steepness and panels (c, d) use two locally-defined steepness measures at incipient breaking. Vertical green lines in (c) indicate the breaking-onset threshold steepness identified by C23 (Cao et a… view at source ↗
Figure 8
Figure 8. Figure 8: As of figure 7 but now for breaking waves from BUBER (no wind) and EURUS (Cp/U10 ≈ 0.6 and 0.3) datasets. Results from SIREN are shown as background data in (d) for comparison. the local crest-front shape is taken into account. Indeed, when we use Sfront(tb) the influence of wind forcing is largely removed (figure 8d), and the wind-forced cases follow trends that are consistent with the wind-unforced data … view at source ↗
Figure 9
Figure 9. Figure 9: Illustration of the analysis of hydrophone acoustic outputs during a typical breaking event from the SIREN dataset (γ = 3, Tp = 1.2 s, A = 110 mm). (a) Original pressure time history and the corresponding signal after applying a 200 Hz high-pass filter (scaled by a factor of 10 for visual comparison). Region inclosed by the black lines denotes the acoustically-active phase with duration τb. (b) Spectrogram… view at source ↗
Figure 10
Figure 10. Figure 10: Energy dissipation rates of individual breaking waves ϵb plotted as a function of two S measures where (a, c) Ss(xI ) and (b, d) Sfront(tb). In all panels, the steepness measures are rescaled by E(xI ) T −1 b . Data from SIREN are shown in panels (a, b) and are included as background points in panels (c, d), where additional results from BUBER, EURUS, T10 (Tian et al., 2010), and DK16 (Derakhti and Kirby,… view at source ↗
Figure 11
Figure 11. Figure 11: Comparison between fractional energy dissipation and the scaling set (λb/L′ ) Sb (τb/Tb) for all breaking waves from the SIREN, BUBER and EURUS datasets. The yellow line shows the linear best fit forced through the origin, with a gradient of 0.024(±0.001) and r 2 = 0.8. 4.3 Discussion on the implications of Sfront(tb) and (22) for constraining ∆Ebr/E0 As discussed in §4.1.4, among the various steepness me… view at source ↗
Figure 12
Figure 12. Figure 12: Relationships between the breaking strength parameter bb calculated using equation (1), and different measures of (a–c) wave group steepness and (d–f) local steepness. The grey line in (a) shows the S 5/2 n scaling. Results derived from T10, DK16 and C23 are shown as different markers in (b), together with the linear best fit applied to the present dataset (green line). Panel (c) replots the same linear f… view at source ↗
Figure 13
Figure 13. Figure 13: Comparison of calculated values of (a)–(c) breaking-induced energy dissipation ∆Ebr and (d)–(f) J based upon different methods outlined in §3 for a selection of SIREN breaking waves with γ = 2 and Tp = 1.1 s (left panels), 1.3 s (middle panels) and 1.5 s (right panels). The values of E(xI )|max/E(xI )|min displayed in (a)–(c) indicate the ratio of the maximum to the minimum E(xI ) for a given (Tp, γ) comb… view at source ↗
Figure 14
Figure 14. Figure 14: Bin-averaged values of the breaking-induced fractional energy dissipation plotted against (a) wave group steepness Sn(xI ) and (b) crest-front steepness at incipient breaking Sfront(tb) for bulk SIREN breaking wave groups, calculated using the original, halved, and doubled breaking distances |xb1 − xb2|. Vertical lines indicate ±1 standard deviation intervals of the bin averages computed using the origina… view at source ↗
read the original abstract

Breaking-induced energy dissipation is studied for individual unsteady breaking waves using laboratory measurements of unidirectional surface gravity wave groups across a range of wave scales and wind stresses. A refined framework to estimate breaking-induced dissipation $\Delta E_{br}$ is proposed that accounts for background dissipation from non-breaking processes. Using this framework, we show that variations in wave scale primarily influence breaking energetics, such as fractional dissipation $\Delta E_{br}/E_0$ and dissipation rate $\epsilon_b$, by modifying the breaking onset threshold. Also, co-flowing wind systematically reduces both $\Delta E_{br}/E_0$ and $\epsilon_b$ relative to unforced conditions, as wind-forced waves break earlier with reduced crest forward-leaning. Exploiting the crest-front steepness at incipient breaking $\mathcal{S}_{\text{front}}(t_b)$ to characterise breaking onset and local crest geometry, we formulate a scaling for $\epsilon_b$ based on this local measure. This then yields $\Delta E_{br}/E_0 \propto \beta^{*}\,\mathcal{S}_b\,(\tau_b/T_b)$, where $\beta^{*}$ is crest forward leaning, $\mathcal{S}_b$ local steepness, and $\tau_b/T_b$ non-dimensional breaking duration. This scaling highlights the important roles of crest asymmetry and breaking duration in setting the breaking energy dissipation. Finally, we consider the breaking strength parameter $b$ by assessing existing steepness-based scaling laws, and relate $b$ to $\mathcal{S}_{\text{front}}(t_b)$, yielding an approximately linear dependence once the breaking-onset threshold is considered.

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 reports laboratory experiments on unidirectional surface gravity wave groups across a range of scales and co-flowing wind stresses. It proposes a refined framework to isolate breaking-induced dissipation ΔE_br from background non-breaking dissipation, demonstrates that scale and wind primarily modulate fractional dissipation ΔE_br/E0 and rate ε_b via shifts in the breaking onset threshold, and introduces a scaling ΔE_br/E0 ∝ β* S_b (τ_b/T_b) based on crest-front steepness S_front(tb) at incipient breaking. It further reports an approximately linear relation between the breaking strength parameter b and S_front(tb) once the threshold is accounted for.

Significance. If the central scaling and threshold-modification narrative hold, the work supplies a physically grounded parameterization for breaking dissipation that incorporates crest asymmetry (β*), local steepness, and duration, with potential value for spectral wave models and air-sea interaction studies. The controlled variation of scale and wind in the laboratory and the attempt to refine the dissipation estimation framework constitute clear strengths.

major comments (2)
  1. [Abstract and results presenting the scaling] The claim that scale and wind act primarily by modifying the onset threshold characterized by S_front(tb) is load-bearing for the scaling and the narrative that other influences are captured by β*, S_b, and τ_b/T_b. The abstract presents the linear b vs. S_front(tb) relation only after threshold adjustment but does not indicate whether residual scatter or systematic dependence on peak frequency or wind stress remains; explicit tests for independent effects (e.g., data collapse at fixed S_front(tb)) are required to confirm sufficiency of this local measure.
  2. [Methodology and results sections] The refined framework for estimating ΔE_br is central to all quantitative claims, yet the abstract provides no quantitative details on background subtraction procedure, error bars on ΔE_br/E0 and ε_b, or validation against independent dissipation estimates. Without these, the strength of evidence supporting the scaling and the threshold-modification interpretation cannot be fully assessed.
minor comments (1)
  1. Notation for quantities such as β*, S_b, and τ_b/T_b should be defined explicitly at first use with reference to the relevant equations or figures.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their thoughtful and constructive report. The comments highlight important aspects of presentation and evidence strength that we address point-by-point below. We propose targeted revisions to improve clarity without altering the core findings.

read point-by-point responses

  1. Referee: [Abstract and results presenting the scaling] The claim that scale and wind act primarily by modifying the onset threshold characterized by S_front(tb) is load-bearing for the scaling and the narrative that other influences are captured by β*, S_b, and τ_b/T_b. The abstract presents the linear b vs. S_front(tb) relation only after threshold adjustment but does not indicate whether residual scatter or systematic dependence on peak frequency or wind stress remains; explicit tests for independent effects (e.g., data collapse at fixed S_front(tb)) are required to confirm sufficiency of this local measure.

    Authors: We agree that explicit verification of data collapse and residual analysis strengthens the claim. The manuscript already demonstrates collapse of ΔE_br/E0 and ε_b across scales and wind conditions when plotted against the threshold-adjusted S_front(tb), with the scaling parameters β*, S_b and τ_b/T_b accounting for remaining variation. However, to directly address potential residual dependence, the revised manuscript will include (i) a supplementary figure showing fractional dissipation and b versus S_front(tb) stratified by peak frequency and wind stress, and (ii) a quantitative residual analysis confirming no systematic trends remain once the threshold is accounted for. These additions will be referenced in an updated abstract sentence. revision: yes

  2. Referee: [Methodology and results sections] The refined framework for estimating ΔE_br is central to all quantitative claims, yet the abstract provides no quantitative details on background subtraction procedure, error bars on ΔE_br/E0 and ε_b, or validation against independent dissipation estimates. Without these, the strength of evidence supporting the scaling and the threshold-modification interpretation cannot be fully assessed.

    Authors: The background-subtraction procedure, including the decomposition into breaking and non-breaking contributions and the associated uncertainty quantification from ensemble repeats, is fully detailed in Section 3.2, with error bars reported on all ΔE_br/E0 and ε_b values in Figures 4–7. Validation against independent spectral dissipation estimates appears in Section 4.1. We acknowledge that the abstract is necessarily concise and omits these specifics. In revision we will add one sentence to the abstract summarizing the error estimation and validation approach while respecting length limits; no changes to the underlying methodology are required. revision: partial

Circularity Check

0 steps flagged

No significant circularity; scaling derived from independent experimental measurements

full rationale

The paper's central scaling ΔE_br/E0 ∝ β* S_b (τ_b/T_b) and the relation for b are formulated from laboratory data on wave groups, using measured crest-front steepness S_front(tb) at incipient breaking to characterize onset. The framework refines dissipation estimates by subtracting background processes, and the proportionality is presented as an empirical outcome after observing how scale and wind modify the threshold. No equations reduce by construction to fitted inputs, no load-bearing self-citations are invoked, and the derivation chain remains self-contained against the reported observations without renaming or smuggling ansatzes.

Axiom & Free-Parameter Ledger

1 free parameters · 2 axioms · 0 invented entities

As an experimental study, the central claims rest on measurement protocols, data interpretation assumptions, and standard fluid dynamics principles rather than new theoretical constructs or entities.

free parameters (1)
  • proportionality constant in scaling
    The scaling is stated as proportional, implying an implicit constant potentially determined from data but not specified in the abstract.
axioms (2)
  • standard math Incompressible, irrotational flow assumptions for surface gravity waves prior to breaking
    Standard background for wave theory and breaking analysis.
  • domain assumption Background dissipation from non-breaking processes can be independently estimated and subtracted to isolate breaking effects
    Central to the refined framework proposed in the abstract.

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Reference graph

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