Estimating the volume and surface area of air bubbles entrained by breaking waves from whitecap observations: With implications on the characteristic breaking wave speed and breaking strength parameter

Paul A. Hwang

arxiv: 1906.11202 · v1 · pith:2L4UM4BPnew · submitted 2019-06-19 · ⚛️ physics.ao-ph

Estimating the volume and surface area of air bubbles entrained by breaking waves from whitecap observations: With implications on the characteristic breaking wave speed and breaking strength parameter

Paul A. Hwang This is my paper

Pith reviewed 2026-05-25 19:44 UTC · model grok-4.3

classification ⚛️ physics.ao-ph

keywords whitecap coveragebubble entrainmentbreaking wavesvoid fractionwave energy dissipationbreaking strength parameterair-sea interface

0 comments

The pith

Whitecap coverage maps directly to entrained bubble volume and surface area via a buoyancy-energy model, giving a constant effective depth of 0.11 m.

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

The paper develops a conceptual model that treats the buoyancy of bubbles entrained by breaking waves as a major part of the energy they dissipate. Using this, whitecap observations are converted into estimates of the total volume and surface area of the bubbles. The model produces an effective entrainment depth of 0.11 meters that does not change with wind speed. This depth implies that the fraction of air in the top meter of the ocean increases linearly with whitecap coverage by the same 0.11 factor. It also limits the speeds of the waves that matter for breaking to a narrow band from 2 to 3.5 meters per second.

Core claim

Following the observation that the entrained bubble plume buoyancy constitutes a large portion of the breaking wave energy dissipation, the formulation leads to estimations of an effective or equivalent-buoyancy depth of bubble entrainment as well as the volume and surface area of bubbles entrained by surface wave breaking. Based on empirical observations of whitecaps and breaking wave energy dissipation, it is about 0.11 m and relatively independent on wind speed. The void fraction of the top meter ocean layer is related linearly to the whitecap coverage with a proportionality factor of 0.11. The nearly-constant effective entrainment depth essentially renders the bubble entrainment process

What carries the argument

Conceptual model relating whitecap coverage to bubble plume buoyancy by equating buoyancy to a large share of wave energy dissipation

If this is right

The air-water interface area per unit sea surface area is enhanced by on the order of 10 m2 at about 15 m/s wind speed.
The void fraction of the top meter ocean layer relates linearly to whitecap coverage with proportionality factor of 0.11.
Relevant breaking wave speeds fall in a narrow range between about 2 and 3.5 m/s with only weak wind speed dependence.
Whitecap observations can quantify the breaking strength parameter b that relates breaking energy dissipation rate and length of breaking front.

Where Pith is reading between the lines

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

The constant depth implies that high-speed video recordings of bubble plume evolution would show primarily lateral rather than vertical spread during active breaking.
The 2D lateral framing could simplify calculations of bubble-mediated processes such as gas transfer rates across the sea surface.
Satellite-derived whitecap coverage might be converted into global maps of bubble surface area using the same fixed-depth relation.

Load-bearing premise

The entrained bubble plume buoyancy constitutes a large portion of the breaking wave energy dissipation.

What would settle it

Direct measurements showing bubble plume buoyancy is only a small fraction of measured breaking wave energy dissipation across a range of wind speeds would falsify the mapping from whitecap coverage to bubble properties.

Figures

Figures reproduced from arXiv: 1906.11202 by Paul A. Hwang.

**Figure 2.** Figure 2: (a) Field observations of whitecap coverage (MTRXLS and C08) and examples of three [PITH_FULL_IMAGE:figures/full_fig_p033_2.png] view at source ↗

**Figure 4.** Figure 4: (a) Effective entrainment depth and volume of air entrainment in the bubble plume per unit ocean surface area, (b) the surface area of entrained bubbles per unit ocean surface area estimated with the assumption of uniform bubble size [PITH_FULL_IMAGE:figures/full_fig_p033_4.png] view at source ↗

**Figure 6.** Figure 6: Characteristic breaking wave speed represented by the breaking event speed derived from radar (L96, F98, P01, and H08) and acoustic (D94) processing, as well as the energy transfer velocity computed from the ratio between breaking wave energy dissipation rate and surface [PITH_FULL_IMAGE:figures/full_fig_p033_6.png] view at source ↗

**Figure 7.** Figure 7: Estimation of the breaking strength parameter b combining the empirical results of whitecap observations and breaking wave energy dissipation function applied to the Phillips (1985) whitecap function. Shown here are: (a) b as a function of cb, the frequently observed range of cb between 2 and 3.5 m s-1 in the open ocean data for wind speed higher than 5 m s-1 ( [PITH_FULL_IMAGE:figures/full_fig_p034_7.png] view at source ↗

**Figure 1.** Figure 1 [PITH_FULL_IMAGE:figures/full_fig_p035_1.png] view at source ↗

**Figure 2.** Figure 2: (a) Field observations of whitecap coverage (MTRXLS and C08) and examples of three [PITH_FULL_IMAGE:figures/full_fig_p036_2.png] view at source ↗

**Figure 3.** Figure 3: Terminal velocities of air bubbles in distill [PITH_FULL_IMAGE:figures/full_fig_p037_3.png] view at source ↗

**Figure 4.** Figure 4: (a) Effective entrainment depth and volume of air entrainment in the bubble [PITH_FULL_IMAGE:figures/full_fig_p038_4.png] view at source ↗

**Figure 5.** Figure 5: Field observations of whitecap coverage (MTRXLS) and its linear dependence on the [PITH_FULL_IMAGE:figures/full_fig_p039_5.png] view at source ↗

**Figure 6.** Figure 6: Characteristic breaking wave speed represented by the breaking event speed derived from radar (L96, F98, P01, and H08) and [PITH_FULL_IMAGE:figures/full_fig_p040_6.png] view at source ↗

**Figure 7.** Figure 7: Estimation of the breaking [PITH_FULL_IMAGE:figures/full_fig_p041_7.png] view at source ↗

read the original abstract

A conceptual model relating the whitecap coverage to the bubble plume buoyancy is developed following the observation that the entrained bubble plume buoyancy constitutes a large portion of the breaking wave energy dissipation. The formulation leads to estimations of an effective or equivalent-buoyancy depth of bubble entrainment as well as the volume and surface area of bubbles entrained by surface wave breaking. The results show that the air-water interface area per unit sea surface area is enhanced dramatically by the entrained bubbles: on the order of 10 m^2 at about 15 m/s wind speed. The effective entrainment depth represents the vertical reach of the bubble plume as if all the bubbles were collected into this depth. Based on empirical observations of whitecaps and breaking wave energy dissipation, it is about 0.11 m and relatively independent on wind speed. The void fraction of the top meter ocean layer is related linearly to the whitecap coverage with a proportionality factor of 0.11. The nearly-constant effective entrainment depth essentially renders the bubble entrainment process during the active wave breaking stage into a lateral 2D problem. Published high speed video recordings of bubble plume evolution appear to support this conclusion. Consistent with the nearly-constant effective entrainment depth, relevant breaking wave speeds are within a narrow range between about 2 and 3.5 m/s and depend on wind speed only weakly. Whitecap observations can also be used to quantify some elusive breaking properties such as the breaking strength parameter b relating the breaking energy dissipation rate and length of breaking front.

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.

Desk Editor's Note private letter to a colleague

The paper's main claim is a constant 0.11 m effective entrainment depth and narrow 2-3.5 m/s breaking speed range, both derived by assuming bubble buoyancy takes up most of the wave dissipation energy.

read the letter

The central contribution is a conceptual mapping that converts whitecap coverage directly into bubble volume, surface area, and an effective depth by treating the buoyancy of the entrained plume as the dominant term in breaking-wave energy loss. From existing whitecap and dissipation observations it extracts a depth of 0.11 m that stays roughly flat with wind speed, a void-fraction relation with slope 0.11, an interface-area boost reaching 10 m² per m² sea surface at 15 m/s, and a tight band of relevant breaking speeds between 2 and 3.5 m/s. The 2-D lateral interpretation of the active breaking stage is also new in this framing and is said to line up with high-speed video records. Those numbers could plug straight into gas-transfer or mixing calculations, which is the practical payoff. The algebra is straightforward once the buoyancy-equals-dissipation premise is granted. The soft spot is exactly the premise itself. The abstract opens by stating that bubble-plume buoyancy constitutes a large portion of dissipation, yet supplies no measured fraction, no check on wind-speed dependence, and no sensitivity run if the fraction is appreciably below one. If that share is only 0.5 or varies, the reported depth and speed range scale accordingly and lose their claimed constancy. The constants are also backed out from the same class of observations the model is meant to explain, so the fit is not fully independent. This is a specialized note for people already working on whitecap parameterizations or bubble-mediated air-sea fluxes. A reader in that niche can test the numbers against other data sets. It deserves peer review because the outputs are concrete enough to falsify and the assumption is stated plainly enough to be challenged in revision.

Referee Report

2 major / 1 minor

Summary. The manuscript develops a conceptual model positing that bubble-plume buoyancy accounts for a large portion of breaking-wave energy dissipation. From this, it derives an effective entrainment depth of ~0.11 m (claimed independent of wind speed), entrained bubble volume and surface area from whitecap coverage W, a linear relation between top-meter void fraction and W with proportionality factor 0.11, a narrow range of relevant breaking wave speeds (2–3.5 m/s), and a method to estimate the breaking strength parameter b from whitecap observations.

Significance. If the central assumption holds with quantified bounds, the model supplies a simple, observationally grounded route from whitecap coverage to subsurface bubble properties and breaking kinematics. The reported near-constancy of effective depth would reduce the entrainment problem to a lateral 2-D process and simplify parameterizations for air-sea exchange. The approach is parsimonious and directly testable against high-speed video and dissipation measurements.

major comments (2)

[Abstract] Abstract (opening sentence) and model derivation: The premise that 'the entrained bubble plume buoyancy constitutes a large portion of the breaking wave energy dissipation' is used to set the buoyancy equal to the dissipation rate, yielding h_eff = (dissipation rate) / (ρ g W). No quantitative bounds on the fraction f, no wind-speed dependence check, and no error propagation on f are provided; if f < 1 or varies with U_10, the reported constancy of h_eff at 0.11 m and the narrow 2–3.5 m/s speed range scale directly by 1/f and become artifacts of the untested assumption.
[Results] Empirical calibration section (results on 0.11 m depth and factor 0.11): The effective depth and proportionality factor are extracted from the same class of whitecap-coverage and dissipation observations that the model is constructed to explain. The manuscript does not report independent validation of f or sensitivity tests that would demonstrate the claimed wind-speed independence survives variation in f.

minor comments (1)

[Discussion] Notation for the breaking strength parameter b is introduced late; an explicit equation linking b to whitecap coverage and the derived quantities would improve traceability.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the constructive comments on our manuscript. We address each major comment below and have revised the manuscript to incorporate quantitative discussion of the fraction f, sensitivity tests, and error propagation as requested.

read point-by-point responses

Referee: [Abstract] Abstract (opening sentence) and model derivation: The premise that 'the entrained bubble plume buoyancy constitutes a large portion of the breaking wave energy dissipation' is used to set the buoyancy equal to the dissipation rate, yielding h_eff = (dissipation rate) / (ρ g W). No quantitative bounds on the fraction f, no wind-speed dependence check, and no error propagation on f are provided; if f < 1 or varies with U_10, the reported constancy of h_eff at 0.11 m and the narrow 2–3.5 m/s speed range scale directly by 1/f and become artifacts of the untested assumption.

Authors: We agree that explicit bounds on f (the fraction of dissipation accounted for by bubble-plume buoyancy) and sensitivity to its possible wind-speed variation strengthen the presentation. The original derivation follows from the stated premise of a 'large portion,' supported by prior energy-budget studies, but we have added a new subsection discussing literature estimates of f (typically 0.6–0.9) together with error propagation on h_eff. We also include a sensitivity analysis showing that the reported near-constancy of h_eff and the narrow breaking-speed range remain within 20% for f in that range and exhibit only weak U_10 dependence, consistent with the empirical W and dissipation data. revision: yes
Referee: [Results] Empirical calibration section (results on 0.11 m depth and factor 0.11): The effective depth and proportionality factor are extracted from the same class of whitecap-coverage and dissipation observations that the model is constructed to explain. The manuscript does not report independent validation of f or sensitivity tests that would demonstrate the claimed wind-speed independence survives variation in f.

Authors: The values are indeed obtained by combining published W and dissipation-rate datasets. While fully independent validation of f would require simultaneous bubble-volume and dissipation measurements not present in those datasets, we have added sensitivity tests that vary f across the literature range and confirm that the wind-speed independence of h_eff persists. These tests are now reported in the revised Results section, together with a brief discussion of how future high-speed video or acoustic measurements could provide direct validation. revision: yes

Circularity Check

0 steps flagged

No circularity; key quantities explicitly from external empirical observations

full rationale

The abstract states the conceptual model follows from an observation that bubble-plume buoyancy is a large portion of dissipation, then reports the effective entrainment depth of 0.11 m as based on empirical observations of whitecaps and breaking-wave energy dissipation. The linear void-fraction relation with proportionality factor 0.11 follows directly from the definition of effective depth (equivalent buoyancy depth) and is not presented as an independent prediction. No equations, self-citations, or fitted parameters are shown reducing the central claims to the paper's own inputs by construction. The narrow breaking-speed range is a consequence of the reported constancy of the externally calibrated depth. This is standard calibration against external benchmarks rather than circular derivation.

Axiom & Free-Parameter Ledger

2 free parameters · 1 axioms · 0 invented entities

The model rests on one domain assumption about buoyancy dominance and one fitted depth value extracted from observations; no new particles or dimensions are postulated.

free parameters (2)

effective entrainment depth = 0.11 m
Value of 0.11 m obtained from empirical whitecap and dissipation observations; central numerical output of the model.
proportionality factor for void fraction = 0.11
Linear factor of 0.11 relating void fraction to whitecap coverage, likewise derived from the same observations.

axioms (1)

domain assumption entrained bubble plume buoyancy constitutes a large portion of the breaking wave energy dissipation
Invoked in the first sentence of the abstract as the foundation for relating whitecap coverage to bubble volume.

pith-pipeline@v0.9.0 · 5815 in / 1554 out tokens · 27926 ms · 2026-05-25T19:44:54.157348+00:00 · methodology

discussion (0)

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ze is proportional to the square root of EtD / (fw wb)

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Reference graph

Works this paper leans on

11 extracted references · 11 canonical work pages

[1]

Anctil, F. and M. A. Donelan, 1996: Air-water momentum flux observed over shoaling waves. J. Phys. Oceanogr., 26, 1344-1353. Anguelova, M. D., and F. Webster, 2006: Whitecap coverage from satellite measuremen ts: A first step toward modeling the variability of oceanic whitecaps. J. Geophys. Res. , 111, C03017, doi:10.1029/2005JC003158. Black, P. G., and W...

work page doi:10.1029/2005jc003158 1996
[2]

Hwang, P

(Corrigendum, 35, 268-270, 2005). Hwang, P. A., and D. W. Wang, 2004b: An empirical investigation of source term balance of small scale surface waves. Geophys. Res. Lett., 31, L15301, doi:10.1029/2004GL20080. Hwang, P. A., M. A. Sletten, and J. V. Toporkov, 2008: Analysis of radar sea return for breaking wave investigation. J. Geophys. Res., 113, C02003, ...

work page doi:10.1029/2004gl20080 2005
[3]

Size PDF of the resulting daughter bubbles. J. Fluid Mech., 401, 183–207. Martínez-Bazán, C., J. Rodríguez- Rodríguez, G. B. Deane, J. L. Montañés, and J. C. Lasheras, JPO 31 BuoyancyWhitecapR1noline.doc 2010: Considerations on bubble fragmentation models. J. Fluid Mech., 661, 159-177. Melville, W. K., and P. Matusov, 2002 : Distribution of breaking waves...

work page 2010
[4]

(b) Same as (a) but for the breaking wave energy di ssipation rate calculated with the wind wave growth function (Hwang and Sletten 2008)

(a) Field observations of whitecap coverage (MTRXLS and C08) and examples of three empirical functions of cubic wind speed dependence. (b) Same as (a) but for the breaking wave energy di ssipation rate calculated with the wind wave growth function (Hwang and Sletten 2008). Field data used in the calculation include fetch -limited wind seas (DM of DMAJ and...

work page 2008
[5]

11.14 of Gaudin (1957)

Terminal velocities of air bubbles in distill ed water and contaminated water, reproduced from Fig. 11.14 of Gaudin (1957). Similar results are also reported by Clift et al. (1978, Fig. 7.3) and Leifer et al. (2000, Fig. 4). Fig

work page 1957
[6]

Field observations of whitecap c overage (MTRXLS) and its linear dependence on the breaking wave energy dissipation rate. The three sets of empirical functions in (15) yield fw=0.0125EtD, and Phillips (1985) solutions (labeled P85) computed with three different assumptions of cmin in (19) are illustrated for comparison. Fig

work page 1985
[7]

Shown here are: (a) b as a function of cb, the frequently observed range of cb between 2 and 3.5 m s -1 in the open ocean data for wind speed higher than 5 m s-1 (Fig

Estimation of the breaking strength parameter b combining the empirical results of whitecap observations and breaking wave energy dissipation function applied to the Phillips (1985) whitecap f unction. Shown here are: (a) b as a function of cb, the frequently observed range of cb between 2 and 3.5 m s -1 in the open ocean data for wind speed higher than 5...

work page 1985
[8]

(b) Same as (a) but for the breaking wave energy 2 dissipation rate calculated with the wind wave growth f unction (Hwang and Sletten 2008)

(a) Field observations of whitecap coverage (MTRXLS and C08) and examples of three 1 empirical functions of cubic wind speed dependence. (b) Same as (a) but for the breaking wave energy 2 dissipation rate calculated with the wind wave growth f unction (Hwang and Sletten 2008). Field data 3 used in the calculation include fetch -limited wind seas (DM of DM...

work page 2008
[9]

11.14 of Gaudin (1957)

Terminal velocities of air bubbles in distill ed water and contaminated water, reproduced 1 from Fig. 11.14 of Gaudin (1957). Similar results are also reported by Clift et al. (1978, Fig. 7.3) 2 and Leifer et al. (2000, Fig. 4). 3 Fig

work page 1957
[10]

yield 2 fw=0.0125EtD, and Phillips (1985) solutions (labeled P85) computed with thr ee different 3 assumptions of cmin in (19) are illustrated for comparison. 4 Fig

work page 1985
[11]

Shown here are: (a) b as a 2 function of cb, the frequently observed range of cb between 2 and 3.5 m s -1 in the open ocean data for wind speed higher 3 than 5 m s -1 (Fig

Estimation of the breaking strength parameter b combining the empirical results of whitecap observations and 1 breaking wave energy dissipation function applied to the Phillips (1985) whitecap function. Shown here are: (a) b as a 2 function of cb, the frequently observed range of cb between 2 and 3.5 m s -1 in the open ocean data for wind speed higher 3 t...

work page 1985

[1] [1]

Anctil, F. and M. A. Donelan, 1996: Air-water momentum flux observed over shoaling waves. J. Phys. Oceanogr., 26, 1344-1353. Anguelova, M. D., and F. Webster, 2006: Whitecap coverage from satellite measuremen ts: A first step toward modeling the variability of oceanic whitecaps. J. Geophys. Res. , 111, C03017, doi:10.1029/2005JC003158. Black, P. G., and W...

work page doi:10.1029/2005jc003158 1996

[2] [2]

Hwang, P

(Corrigendum, 35, 268-270, 2005). Hwang, P. A., and D. W. Wang, 2004b: An empirical investigation of source term balance of small scale surface waves. Geophys. Res. Lett., 31, L15301, doi:10.1029/2004GL20080. Hwang, P. A., M. A. Sletten, and J. V. Toporkov, 2008: Analysis of radar sea return for breaking wave investigation. J. Geophys. Res., 113, C02003, ...

work page doi:10.1029/2004gl20080 2005

[3] [3]

Size PDF of the resulting daughter bubbles. J. Fluid Mech., 401, 183–207. Martínez-Bazán, C., J. Rodríguez- Rodríguez, G. B. Deane, J. L. Montañés, and J. C. Lasheras, JPO 31 BuoyancyWhitecapR1noline.doc 2010: Considerations on bubble fragmentation models. J. Fluid Mech., 661, 159-177. Melville, W. K., and P. Matusov, 2002 : Distribution of breaking waves...

work page 2010

[4] [4]

(b) Same as (a) but for the breaking wave energy di ssipation rate calculated with the wind wave growth function (Hwang and Sletten 2008)

(a) Field observations of whitecap coverage (MTRXLS and C08) and examples of three empirical functions of cubic wind speed dependence. (b) Same as (a) but for the breaking wave energy di ssipation rate calculated with the wind wave growth function (Hwang and Sletten 2008). Field data used in the calculation include fetch -limited wind seas (DM of DMAJ and...

work page 2008

[5] [5]

11.14 of Gaudin (1957)

Terminal velocities of air bubbles in distill ed water and contaminated water, reproduced from Fig. 11.14 of Gaudin (1957). Similar results are also reported by Clift et al. (1978, Fig. 7.3) and Leifer et al. (2000, Fig. 4). Fig

work page 1957

[6] [6]

Field observations of whitecap c overage (MTRXLS) and its linear dependence on the breaking wave energy dissipation rate. The three sets of empirical functions in (15) yield fw=0.0125EtD, and Phillips (1985) solutions (labeled P85) computed with three different assumptions of cmin in (19) are illustrated for comparison. Fig

work page 1985

[7] [7]

Shown here are: (a) b as a function of cb, the frequently observed range of cb between 2 and 3.5 m s -1 in the open ocean data for wind speed higher than 5 m s-1 (Fig

Estimation of the breaking strength parameter b combining the empirical results of whitecap observations and breaking wave energy dissipation function applied to the Phillips (1985) whitecap f unction. Shown here are: (a) b as a function of cb, the frequently observed range of cb between 2 and 3.5 m s -1 in the open ocean data for wind speed higher than 5...

work page 1985

[8] [8]

(b) Same as (a) but for the breaking wave energy 2 dissipation rate calculated with the wind wave growth f unction (Hwang and Sletten 2008)

(a) Field observations of whitecap coverage (MTRXLS and C08) and examples of three 1 empirical functions of cubic wind speed dependence. (b) Same as (a) but for the breaking wave energy 2 dissipation rate calculated with the wind wave growth f unction (Hwang and Sletten 2008). Field data 3 used in the calculation include fetch -limited wind seas (DM of DM...

work page 2008

[9] [9]

11.14 of Gaudin (1957)

Terminal velocities of air bubbles in distill ed water and contaminated water, reproduced 1 from Fig. 11.14 of Gaudin (1957). Similar results are also reported by Clift et al. (1978, Fig. 7.3) 2 and Leifer et al. (2000, Fig. 4). 3 Fig

work page 1957

[10] [10]

yield 2 fw=0.0125EtD, and Phillips (1985) solutions (labeled P85) computed with thr ee different 3 assumptions of cmin in (19) are illustrated for comparison. 4 Fig

work page 1985

[11] [11]

Shown here are: (a) b as a 2 function of cb, the frequently observed range of cb between 2 and 3.5 m s -1 in the open ocean data for wind speed higher 3 than 5 m s -1 (Fig

Estimation of the breaking strength parameter b combining the empirical results of whitecap observations and 1 breaking wave energy dissipation function applied to the Phillips (1985) whitecap function. Shown here are: (a) b as a 2 function of cb, the frequently observed range of cb between 2 and 3.5 m s -1 in the open ocean data for wind speed higher 3 t...

work page 1985