Bubble size distribution in surface wave breaking entraining process

Science in China Series D: Earth Sciences❽2007 SCIENCE IN CHINA PRESS旦SrinerBubble size distribution in surface wave breakingentraining processHAN Lei1.2t & YUAN YeLi?2↑Department of Ocean Environment, Ocean University of China, Qingdao 266003, China;2The First Institute of Oceanography, State Ocean Administration, Qingdao 266061, ChinaFrom the simlarlty theorem, an expression of bubble populatlon is derived as a functlon of the alr en-trainment rate, the turbulent kInetic energy (TKE) spectrum density and the surtace tension. The bubblesize spectrum that we obtain has a dependence of a2 5+rnd on the bubble radius, in which ng is positiveand dependent on the form of TKE spectrum within the viscous dissipation range. To relate the bubblepopulation with wave parameters, an expression about the alr entrainment rate is deduced by intro-ducing two statistical relations to wave breaking. The bubble populatlon vertical distribution is alsoderived, based on two assumptions from two typical observation results.wave breaking, bubble cloud, bubble sie spectrum1 Introductionbreaking waves, which may provide clues to establishthe relationship between the waves and turbulence".Surface wave breaking is an important process in theThis paper is dedicated to the bubble population dis-upper ocean. Wave breaking is accompanied by splashtribution. The bubble distribution is one of the majorof water droplets and entrainment of air bubbles. Thecontents in bubble research, because it is the basis forformer process forms water droplet cloud above the wa-evaluating the effects of bubbles on the upper ocean dy-ter surface, and the latter produces bubble cloud belownamics. In this paper, the bubble distribution includesthe water surface. Both of them are important outcomestwo aspects: the size distribution and the spatial distribu-of wave breaking and can potentially accelerate the gastion. The bubble size ditribution, called the bubble sizeand heat exchange between the atmosphere and thespectrum, can be expressed as N ~ d", where N denotesocean. Bubble cloud is an important factor in the re~the bubble population, and a the bubble radius. The keysearch on air-sea interface processes: Air bubbles form aissue of bubble size distribution research is to detemineremarkable information source in optical remote sensing.the index n. A lot of observations about empirical valuesBubbles can effectively scatter light, so that their exis-of n have been given since half a century which spreadtence remarkably influences total scattering ratio andin a wide range between -1 and - -5 according to differ-back scattering ratio2. Meanwhile, the sound gener-ent observersl-。 In section 2, we introduce some ob-ated by bubble formation and oscillation is one of theservation results about the bubble size distribution,most important sources for ocean noisel5- 5. Air bubblesThere are also theoretical results on the bubble size dis-also provide an effective way for all gases with lowtribution'. Considering their deficiencies, we introducesolubility in sea water to transfer from the atmosphere tonew govemning variables and derive a new expressionthe oceanb. Besides, the whitecap, one of the most im-form for the bubble size spectrum which is comparedportant characteristics of breaking wave is formed due togathering of air bubbles at the sea surface. Air bubbles中国煤化工could also be regarded as good tracers introduced byHCNMHGwww.scichina.com ww.springerink.comSct China Ser D-Earth Scil Nov.2007 |vol.501no. 11 11754-1760with the previous results in Section 3. In Section 4, we - -3.3. The result can be expressed aspropose two assumptions about the vertical distributionN(a)~Qe-"3a-10/3,(2.1)of bubble population. These two assumptions are ab-where Q is the air volume entrained by per volume ofstracted from two experimental results from Terrill etwater per second, and E the turbulent kinetic energy disal.I5] and Hwang et al.!". With these two assumptions, weextend the bubble population associated with radius N(a)sipation rate ( hereinafter TKED).Deane and Stokes 14 found out in their experimentto that associated with the radius and the depth Na, z).that bubbles with different size follows different size2 Previous research on bubble size spec-spectra. So they divided the bubble range into two sec-tions. For large bubbles, they adopted the analytical re-trumsult of Garret et al.U5!. For small bubbles, they intro-2.1 Observationsduced a jet velocity V and surface tension T/pw, andHaines and JohnsonI71 reviewed the results on bubblethen derived the bubble size spectrum with slope n =size spectrum's slope n in the past decadesl6.-. 251 (Ta--1.5.ble 1) and summarized the bubble size spetral slope nN(a) ~ Q(T1ρw )-312v2a312.(2.2)lies from-1.5 to -4.7. Such a big difference is mainlyThe above theoretical relationships match well withdue to two reasons: one is the different observation do-experimental results. However, there still exist problemsmains. Baldy' A suggested that the value ofn is approxi- in them. Although (2.1) assumes that turbulence playsmately -2 in‘bubble generation' zone, and -4 in deeperin important role in bubble formation during air en-‘bubble dispersion' zone, which is because the risingtrainment, the turbulent characteristic variable intro-speed of large bubble exceeds water entrained speed, Soduced in it is TKED E which is irelevant to the lengththat there are less large bubbles in the deeper zone. Thescale. Thus, the TKED cannot be used to account forother is probably the temporal evolution of bubble sizepopulation distribution for different sized bubbles. As aspectrum. Deane and Stokes"4 showed that the bubbleresult, it seems to be inappropriate to involve TKED insize spectral slope n has turmed to - 6 from -3.3 in onethe dimensional analysis of bubble size spectrum. Fur-and a half seconds. The comparison of bubble size spec-thermore, the bubble population is proportional to ε -I/3tra under different air void faction presented by Terrill etin (2.1), which means that when turbulent dissipationa1.!5] also shows quick variation of bubble size spectralincreases, the bubble population decreases instead ofslope with time. Therefore, during experiments in whichincreasing with stronger turbulence. To take the extremethe bubble entrainment is not an instantaneous process,situation as example, the bubble population approachesthe measured bubble size spectrum is actually a sort ofinfinity while there is no turbulence. This obviouslymixture of spectra at different evolving phases. Since thecontradicts with common sense,bubble size spectral slope changes rapidly with time,Deane and Stokesl4 noticed the fact that the bubbleslope of the 'mixture' is highly indefinite.size spectra have different slopes at the different bubble.2 Theoretical studiessize. Thus, they divided the bubble radius range into twoGarrett et al.!'3] deduced a bubble size spectrum with n =parts, and assumed different mechanisms in the twoTable 1 Sample lteraure on determinarions of bubble spectral slopetReferenceDiameter range (um)Peak (um)Depth (cm)[16]lab(F)/wind wave800-30009004-10-2,-4[18]near shore seabreaking wave750- 1500none0-4.7lab(S,F/watrfall160-40005007.5-14.5-3[20]open ocean30-640140150-800-3.5(21]coastal sea/background34-61010075-4.5[22]lab(S)ycontinuous waterfall50-8000100-300-1.5.-4 .[2311ab(F)/wind wave20- 100B02024]60- -3000中国煤化工-2,4[25]open ocean/breaking wave60-480-2.5,-4HHCNMHGHAN Lei et al. Sci China Ser D-Earth Scil Nov. 2007 Ivol. 50 Ino.11 175417601755separate ranges. While the two ranges are so adjacent todeformed bubble back to round. The following fiveeach other that it is improper to assume two kinds ofvariables are considered to control the bubble formationtotally different bubble generating mechanisms. Other-process: N: the number of bubbles per m' sea surface perwise there will be some kind of much more complexμm radius increment, whose dimension is L'; Sp: themechanism integrating both mechanisms above at leastturbulent kinetic energy spectral density, whose dimen-around the edge zone between the two ranges, becausesion is L'T ; Q: the volume of air entrainment per m'the effects by both mechanisms are comparable in such asea surface per second, whose dimension is LT-'; a: thezone. In other words, it is impossible for two adjacentrranges to be controlled by two independent mechanisms.bubble radius, whose dimension is L; - 二: the surfaceP.aMeanwhile the introduction of jet velocity v seems to betension, whose dimension is Lr-2.redundant. This velocity was introduced to account forthe fact that low-velocity jet air entrainment rates areThus, we can obtain three dimensionally consistentrelations from the above five variables:known to scale with 1Nevertheless, the quantity Qhas already included the meaning of air entrainment rateN~a3(3.1)in (2.2). The bubble size spectral slope has been shownS,~Q'a, .(3.2)by many experimental results to vary with the bubblesize consecutively not piecewise (Figure 1).Pwa~Q",(3.3)10 rwhich can be witten as three nondimensional variables:瓜= Na',(3.4)Sk(3.5)Q2a’0-β: 10-7-10*(3.6)pwQ2a”-: 10*-10-5According to the similarity theorem, the three nondi--B. 10~s- 10~4mensional variables must satisfy some functional rela--β> 10→tionship. So it might be written as some kind of product10of power functions such asBubble radius (μm)Figure 1 Calculation of the average bubble size spectrums with dferent(TIP(3.7)void faction pl16l.~-(Q'a )orAccording to the discussion above, we find that thereare a lot of problems with current analytical results ofN~ S."(a):(T/P.nL.a-3.(3.8)bubble size spectra. Thus, we will propose a new(Q2a)+tmethod for deriving the bubble size spectrum and try towhere m and I are undetermined coefficients.solve the problems mentioned above.The relation of bubble population N and the air en-trainment rate Q is derived below. The total air volume3 A new bubble size spectrum modelcontained in air bubble cloud can be witen as3.1 Derivation of the theoretical relationQo= s"N -r'd, .(3.9)Considering the bubble generation process, we assumewhich is supposed to be proportional to the air entrain-that the air entrainment is the origin of air bubbles, theturbulent intensity with the different scale is the cause ofment rate Q, i.e.,forming dfferent size bubbles, and the surface tension is~x.4mn3dq(3.10)the resistance preventing larger bubbles being tom into中国煤化工”smaller bubbles for the suface tension always pulls theMYHCNMHG1756HAN Lei et al Sci China Ser D Eath Scil Nov. 2007 Ivol. 501no. 11 I 1754-1760Q∞N(Q) I(3.11)Since there is only one relation, we can obtain only onenondimensional variable. Thus, the following relation isthen we derivederived:N(Q)xQ,(3.12)(3.15)which means m+l=-1/2 must hold true in (3.8).Thus, the bubble size spectrum can be witten further asFrom (3.15), air entrainment rate is irelevant to theN~S."(a)-(TIp.w)~m~0SQ.a2S. (3.13)breaking area ratio. Substituting (3.15) into (3.13) yieldsSince the bubble population increases with the turbulentN(a)~-S"(a).a-25.(3.16)intensity, the exponent m must be a positive value. So mp.c;2 T1p,.,m5+ 0.5 must be positive too, implying that the strongerRelation (3.16) is the expression of bubble size spectrumsurface tension, the larger bubble population. Fromwe derive here, in which m is an undetermined positive(3.13) we can see that bubble spectral slope is affectedcofficient.by the TKE spectral density, but not by the surface ten-3.2 Discussionsion. The surface tension only influences the number ofbubbles. For example, it is inclined to produce moreThe bubble size spectrum (3.16) appears to be morebubbles in sea water than in fresh water as a larger sur-reasonable compared to the previous results in the fol-face tension coefficient is larger.lowing several respects.To further derive the expression of air entrainmentFirst, we introduce the TKE spectral density insteadrate Q, we analyze the air entrainment process still usingof the TKE dissipation rate as the characteristic parame-the similarity theorem. Assume that there are four com-ter of turbulent intensity. It explains the reason why dif-ferent sized bubbles have different populations, i.e, thebinations of variables controlling this process:2.thegeneration of some certain sized bubbles is controlled bythe turbulent intensity in the similar scale as this bubbleentrained air volume per breaking area, whose dimen-size.sion is L;: the energy loss per breaking area, whoseSecond, the surface tension is thought to be the mainS;resistance in the bubble generation process for the rea-M，son that surface tension always pulls back the deformeddimension is学; Cg; the group velocity of breakingbubble into its original round shape. Therefore we applythe surface tension to all the bubbles not only to smallwave, whose dimension is ; Pw: the sea water den-ones. As a result, the bubble size spectrum for both largesity, whose dimension isMand small bubbles can be expressed in a uniform expres-sion.Third, the bubble size spectrum we derived hereThe air entrainment rate Q still denotes entrained airshows larger population with stronger turbulence, andvolume per sea surface area per second, E, denotes theenergy loss due to breaking per sea surface area perwith weaker surface tension. This coincides with theground truth. For example, Scott!27] found in his experi-second, and S; denotes the breaking area per sea surfacement about air entrainment that the bubble populationarea per second. The reason why we introduce a groupformed in sea water is far more than that in fresh watervelocity rather than a phase speed is that wave breakingis more like a property of wave group not a singlewith larger surface tension coefficient.Fourth, the bubble size spectral slope will graduallywavel20o. Here we obtain three basic dimensions: lengthdimension L, time dimension T and mass dimension M.become smaller or steeper with the time. While the bub-ble spectrum (3.16) only focuses on the initial stage ofFirst, we select three basic variables E/S, C& Pw and use)ubble generation, not considering the bubble cloudsthem to express the remaining variable QIS, then deriveevolving stage. If we assume the turbulent kinetic en-a dimensionally consistent relation:ergy distribution S(k) in eq, (3.16) has the dependency2._ 1 (旦)(3.14)中国煤化工)~a (nu>0), theS, p.c2(s,)slope|YCN M H Gnes -2.5 + ns, whereHAN Lei et al. Sci China Ser D -Earth Scil Nov. 2007 vol. 50 Ino. 11 I 1754-17601757nd depends on the form of the turbulent kinetic energy4 Vertical distribution of bubble popula-distributed over the turbulent dissipative region, whichtionagrees with the experimental result of Hwang et al.1o),Baldyl24，Medwin and Breitz45] (see Table 1). Further-In this section, we will propose two assumptions ac~more, our results suggest a better agreement with theo-cording to the two experimental results about the bubbleretical result by Baldyl281 which gives the bubble sizepopulation at different depths. Based on these two as-spectral slope equal to -2 in the initial stage of bubblesumptions, we will extend the bubble size spectrum orgeneration.bubble population varying with the bubble radius N(a)Fifth, the change of the spectral slope n with the bub-to that varying with both the bubble radius and the depthble radius has been revealed in many observa-N(a, z).tions'1,1,1,1While according to (3.16), this property ofAssumption 1: Bubble size spectra at the differentbubble size spectrum may be explained by the change indepth share the same distribution patterms. This assump-the TKE spectral slope with the scale,tion was derived from Temill et al. [151 (Figure 3). FromMoreover, it is worth mentioning about the conver-Figure 3 we can see that the bubble size spectra are al-sion process between (3.16) and (2.1)5. If we adopt themost parallel to one another. Thus we assume that theclassic‘- 5/3' law for the turbulent spectrum and neglectbubble size spectrum is independent of the depth, in thethe surface tension, the bubble size spectral slope will beform ofN(a,z)= 8(z).f(a),(4.1)reduced to -10/3, which is the same value as that ofGarrett et al."5. However, this conversion process can-where g (z) andf(a) are two arbitrary functions.not be applied here. The inertial subrange for which‘- 5/3’law291 is applied is approximately lying between0(1) cm and 0(100) cm for marine turbulence (Figure●0.7m●1.3m2), while the bubbles entrained by wave breaking aremostly of the size between 0(10) μm and 0(10) mml14]104口4.1 m .There is almost no overlap for both ranges. While insection 3.2, we assume that the bubbles' formation de-pends on turbulent kinetic energy S(k), where kt is con-nected with bubble size. Since bubble size is not locatedwithin the inertial subrange, S(k) cannot take the form of1510020*Bubble radius (μm)-5/3 power law applied in inertial subrange, whichmakes the application of Kolmogorov's“-5/3' law inap-Figure 3 The bobble size distributions simultancously measured in tbefield a four dfferenet depths during a large breaking evealelsplicable in deriving Garett et al's resul').Assumption 2: The total bubble population follows anexponential distribution with the normalized depth. Thisassumption is derived according to Huang et al.1"6] (Fig-0-ure 4). The figure shows a good agreement between theexponential distribution and observations at variouswind speeds. Thus, assumption 2 can be written as." "N(a,z)da=e-lH. .”"N(a,)da, (4.2)-5/07where H, is the significant wave height.Based on assumptions (4. 1) and (4.2), the expressionof bubble population N(a, z) can be derived from the10-8following procedures:10-、10x(cm")Interating both sides of (4.1) vieldsFIgure2 Ocean wurbulent kinetic spectrum!ol.中国煤化工. f(a)da(4.3)TYHCNMHG1758HAN Lei et al. Sci China Ser D-Earth Sci1l Nov. 2007 Ivol. 50 I no. 11 11754-176010namics. In this study, we propose a new set of control-Wind velocityling variables to model the processes compared with●10m/s变。II m/sprevious investigations. The essential changes are: in-。12 m/s●13 m/stroducing the TKE spectrum instead of the TKE dissipa-●I4 m/stive rate to describe the turbulent intensity, and applying●I5 m/sthe surface tension for all the bubble size not only forsmall bubbles. We derived the bubble size spectrumwhich may be applied for all the bubble sizes. The spec-tral slope can be expressed as -2.5+ nd, in which ns de-pends on the TKE spectral density within viscous dissi-pative range, and is usually a function of length scale a.Normalized depth, h,The results we derived here fix some problems in theFigure 4 Normalized vertical distributions of bubble populations atprevious analytical results, also agree well with observa-various wind speesl.[24,251and theoretical results!An important quantity in the bubble size spectrum is01the air entrainment rate Q. There is no specific discus-f"N(a,2)dasion about this quantity in the literature. We derive theg(z)=(4.4)expression for air entrainment rate with two wave5”f(adabreaking related statistical variables introduced by YuanSubstituting (4.4) into (4.1) yieldset al.bIl in their wave breaking model. The result (3.15)N(a,z)=e~ z/h 8(0)f(a)=N(a,0)e -1作。(4.5)shows that the air entrainment rate is proportional to theIntegrating (4.5) at bubble entrainment depth he yieldsenergy loss due to wave breaking, and is independent ofthe total underwater bubble population:the wave breaking area per m' of sea surface. The ex-pression of air entrainment rate correlates the bubbleN(a)= N(a,0)+(1-e h )h,(4.6)population with wave parameters and wind speedthrough the model by Yuan et al.5nI. Our expression isSubstuting (4.6) into (4.5) yieldslooking forward to being venified by more observations.The spatial distribution of bubble population's isN(a,z)= N(a)(l-e(4.7)mainly about the distribution with the depth. We obtainNow we have derived the bubble population at a certaintwo assumptions on the vertical distribution of bubbledepth N(a, z) as a function of that at all the depths N(a).population from observations, and derive the relation-ship between the bubble population at a certain depthSubstituting (3.16) into (4.7) yieldsN(a, z) and that at all depths N(a). Thus, we extend theN(a,z)=C-E，_ S,"(a) _ax2s(-e”厂bubble population formula with bubble radius to thatP.c,2 (TIp:)m+t05h,varying with both bubble radius and water depth.The observations show that bubble size spectral slope(4.8)becomes smaller (larger in absolute value) with time, orwhere m is a positive coffcient to be undetermined, Cin other words, the bubble size spectrum becomesis a proportional factor. (4.8) is the expression of wavesteeper with time. According to Terill et al!tl, the bub-breaking-entrained bubble population as a function ofble size spectral slope n turms to -5 from - -3 in a certainbubble radius and water depth, which is also the mainperiod. The reason for this change is that larger bubblesanalytical result of this paper.rise more quickly and disappear earlier in bubble clouds.The temporal evolution of the bubble size spectrum of5 Conclusions and discussionsthis kind can explain the fact that different observationsThe generation process and distribution of bubble cloudsof bubble size spectrum have relatively large differencesentrained underwater by wave breaking is the premisein th中国煤化工xcause the governingfor the effects of bubble clouds on the upper ocean dy-varial;YHC N M H G analysis mainly reHAN Lel et al. Sci Chine Ser D-Earth Scil Nov. 2007 Ivol.50 Ino. 11 11754-17601759lates with bubble entrainment process, not including thestage of bubble cloud evolution, such as rising by buoy-**。。Moeled from TKE sectrum in Figure2.0# Observation from Deane and Sokes (2002)ancy, the result we derive in this paper is closer to ob-servation result of bubble size spectrum at the initialstage of bubble formation. The evolving stage needsfurther study in the future.叫The bubble size spectral slope we derive here is de-opendent on the TKE spectral density in the length scaleof bubble size. Figure 5 demonstrates model results ofbubble size spectrum from the TKE spectral density in102ocean (shown in Figure 2). The model constant mtakes a value of 0.4, which suggests that the range of10Bubble radius (μum)model result cannot cover that of the bubble size spec-Figure 5 Model and Observation resuts for bubble size spectum.trum because of lacking observation result in the mi-cro-scale range of the TKE spectral density in ocean,Therefore more intensive acquaintance is required toWe would like to thank Prof. Zherg Quanan from University of Marylandcompare the model result with direct observation ofand Xu Vifrom University of Maine for thoroughly revising the originalmanuscript. We also thank the anornymnous reviewers for careful reviewsbubble size spectrum.and constructivte coments on improving our original manuseript.1 Stranski D. Gas Microbubbles: An Asessmeat of their Sgnifcancewind waves : a laboratory study. JPhys Oceanogr, 1990, 20: 19-28to Light Sattering in Quiescent Seas. In: Jafe J s, ed. Oocean Optics7 Haines M A. Johnson B D. Injected bubble in seawater and freshwalerXII. Washington: Int Soc for Opr Eng, Belingham, 1994. 704- -710measured by a photographic method. J Geopbys Res., 1995, 100:2 Stramski D, Tegowski J. Effects of Intermittent Entrainment of Air7057- 7068Bubbles by Breaking. J Gcophys Res, 2001, 106: 31345 -3136018 Blanchard D C. Woodecock A H Bubble formation and modifcation in3 Pumphery HC, Crum L A. Acoustic Emissions Associated wth Dropthe sca and its meteorological significance. Tellus, 1957, 9: 145- 158Impacts. In: Kerman B R ed. Sea Surface. Dordrecht; Kluwer Aca-9 Monahan E C, Zictlow C R Laboratory comparisons of freshwaterdemic Publishers, 1988. 151- 172and saltwater whitecaps. J Geophys Res, 1969, 74: 6961 -69664 Lu N Q, Prospereti A, Yoon S W. Underwater Noise Emissins from0 Kolovayev P A. Investigation of the concentation and satistical sizeBubble Clouds. IEEEJ Ocean Engineerig. 1990, 15: 275- 281distribution of wind-produced bubbles in the near-surface ocean layer.5 Yoon s W, Crum L A. Properecti A et al. An invesigation of theOceanol Engl Thans, 1976. 15: 659-661cllcive oslations of 8 bubble cloud. J Acoust Soc Am, 1991, 89:21 Johnson B D, Cooke R C. Bubblepopulations and Spectma in CoasalWaler:a Photographic Approach. J Geopbys Res, 1979, 84:6 Woolf D K, Thorpe s A. Bubbles and the air-sea exchange of gases in3761 -3766near- saturation conditions. J Mar Res, 1991, 49: 435- 46622 Cipriano R Blanchard D C. Bubble and aerosol spectrs produced bya7 Thope S A. Some factors afecting the size distribution of oceanlaboratory breaking wave. J Geophys Res, 1981, 86: 8085- 8092bubbles. J Plhys Oceanogr, 1992, 22: 382-3893 Broccker H Ch, Siems w. The role of bubbles for gas transfer fromMedwin H In siu acoustic measurements of bubble populations inwater to air at higher wind speeds. Experiments in the wind-wave fa-coastal waters. J Geophys Res, 1970, 75: 599- -611cility in Hamburg. In: Brutsaert w, Jirka G H, eds. Gas Transfer atThorpe s A. On the clouds of bubbles formed by breakingWater Surfaces. Norwell: D. Reidel, 1984. 229- -236wind-waves in deep water, and their role in air-sca gas transfer. Phil24 Baldy s. Bubbles in the close vicinity of breaking wvestaisticalTrans R Soc Lond, 1982, A304: 155-210charnctrstics of the generation and dispersion mechanism. J Geo-10 Farmer D M, Vagle S. On the determination of breaking surface wavephys Res, 1988, 92: 2919-2929disuributions using ambient sound. J Gcophys Res, 1988， 93:25 Medwin H, Breitz N D. Ambient and transient bubble spetral densi-3591 -3600ties in Quiescent Scas and under siling breakers. J Geopbys Res,11 Famner D M, Vagle S, Booth A D. A re-fooding acoustical reso-1989, 94: 12751- 12759nator for measurement of bubble size distribution. J Atmos Oceanic26 Rapp R J, Melville W K. Laboratory measurements of deep-waterTech, 1998, I5: 1132- 1146breaking waves. Phil Trans R Soc London, 1990, A331: 735-80012 Terill EJ, Melville W K. A broadband acoustic technigue for meas-27 ScottJ C. The role of salt in whitecap pristance. Deep Sea Research,uring bubble size distribution:laboratory and shallow water meas-1975. 22: 653- -657urements. J Atmos Oceanic Tech, 2000, 17: 220- -23928 Baldy S. A generation-dispersion model of ambient and transicnt1:Garret C, Li M, Farmer D. The conncction between bubble sizebubbles in the close vicinity of breaking wave. J Geophys Res, 1993,spectra and energy dissipation rates in the upper occan. J Phys98: 18277- 18293Oceanogr, 2000, 30: 2163- -21719 Kolmogorov A N. Enery dssipation in loallyl isotropic turbulence.14 Deane G B, Stokes M D. Scale Dependence of Bubble CreationDoklady AN SSSR, 1941,32: 19-21Mechanisms in Breaking Waves. Nature, 2002, 418: 839 -84430 Paka v T, Nabatov v N, Lozovatsky1D, et al. Oceanic microstructure15 TerillE 1, Melille W K, Stramski D. Bubble entainment bymeasurements by "Baklan" and“Grif",J Atmos Oceanic Tech, 1999,breaking waves and their infuence on optical satteing in the upperocean. J Geophys Res, 2001, 106(C8): 16815- 1682316 HwangPA, Hsu Y H L, Wu J. Air bubbles produced by breakingHC N M H 932457-51760HANLei et al Sci China Ser D-Earth Sci| Nov.2007 Ivol.50 Ino.11 1754-1760