Asymptotic light field in the presence of a bubble-layer

Asymptoticlightfieldinthepresenceofa bubble-layer Piotr J. Flatau Scripps Institution of Oceanography, University of California, San Diego, La Jolla, California 92093-0221 pflatau@ucsd.edu Jacek Piskozub Institute of Oceanology, Polish Academy of Sciences, 81712 Sopot, Poland piskozub@iopan.gda.pl J. Ronald V. Zaneveld Oregon State University, College of Ocean and Atmospheric Science, Corvallis, OR 97331 zaneveld@oce.orst.edu Abstract: Wereportthatthesubmergedmicrobubblesareanefficient sourceofdiffuseradianceandmaycontributetoarapidtransitionto thediffuseasymptoticregime.Inthisasymptoticregimeanaverage cosineiseasilypredictableandmeasurable. c⃝1999OpticalSocietyofAmerica OCIS codes: (010.4450)Oceanoptics;(290.4210)MultipleScattering References and links 1. J.R.V.Zaneveld.“Anasymptoticclosuretheoryforirradianceintheseaanditsinversionto obtaintheinherentopticalproperties,”Limnol.Oceanogr. 34,1442–1452(1989). 2. N.J.McCormick.“Mathematicalmodelsforthemeancosineofirradianceandthediffuseatten- uationcoefficient,”Limnol.Oceanogr.40,1013–1018(1995). 3. T.T.Bannister.“Modelofthemeancosineofunderwaterradianceandestimationofunderwater scalarirradiance,”Limnol.Oceanogr.37.773–780(1992). 4. J.Berwald, D.Stramski,C. D.Mobley, and D.A.Kiefer.“Influences of absorption and scat- teringonvertical changes inthe average cosineof the underwater light field,”Limnologyand Oceanography 40,1347–1357(1995). 5. D.Stramski.Gasmicrobubbles:Anassessmentoftheirsignificancetolightscatteringinquiescent seas.In OceanopticsXII:13-15June1994,Bergen,Norway ,JulesS.Jaffe,editor,Proc.SPIE v.2258,704–710(Bellingham,Wash.,USA,1994). 6. R.Frouin,M.Schwindling,andP.-Y.Deschamps.“Spectralreflectanceofseafoaminthevisible andnear-infrared:Insitumeasurementsandremotesensingimplications,”J.Geophys.Res. 101, 14361–14371(1996). 7. CurtisD.Mobley. Lightandwater:radiativetransferinnaturalwaters (AcademicPress,San Diego,1994). 8. P. J. Flatau, M. K. Flatau, J. R. V. Zaneveld, and C. Mobley. “Remote sensing of clouds of bubblesinseawater,” Q.J.Roy.Met.Soc.(1999)(tobepublished). 9. D.M.FarmerandD.D.Lemon.“Theinfluenceofbubblesonambientnoiseintheoceanathigh windspeeds,”J.Phys.Oceanogr. 14,1762–1778(1984). 10. S.A.Thorpe.“Dynamicalprocessesoftransferattheseasurface,”Prog.Oceanogr. 35,315–352 (1995). 11. B. D. Johnson and P. J. Wangersky. “Microbubbles: stabilization by monolayers of adsorbed particles,”J.Geophys.Res.92,14641–14647(1987). 12. H.Medwin.“Insituacousticmeasurementsofmicrobubblesatsea,”J.Geophys.Res.82,971–976, (1977). (C) 1999 OSA 30 August 1999 / Vol. 5, No. 5 / OPTICS EXPRESS 120 #11948 - $15.00 US Received July 23, 1999; Revised August 26, 1999

13. K.Isao,S.Hara,K.Terauchi,andK.Kogure.“Roleofsub-micrometreparticlesintheocean,” Nature 345,242–244(1990). 14. S.A.Thorpe,P.Bowyer,andD.K.Woolf.“Somefactorsaffectingthesizedistributionsofoceanic bubbles,”J.Phys.Oceanogr.22,382–389(1992). 15. P.J.Mulhearn.“Distributionofmicrobubblesincoastalwaters,”J.Geophys.Res. 86,6429–6434 (1981). 16. J.Piskozub. Effectsofsurfacewavesandseabottomonself-shadingofin-wateropticalinstru- ments.In OceanopticsXII:13-15June1994,Bergen,Norway ,JulesS.Jaffe,editor,Proc.SPIE v.2258,300–308(Bellingham,Wash.,USA,1994). 1 Introduction Theverticalstructureoflightintheseaisimportantinmarinebio-optics.Sunlightis theenergysourceforthebiologicalfoodchain,andtheamountandspectrumofsolar energyavailableatagivendepthmustbeknownifaccurateproductivitycalculations aretobemade.Theverticalstructureofthediffuseattenuationcoefficientinthenear surfaceregimeisimportantinunderstandingopticalremotesensing.Thebehaviorof thelightfieldintheseaisdescribedbytheequationofradiativetransfer,whichre- latesthelightfieldtotheinherentopticalpropertiesofwateranditsconstituents[1]. Oncethisequationissolvedcertainmomentsofthelightfieldcanbederivedandcom- paredwithmeasurements.Ofparticularinterestistheaveragecosineofirradianceat depthzwhichisdefinedby ¯µ(z)= E(z)/E 0(z),whereE0(z)isthescalarirradiance andE(z)isthenet(vector)irradiance,bothwithunitsWm −2.Thedownwellingav- eragecosineis ¯µd(z)= Ed/E0,where Ed isthedownwellingirradiance.Theaverage cosine,whichcanvarybetween-1and1,givesdirectionalinformationabouttheradi- ancedistribution.Theaveragecosineisalsorelatedtothediffuseattenuationcoefficient K(z)= −1/E(z)dE(z)/dz.Integrationoftheradiativetransferequationoveralldirec- tionsleadsto dE(z)/dz= −a(z)E 0(z)sothat¯µ(z)= a(z)/K(z),whichisknownas Gershun’sequation.Wecannowmeasurea(z)and K(z)routinely,sothatitispossible toverifypredictionsoftheverticaldistributionof¯µonthebasisofobservations.Anum- berofmodelsoftheverticalstructureoftheaveragecosineofthelightfieldhavebeen presented[1-4]buttheyhaveallbeenlimitedtohomogeneousmediawithaflatsurface. Ontheotherhand,usingthedatafromtheGulfofCaliforniaandtheEastCoastshelf oneofus(JRVZ,unpublished)concludedthattheaveragecosinemodelsdonotfitthe determinationof ¯µ(z)from K(z)and a(z).Itisspeculatedthatthisdeparturefrom theoryisduetoverticalinhomogeneitiesintheinherentopticalproperties(IOPs)and thepossibilitythatthelightfieldinthesurfacelayeroftheoceanmaybemorediffuse thantheoreticalmodelsassume.Thisdiffuselightfieldmaybeduetohazeandclouds intheatmosphere,relativelylowsolarzenithangles,surfacewaves,andmicrobubbles inthesurfacelayer.Acombinationofthesefactorscontributestothelightfieldbeing quitediffusethroughoutthewatercolumn.Inthisstudywedemonstratethecapacity ofairbubblestobeanefficientscatterer. 2R e s u l t s Thereislimitedknowledge[5-7]abouttheradiativetransferpropertiesofbubbleclouds, theirinherentopticalproperties,andtheirglobalclimatology.Recently,wereported[8] ontheinfluenceofsubmergedbubblecloudswithinthewaterontheremotesensing reflectance.Individualbubblecloudspersistforseveralminutesandaregeneratedby breakingwaves.Thereisevidencethatathighwindspeeds,separatebubbleclouds nearthesurfacecoalesce,producingastratuslayer[9,10].Themajorityofbubblesin- jectedintothesurfacelayersofnaturalwatersareunstable,eitherdissolvingdueto enhancedsurfacetensionandhydrostaticpressuresorrisingtotheair-waterinterface (C) 1999 OSA 30 August 1999 / Vol. 5, No. 5 / OPTICS EXPRESS 121 #11948 - $15.00 US Received July 23, 1999; Revised August 26, 1999

Wind Velocity 8 m/s Bubble Clouds 0m -6m -3m Fig. 1. Bubble-stratus layer generated by breaking waves. Bubble population is continually replenishedbywave activity (breaking waves atthe surface)leading eventuallytosemi-homogeneouslayer. wheretheybreak[11].However,bubbleswithlongresidencetimes,i.e.stablemicrobub- bleshavebeenobserved[12-14].Oneofthestabilizationmechanisms[15,11]assumes thatthesurfactantmaterialisanaturaldegradationproductofchlorophyll,presentin photosynthesizingalgae. Weconsiderthreesimplesituations:(1)“infinitely”deephomogeneousoceancom- posedofCDOM,water,andparticulates;(2)“infinitely”deephomogeneousoceancom- posedofthesameCDOM,water,andparticulatesbutalsowithbubbles;(3)twometer layerofbubblesand“background”CDOM,water,andparticulates.InFig.1thetwo- layercaseispresented.(basedon[10]). Thebackgroundwater,CDOM,andparticulatesarecharacterizedbyabsorption andscatteringcoefficientsa=0.1801m −1 andb=1.2525m−1 (thusweincludewater aandb).Itcorrespondstoachlorophyllconcentrationof10mgm −3 atawavelength λ=550nm.TheparticulatephasefunctionisthatofPetzoldforturbidwater[7].The phasefunctionforbubbleswascalculatedbyaveragingfromsizeparameter x=10 −350, wherex=2πr/λ,risradius.Thesizedistributionfollowsa r−4 law.Aphasefunc- tionwasderivedusingtheexactmethodforspheresandweassumearefractiveindex n=3/4.Thescatteringcoefficientforbubblesisassumedtobe b bubble=0.6181m−1.In ourrecentpaper[8]wediscussindetailtheclimatologyofbubblelayerdepth,vertical distribution,anddependenceofdepthonthewindspeed.Assumptionsmadehereare typicalfor10m /swindsinbubble-stratusregime.Theradiativetransfercalculations wereperformedusingtheMonte-Carlotechnique[16].Themainresultsofthisnote arepresentedinFigs.2-3.Fig.2showstheaveragecosineforthedownwellingradia- tionforthethreecasesdiscussedabove.Inthisfigure“circles”areforthetwo-layer system.Atfirstthereaisrapiddecreaseinthevalueoftheaveragecosineasinitially collimatedphotonsareefficientlyscattered.Afterexitingthe“bubble”layerthelightis alreadydiffuseandonlyasmalladjustmentisneededtoattainanasymptoticregime forthe“nobubbles”situation.Thisdouble-exponentialtransitionmechanismseemsto beparticularlyefficientinestablishingthenear-surfacediffuselightfield.Clearly,the depthofthebubblelayerandtheinherentopticalpropertieswillfurtherdeterminethis efficiency.Fig3shows¯µ(z),whichcontainscontributionfromtheupwellingradiation aswell.Theupwellingradiancefieldadjustsgraduallywhenphotonsinteractwiththe “bubble”layer.Thereispreferentiallossofphotonstravelinginthehorizontaldirection. Thisleadstomorecollimatedlightandincreasedaveragecosineclosetothesurface.In reference[8]wediscussinmoredetailtheclimatologyofbubbleclouds,whichstrongly dependsonwindspeed.Theimportanceofbubblesonanasymptoticlightfielddepends onwindspeed,windgustiness,andmicrophysicalpropertiesofbubbles.Incomingfield projectsandlaboratorystudiesshouldgiveacleareranswerontheinfluenceofbubbles onmarinelightfields. Insummaryweshowthatthesubmergedmicrobubblesarean efficientsourceofdiffuseradianceand,ifpresent,contributetorapidtransitiontothe (C) 1999 OSA 30 August 1999 / Vol. 5, No. 5 / OPTICS EXPRESS 122 #11948 - $15.00 US Received July 23, 1999; Revised August 26, 1999

0.70 0.75 0.80 0.85 0.9 z [m] 0 2 4 6 8 No bubbles 2m bubble layer Inf. bubble layer average cosine (down) Fig. 2. Average cosine for downwelling radiation ¯µd(z) for (1) “infinitely” deep homogeneousoceancomposedofCDOM,purewater,andparticulatescorrespond- ingto chl= 10mgm −3 (triangles); (2) “infinitely” deep homogeneous ocean with bubbles(cubes);(3)two-layersystemcomposedofbackgroundCDOM,water,and particulatesand2mlayerofsubmergedbubblesclosetothesurface(circles). average cosine 0.60 0.65 0.70 0.75 0.80 z [m] 0 2 4 6 No bubbles 2m bubble layer Inf. bubble layer Fig.3.Sameas2butfortheaveragecosine¯ µ(z). (C) 1999 OSA 30 August 1999 / Vol. 5, No. 5 / OPTICS EXPRESS 123 #11948 - $15.00 US Received July 23, 1999; Revised August 26, 1999

diffuseasymptoticregime. Acknowledgements P.J.FlatauwassupportedinpartbytheOfficeofNavalResearchYoungInvestigator Program.J.PiskozubhasbeensupportedinpartbytheInstituteofOceanologyPAS andbyOfficeofNavalResearchEuropeVisitingScientistProgram(VSP).J.Ronald V.ZaneveldacknowledgessupportoftheEnvironmentalOpticsprogramoftheOffice ofNavalResearchandtheBiogeochemistryprogramofNASA. (C) 1999 OSA 30 August 1999 / Vol. 5, No. 5 / OPTICS EXPRESS 124 #11948 - $15.00 US Received July 23, 1999; Revised August 26, 1999