# gupea_2077_41805_2

THESISFORTHEDEGREEOFDOCTOROFPHILOSOPHY EstimatingtheAir‐Water GasTransferVelocityduring LowWindConditions SAMTOREFREDRIKSSON FACULTYOFSCIENCE DOCTORALTHESIS UNIVERSITYOFGOTHENBURG DEPARTMENTOFMARINESCIENCES GOTHENBURG,SWEDEN2016 ISBN(Print):978‐91‐628‐9798‐7 ISBN(PDF):978‐91‐628‐9799‐4 SamToreFredriksson EstimatingtheAir‐WaterGasTransferVelocityduringLowWindConditions ISBN(Print):978‐91‐628‐9798‐7 ISBN(PDF):978‐91‐628‐9799‐4 Internet‐ID:http://hdl.handle.net/2077/41805 PrintedbyInekoAB Copyright©SamT.Fredriksson,2016 Distribution:DepartmentofMarineSciences,UniversityofGothenburg,Sweden II Abstract The abundances of atmospheric carbon dioxide, CO , and methane, CH , are increasing. These increasesaffecte.g.,theglobalcarboncycleandtheclimatebothregionallyandglobally.Tobetter understand the present and future atmospheric CO and CH concentrations and their climate impact, the gas exchange between water and the atmosphere is important. This exchange can occurintwodirections.OceanstakeupapproximatelyonethirdoftheanthropogenicCO release (theoceancarbonsink).Atthesametimecoastalwatersandinlandwatersemitlargeamountsof CO andCH ,altogethercorrespondingtoasimilaramountastheoceansink. Theinterfacialgas‐fluxforCO andCH iscontrolledbythewater‐side.Thegas‐flux, ,isfor where isthegastransfervelocity, suchgasestypicallyestimatedas and are the gas concentrations in the water bulk and in the air at the surface, and is the dimensionlessOstwaldsolubilitycoefficient.Thesubjectofthisthesisistodescribeandestimate for gases that have a water‐side controlled gas‐flux (e.g., CO , and CH ). Besides being importantforthegeophysicalsciences, isalsousedtodesignandoptimizemanyapplications ine.g.,chemicalandenvironmentalengineering. The transfer velocity is influenced by interfacial shear stress from wind, natural convection duetosurfaceheatflux,microscalebreakingwavesatmoderatewindspeeds,breakingwavesat high wind speeds, bubbles, surfactants, and rain. This thesis focuses on the low wind condition where the forcings due to shear stress, natural convection, and surfactants are important. The relative importance of buoyancy and shear forcing is characterized via a Richardson number ⁄ ∗ . Here , , and ∗ are the buoyancy flux, kinematic viscosity, and friction velocity, respectively. The thesis summarizes three papers where has been studied numerically with directnumericalsimulations(DNS)andonepaperwherefieldobservationshavebeenused. Theresultsfromthefieldmeasurementsshowcloserelationshipsforthemethodusing flux‐chambers and the parameterization using the rate of turbulent kinetic energy dissi‐ pation, and the quantities surface rms velocity and the significant wave height. A para‐ meterizationofarea‐integratedvaluesof fromwavemeasurementswasproposed. The DNS comprise flow conditions ranging from convection‐dominated to shear‐dominated cases. The results are used to: (i) evaluate different parameterizations of the air‐water gas‐ exchange,(ii)determine,foragivenbuoyancyflux,thewindspeedatwhichgastransferbecomes primarily shear driven, (iii) find an expression for the gas‐transfer velocity for flows driven by both convection and shear, and (iv) investigate the influence of surfactants on gas transfer velocity. Parameterizations using either the rate of turbulent kinetic energy dissipation or the horizontal surface flow‐divergence show a larger disadvantageous dependence on the type of forcingthantheparameterizationusingthesurface‐normalheat‐flux.Twoparametrizationsusing thewind‐speedabovethesurfacegivereasonableestimatesforthetransfer‐velocity,depending however on the surface heat‐flux. The transition from convection‐ to shear‐dominated gas‐ transfer‐velocity is shown to be at 0.004. This means that buoyancy fluxes in natural above approximately conditions are not important for gas exchange at wind velocities 3 .Belowthiswindspeedthebuoyancyfluxesshouldbetakenintoaccount. The transfer velocity is shown to be well represented by two different approaches: ⁄ ⁄ 1 ⁄ , where is a (i) Additive forcing as , ∗ critical Richardson number, and (ii) either buoyancy or shear‐stress forcing that gives ⁄ for and for . Here 0.4 and ∗ ⁄ istheSchmidtnumber, isthegasdiffusivityinwater,and 0.1areconstants, isanexponentthatdependsonthewater‐surfacecharacteristics. Keywords: air‐sea gas exchange, turbulence, heat flux, natural convection, shear, direct numerical simulations,gastransfervelocity,IR,flux‐chambers III Populärvetenskapligsammanfattning Mängdenväxthusgasersåsomkoldioxidochmetanökariatmosfären.Ökningenpåverkarbland annatvårtklimatsåvälglobaltsomregionaltochävenjordenskretsloppavkol. Eftersomnaturensträvarefterjämviktlederskillnadenikoncentrationmellanhavochlufttill en transport i riktning mot koncentrationsunderskottet. En ökning av koldioxidhalten i atmosfärenlederalltsåtillenökningavmängdenkolijordensvattendragochviceversa. Mängdenkolpåverkarisinturexempelvisvattnetssurhetvilketgörattlivetihavetförändras ochpåverkardeorganismersomfinnsihavetochdessnärhet. Förändringar i klimatet leder till förändrade temperaturer i hav och sjöar vilket i sin tur växelverkarmedbådeklimatetochmedhursnabbtkoncentrationsutjämningenmellanatmosfär ochhavsker.Dennaväxelverkanärkomplexochförattökaförståelsenförhurökadehalterav växthusgaserpåverkarjordenärmodelleristorochlitenskalabrahjälpmedel. För att ge trovärdiga och användbara resultat krävs att viktiga processer är modellerade så korrekt som möjligt. En sådan viktig process är med vilken hastighet en ojämvikt i koncentrationen mellan luft och vatten utjämnas. Denna hastighet beskrivs ofta som en övergångshastighet för olika gaser. Denna avhandling handlar om att kunna beskriva och modellera denna övergångshastighet (för vattenlösliga gaser för vilka gasflödet kontrolleras av vattensidan,såsomkoldioxidochmetan)såkorrektsommöjligt. Övergångenmellanvattenlöstkoldioxidochkoldioxidiluftenskergenommolekylärdiffusion och kan beskrivas med hjälp av Ficks lag där flödet bestäms av gasens diffusionskoefficient multipliceradmedgasenskoncentrationsgradientprecisundervattenytan. Dådetgällerkoldioxidtransportgenomenvattenytaberorkoncentrationsgradiententillstor delpåhureffektivtransportenavkoldioxidenärivattnet,vilketberorpåattkoldioxidblandarsig mycketlångsammareivattenäniluft.Gasflödetsflaskhalsbliralltsåvattnet. Gasens koncentrationsutjämning med hjälp av enbart diffusion, som kan ses som koncentrationsutjämning i en helt stillastående vätska, är en mycket långsam process medan koncentrationsutjämning med hjälp av rörelser (turbulens) i vattnet är betydligt snabbare. Turbulensens intensitet påverkas av konvektion på grund av temperaturskillnader i vattnet, vågor, hastighetsvariationer i vattnet på grund av vind, strömmar i vattnet, regn, bubblor, bottenegenskaperocheventuellytfilmvilkenframföralltpåverkarvattnetsytspänning. I detta arbete studerar vi övergångshastigheten som funktion av naturlig konvektion, vind (vid relativt låga vindhastigheter), djup, ytfilm och vågor. Studien har utförts med hjälp av fältmätningar vid Bornö forskningsstation i Gullmarsfjorden, men framförallt genom numerisk modellering. Resultaten kan sammanfattas i att övergångshastigheten vid låga vindhastigheter kan beskrivas med hjälp av naturlig konvektion i vattnet och den skjuvkraft med vilken vinden påverkarvattenytan.ResultatenvisarvidareattettRichardson‐talkananvändsförattbestämma om det är drivningen från konvektion eller skjuvning som är dominerande för drivningen av gasflödet.Konvektionenpåverkarövergångshastighetenupptillcirka3m/s.Resultatenbekräftar ocksåetttydligtsambandmellanövergångshastigheten,gasensdiffusionshastighetivattnetoch ytfilmensegenskaper. IV Preface ThisthesisconsistsofasynthesisinPartAandfourappendedpublicationsinPartB. ThepublicationsarereferredtointhetextbytheirRomannumbers.Thepublishedor acceptedpublicationsarereprintedwithpermissionfromrespectivejournal. I. Fredriksson,S.T.,Handler,R.A.,Nilsson,H.,Zhang,Q.,andArneborg,L.(2016) AnEvaluationofGasTransferVelocityParameterizationsDuringNatural Convection using DNS. Journal of Geophysical Research: Oceans. doi: 10.1002/2015JC011112 Fredrikssonperformedthemodellingandalldataanalysis,hadaleadingrolein writingthetextandpreparedallfigures. II. Zhang, Q., Handler, R.A., and Fredriksson, S.T. (2013) Direct numerical simulation of turbulent free convection in the presence of a surfactant. International Journal of Heat and Mass Transfer. doi: 10.1016/ j.ijheatmasstransfer.2013.01.031 Fredriksson had a minor role in writing, discussed the conclusions, and commentedthetext III. Fredriksson, S.T., Handler, R.A., Nilsson, H., and Arneborg, L. (2016) Surface Shear Stress Dependence of Gas Transfer Velocity Parameterizations usingDNS.Submitted2016. Fredrikssonperformedthemodellingandalldataanalysis,hadaleadingroleinwritingthetext andpreparedallfigures. IV. Gålfalk, M., Bastviken, D., Fredriksson, S.T., and Arneborg, L. (2013) Determination of the piston velocity for water‐air interfaces using flux chambers, acoustic Doppler velocimetry, and IR imaging of the water surface. Journal of Geophysical Research: Biogeosciences, doi: 10.1002/jgrg.20064 Fredrikssonhadaminorroleinwriting,discussedtheconclusions,andcommentedthetext Peerreviewedpublicationsnotincludedinthisthesis: Andric,J.,Fredriksson,S.T.,Lindstrom,S.B.,Sasic,S.,andNilsson,H.(2013)Astudyof a flexible fiber model and its behavior in DNS of turbulent channel flow. Acta Mechanica.doi:10.1007/s00707‐013‐0918‐y Fredriksson, S.T., Arneborg, L., Nilsson H., and Handler, R.A. (2015). Near‐surface physicsduringconvectionaffectingtheair‐watergastransfer.Proceedingsof7th InternationalSymposiumonGasTransferatWaterSurfaces,Seattle,USA. V TABLEOFCONTENTS PartA.Synthesis 1 INTRODUCTION................................................................................................................................1 1.1 1.2 1.3 1.4 Backgroundandthesismotivation...........................................................................2 Gasexchangeprinciples................................................................................................4 Forcingbywindandnaturalconvection...............................................................4 Gastransfervelocity........................................................................................................6 1.4.1 1.4.2 1.4.3 1.4.4 1.4.5 Estimationsof basedonwindspeed...........................................................7 Estimationsof basedonsurfaceheatflux................................................8 Estimationsof basedondissipation............................................................9 Estimationsof basedonsurfaceflowdivergence.................................9 Influenceofsurfactants............................................................................................9 2 METHODS.........................................................................................................................................10 2.1 2.2 Directnumericalsimulations...................................................................................10 Fieldmeasurements.....................................................................................................12 2.2.1 2.2.2 2.2.3 Fluxchambermethod............................................................................................13 Dissipationrateparameterization...................................................................14 SurfacedivergenceparameterizationviaIRandPIV..............................14 3 CONTRIBUTIONS...........................................................................................................................15 3.1 3.2 3.3 3.4 3.5 3.6 3.7 Generalgas‐exchangecharacteristicsforbuoyancydrivenflow............15 Scaling.................................................................................................................................16 Influenceofnaturalconvectiveforcingandsurfacemixedlayer thicknessongastransfervelocity..........................................................................18 Influenceofgasdiffusivityandsurfactantsonthegastransfer velocity...............................................................................................................................20 Influenceofcombinedwindandbuoyancyforcingonthegastransfer velocity...............................................................................................................................24 Observationalresultsathigherwindspeeds,andspeculations abouttheinfluenceofwaves....................................................................................27 Parameterizationsofthegastransfervelocity................................................29 3.7.1 3.7.2 3.7.3 Basedondiffusivity,surfactants,andshear‐stressand buoyancyforcing......................................................................................................29 Basedondissipation,divergence,orheatflux...........................................30 Basedonthemeanwindspeed .................................................................31 4 SUMMARYANDCONCLUSIONS..............................................................................................32 5 FUTUREPERSPECTIVES............................................................................................................34 ACKNOWLEDGEMENTS.....................................................................................................................35 REFERENCES..........................................................................................................................................36 PartB.PaperI‐IV VI PartA Synthesis SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 1.INTRODUCTION 1 INTRODUCTION The concentrations of atmospheric carbon dioxide (CO ) and methane (CH ) are increasing.Thisaffectse.g.,theclimateoftheearthandresults,throughair‐watergas exchange, also in an acidification of aquatic systems such as oceans, lakes, and watercourses.Thegasconcentrationandthegasexchangevary,however,largelyboth temporallyandspatially(Figure1)andanincreasedknowledgeofthetransportand accumulationprocessesofCO andCH isincreasinglyimportantinordertobeableto make more precise predictions of the future climate and the aquatic environment. Recent research has e.g., updated the global carbon cycle estimates (Figure 2), resultingintheinsightthatthegas‐exchangefrominlandwatersplaysamuchlarger role than previously believed [Bastviken et al., 2011; Ciais et al., 2013; Tranvik et al., 2009]. These predictions are often based on numerical global and regional models wherethegasflux usuallyareestimatedasaproductofagastransfervelocity, , andthegasconcentrationdifferencebetweenthewaterandair.Theuncertaintyinthe estimationsof isthoughstilllargeforlowwindconditions,typicallyfoundininland watersandoccasionallyintheoceans.Typicalareaswithlowaveragewindspeedsin the oceans are found e.g., along the equator [Monahan, 2006]). This thesis discusses what affects during low wind conditions and presents new parameterizations whichcanbeusedtoestimateit. Figure1. Estimatedcarbondioxidefluxaveraged overyear2000. (Figure13in[Takahashiet al.,2009]) The introduction will continue with a more thorough background description, and a motivation for the need of further understanding of the gas‐exchange. Then the gas transfervelocitywillbedefinedandtheprocessesthataffectitwillbediscussed.The introduction is closed by presenting some of the parametrizations presently being used. Section 2 describes the numerical and field‐measurement methods used in the papersthatconstitutethisthesis.Themaincontributionsoftheworkarepresentedin section3.Finally,theconclusionsandfutureperspectivesaregiveninsection4. 1 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 1.INTRODUCTION 1.1 Backgroundandthesismotivation Figure3showsaclassicdiagraminthecontextofclimatechangeandtheabundance of greenhouse gases. The measurements are carried out at Mauna Loa at Big Island, Hawaii. It represents the longest continuous measurement series of the atmospheric concentrationofCO anditshowsacontinuousincreaseoftheannualaveragesallthe way from the beginning of the measurements. The black curve is the yearly mean while the blue curve shows the yearly variation. Similarly, recordings of the concentration ofatmosphericCH showasteadyincrease(approximately 10%since 1988). An increase of atmospheric CO affects the aquatic systems by changing the balanceofdissolvedCO inthewater.Althoughtheincrease naturallyisaffectsland basedprocessesaswell,thesearenotfurthertreatedinthisthesis.Furthermore,itis affectingtheglobalclimate.Differentregionalandglobalmodelsareusedtoenhance theknowledgeofhowthisincreaseaffectstheworldnowandinthefuture.Figures1 and 2 show two examples of results from global models where Figure 1 shows the spatialdistributionofthemeancarbondioxidefluxfortheyear2000[Takahashietal., 2009]andFigure2showsanestimateoftheglobalcarboncyclewheretheredarrows manifest the anthropogenically changed carbon fluxes and reservoir masses [Ciais et al.,2013]. Figure2.“Simplifiedschematicoftheglobalcarboncycle.Numbersrepresentreservoirmass, alsocalled’carbonstocks’inPgC(1 10 )andannualcarbonexchangefluxes(in ).Blacknumbersandarrowsindicatereservoirmassandexchangefluxesestimated forthetimepriortotheindustrialEra,about1750...Redarrowsandnumbersindicateannual ’anthropogenic’fluxesaveragedoverthe2000‐2009timeperiod…”(Figure6.1inIPCC2013 [Ciaisetal.,2013]). 2 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 1.INTRODUCTION The work summarized in this thesis aims at improving: (i) numerical model performance, (ii) monitoring of gas exchange in water bodies, and (iii) the understanding of the processes involved in interfacial gas exchange. The common methodtoestimatetheair‐watergas‐exchangeinbothregionalandglobalmodelsis tomultiply agastransfervelocitywith the gasconcentrationdifference inthewater and in the atmosphere. In turn, the gas transfer velocity is often estimated as a function of the wind speed. Although the wind speed is important for the gas‐ exchange,itcannot,especiallyduringlowwindconditions,beusedalonetoestimate the transfervelocitywithout missingoutotherimportant factors,suchaswater‐side natural convection and surfactants. This is also manifested through a widely varying magnitude of between different parameterizations [Bade, 2009; Takahashi et al., 2009;Wanninkhofetal.,2009].Thefirstgoaloftheworksummarizedinthisthesisis thereforetoprovideabetterparameterizationofthegastransfervelocitytoregional andglobalmodels. On a smaller scale, it is also important to be able to understand the processes in smallerwaterbodiese.g.,lakes,streams,andcoastalwaters.Thegasfluxisdifficultto measure directly, whereas measuring secondary quantities (to be used for the flux estimation) such as the gas concentration in the water, surface heat flux, and wind speedareeasier.Thesecondgoalisthereforetoimprovemethodsformonitoringand estimating fluxes based on secondary quantities. These estimates can then beside improvingtheunderstandingoftheactualwaterbodyalsobeusedtoaggregatebetter estimates of the global gas‐exchange. It can e.g., be noted that the freshwater outgassing that was not included in the previous assessment by IPCC is of the same orderofmagnitudeasthenetoceangasflux(Figure2). The third goal with this thesis is to enhance the understanding of the small‐scale processespresentinthevicinityoftheair‐waterinterface.Thisunderstandingofair‐ watergastransfercanalsobeusedoutsidethegeophysicalsciences,e.g.,inchemical andenvironmentalengineering[JahneandHaussecker,1998]. Figure 3. Recorded concentrations of atmospheric carbon dioxide from 1958 up to now at Mauna Loa, Hawaii, USA. [Dr. Pieter Tans, NOAA/ESRL (www.esrl.noaa.gov /gmd/ccgg/trends/) and Dr. Ralph Keeling, Scripps Institution of Oceanography (scrippsco2.ucsd.edu/)] 3 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 1.INTRODUCTION 1.2 Gasexchangeprinciples The factors that influence the gas exchange across an air‐water interface can be divided into (i) physical factors, i.e., advective/turbulent and molecular transport processesand(ii)biochemicalfactors(however,notconsideredinthisthesis)which typicallyarebiochemicalorbiologicalprocessesthateitherproduceorconsumegas. ThetransfervelocityforthetwogreenhousegasesCO andCH iscontrolled(limited) by the water side meaning that the flux is mainly limited by physical transport processes in the water [Jahne and Haussecker, 1998]. As a note it can be mentioned that the transfer of many other common properties such as heat, momentum and watervaporarecontrolledbytheairside.Thepresentworkdoesonlyconsiderwater‐ side‐controlled gases and hence only the transport processes at the water side. The physical factors that influence the exchange of these gases (controlled by the water‐ side)comprisee.g.,interfacialshearduetowindforcing,microscalebreakingwavesat moderate wind speeds, breaking waves at high wind speeds, bubbles, raindrops, surfactants,andconvectionduetosurfaceheatloss[Macintyreetal.,2002]. The actual interfacial gas‐exchange is, neglecting the effects of bubbles or raindrops, maintainedbypuremoleculardiffusiondrivenbythegasconcentrationgradientjust belowtheair‐waterinterface.Adiffusiveboundarylayer(Figure4)isformedabovea turbulent layer where the turbulent motions are attenuated due to viscous damping and the presence of the surface. This diffusive gas exchange in this layer can be estimatedbyFick’slaw , (1) where isthemoleculardiffusivityand ⁄ istheverticalconcentrationgradient (Figure 4). Even though the gas exchange is a molecular diffusive process in the diffusiveboundarylayer,themagnitudeofthediffusiveexchangeishighlydependent on the turbulence below. This is since besides being the main transport agent in the turbulent layer, the turbulence is also to a large degree influencing the molecular diffusive transport by affecting the diffusive boundary layer thickness, . Intense turbulenceandadvectivemotionsresultinathinner ,alargerconcentrationgradient ⁄ andhenceahighergasexchange. 1.3 Forcingbywindandnaturalconvection In this thesis we focus on the forcing from interfacial shear due to wind shear and naturalconvectionduetosurfaceheatlosssincetheyareimportantduringlowwind conditions.AschematicoftheseforcingsisshowninFigure4. The air‐water velocity difference causes a shear stress, , at the air‐water interface. Thisshearstressresultsinamomentumexchangebetweentheairandwater,thereby affecting the velocity and turbulence intensity in both. This exchange is in the numericalsimulations modeledwithaconstantshearstressat thesurfaceboundary assumingsteadywindconditionsandnowaves,spray,bubbles,orrain. 4 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 1.INTRODUCTION Clouds z z QR,SW U10 Reflection QR,LW Air ϑCab Turbulent layer QS , QL Diffusive layers ϑCas Water Cws Turbulent layer Absorption Cwb Mixed layer depth Velocity Gas conc. Figure 4. Conceptual diagram of air and water velocity, heat fluxes, and mean gas concentration in the air and water. Sensible heat flux, , latent heat flux, , and net longwave radiation, , , originate from the water surface whereas the shortwave solar isthewindspeed10 abovethewatersurface. radiation, , ,penetratesthesurface. , , , and are the bulk and surface gas‐concentrations in the water and air, respectively. is the dimensionless Ostwald solubility coefficient. Advective transport dominates in the turbulent layers, whereas diffusive transport dominates in the very thin diffusive boundary layers (note the different vertical scale for the velocity gradient and the gasconcentrationrepresentedwithabroken ‐axisintheverticaldirection,e.g.,thediffusive boundarylayerthicknessfor and inwateristypicallyoftheorderof1mmorless). Thewindspeedisoftenreferredtoas whichisthewindspeedat10 abovethe watersurface.Inordertoexpresstheapplied in ,theequation ∙ ∗ κ ln 5.7 (2) ∗ canbeusedforneutralconditions[Csanady,2001].Herethesubscript denotetheair side, istheheightabovetheair‐waterinterface, isthevonKarmanconstant,and is the air‐side kinematic viscosity. The water‐side and air‐side friction velocity is ⁄ ⁄ where and are the densities of air and water. The related as ∗ ∗ appliedshearstresses, ∗ ,intheDNScasesinthepresentworkcorrespondto up to approximately 2 according to Equation (2). Without going into too muchdetailitcanbesaidthatstableairconditions(meaninganegativebuoyancyflux intheair,i.e.,decreasingdensitywithheight)decreasetheturbulenceintensityinthe air,whileanunstableincreasesit.Thisresultsinthatahigher isrequiredinorder to maintain the same ∗ for stable compared to neutral conditions [Csanady, 2001; Garratt,1992].Similarlyalower isrequiredinordertomaintainthesame ∗ for unstablecomparedtoneutralconditions. 5 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 1.INTRODUCTION Thenetheatfluxattheair‐waterinterfacecanbewrittenas , , (3) where the sensible heat flux QS is driven by the temperature difference between the waterandair,thelatentheatfluxQLbywaterevaporation,andtheradiativeheatflux QR,LWbylongwaveradiativetransfer.Theradiativeheatfluxisheredecomposedinto shortwave,QR,SW,originatingfromthesolarirradianceandlongwaveradiativefluxes, QR,LW. QS, QL, and QR,LW originate from the uppermost molecular layers of the water whereas QR,SW penetrates the surface. The penetration depth of QR,SW depends on the radiative power, the wave length, and water characteristics OBS [Fairall et al., 1996; Jerlov,1976;Ohlmannetal.,2000;Wicketal.,2005]. Apositivenetheatflux, ,isdefinedinthepositivedirectionofthe –axis(upwards from the interface). QS can be either positive or negative whereas QL and QR,LW are typically positive (upwards). is usually positive which results in a cooling of the surface, especially during nighttime resulting in a so‐called cool skin at the surface [Fairalletal.,1996;SolovievandSchlussel,1994].Theannualmeanisintherangeof 40 230 [Stewart,2008].Theshortwaveradiationcanespeciallyondays without clouds, depending on the vertical distribution of the radiative absorption, influence the buoyancy flux and thereby the cool skin. The heat flux condition in the present work represents a situation with an even vertical distribution of the absorptionofQR,SW.ThisconditionincludesthecasewithlimitedQR,SWasduringnight time.Notealsothat thebuoyancyfluxin theatmosphericboundarylayerconsistsof thesensibleandlatentheatfluxesonly,whichimpliesthatthebuoyancyfluxesabove andbelowthesurfacearedifferent. 1.4 Gastransfervelocity Equation (1) is difficult to use since it is difficult to measure the gas concentration gradientintheverythindiffusiveboundarylayer,whichhasthickness intheorder of 1 mm or less. For estimations of the air‐water gas exchange, equation (1) is thereforeoftenrestatedinto , (4) in order to be able to use the more easily measured bulk and air gas‐concentrations instead.Here isthegastransfervelocity, isthegasconcentrationinthewater underthediffusiveboundarylayer, isthegasconcentrationintheairatthewater surface, and is the dimensionless Ostwald solubility coefficient (Figure 5). Even thoughitiseasiertouse and insteadofmeasuringtheconcentrationgradient, itisnowinsteadachallengetoestimate . 6 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 1.INTRODUCTION Diffusive layer thickness, δ ϑCas Turbulent layer Water Air z ∆C ∆C Cwb Figure5.Aschematicsnapshotofthegasconcentration attwodifferentlocationsforagas with a gas‐flux that is controlled by the water side. The gas concentration gradients are assumed to be small in the air. The momentary gas concentration, , and the diffusive boundarylayerthickness, ,aregiveninredwhilethemeanquantitiesaregiveningrey. isthebulkgas‐concentrationinthewaterand isthesurfacegas‐concentrationintheair. is the dimensionless Ostwald solubility coefficient. The water‐side gas concentration gradient ⁄ variesduetothevaryingdiffusiveboundarylayerthickness. In sections 1.4.1‐1.4.4 four commonly used parameterizations of estimating , are described, i.e., (1) based on wind speed, (2) based on surfaceheat flux, (3) based on rateofturbulentkineticenergydissipation,and(4)basedonsurfaceflowdivergence. All these parameterizations have a term that takes the influence of the molecular diffusivityandtheabundanceofsurfactantsatthesurfaceintoaccount.Thisinfluence is therefore generally discussed in section 1.4.5. Although there are other processes thatinfluencethegas‐exchange,suchaswhitecapping,seaspray,bubbles,rain,these are not taken into account in this work since white capping, sea spray, and bubbles from waves are not present during low wind conditions, and ebullition is not controlledbythesurfacediffusiveboundary,andrainisdifficulttostudyinDNS. 1.4.1 Estimationsof basedonwindspeed There has been many attempts to estimate based on the wind speed 10 meters above thesurface,referredtoas [e.g.,Bade, 2009;Wanninkhofetal.,2009].Two parameterizationsoftenusedare 0.215 . 2.07, (5) , , 7 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 1.INTRODUCTION presentedbyColeandCaraco[1998]forinlandwaters,and 0.1 0.064 0.011 3, , , (6) presentedbyWanninkhofetal.[2009]foroceanconditions.Herethetransfervelocity is given in ( ) and the wind speed is given in ( ). It can be seen that both these equations have constants implying that there is a gas flux also for zero‐wind conditions. This gas flux (and gas transfer velocity) must then be due to processes enhanced by other means than wind speed, e.g., buoyancy flux. This is of course problematicsincethevariationine.g.,thebuoyancyfluxisnotbedescribedinthese equations. Equations(5)and(6)aregivenforSchmidtnumbers 600and660representing ⁄ expresstheratio at20°Cinfreshwaterandseawaterrespectively.Here of the kinematic viscosity and molecular diffusivity. The relation between two transfer velocities with different gas‐water properties are generally expressed via theirdifferentSchmidtnumbersas , , , (7) where is an exponent that depends on the surface characteristics. The exponent is usually between 1⁄2 and 2⁄3 and represents the Schmidt number dependency and therebythemoleculardiffusivitydependencyonthetransfervelocity. 1.4.2 Estimationsof basedonsurfaceheatflux The heat transfer velocity, , and an expression for the conversion between heat and gas through their and Prandtl number, , have been used [e.g., Frew et al., 2004;Garbeetal.,2003;Hausseckeretal.,1998]toestimatethegastransfervelocity as , , . (8) ⁄ istheSchmidtnumberforheatusingthethermaldiffusivity instead Here of the molecular diffusivity used in . is a transfer velocity constant for the parameterizationbasedonheatflux where isthespecificheatcapacity,and is the surface skin‐bulk temperature difference across the thermal boundary layer. Equation(8)isbasedontheassumptionthatthethermaldiffusiveboundarylayeris controlled by the same processes as the gas diffusive boundary layers. There are, however, three main differences: (i) Heat influences the buoyancy and thereby the turbulentmotionsbelowthesurface,(ii)thesurfaceboundaryconditionsforgasand heatdiffersbecausethetransportofgasiscontrolledbythewatersideandheatbythe airside,and(iii)thediffusivitiescandifferbyordersofmagnitudewith 10 and 10 dependingonwhichgasitis.Inspiteofthesedifferences,e.g.,Jahne etal.[1989]haveshownagoodagreementforoxygen( 10 )betweendirectly measured transfer velocities and transfer velocities extrapolated from heat transfer velocities. 8 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 1.INTRODUCTION 1.4.3 Estimationsof basedondissipation Theparameterizationusingtherateofturbulentkineticenergydissipationisbasedon the theoretical framework of the eddy cell model [Fortescue and Pearson, 1967] but withtheassumptionthatitratheristhesmallscaledissipativeeddiesthanthelarge scaleeddiesthatarethemaintransportationagents[Banerjeeetal.,1968;Lamontand Scott, 1970]. This assumption and the assumption that the turbulence can be describedwithastandardturbulencespectrumyieldsthegastransfervelocityas ⁄ . (9) , Here isatransfervelocityconstantforthedissipationparameterizationand is therateofkineticenergydissipation.LamontandScott[1970]foundtheappropriate valuestobe 1⁄2and2⁄3forfreefluidandsolidsurfacesrespectively.Inspiteof some problems with the assumptions during natural conditions, the dissipation parameterizationhasperformedwellinmanycasese.g.,PaperIV,Zappaetal.[2003], andZappaetal.[2007]. 1.4.4 Estimationsof basedonsurfaceflowdivergence The parameterization using the horizontal flow divergence, thesurfaceisgivenby ⁄ , ⁄ ⁄ , at (10) where istheroot‐mean‐square,rms,ofthesurfaceflowdivergenceand and isatransfer arethehorizontalvelocitiesinthe ‐and ‐directionsatthesurface. velocityconstantfortheparameterizationbasedondivergence.Theparameterization has been used in many studies [e.g., Banerjee et al., 2004; Calmet and Magnaudet, 1998;McKennaandMcGillis,2004]andtheoreticallyderivedbye.g.,Ledwell[1984]. 1.4.5 Influenceofsurfactants Surfactantsaresurface‐activechemicalagentsthatgenerallyreducegasexchange[e.g., Bade, 2009; McKenna and McGillis, 2004; Wanninkhof et al., 2009]. They are almost always present in natural waters and occasionally the amount of surfactants even forms a surface film and then requires an extra layer to represent the surfactants when estimating the gas‐exchange. Also in smaller abundances, which is the usual situation, they act to change the hydrodynamic conditions of the air‐water interface [McKennaandMcGillis,2004].Theflowatthesurface,includingeddies,redistributes the surfactant concentration and makes it patchy. A surfactant lowers the surface tension,andsincethesurfactantconcentrationnowispatchy,thesurfacetensionwill varyoverthesurface.Avaryingsurfacetensionresultsinelasticforcesthatattenuate theturbulenteddies.Thisattenuationinfluencesthegasexchange,inparticularforthe gases with thin diffusive boundary layers, i.e. those with high Schmidt numbers. The surfactantinfluenceonthegas‐fluxanditsattenuationcanbediscussedinthelightof thetheoreticalworkofLedwell[1984]wheretheinfluenceofmoleculardiffusivityon the gas transfer velocity is studied and compared for both slip and no‐slip boundary conditions.ThisgivesSchmidtnumberexponents 1⁄2foracleansurfaceand2⁄3 for a surface with surfactants, if a clean surface is represented by a slip boundary condition (no attenuation of the horizontal components of the surface flow), and a 9 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 2.METHODS surface with a large abundance of surfactants is represented by a no‐slip boundary condition(horizontalcomponentsofthesurfaceflowiszero). 2 METHODS 2.1 Directnumericalsimulations PaperI‐IIIusedirectnumericalsimulations,DNS,tostudyhowtheturbulenceandthe heat‐andgas‐transportsdependondifferentflowconditions.Thegasismodeledasa passivescalarwhichcanbeseenasaninertgas.Theflowconditionsarevariedvia(i) different surface boundary conditions for the velocity (including shear and surfactants) and the temperature (surface heat flux), (ii) different depths, and (iii) differentmoleculardiffusivitiesforthescalar. ThecomputationaldomaincanschematicallybeseeninFigure6.Thisschematicisin general valid for the Papers I‐III. However, the horizontal plane in Paper I‐II is quadratic since all the boundary conditions are identical in the spanwise and streamwise directions in these papers (no surface shear‐stress). Furthermore, the ‐ and ‐directionsaredefinedintheverticalandthespanwisedirectionsinPaperII. Figure 6. Computational domain for the cases with combined buoyancy and shear stress 0.1204 , 3 and inthedepth, forcing.Thedomainsizeisgivenby streamwise, and spanwise direction respectively. The surface is subject to a constant outward‐going heat flux, , and a constant scalar concentration, , while the bottom is subjecttozerofluxboundaryconditions.Thevelocityboundaryconditionsareeitherslip,no‐ slip or constant shear stress, , at the surface boundary and slip at the bottom boundary. Periodic(cyclic)boundaryconditionsareusedforallvariablesinthehorizontal( ‐and ‐) directions.Thetemperaturefieldisasnapshotfromcase240 ( ∗ 240withbuoyancy, seesection3).(RedrawnfromFigure1inPaperIII). 10 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 2.METHODS DNSimplythatthereisnoturbulencemodelthatmodelsorcreatesturbulenceinthe computation. The turbulence is instead “naturally” invoked due to flow instabilities thatariseduetotheflowforcing.TheDNSmethodisusedinordertobeabletostudy theactualturbulenceanditsinfluenceonthegasexchangebysolvingthesmall‐scale turbulent motions instead of using a turbulence model. The forcing is in Paper I‐III eithernaturalconvection(buoyancy)orsurfaceshearstressoracombinationofboth. The lack of a turbulence model sets very high requirements on the computational mesh resolution since all the fluid motions and eddies must be resolved all the way down to the sizes where the turbulence dissipates to heat. An extensive mesh resolution and domain aspect‐ratio study was therefore performed in Paper I. The resulting mesh resolution for Paper I and III can schematically be seen in the front cornerofthedomaininFigure6.Hereonlyeveryfourthgridlineareplottedsincethe mesh is very fine and the grid lines otherwise would be difficult to see. The mesh is equidistant in the horizontal plane, and in Paper I and III densified towards the surface. In Paper II the mesh is densified towards both the surface and bottom. The verticalmeshspacingclosetothesurfaceis,though,thesameforthemeshesusedin PaperI‐III. InPaperIandIIIafinite‐volumemethodisusedwhereasapseudo‐spectralmethodis used in Paper II. Please also refer to Paper I‐III where these methods as well as the spaceandtimediscretizationsarediscussedinmoredetail. InalltheDNS‐papers(I‐III)theNavier‐Stokesequations U k U∙ U Π U (11) and ∙U 0 (12) using the Boussinesq approximation are solved in conjunction with the transport equation (13) U∙ forthetemperature .InPaperIandIIIatransportequation U∙ S (14) forapassivescalar issolvedaswell.HereU= , , isthefluidvelocitywherethe components are given in ‐, ‐, and ‐directions respectively and is the time. ⁄ Π , is the pressure, is a reference density, g is the acceleration of gravity, k is a unit vector in the vertical direction, is the kinematic viscosity, ⁄ | isthethermalexpansioncoefficient, isthedensity, isthe 1⁄ temperature, isareferencetemperature, isthethermaldiffusivity, isthescalar concentration, and is the molecular diffusivity of the scalar in water. In order to sustainaconstantmeantemperatureinthedomainanevenlydistributedheatsource isaddedtoequation(13)tobalancetheheatfluxthroughthesurface.Similarlyan evenly distributed pressure gradient (not shown) is added to equation (11) in the direction opposite to the surface shear stress cases in Paper III. The scalar source ⁄ isusedtoimposethescalarfluxthroughthesurfaceand isthearea‐ 11 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 2.METHODS averagedmeanscalarfluxoncesteady‐stateisachieved.Inthefollowing, , , , , and areusedtorepresentthefluctuatingpartsof , , , ,and respectively. The surface boundary is flat, assuming that the surface deflection is negligible. The surface boundary condition for the vertical velocity is therefore 0 for all cases. The boundary conditions for the horizontal velocities are for the pure convection ⁄ drivencaseswithoutsurfactants(PaperI)eitheraslip( ⁄ 0 orano‐ slip ( 0 boundary condition. In Paper II the surfactant boundary conditions ⁄ ⁄ and ⁄ ⁄ where for the horizontal velocities are ⁄ . The surface tension is a function of the surfactant concentration Γ andislinearizedaroundtheinitialsurface‐tension .ForthecasesinPaperIIIdriven by pure surface shear‐stress or by a combination of natural convection and surface shear‐stress there is a slip boundary condition in the y‐direction and a shear‐stress ⁄ ). Here is the surface shear boundary condition in the ‐direction ( ⁄ stress.Thebottomboundaryisassumedtobestress‐freeandismodelledwithaslip boundaryconditionforallcases. ⁄ assuming a The surface boundary condition for the temperature is ⁄ constant surface heat flux, [Soloviev and Schlussel, 1994]. Here λ is the thermal conductivity. The surface boundary condition for the scalar is a constant scalar concentration assuming that the air‐water gas exchange is controlled by the water‐side [Jahne and Haussecker, 1998]. The bottom boundary conditions for the ⁄ temperature and scalar are ⁄ 0 assuming no heat or gas exchange throughthatboundary. Periodic (cyclic) boundary conditions are used for all variables in the horizontal ( ‐ and ‐)directions. 2.2 Fieldmeasurements The field measurements reported in Paper IV were performed in order to compare different methods of estimating in the field. These methods comprise (i) flux chamber measurements, (ii) parameterization of as a function of the rate of turbulent kinetic energy dissipation, and (iii) parameterization of as a function of thesurfacedivergence.Herethesurfacedivergencewasestimatedviaparticleimage velocimetry (PIV) of the surface temperature structures recorded by IR imagery. In addition to these parameterizations a number of environmental parameters were measured in an attempt to find other methods of estimating . These parameters comprise e.g., bulk temperature (mean and rms), surface temperature (mean and rms), IR surface velocity mapping (mean and rms), mean IR coherent structure size, and wave height. The measurements were performed in August 17‐18 2010 in Gullmarsfjord at the Bornö marine research station close to Lysekil, north‐west of Gothenburg, Sweden. The measurement setup at the suspension bridge at Bornö, where the depth is about 33 m, is sketched in Figure 7. The simultaneous measurementswererecordedduringonediurnalcycle. 12 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 2.METHODS Figure 7. Illustration of the instrument setup at the Bornö suspension bridge. Three instruments were used simultaneously to measure : An IR camera, an ADV, and a flux chamberwithtubesforcontinuousmeasurements.(Figure1inPaperIV). 2.2.1 Fluxchambermethod Here wasestimatedbymeasurementsofthegasfluxacrosstheair‐waterinterface and the gas concentration in the water and air inside the chamber. The gas flux was estimated by measuring the concentration change ⁄ inside a round lightweight chamber that was placed at the water surface. The concentration change was then transformedintofluxusingthecommongaslaw,thevolumeofthechamber,andthe surface water area covered by the chamber. The implicit assumption is that the concentrationchangeiscausedbyfluxesthroughthesurface,andthatthefluxinside the chamber is similar to that outside the chamber. The concentration at the surface ( inequation(4))wasestimatedastheequilibriumconcentrationofthegasatthe surfaceusingtheinitialgasconcentrationinthechamberandHenry’slaw.Here,the assumption is that the gas concentration outside the chamber is equal to the initial concentrationinthechamber.Thebulkgasconcentrationinthesurfacewater( in equation (4)) was measured at the start and end of each measurement period at a depthofapproximately40cm.AlthoughfluxesofbothCO andCH weremeasured, onlyCH ‐fluxeswereusedforestimating .Therationaleisthattherewasalwaysa significant positive net flux of CH into the chamber while the fluxes of CO showed more variation with both release from and uptake to the water with intermediate periodsofnosignificantfluxduringthemeasurementperiod.Thetransfervelocityfor accordingtoequation(7)with 1⁄2. CH wastransformedinto The edges of the chamber was submerged only 2.5 cm into the water due to the lightweight chamber construction and the chamber was attached to the bridge with 13 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 2.METHODS thin, slack strings enabling it to move with the water as freely as possible. This has previouslybeenshowntobesuccessfulforchamberperformance[Coleetal.,2010]. 2.2.2 Dissipationrateparameterization The rate of the turbulent kinetic energy dissipation, , was estimated from the turbulence spectra at approximately 0.3 m depth. The transfer velocity was then calculated with equation (9) with 0.42 as found by Zappa et al. [2007]. The spectra were calculated from 3D velocity vector time series recorded by the use of acousticdopplervelocimetry(ADV).TheADVwasmountedlookingupwardsonataut linehangingfromthebridge.AfinontheADVensuredthatthesensorwasupstream oftheline(undisturbed). 2.2.3 SurfacedivergenceparameterizationviaIRandPIV Ithasbeenshowninlaboratoryexperimentsthatparticleimagevelocimetry(PIV)can be used to determine the surface divergence. Veron et al. [2008] extended this idea, and used infrared (IR) imagery of the ocean surface to estimate surface velocity, vorticity, and surface divergence. The idea is that the surface temperature (heat pattern)measuredwithanIRcameracanbeusedasafluidflowtracer,seeFigure8. This was in Paper IV also confirmed to be the case by comparing (directly in the IR imagesequences)themotionofthebubblesandfoamatthesurfacewiththemotion of the heat pattern. It should, however, be noted that thermal IR radiation has very short transmission lengths in water (order of 1 100 , [e.g., Garbe, 2001]) which meansthatPIVofheat‐patternsestimatesthevelocitiesintheverytopsurfacewater (skin)whichinturnmaybedifferentfromthevelocitiesjustbelowthesurface[Volino andSmith,1999].Theuseoftheheatpatternasafluidflowtracercanbeproblematic for certain flow conditions, e.g., pure natural convection where there can be a horizontal flow inside a more or less stationary plume as will be seen in section 3.1. That type of flow would not be detected with IR/PIV, which only recognizes the motionoftemperatureanomaliessuchasthestreaksofthecoldwaterenclosingthe warm plume. This problem is decreasing for cases where there is more than pure natural convection forcing. It should also be noted that the temporal and spatial resolutionoftheIRimageryandthePIVmustbehighenoughtoestimatethesurface divergenceproperly. Figure 8. Example of a temperature field from an IR image. The average field of view is 95 88 .(PartofFigure4inPaperIV). 14 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 3.CONTRIBUTIONS 3 CONTRIBUTIONS InPaperI‐IVavastamount ofresultsare presented,discussionsareperformed,and conclusions are drawn. In this section of the thesis an effort to synthesize all this in majorcontributionshasbeenmade.Inordertodosoanewnumberingsystemforall the numerical cases could have been presented. This could, however, also lead to misinterpretations since all the cases are more thoroughly described in each paper thaninthissynthetizationoftheresults.Hencethecaseswerechosentobepresented herewiththesamelabelingasinthepapersenclosed. The cases in Paper I are named with (i) a letter for lip or for o‐slip surface velocity boundary conditions, (ii) a number describing the computational domain aspectratio,(iii)aletter or forthe asecaseora inemeshresolution,(iv)anda letter or fora eepora hallowdomainor or fora owor ighsurfaceheat flux.Thecasename 2 representsi.e.,acasewithaslipsurfacevelocityboundary condition,adomainaspectratioof2,abasecasemeshresolution,andahighsurface heatflux. ThewindforcingisinPaperIIImodeledasafixedshearstress ∗ .Thecasesare named with (i) a number for the shear‐based Reynolds number for ∗ ∗ ⁄ , which describes the ratio between inertial and viscous forces, and (ii) a letter for Buoyancy or letters for No‐Buoyancy. In the no‐buoyancy cases the gravitational acceleration was set to zero, so the temperature acted as a passive tracer only. The domainaspectratio,domaindepth,andthemeshresolutionwasthesameforallcases. Furthermore,thecase0 inPaperIIIisidenticalwiththecase 2 inPaperI. Intheplotstocome, ̅and〈 〉denotevolumeandhorizontalareaensemblemeanof anarbitraryvariable ,respectively. 3.1 Generalgas‐exchangecharacteristicsforbuoyancydrivenflow InFigure9asnapshotofcase0 (slipboundaryconditionandbasecaseheatfluxand meshresolution)ispresentedinordertogiveanintroductiontomanyoftheresults presentedlaterinthissection. The surface heat flux cools the surface water and thereby makes it denser. Due to instabilitiesthisdenserwaterthenstartstodescendinthinplumessketchedasblue arrows.Thedescendingplumessetupahorizontalflowsketchedwithyellowarrows and an ascending flow of warmer water between the descending plumes. The descending plumes are typically much thinner than the ascending warm water. The horizontal and ascending flow typically stretch and squeeze the diffusive boundary layer and thereby make it thinner. The patches with thin diffusive layer thicknesses areherevisibleas“islands”penetratingthemeanthickness,separatedwithtrenches oflimitedverticaldiffusivetransport. 15 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 3.CONTRIBUTIONS a) d) F ⁄F 0 2 γ⁄γ -2.5 c) 1 b) ̅ T -4 T ⁄T Depth mm 0.5 0 10 Figure 9. Snapshot for the case 0 (pure buoyancy forcing with slip surface boundary ⁄ and the surface water condition). (a) The normalized temperature field velocities. Blue and yellow arrows schematically represent the descending plumes and the horizontalflowclosetothesurface.(b)Themeanandmomentarydiffusiveboundarylayer thickness, ̅and ,wherethethicknessisdefinedaswhere5%ofthetotalverticaltransport is diffusive. The iso‐surface of the momentary diffusive layer is colored by the actual layer thicknesswherecoldercolormeanssmallerthicknesses.Theverticaldimensionintheplotis scaledbyfiveforbettervisibility.(c)Thetemperaturefieldandcontoursofthenormalized horizontal flow divergence, ⁄ , at the surface. (d) The surface‐normal scalar transport, ⁄ ,for 7acrossthesurface. Figure5sketchesavaryingdiffusivelayerdepthsimilartowhatwouldbetheresultsif Figure9bwasslicedwithaverticalplane.ItisseeninFigure5thatasmalllayerdepth results in a higher concentration gradient, which in turn results in an increased surface‐normal diffusive transport (Figure 9d). This increased gas flux can now in Figure9dbeseentocoincidewiththe“islands”withthinboundarylayerthicknessin Figure 9b. This is the reason for the good correlation between the temperature, and horizontalflowdivergencefieldsthatcanbeseeninFigures9a,candeventuallyalso between the momentary diffusive layer depth and surface‐normal scalar flux seen in Figures9b,d. 3.2 Scaling Nondimensional numbers and scales for e.g., length, time, velocity, temperature, and scalar concentration can be used as tools to facilitate the understanding of which processes and scales that are important in determining the transfer velocity. Paper I presents scales that are appropriate for the analysis of flows driven by pure natural convection(buoyancyforcing).Thesescales,presentedinTable1,aredividedininner 16 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 3.CONTRIBUTIONS andouterscales.Theinnerscalesarebasedontheassumptionthatitistheprocesses inthevicinityofthesurfacethatcontrolthegastransferwhereastheouterscalesare based on the assumption that it is the whole domain (surface mixed layer) that controlsthegastransfer.Theinnerscaleswillbeusedtodiscusstheresultsinsections 3.3 and 3.4 since these scales were found in Paper I to best scale the processes of importanceforthegastransfervelocity. Table1Scalingschemes Scheme Length, Velocity, Time, Temp., ⁄ Inner Outer ∗ / ∗ ⁄ Scalar, ⁄ / ⁄ ∗ ∗ ⁄ Div.,γ / ∗ ⁄ 1⁄ 1⁄ ∗ Themixinglayerdepthisinthesesimulationsassumedtobethedomaindepth . Itismoreintriguingtofindsuitablescalesforthecaseswithcombinedforcing.Aflow dominatedbybuoyancyforcingshouldusetheabovescaleswhileaflowdominatedby shearforcingshouldusetheshearscalespresentedinPaperIII.Thesescalesandthe transitionfromonesetofscalestotheotherarediscussedinmoredetailinPaperIII. Furthermore,dimensionalanalysisinpaperIIIshowsthatthegastransfervelocityfor acasewithcombinedforcingfrombothbuoyancyandshearstresscanbeexpressedin non‐dimensionalrelationshipsaccordingto , , ∗, . (15) ∗ where havebeendefinedabove.TheReynoldsnumber and ∗ ∗ (16) ⁄ , represents the ratio between the inertial based on the friction velocity, ∗ and the viscous forces. Here, , is the depth of the surface mixed‐layer (generally considered as a quasi‐homogenous region in the upper ocean characterized of little variation in density and temperature with depth [Kara et al., 2000]). It is here assumedtoberepresentedbythecomputationaldomaindepth .Thevalidityofthis assumption is discussed in section 3.3. The fourth variable is a Richardson number definedas , (17) ∗ ⁄ is representingtheratiobetweenthebuoyancyandshearforcing.Here thebuoyancyfluxjustbelowthesurface. willbeusedinsection3.5asameasureof thetransitionfrombuoyancy‐toshear‐dominatedgastransfer. 17 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 3.CONTRIBUTIONS 3.3 Influenceofnaturalconvectiveforcingandsurfacemixedlayer thicknessongastransfervelocity The momentary temperature fields at the surface for three different surface heat fluxes aregiveninFigure10.Theupper row(Figures10a‐c)showsthetemperature variation in dimensional units (°C) whereas the temperature has been normalized withtheinnertemperaturescale inthelowerrow(Figures10d‐f).Itcaninbeseen thattheflowfeaturestypicallybecomesmallerforstrongernaturalconvectionforcing. The temperature variation is increasing at the same time (i.e., a higher surface heat flux gives higher buoyancy flux which results in smaller flow features and larger temperature variation). The larger temperature variation is seen as an increasing contrast in Figures 10a to 10c The strength and validation of the scaling practice is showninFigures10d‐fwhereitisseenthatthetemperaturevariationscaleswellwith ⁄ theinnertemperaturescale ∝ (approximatelythesamecontrast,whichhere representsthenormalizedtemperaturevariation). Statistics for temperature and scalar concentrations for these cases with a heat flux variationaswellasforthecaseswithadepthvariationarepresentedinFigure11.It showsthehorizontallyaveragedrmsandmeanvaluesofthetemperatureandscalar concentrations. The convective inner scales for length, temperature, and scalar concentration have been used for the normalization. It can be seen that these scales collapsetheresultsforboththetemperatureandscalarconcentrationverywell.This isinterestingsincealloftheinnerscales( , ,and )thatareusedinFigure11, arefunctionsofeithertheheatorscalarflux,and , , ,and butnotthevertical dimensionofthecomputationaldomain.ThisisalsomanifestedinFigure9inPaperI where the transfer velocity dependence of the buoyancy flux and domain depth is presented.Thereisacleartransfervelocitydependenceofthebuoyancyflux ∝ ⁄ butonlyaverylimiteddependenceofthedomaindepth.Furthermore,itisshownin PaperIthatouterscaling,thatincludesthedomaindepthforbuoyancydrivenflows, doesnotscalethenear‐surfaceprocessesasaptastheinnerscaling.Thisimpliesthat, (i) for large enough depths, the surface mixed layer thickness does not influence the gas transfer velocity, (ii) the gas transfer velocity is a function of the buoyancy flux, and(iii)thedomaindepthinthesimulationsislargeenoughtomodeltheprocessesin thesurfacemixedlayerfornear‐surfaceprocessesastheinterfacialgas‐fluxprocesses. 18 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 3.CONTRIBUTIONS a) d) T T b) Q 50 Wm c) 100 Wm Q Q 200 Wm f) e) T T ⁄T Figure10.Momentarytemperaturefieldsatthesurfaceforthreedifferentsurfaceheatfluxes where 100 isthebasecaseheatflux.Theupperrow(a‐c)showsthetemperature ⁄ withacommon andthelowerrow(d‐f)showsthenormalizedtemperature ⁄ is the convective inner scale. is the domain mean temperature and scale.Thesecasesarepresentedas 2 , 2 ,and 2 inPaperI. Furthermore,Figure11showsthat and differclosetothesurfaceduetothe twodifferentsurfaceboundaryconditionsforthetemperature(constantheatfluxi.e. constanttemperaturegradient)andscalar(constantconcentration).Nevertheless,the figurealsoshowsthatthemeangradientsareverysimilar.Thisisinterestingfroma gas‐transfer‐velocity point of view, since the mean temperature magnitudes, as a function of the depth, are used in the parameterization based on the heat flux (equation (8)). This high similarity therefore gives reason to believe that the parameterizationbasedontheheatfluxwillwork,asdiscussedfurtherinSection3.7. 19 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 3.CONTRIBUTIONS 0 0 −10 −2 −20 mean rms −4 −30 −6 z/L+ −40 −8 −50 −60 −10 −3 −2 −1 0 1 Base −70 Low Q0 −80 High Q0 Shallow −90 Deep −100 −3 −2 〈T〉 T ⁄T , 〈s〉 −1 s̅ ⁄s , T 0 ⁄T , s ⁄s 1 Figure 11. Mean and rms temperature (black) and scalar concentration (blue) normalized withconvectiveinnerscales, and .Thedepthisnormalizedwith .Theinsetisazoom ofthenear‐surfacearea.Thedifferentcasesaredifficulttodistinguishintheplotsincethe normalizationcollapsetheresultsverywell.Thesecasesarepresentedas 2 , 2 , 2 , 2 ,and 2 inPaperI. 3.4 Influenceofgasdiffusivityandsurfactantsonthegastransfer velocity TheinfluenceofgasdiffusivitywasstudiedinPaperIbymodelingscalarswith 7, ⁄ numberswerechosenas(i)oneequivalent 150,and600respectively.These ⁄ numberwhichcanbeseenasthe numberforheat,(ii)onethat withthe isequaltothe numberusuallyusedforCO infreshwater,and(iii)onein‐between. AccordingtoFickslawgiveninequation(1),thediffusivegasfluxdependslinearlyon thediffusivity.Thismeansingeneralthat,inordertomaintainthesamegasflux,the gasconcentrationgradientmustbelargerforagaswithlowmoleculardiffusivitythan for a gas with high molecular diffusivity. A larger concentration gradient can be achieved by either a thinner boundary layer or a larger concentration difference across the layer. It is shown in Figures 12a‐b to be both. Here both the diffusive boundarylayerthickness, ,isthinnerandtheconcentrationdifferenceislargerfor ascalar with 600thanforascalarwith 150andevenmoresofor 7. This tendency can be seen to be amplified for the no‐slip condition (Figure 12b) compared to the slip condition (Figure 12a). Both the changing boundary‐thickness andconcentration‐differencescalewellwith and with 1⁄2and 2⁄3for slipandno‐slip,respectively(Figures12c‐d). 20 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 3.CONTRIBUTIONS 0 −1 δ600 δ150 a) b) δ600 δ150 depth[mm] −2 −3 −4 δ7 −5 −6 −7 −2 Sc = 7 Sc = 7 Sc = 150 Sc = 150 Sc = 600 Sc = 600 −1.5 −1 〈s〉 −0.5 0 −6 −5 −4 −3 〈s〉 s̅ δ7 −2 −1 0 s̅ 0 −1 c) d) −2 −3 −4 z/L + −5 −6 −7 δ600 Sc = 7 −8 −9 −10 −3 δ150 δ600 δ7 δ7 δ150 −2.5 Sc = 7 Sc = 150 Sc = 150 Sc = 600 Sc = 600 −2 −1.5 〈s〉 −1 s̅ ⁄s −0.5 0 −4 −3 −2 〈s〉 −1 0 s̅ ⁄s Figure12. Meanscalarconcentrationforslipandno‐slipboundaryconditions.Concentrations andthesublayerthicknesses, ,aregivenforscalarswith numbersequalto7,150and600. (a) Slip boundary conditions.The results for 2 (basecasemeshresolution)arepresented withafulllinewhiletheresultsfor 2 (finehorizontalmeshresolution)arepresentedwitha dash‐dottedline.Theresultsareverysimilarandthereforethefullanddash‐dottedlinesare difficult to distinguish in the plot. (b) No‐slip boundary conditions, 2 . (c‐d) as (a‐b) but scaled with inner scales and with 1⁄2 and 2⁄3 for slip and no‐slip boundary conditions,respectively.(PartofFigure13inPaperI) Ahigher number(i.e.lowerdiffusivity)requiresingeneralafinermeshresolution inordertobefullyresolved.Theeffectofatoocoarsemeshcanbeseen(Figure12in PaperI)asoscillationsintheconcentrationforthehigher numbers.Itis,however, arguedwithsupportfromthemeshsensitivityanalysis,alsoperformedinPaperI,that theuncertaintyintheresultsduetotoocoarsemeshresolutionisacceptableforthe averagedflowquantitiesofinterestforthegas‐exchangeevaluation.Itcane.g.,beseen 21 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 3.CONTRIBUTIONS inFigure12athatcase 2 and 2 givevirtuallythesamescalarmeanconcentration although 2 hasafinermeshresolutionthanthebasecase 2 . Theinfluenceoftheabundanceofsurfactantsforflowsdrivenbynaturalconvectionis studiedinPaperII.Theresultsshowthatthesurfactantinfluencedependsbothonthe meansurfactantabundanceatthesurfaceandontheturbulentconditionsintheflow underneath the surface. Furthermore, it is possible to estimate this influence by a turbulence‐surfactantparameter expressingtheratioofelastictoinertialforces.By increasingthesurfactantabundanceitisalsofoundthatthereeventuallyisasaturated surfactant abundance. At this abundance, the studied flow parameters (i.e. rms velocity, rms and mean temperature, dissipation and mean absolute divergence) are not affected further for an increased abundance. It is further shown that there is a smoothtransitionfortheseflowparametersfromthecleantothesaturatedsurfactant condition(Figure4inPaperI). InPaperItheresultsfromPaperIIarecomparedtotheresultswithaslipandano‐ slip boundary condition. This is done in order to find out if the less resource‐ demanding no‐slip boundary condition can be used to study flow conditions with surfactants. It is shown that the results for the no‐slip boundary conditions are very similartothesaturatedsurfactantcaseexceptforthermshorizontalvelocitycloseto thesurface.Thisiszeroforno‐slipandnon‐zeroforsurfactantboundaryconditions. Atafirstglance,itissurprisingthattheflowdivergence(whichisimportantforgas‐ exchange) can have a similar behavior for the two different boundary conditions althoughthermshorizontalvelocitiesdifferclosetothesurface.Thesimilarityinflow divergence can be explained by a decomposition of the horizontal flow into a solenoidal and an irrotational component (see Figure 13) [Hasegawa and Kasagi, 2008]. Asurfactant boundarycondition mainlydampenstheirrotationalcomponent, which is the dominating contributor to the horizontal flow divergence while the solenoidal component is less dampened and still contributes to the rms horizontal velocity. To summarize, it is therefore found that a slip and a no‐slip boundary condition can be used to model a clean and a saturated surfactant condition, respectively,whenstudyinggastransfer. Figure 13. Decomposition of the interfacial velocity vector into (a) solenoidal and (b) irrotationalcomponents.(Figure4inHasegawaandKasagi[2008]) The resulting transfer velocities, , for slip and no‐slip boundary conditions for scalars with 7, 150, and 600 are shown in Figure 14. The transfer velocity decreases by a factor of approximately 3 for a no‐slip (surfactant‐saturated) comparedtoaslip(clean)boundaryconditions,whichmatcheslaboratoryresultsfor gas transfer velocities for clean and contaminated surfaces [McKenna and McGillis, 22 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 3.CONTRIBUTIONS 2004]. The difference is decreasing with decreasing and is approximately 1.5 for 7. Furthermore, it is seen that the results for the gas transfer dependence of the numbers closely follow theoretical derivations [e.g., Ledwell, 1984]. The DNS give ⁄600 with 0.521and 0.668,whilst 1⁄2and 2⁄3inthe theoreticalderivationforslipandno‐slip,respectively.Thisisthefirsttimetothebest of our knowledge that this Sc dependency has been confirmed with DNS. The close agreementwiththeresultofthetheoreticalderivationisinterestingsinceitisbased onthe inthevicinityofthesurface,andtheassumptionthat ∝ fora ∝ for a surface with surfactants (no‐slip). This clean surface (slip) and assumption is as shown in Paper I only valid for the innermost part of the diffusive boundarylayerfor 7,whichinturn,impliesthatitactuallyistheprocessesinthe very vicinity of the water surface that controls the gas exchange during natural convection. −2 10 n = 1/2 n = 2/3 k s2F k s2B k n2B −3 10 ks ms-1 s2F n7−600 = 0.519 s2B n7−600 = 0.521 −4 10 −5 10 n2B n7−600 = 0.668 −6 10 0 1 10 2 10 10 Sc 3 10 Figure14. Scalartransfervelocityforcases 2 , 2 (bothslip,clean surface)and 2 (no‐ slip,saturatedsurfactant)forthreepassivescalarswithSchmidtnumber 7, 150, and 600, respectively. The results for thebasecase 2 andthecasewithfinermesh resolution, 2 , are difficult to distinguish since they are very similar. Dashed and dotted linescorrespondto 1⁄2and 2⁄3andoriginatefrom , fortheslipandno‐slip boundaryconditioncaserespectively. 23 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 3.CONTRIBUTIONS Heretheparameterizationofthetransfervelocityforpurenaturalconvection ⁄ , (18) canbepresentedbeforetheresultsforthecaseswithcombinedbuoyancyandshear‐ 0.4isatransfervelocity stressforcingarepresentedinthesectiontocome. coefficientand 1⁄2forslip(clean)and 2⁄3forno‐slip(saturatedsurfactant) boundaryconditionsatthesurface. 3.5 Influenceofcombinedwindandbuoyancyforcingonthegas transfervelocity One of the major contributions of this thesis is to find at which conditions the buoyancy‐ and wind‐forcing dominate the gas transfer velocity. The results for low wind speed conditions ( 2 ) from the numerical study in Paper III are discussedinthissectionwhiletheresultsforintermediatewindspeedfromthefield measurementspresentedinPaperIVwillbediscussedinsection3.6. Inordertohelpthereaderwerepeatthatthecasesarenamedwith(i)anumberfor the shear‐based Reynolds number ∗ ∗ ⁄ , which describes the ratio between inertialandviscousforces,and(ii)a forBuoyancyor forNo‐Buoyancy.Thecase 0 inPaperIIIisidenticalwiththecase 2 inPaperI. The flow pattern for buoyancy driven flows is characterized by thin descending plumesofcolddensewater,warmwiderascendingplumes,andoccasionallysurface‐ normalvortices.Thesurfacenormalscalarfluxfollowthispattern(Figures9aandd). Figure 15 shows snapshots of the surface normal scalar flux fields at the surface for the case 0 (pure convective forcing) and the cases with combined and pure shear forcing.Itisseenthatoncetheshearstressisappliedtothesurface(inthe ‐direction towardsrightinthefigures),thepatternandvorticesstarttobebendedandstretched andafish‐scalepatternbecomesvisible. Wall‐bounded flows have been shown to typically create streaky structures in the vicinityofthewallwithaspanwisespacingofabout100 ∗ [e.g.,KimandMoin,1989; Kim et al., 1987]. Later it was shown to be valid for temperature fields and slip boundary conditions as well [e.g., Handler et al., 2001]. It can now be seen that the coherent structures in the fish‐scale patterns follows this streak spacing length scale well for cases driven by pure shear‐stress (120 and 180 in Figure 15). It can, however, also be seen that these coherent structures typically are finer with than without buoyancy comparing 120 with 120 and 180 with 180 . The scalar fluxvariationisincreasingwithincreasingshearforcingandthevariationishigherfor pure shear forcing than for combined forcing. These difference between cases with pure shear and combined forcing decrease with increasing shear‐stress (increasing ),indicatingthatthebuoyancyforcingbecomeslessimportant.Itisinthefollowing shown that this transition in the scalar flux pattern (decreasing difference) is accompaniedbytransitionsinmanyotherflowcharacteristicsandeventuallythegas transfervelocity. 24 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 3.CONTRIBUTIONS Increasing shear forcing Buoyancy and shear forcing Pure shear forcing 0 60 100 ∗ 120 120 100 ∗ 100 ∗ 180 180 100 ∗ 100 ∗ 240 2 100 ∗ 0 F ⁄F 4 5 Figure 15. Normalized surface‐normal scalar flux fields. The cases are named as the for o‐ uoyancy.Thesamescalingisusedforall ∗ ∗ ⁄ and for ouyancyand ⁄ ∗ ,isindicatedinthesubplotsforcaseswith subplots.Thelengthscale100 ∗ ,where ∗ 0. Case 60B is not shown in Paper III since it needs further sampling time before ∗ statisticalpostprocessing.Thesimulationhas,however,reacheditssteadystateconditionso thatasnapshotcanbeusedasisdoneinthisfigure. Figure16showsthesurface‐normalscalarfluxatthesurface,thescalarconcentration intheinteriorandiso‐surfacesofthepositiveandnegativestreamwisevorticityΩ .By comparing Figure 9 with Figure 16, the transition from buoyancy to shear stress dominatedflowcanbeseenwithintheinterioroftheflowaswell.Itisshownthatthe vorticity cores are more elongated and surface‐centered for increasing shear‐stress. Furthermore,itcanbeseenthatthevariationofthenormalizedsurface‐normalfluxis increasing (note the different scales) with streaks of intense flux (red) as the shear stress is increased. The zoomed square shows the interplay of streamwise vorticity, scalarconcentrationandsurface‐normalscalarfluxclearer.Hereitisseenthatwater with low scalar concentration is drawn downwards (downwelling) between the vorticity‐cores shifting from positive to negative in the ‐direction (diagonally from thelefttotheright).Thisphenomenonissimilartothethinplumesofdensewaterin the case with no shear shown in Figure 9. Concurrently, areas of thin diffusive boundary‐layersareformedbetweenthevorticity‐coreschangingsignfromnegative to positive, and these areas then coincide as expected with areas of high surface‐ normalscalar‐flux(upwelling). 25 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 3.CONTRIBUTIONS 60 240 F ⁄F F ⁄F 0 s⁄F ∙ 10 sm 0 2.5 0 4 s⁄F ∙ 10 sm 0 1.8 1.5 Figure16.Snapshotsforcase60 (lowleft)and240 (lowright)andablow‐upfromcase 240 .Surface‐normalscalarflux ⁄ atthesurfaceandscalarconcentration ⁄ inthe interiorofthedomain.Isosurfacesofnormalizedstreamwisevorticity ⁄ ∗ ⁄ equal0.25 and 0.25coloredredandblue,respectively.(Figure3inPaperIII). Thescalartransfervelocities , increaselinearlywith ∗ forcaseswithpureshear‐ stress forcing (Figure 17a). These results are close to the measurements of gas transfervelocitiesinawindtank[Jahneetal.,1987] , 8.9 , (19) ∗ given in the same figure. Combined forcing gives on the other hand a more or less constant , for low ∗ , and then , seems to connect to the linear trend as ∗ increases.AnotherwayofexpressingthiscanbeseeninFigure17bwhere , ⁄ ∗ asa functionof ispresentedfollowingequation(15).Here , ⁄ ∗ isdecliningdowntoa limiting magnitude for decreasing Ri. This limiting magnitude is set by the no‐ buoyancy cases. A Richardson number 0.004 is found to express the conditions whenthescalartransferstartstochangefrombeingdominatedbybuoyancyforcing toshear‐stressforcingwhichisrelevantfordeterminingthebuoyancyinfluence[e.g., Macintyreetal.,2002;Readetal.,2012;RutgerssonandSmedman,2010]. 26 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 3.CONTRIBUTIONS −4 a) 1 b) x 10 0.08 B 0.9 0.07 0.8 0.06 0.6 ks,7/u* ks,7 ms-1 0.7 0.5 0.4 B 0.3 NB 0.05 0.04 kg,W2009 0.2 0.03 kg,CC1998 0.1 0 180NB kg,J1987 0 0.5 1 1.5 u∗ ∙ 10 ms 2 0.02 −3 10 2.5 −2 10 −1 10 Ri Figure 17. (a) The scalar transfer velocity , . Circles denote cases with , ⁄ buoyancy and squares denote no‐buoyancy cases. The dashed and dash‐dotted lines denote the wind parameterizations with 1⁄2 according to equations (5) and (6) respectively. The solid line denotes a linear increase of the transfer velocity as a function of the friction velocity[Jahneetal.,1987].(b)Transfervelocityconstant , ⁄ ∗ accordingtoequation(15) asafunctionof .(Figure6and7binPaperIII) 3.6 Observationalresultsathigherwindspeeds,andspeculations abouttheinfluenceofwaves The transfer velocity estimations based on the flux‐chamber and dissipation parameterization, and the quantities significant wave height ( ), and IR surface ) show a close relationship (Figure 18). The uncertainty in the velocity ( dissipation parameterization, however, increased for low dissipation rates since the turbulencesignalthenwasdominatedbywavesandinstrumentnoise.Hence,transfer arenot velocitiesusingthedissipationparameterizationthatarebelow2 ∙ 10 presented in Figure 18. Nevertheless, a significant linear relationship was found betweenthetransfervelocityestimationsbytheflux‐chambermethodanddissipation parameterizationsandthesignificantwaveheight.Alinearfitgave 1.95 ∙ 10 4.9 ∙ 10 4.9 ∙ 10 1.5 ∙ 10 . (20) , , Nolocalwindmeasurementswereavailablefromtheexperiment,butitisreasonable to assume that the wave heights can be related to the wind speed and fetch, [e.g., Hasselmannetal.,1976].Forafetchofabout3km,whichisreasonablefortheBornö siteatthegivenwinddirection,thisequationcanbetransformedinto 5.4 ∙ 10 3.7 ∙ 10 4.8 ∙ 10 1.5 ∙ 10 . (21) , , 27 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 3.CONTRIBUTIONS Figure 18. Comparison of transfer velocities using the flux‐chamber method (filled circles), dissipation parameterization (plus signs), significant wave height (diamonds), and IR rms surfacevelocity(filledtriangles).(Figure7inPaperIV). Theslopeofequation(21)isclosetotheslopeof , for 1⁄2andratherclose tothetransfervelocityforcedbyshearstressonly,seesection3.7.3.Itcantherefore bearguedthatitactuallyistheshearstressratherthanthewavesthatenhancethegas transfervelocity(elaboratedinPaperIII).Ifso,thesignificantwaveheightcanstillbe agoodproxyfortheintegratedeffectoftheshearstress,andtherebythegasflux,over theareaofinterest.Inordertomakeequation(21)moregeneral,theequationcanbe transformedtoincludethesignificantwaveheightandfetch(seePaperIV). There was also a good correlation between the transfer velocity and the rms horizontal surface velocity estimated from PIV (from IR images). This can be expressedas , , ∙ (22) where and areconstants.Thiscloserelationshipfor isunexpectedsince there is no direct relation between the rms horizontal surface velocity and the rms near‐surfaceverticalvelocitywhichmaybeexpectedtobeimportantforgasexchange (see the discussion regarding solenoidal and irrotational components in section 3.4 and Figure 13). A reasonable explanation for the close relationship may be found in the close relationship between the significant wave height and the transfer velocity, since estimated rms wave orbital velocities agree fairly well with from theIR/PIV‐measurements.Inthatcase,eventhiscorrelationiscausedbytheinfluence ofwindstressonbothgastransferandwaves. One example of the temperature field used for PIV is shown in Figure 8 (from IR images).Itshowsasimilartemperaturepatternasthelowshear‐stresscasesshownin Figure 15. Even though the DNS results show that the surface divergence 28 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 3.CONTRIBUTIONS parameterization works well, the transfer velocity estimation using the surface divergence from the IR/PIV‐measurement failed to give good correlation with the results from the other estimation methods. This is, however, most likely not due to problemswiththedivergenceparameterizationbutrathertheresultofaneedoffiner spatialresolutionofthevelocityfieldintheIR/PIVmethod.Otherwise,theimportant small‐scaledivergenceisnotproperlytakenintoaccountintheestimationofthetotal divergence. 3.7 Parameterizationsofthegastransfervelocity 3.7.1 Basedondiffusivity,surfactants,andshear‐stressandbuoyancy forcing PaperIshowsthatthetransfervelocityforpurebuoyancyforcingiswellrepresented byequation(18).Itishereinterestingtoseetheevidentrelationshiptothebuoyancy fluxandthatthereisnoinfluenceofthedepthforlargeenoughdepths.Itcanalsobe seenthattheSchmidtexponentcloselyfollowsthetheoreticalderivationforslipand no‐slip wall conditions. It should, though, be noted that the kinematic viscosity is included into equation (18) from dimensional considerations only and has not been alteredintheanalysis. Theparameterizationforthecombinedforcingismorecomplextakingboththeshear‐ stress and the buoyancy forcing into account. Paper III presents three different parameterizations.Therationaleforthreeparameterizationsis(i)thattheusersofthe parameterizationsmighthavespecificneedsfortheirimplementationand(ii)thatthe parameterizations represent different ways of interpreting the physics. The first parameterizationisbasedonahypothesisthattheforcingsfrombuoyancyandshear stressareadditiveresultingin k , A u∗ Ri⁄Ri 1 ⁄ Sc , (23) ⁄ where Ri is a critical Richardson number and 0.1 is the transfer velocity coefficient for shear‐stress forcing. It is based on the dissipation parameterization (equation (9)) and makes use of the fact that the forcings from buoyancy and shear scale as and ∗ ⁄ , respectively. This parameterization (equation(23))isfor 50, 100,and200 drawningreeninFigure19. The other two parameterizations are based on the hypothesis that the transfer velocitycanbeseentobeintwodifferentstates.Thefirststateisbuoyancydrivenand the second state is shear‐stress driven. The transition from one state to another is defined by a critical Richardson number. These two parameterizations either use an errorfunctionas , ⁄ ∗ , 1 29 , (24) Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 3.CONTRIBUTIONS orapiecewiselinearfunctionas ⁄ , , ∗ . (25) 50, 100,and200 inredwith , 0.01and Equation(24)isdrawnfor 100 in yellow in Figure 19. All three equation (25) is drawn for parameterizations(23)‐(25)convergeathighaswellaslow . x 10 −4 1.2 1 k7 ms-1 0.8 0.6 B NB 0.4 kg,W2009 kg,CC1998 kg,J1987 kg,sum kg,erf 0.2 kg,tres 0 0 1 2 u∗ ∙ 10 ms 3 Figure 19. The gas transfer velocity constant , according to equations (23) in green and 50, 100,and200 .Thescalartransfervelocitiesforpurebuoyancy (24)inredfor 50and200 aremarkedwithfilledmarkersandwerepresentedin forcingwith PaperI.Thetransfervelocitiesforthepurebuoyancyandshearforcing(equation(25))are indicatedinyellow.(Figure9inPaperIII) 3.7.2 Basedondissipation,divergence,orheatflux It was found in Paper I that the three parameterizations based on heat flux, dissipation,ordivergence(equations(8)‐(10))giveverygoodestimatesofthetransfer velocity for pure buoyancy forcing while varying the surface heat flux and domain depth. The values of the related transfer velocity coefficients are 0.45, 0.57,and 0.90. 30 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 3.CONTRIBUTIONS The results for the parameterizations for the combined forcing in Paper III are, however,notasconvincingastheresultsforbuoyancyforcingonly.Thisgivesreason tobecautiousinusingthedissipationasaunifyingquantity(proxy)thatcanbeused to add different forcings into a total gas transfer parameterization. The relative ⁄ variation is here used to rank the different parameterizations. Thevariationisforthecases0 to240 foundtobeapproximately10%,15%,and more than 35% for the heat, divergence, and the dissipation parameterizations respectively. 3.7.3 Basedonthemeanwindspeed Figure 20 shows a number of parameterizations as functions of . The parameterizationinequation(23)ishereplottedforsurfaceheatfluxesintherangeof 0 400 . The buoyancy flux influences the gas‐transfer velocity up to accordingtoadditiveparameterization,k , . approximately2‐4 Furthermore,itisshownin Figures19and20 that thetwoparameterizations based on the mean wind speed (equation (5) and (6)) give reasonable predictions of the transfer velocity for both the cases with pure natural convection and the combined forcingforacleansurface(slip).Itcanbeseenthatoneparameterization[Wanninkhof et al., 2009] overestimates and the other parameterization [Cole and Caraco, 1998] underestimates the transfer velocity compared to the base case with a surface heat . Congruent transfer velocities using equations (5) and (18) would flux of 100 imply a surface heat flux of approximately 35 , whilst equations (6) and (18) .Thesurfaceheatfluxis,however,inequations wouldimplyapproximately200 (5)and(6)notexplicitlyaccountedforsincethetransfervelocityisafunctionof only. For low wind conditions, it is therefore advisable to use any of equations (23)‐ (25)inordertohaveaparameterizationthattakesthebuoyancyfluxintoaccount.It can further be seen that the two parameterizations (19) from wind tunnel tank test and equation (21) from field measurements give transfer velocities in the same magnitude. Figure20alsoshowsthelargeinfluenceofthesurfactantsonthegasflux.Thereisa factor of approximately 3 between the transfer velocity for a clean surface (slip, 1⁄2) and for a surface that is saturated with surfactants (no‐slip, 2⁄3) for 600. Previous research has found that microscale breaking waves significantly contribute tothemeansquareslopeofwaves,whichinturncanbecorrelatedtothegastransfer velocity [Zappa et al., 2002; 2004]. The gas transfer was found to be enhanced by a factor 3.5 comparing background levels and areas with microscale breakers. Furthermore, it is inferred that the microscale breaking may be the mechanism that enhances heat and mass transfer. The close agreement between the slopes in the parameterizations originating from wind tunnel tank measurements ( , ), using significant wave height ( , ), , and , , and equation (23) for clean conditions( 1⁄2)and 0 ( 0)isthereforeinterestinginaspectsof how large enhancement of the transfer‐velocity waves and microscale breakers may givecomparedtopureshear‐stressforcing,andisrelevanttonoticeforfuturework 31 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 4.SUMMARYANDCONCLUSIONS −5 2.5 x 10 kg,sum n=1/2 n=2/3 kg,sum kg,J1987 n=1/2 kg,W2009 kg,CC1998 kg,G2013 k600 ms-1 2 1.5 1 0.5 0 0 1 2 3 U ms 4 5 Figure 20. Transfer velocity constant , according to equation (23) in green for 0, 50, 100,200and400 and 1⁄2(clean)anddottedline 2⁄3(saturated surfactant).Theparametrizationsinequations(5‐6),(19),and(21)aregivenforreference. The transfer velocity estimated with equation (6) is transformed into 600 using equation(7)and 1⁄2. 4 SUMMARYANDCONCLUSIONS There is a growing need to determine the air‐water gas exchange accurately during lowwindconditionssince(i)thereisageneralincreaseoftheconcentrationandthe interest in greenhouse gases e.g., CO and CH in the atmosphere, (ii) it has become evident that the gas fluxes from fresh‐water bodies (where the wind speed often is low) play an important role in the global carbon cycle, and (iii) often used parameterizations based only on the wind speed cannot take other forcings as e.g., buoyancyintoaccount. Newparameterizationsfor gastransfervelocityestimationshave been developedby use of numerical simulations and field measurements in the ocean. The numerical simulations are performed as direct numerical simulations of fully developed turbulent flow for pure natural convective forcing, and combined convection and wind‐shear forcing. The influence of surface heat flux, mixed layer depth, Schmidt number, and surfactants are evaluated. The field measurements comprise gas flux measurements by gas‐flux chamber, IR/PIV‐recording (e.g., surface flow divergence, rmssurfacevelocity),andrateofturbulentkineticenergydissipationinthewaterby acousticdopplervelocimetry. The temperature and the surface‐normal scalar flux fields at the surface show elongatedstreaksofwarmandcoldwater(highandlimitedscalarflux)forthecases withpureshearstressforcing.Thestreakspacingisoftheorderof100 ⁄ ∗ (forpure 32 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity 4.SUMMARYANDCONCLUSIONS shearstressforcing)whichpreviouslyhasbeenseenforshear‐stressdrivenflows,for both slip and no‐slip surface boundary conditions. The temperature and the surface‐ normalscalar fluxfieldare morecomplexforthecombinedforcedcases.Asshearis “added”toapurenatural‐convectionconditiontheplumesstartstobeelongatedand eventuallyastheshearstressisincreasedthestreaksshowalargersimilaritywiththe pure shear‐driven cases. This and the averaged statistics for all quantities studied indicate a transition from natural‐convection (buoyancy) to shear‐stress dominated flow at 4 ∙ 10 , which means that buoyancy fluxes are not important for gas . exchangeatwindvelocities above3 Parameterizationsusingtherateofturbulentkineticenergydissipation,surfaceflow divergence, and heat flux, estimate the transfer velocity well for the cases with pure natural‐convection, while varying the surface heat flux, domain depth, and Schmidt number. These parameterizations, however, experience some problems for the cases with combined convective and shear stress forcings. The relative variations in the transfervelocityarelargestfortheparameterizationbasedonthedissipationrateand smallest for the parameterization based on the heat flux. The two parameterizations based on wind speed estimate the transfer velocities reasonable well, depending, however, on the surface heat flux. One parameterization [Cole and Caraco, 1998] correspondstoaheatfluxofapproximately35 whiletheother[Wanninkhofet al.,2009]correspondstoaheatfluxofapproximately200 .Furthermore,there isnoincreaseoftransfervelocityasfunctionofincreasingdomaindepth(studiedfor pureconvectionforcedcaseonly). Theproposednewgastransferparameterizationsrepresenttwodifferenthypotheses. Thefirstoneassumesthattheforcingfrombuoyancyandshearstressareadditive.It usestheframeworkofthedissipationparameterizationandsumsthetwodissipation scales for buoyancy and shear stress which results in the expression ⁄ where the critical Richardson number 1 ⁄ , ∗ ⁄ 4 ∙ 10 . The second one assumes that the forcing is either from the buoyancy or the shear stress.Herethetransfervelocityiseithermodelledwithacontinuouserror‐function , ⁄ , ∗ , where , 0.01. orwithapiecewiselinearfunction,thathasaconstanttransfervelocitythatequalsthe ⁄ , for whereafter the transfer due to buoyancy , is transfer velocity due to shear according to the expression , ∗ used. The gas flux is to some extent overestimated by , and underestimated by and , .Themaximumerrorsforthefirsttwoparameterizationsarelessthan , 10% and for the latter approximately 20%. The critical Richardson number in these parameterizations can be seen to express the transition point where the gas‐flux forcingshiftsfrom beingdominated byeither buoyancy orshear‐stress.Thisimplies that the buoyancy flux influence the gas transfer velocity up to approximately 3 fornaturalconditions. 33 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson 5.FUTUREPERSPECTIVES The results from the field measurements show close relationships for the method using flux‐chambers and the parameterization using the rate of turbulent kinetic energy dissipation, and the quantities surface rms velocity and the significant wave height.Theserelationshipscanallbeexpressedas ∝ withaslopeclosetothe resultsintheDNSforpureshearstressforcingandtestsinawindtank[Jahneetal., 1987]. 5 FUTUREPERSPECTIVES Thepresentworkprovidesafirmgroundfortheunderstandingofgasfluxacrossthe air‐waterinterfaceduringlowwindspeedconditions.Thereare,though,stillneedsfor futureworkforhighSchmidtnumbersand higherwindspeeds.Althoughtheresults for high Schmidt number gases in Paper I are consistent regarding the gas transfer velocity,isstillremainstobeconfirmedthatthesearevalidforacomputationalmesh with higher resolution as well. This can be achieved in a similar way as Herlina and Wissink [2014] who used two different meshes with different mesh resolutions for solvingtheflowfieldandthescalarfield. It would also be fruitful to continue the work towards more accurate gas transfer estimationsforhigherwindspeeds.Thenextstepisthentoperformdirectnumerical simulations in order to study the mechanisms of e.g., microscale‐breaking and breaking waves. This has to some extent been done [e.g., Lin et al., 2008; Tsai and Hung, 2007] but there is still a need for a comprehensive parameter study including theinfluenceofsurfactants. Furthermoreitwouldbeinterestingtostudyhowtheconceptofturbulence‐surfactant parameter can be used in field work and how the surfactant concentration and the surfaceelasticitycanbeestimated? Eventually it would be worthwhile to study whether the temperature field recorded withanIRcameracanobtainhighenoughresolutiontocapturetheimportantsmall‐ scaledivergenceatthesurface. 34 SamT.Fredriksson Estimatingtheair‐watergastransfervelocity ACKNOWLEDGEMENTS ACKNOWLEDGEMENTS NowIwouldliketotaketheopportunitytoacknowledgemysupervisors,colleagues, friendsandfamily,whohavesupportedmeduringmystudiesandthisPhDproject. First, I would like to express my sincere gratitude to my supervisor, Lars Arneborg, andco‐supervisor,DavidBastviken.Ihavealwaysfeltaverysupportiveattitudefrom them,andbeenencouraged bytheircompetenceandeagerness tolearnandexplore new things. I would also like to express my sincere gratitude to Robert Handler and Håkan Nilsson who both have been very encouraging during the work leading up to this thesis. The meetings with Robert (by wire, in Texas, or in Gothenburg) have alwaysbeenasourceoflargeinspirationtome.Håkanhasbeenaveryvaluableguide duringmyexplorationandeventuallyprogrammingofOpenFOAM.Iwouldalsoliketo expressmyappreciationtotheremainingco‐authorsofjointlywrittenpapersduring mystudies.Ithase.g.,beeninspiringtoseethefascinatingIR‐imagesbyoneoftheco‐ authors, Magnus Gålfalk. Many thanks also to the group of gas‐flux researchers from theUppsalaUniversityfornicefieldworkandfruitfuldiscussions. I would also like to thank Ola Kalen, a companion at several conferences who I have sharedofficewithforseveralyears.Besidetheworkithasbeeninspiringtotalkabout ourcommoninterestsoutsidetheacademia.Thanksalso;toArdoRobijnforbeingmy fellow researcher at the expeditions to the areas at and around Svalbard, to Anna Wåhlin for making our participating in that research project (PREPARED) possible, andtoallofthemembersofthesememorableresearchcruises.Thelistofcolleagues tothankcouldbeaslongasamoviecredit.However,duetospacelimits,letmejust send my general and genuine thanks to all the colleagues at GU not mentioned here butstillrememberedandallofyouwhohavebecomedearfriendsofmineduringmy studies. Agoodlifeoutsidetheofficeisofgreatimportanceformeandforthesuccessofmy work,andIamluckytohavemanywonderfulpeoplearoundme.FirstIwouldliketo thank Anna for her support, joy, and ultimately even some help with the final formatting of this thesis. I would then like to give my wholehearted thanks; to mom and dad for their caring and unconditional support to me and the family, and to my brotherformanygoodtimestogether.GreatthankstoKristinaandAndersforalltheir journeys to Gothenburg during these years. I am also very grateful for the large and caringringoffriendsthatbringmesomuchfriendshipandenjoyment. EventuallyIwouldliketosendaheapofhugstomylovelychildrenSara,Martin,and Emil.Youhavebeenverypatientduringthisspringandyoualwaysbringmesomuch joy! I would also like to acknowledge Semcon who gave me study leave to fulfil this PhD project. The computations were performed on resources provided by the Swedish National Infrastructure for Computing (SNIC) at C3SE (Chalmers Centre for Computational Science and Engineering) computing resources. My supervisor Lars ArneborgwassupportedbytheSwedishResearchCouncil. 35 Estimatingtheair‐watergastransfervelocity SamT.Fredriksson REFERENCES REFERENCES Bade, D. L. 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