164CHAPTER10PLUMES Figure101AhydrothermalventatthebottomofthePaci cOceandischarginghotwaterupto400CandformingaverticalplumeThedarkcolorgivingrisetothenicknameblacksmokerisduethepresenceofsul ID: 141370
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Chapter10PlumesandThermalsSUMMARY:Thischapterdescribesseveraldistinctstructuresthat\ruidsdevelopinreactiontolocalizedinputsofbuoyancy.Apunctualandsustainedsourceofbuoyancyusuallycreatesacontinuousriseoflighter\ruidthroughtheambientdenser\ruid,withmixingoccurringalongtheway.SuchstructureiscalledaplumeShouldtheprocessbeintermittent,therisingbuoyant\ruidparcelsarecalledther-malsBuoyantjetsareplumeswiththeaddedpropulsionofmomentum,andbuoy-antpu sare\ruidparcelsthatriseunderthecombinedactionofbuoyancyandmomentum.10.1PlumesPlumesarecommonfeaturesinenvironmental\ruids,whichoccurwheneveraper-sistentsourceofbuoyancycreatesarisingmotionofthebuoyant\ruidupwardandawayfromthesource.Theclearestexampleisthatofhydrothermalventsatthebottomoftheocean(Figure10.1).AnotheroccurrenceistherisingoffreshwaterfromthebottomoftheseaatsubmarinespringsinkarsticregionssuchasalongtheDalmatianCoastofCroatia,wheresuchfeaturesarecalledvrulje.Thecom-monurbansmokestackplumeis,however,somewhatdi erentbecausethewarmgasrisesnotonlyunderitsownbuoyancybutalsounderthepropulsionofmomentum(inertia).Suchplumeismoreproperlycategorizedasabuoyantjetorforcedplume.Whatdrivesaplumeisitsheat\rux,de nedastheamountofheat(expressedinjoules)beingdischargedthroughtheexitholeperunittime.Becauseitismorepracticalinlatermathematicaldevelopments,thisquantityisdividedby0Cp(the\ruid'sreferencedensityandheatcapacityatconstantpressure)andthenmultipliedby g(the\ruid'sthermalexpansioncoecientandthegravitationalacceleration),givingrisetothebuoyancy\ruxF163 164CHAPTER10.PLUMES Figure10.1:AhydrothermalventatthebottomofthePaci cOcean,discharginghotwater(upto400C)andformingaverticalplume.Thedarkcolor,givingrisetothenicknameblacksmoker,isduethepresenceofsul desinthewater.[PhotographtakenbyDudleyFoster,courtesyoftheWoodsHoleOceanographicInstitution]F= g 0Cpheat time(10.1)NotethatbecauseheatpertimeisexpressedinJ/s,thebuoyancy\ruxismeasuredinunitsofm4/s3Letusconsiderathree-dimensionalradiallysymmetricplumeprogressingverti-callyfromthebottomthroughahomogeneousandresting\ruid,asshowninFigure10.2.IfwedenotebyT0thetemperatureoftheambient\ruid,thenthetempera-tureinsidetheplumehasthevalueT0+T0,inwhichT0denotesthetemperatureanomaly(positiveinarisingplume,negativeinasinkingplume).Tothistemper-atureanomalycorrespondsadensityanomaly0= 0T0.Fromthelatter,itisconvenienttode nethelocalbuoyancy,orreducedgravity,g0as:g0= g0 0=+ gT0(10.2)Naturally,becauseoftheheterogeneousstructureoftheplume,withentrainmentanddilutiontakingplacealongitssides,thebuoyancyg0andverticalvelocitywwithintheplumedependonbothdistancezabovethesourceandradialdistancerfromthecenterline.Likeforturbulentjets(previouschapter),observationsrevealthattheGaussianpro le(bellcurve)providesarealisticdescriptionofthestatisticalaveragesofg0andwovertheturbulent\ructuations.Beforeusingsuchexpressions,however,weshallinitiallylimitourselvestoconsideringonlycross-plumeaverageswandg0,eachafunctionofz,theheightwithintheplume.Thebuoyancy\ruxFcanbeexpressedastheintegralacrossthesectionoftheplumeoftheproductoftheverticalvelocitywwiththebuoyancyg0.Intermsof 10.1.PLUMES165 Figure10.2:Asmokestackplumerisinginstillair.(Photobytheauthor) Figure10.3:Meanisotherms(left)andstreamlines(right)inandaroundaplumemain-tainedfromapunctualsourceatthebottom.Thenumbersontheisothermsarerelativeval-uesg0=gandthosealongthestreamlinesarerelativevaluesoftheStokesstreamfunction.(FromRouseetal.,1952asshowninTurner,1973) 166CHAPTER10.PLUMESthemeanvaluesacrosstheplume,areasonablygoodapproximationisF=R2wg0=R2 gT0w;(10.3)whereR(z)istheradiusoftheplumeatlevelzAstheplumerises,itentrainsambient\ruid,butthisdoesnotchangetheheat\ruxcarriedbytheplumesincethatambient\ruidcarriesnoheatanomaly.Thus,byvirtueofconservationofheat,thebuoyancy\ruxremainsunchangedwithheightandisthesameatlevelzasitwasatthestartoftheplume.Inotherwords,thequantityFisconstant.Itiswhatdrivestheplume,likemomentumdrivesajet. Figure10.4:Athinslideoftheplumeonwhichtoperformmassandmomentumbudgets.Perfomingamassconservationbudgetoverathinsliceoftheplumeextendingfromlevelztolevelz+dz(Figure10.4),wecanwrite:Massexitingfromthetop=Massenteringthroughthebottom+MassentrainedthroughthesideeR20wz+dz=[R20wz+2Rdz0u;inwhichuisthelateralentrainmentvelocityand2Rdzthelateralareaoftheslice(Figure10.4).Indi erentialform,thisequationbecomesd dz(R2w)=2Ru:(10.4)Likewise,theverticalmomentumbudgetoverthesameslicerequiresMomentumexitingfromthetop=Momentumenteringthroughthebottom+Momentumentrainedthroughtheside+UpwardbuoyancyforceThereisnoverticalmomentumacquiredbylateralentrainmentsincetheambient\ruidisatrest.ByvirtueofArchimedes'principle,theupwardbuoyancyforceisequaltotheweightofthedisplaced\ruid(atdensity0)minustheactualweightoftheplumesegment(atlowerdensity0+0),foratotalof 10.1.PLUMES167Upwardbuoyancyforce=R20gdz R2(0+0)gdz= R20gdz=+R20g0dz;byvirtueof(10.2).Thus,theverticalmomentumbudgettakestheform::R20w2z+dz=[R20w2z+R20g0dz;or,indi erentialform,d dz(R2w2)=R2g0(10.5)Letusstopforamomentandtakestockoftheequationswehave.Therearethreeequations:(10.3)fromtheheatbudget,(10.4)fromthemassbudget,and(10.5)fromthemomentumbudget.And,therearefourunknowns:theradiusRtheaveragevelocityw,theentrainmentvelocityu,andtheaveragedbuoyancyg0eachafunctionoftheelevationz.Thereisthusonemoreunknownthanavailableequations.Toclosetheproblemwithoutsolvingforthedetailsoftheturbulent\row,westateinanalogywiththeturbulentjet,thattheentrainmentvelocityisproportionaltotheshear\rowinducedbytheplume.Inotherwords,weassumeproportionalitybetweenuandwu=aw;(10.6)withconstantdimensionlesscoecientaThesolutionoftheproblemoughttobeexpressedsolelyintermsofthequan-titiesF(inm4/s3)andz(inm),becausethosearetheonlydimensionalvariablesenteringtheequations.Thus,dimensionalconsiderationsleadustoanticipatetheformofthesolution:R=tanzw=bF13 z13g0=cF23 z53whereistheangleoftheconemadebytheplume(Figure10.3).Observations(Turner,1973)indicatethatthisangleisabout8.9,forwhichtan=0157.Substitutionintheequationsatourdisposalyields:Eq.(10.3) !bctan=1Eq.(10.4) !5 3tan=2aEq.(10.5) !4 3b2=c: 168CHAPTER10.PLUMESThevaluesofthedimensionlesscoecientsarefoundtobea=5 6tan=01305b=3 4tan213=214c=4 32tan413=608andtheassembledsolutionisR=0157z(10.7)w=214F13 z13(10.8)u=0130w=0279F13 z13(10.9)g0=608F23 z53(10.10)Letusnowpassfromwandg0averagedacrosstheplumetofunctionswandg0withGaussianpro leacrosstheplume.Forthis,wewrite:w=wmax(z)exp r2 22(10.11)g0=g0max(z)exp r2 22(10.12)withthestandarddeviation(z)beingsuchthat2(z)representstheradiusR(z)oftheplumeatheightz.(Seetheoryfortheturbulentjetinthepreviouschapter.)Thus,=R=2.Thepeakvaluesalongtheplume'scenterlinecanthenberelatedtotheirrespectiveaveragesbyw=1 R2Z10w2rdr(10.13)g0=1 R2Z10g02rdr;(10.14)andweobtain:w=427F13 z13exp 816r2 z2(10.15)g0=122F23 z53exp 816r2 z2(10.16) 10.2.PLUMESINSTRATIFICATION169 Figure10.5:Aplumerisinginanearlymorningwhentheloweratmosphereisstrati ed.Aclueofthisstrati cationisthethinhorizontalbandofcloudontheleftoftheplume(markedinpicture).Notetheinertialovershootoftheplumecloudbeforesettlingatthelevelofneutralbuoyancy.[Photographbytheauthor]Laboratoryexperiments(Turner,1973),indicatethatthefollowingadjustedexpressionsw=47F13 z13exp 96r2 z2(10.17)g0=11F23 z53exp 71r2 z2(10.18)bettermatchtheobservations.Notethatwiththeselastexpressions,thewidthofthevelocitypro leisslightlynarrowerthanthatofthebuoyancy.10.2PlumesinaStrati edEnvironmentWhenaplumerises(orsinks)inastrati edenvironment,itencountersatempera-turebecomingclosertoitsownandprogressivelylosesbuoyancy.Atsomelevel,itwillhavelostallbuoyancyandwillbegintospreadhorizontally.Suchisthecaseofasmokestackplumeinacalm(nowind)andstrati edatmospheretypicaloftheearlymorning(Figure10.5).Theobviousquestiontoaskishowhighdoestheplumereach?Thestrati edambient\ruidischaracterizedbyitsstrati cationfrequencyNde nedfromN2= gdT dz(10.19) 170CHAPTER10.PLUMESwhereT(z)isthetemperaturepro leoftheambient\ruid.Todescribeaplumeinthistypeofenvironment,thesamequantitiesareneededasbefore,namelytheplume'sradiusR,averagedverticalvelocityw,andaveragedbuoyancyg0,eachfunctionoftheelevationz.Thedi erencewiththeprevioussectionisthatnowthebuoyancy\ruxF,de nedin(10.3),isnolongeraconstantalongtheaxisoftheplume.Threeequationsareatourdisposal:Themassbudgetd dz(wR2)=2uR=2awR;(10.20)themomentumbudgetd dz(w2R2)=g0R2(10.21)whichremainunchanged,andtheheatbudget,whichisd dz(wTplumeR2)=uT(z)2R:(10.22)Usingthebuoyancylocallyexperiencedbytheplume,g0= ggTplume T(z)],thelastequationcanberecastasd dz(g0wR2)+d dzz gT(z)wR2]=2uR gT(z)WithEquation(10.20)andde nition gdT(z)=dz=N2,itcanbereducedtod dz(g0wR2)= N2wR2(10.23)Thesolutiontothissetofequationsdoesnotexhibitsimilarity,buttheequationscaneasilybeintegratednumerically.Startingwithinitialconditions(atlevelz=0)suchthatthemomentumandvolumetric\rowarenilbutbuoyancy\rux nite,one ndstheresultsshowninFigure10.6.Onthat gure,thevariablesaremadedimensionlessbyscalingasfollows:zandRby(F=N3)14,wby(F=N)14,andg0by(F=N5)14,whereFisthestartingbuoyancy\rux(atz=0).Theentrainmentparameterawastakenas0125.(Forasimilarnumericalintegration,seeMorton,TaylorandTurner,1956).WenoteonFigure10.6thatthebuoyancycrosseszeroatz=298(F=N3)14buttheresidualverticalvelocityatthatlevelmakestheplumeovershoot,uptoz=392(F=N3)14,bywhichleveltheverticalvelocityvanishesandtheradiusbecomesin nite.Laboratoryexperimentsand eldobservationscon rmandtweakthistheoreti-calprediction(Figure10.7).Briggs(1969)giveszmax=50F N314=376F N314(10.24) 10.3.THERMALS171 Figure10.6:Numericalinte-grationofEquations(10.20),(10.21)and(10.23),witha=0125,tracingtheverticalstruc-tureofabuoyantplumeasitrisesinastrati edenviron-ment.Thebuoyancycrosseszeroatz=298,abovewhichtheplumebecomesnegativelybuoyant.Theverticalveloc-ityvanishesatz=392andtheradiusbecomesin nite.Seetextfordetailsofthenon-dimensionalizationemployedinmakingthegraph.10.3ThermalsAthermalisa niteparcelof\ruidconsistingofthesame\ruidasitssurroundingsbutatadi erenttemperature.Becauseofitsbuoyancy,acoldthermalsinks(neg-ativebuoyancy),whileawarmthermalrises(positivebuoyancy).Thenamewasgivenbygliderpilotstowhattheyperceivedasregionsofwarmairrisingaboveaheatedgroundinwhichtheycouldsoar.Convectionintheatmospheredoesindeedproceedbymeansofrisingthermals(Priestley,1959).Thesituation,however,canbequitechaotic,withacollectionofthermalsrisinghereandthereatvarioustimes,someofthemsmallerandslower,andotherslargerandfaster.Here,forthesakeofunderstandingthebasicmechanism,weshallbeconcernedwithasinglethermalimmersedinanin nitehomogeneous\ruidatrest.Experimentshavebeenconductedinthelaboratory(Figure10.8),andithasbeenfoundthatallthermalsroughlybehaveinsimilarways:astheyrise(orsink),theyentrainsurrounding\ruidandbecomemoredilute,therebyslowingdownintheirascent(ordescent).Theactualshapeofathermal,however,canvaryconsid-erablyfromonesetofobservationstoanother.Here,basicdynamicssupplementedbyafewdimensionlessnumbersgleanedfromexperimentswillbeusedtoestablishasimpletheoryforthepredictionofathermal'sbehaviorovertime.Thekeypropertyofathermalisitstotalbuoyancy,de nedasB= gT0V=g0V(10.25)inwhichVisthevolumeofthethermal,T0itstemperatureanomaly,andg0= gT0thereducedgravityitexperiences.Thistotalbuoyancyisaconservedquantityasthethermalrises(orsinks)because,whileitentrainssurrounding\ruid,itstem-peratureanomalydecreasesbydilutioninproportiontoitsvolumeincrease,thuskeepingtheproductT0Vconstantduringthethermal'slife. 172CHAPTER10.PLUMES Figure10.7:Measurementsofplumeriseincalmstrati edsurroundings,revealingthattheultimateheightreachedbyaplumefollowsEquation(10.24).(AdaptedfromBriggs,1969)ThevolumeofathermalcanbeexpressedasV=mR3(10.26)whereRistheradiusofthethermalseenfromabove,andmisacoecientlessthat4=3=42(valueforasphericalvolume)becauseathermalhasaslightly\rattenedshape.Thevalueofmisnotoriouslydiculttomeasure,andsomeindirectmeasurementisinorder,asweshallseelater.MassconservationovertimecanbeexpressedasdV dt=Au;inwhichAistheenclosingsurfaceareaofthethermalandutheaverageentrainmentvelocityacrossthatsurface(Figure10.9).TakingtheareaAasproportionaltoR2thesquareofthethermal'sradius,andtheentrainmentvelocityuasproportionaltothethermal'sverticalvelocityw,wecanexpresstheprecedingequationasdV dt=aR2w;(10.27)inwhichthecoecientaoughttobeadimensionlessconstant,tobedeterminedfromexperimentsorobservations.Using(10.26),thisequationcanbereducedto:dR dt=a 3mw:(10.28)Themomentumbudgetovertimetakestheformd dt3 2thermalVw=Upwardbuoyancyforce Downwardweight 10.3.THERMALS173 Figure10.8:Descendingthermalsinalaboratoryexperiment.Thesethermalsinwateraremadevisiblebybariumsulfate.Thestemleftbehindbyeachthermalisduetothemannerasphericalcapwasrotatedtoprovoketherelease.Thesecondthermal(bottomrow)hasalargernegativebuoyancythanthe rst(toprow).[FromScorer,1997]=ambientVg thermalVg=thermal T0Vg=thermalg0V;inwhichthefactor3=2ontheleftisduetotheadded-masse ect.Physically,thethermalissubjecttoitsownacceleration(timederivativeofonetimethermalVw),butitschangingpacealsocausesaccelerationofthesurrounding\ruiddivertedbyitspassage,e ectivelyaccelerating50%more\ruidmass,hencethefactor3=2=15.Divisionbythermal,whichatalltimesremainsclosetothereferencedensityofthe\ruid,yieldsd dt(Vw)=2 3g0V:(10.29)Eliminationoftheproductg0VbyvirtueofEquation(10.25)indicatesthattheright-handsideoftheprecedingequationisaconstant,leadingtoanimmediateintegration:Vw=2 3Bt;(10.30)forwhicht=0marksthetimewhenthethermalhadzeromomentum.Next, 174CHAPTER10.PLUMES Figure10.9:TheanatomyofarisingthermalaccordingtoScorer(1997).Fluidwithinaconeofabout12isentrainedbythetopofthethermal,while\ruidoustideofthisandwithinawiderconeof15isentrainedintherear.Therestoftheambi-ent\ruidismerelyde\rectedbythepassageofthethermal.Theobliqueredlinetracestheouteredgeofthethermalovertime,forminganangleofabout14fromthevertical.(FromScorer,1997)solvingforw(w=2Bt=3V=2Bt=3mR3)andreplacinginEquation(10.28),weobtainasingleequationfortheradiusRofthethermal:dR dt=2a 9m2Bt R3Thesolutionofthisequationis:R=4a 9m214B14t12(10.31)Nowknowingtheradiusasafunctionoftime,wecanreadilysolvefortheotherquantities,namelyvolumeV,verticalvelocitywandreducedgravityg0V=64a3 729m214B34t32(10.32)w=9m2 4a314B14 t12(10.33)g0=729m2 64a314B14 t32(10.34) 10.3.THERMALS175Intheseexpressions,itisclearthatthetimeoriginactuallyreferstoavirtualstageinwhichthethermalhadzerovolume,in nitevelocityandin nitetemperatureanomaly.Obviously,theactuallifeofthethermalstartedsome nitetimeafterthis,witha nitevolume, nitevelocityand nitetemperatureanomaly.Notethatthecompletesolutiondependsontwodimensionlessparameters,aandm.Sinceneitheriseasytodeterminedirectly,itiswisetoseektheirvalueindirectlybymatchingthermal'spropertiesthataremorereadilyobserved.Onesuchpropertyisthemannerinwhichthethermal'sradiusgrowswithdistance.Forthis,weintegratedz=dt=wtoobtainthethermal'selevationasafunctionoftime.Theresultis:z=36m2 a314B14t12(10.35)ItappearsthatbothelevationzandradiusRgrowatsimilarrates,yieldingaconstantratio:R z=a 3m(10.36)Laboratoryobservations(Figure10.9)revealthatthisisindeedthecasethatther-malsbehaveinaself-similarway,andthattheratioofRtozisabouttan14=025.Thus,R=025z;(10.37)anda=075mTheotherreliableobservationisthattheratioz2=t(atimeconstantaspredictedbythetheory)variesfromexperimenttoexperimentinproportiontop B(Figure10.10).Thetheoreticalcoecientofproportionalityisp 36m2=a3,andexperimentsgiveitavalueof5.80.Solvinga=075mtogetherwithp 36m2=a3=580yields:a=190andm=254.Fromthisfollowallothercoecients:R=060B14t12(10.38)V=055B34t32(10.39)w=120B14 t12(10.40)z=241B14t12(10.41)g0=181B14 t32(10.42) 176CHAPTER10.PLUMES Figure10.10:Plotofthequan-tityz2=t(atimeconstantdur-ingthelifeofthermal)versusthesquarerootofthethermal'sbuoyancy.Eachnumbereddotreferstoadi erentlaboratoryexperiment,andthesolidlineshowsthebestlinear t.(FromScorer,1997)10.4ThermalsinaStrati edEnvironmentWhenathermalrises(orsinks)inastrati edenvironment,itprogressivelyencoun-tersatemperatureclosertoitsownandthereforelosesitsbuoyancy.Ultimately,itwillreachalevelofnobuoyancyandbegintospreadlaterally(Figure10.11).Withnothermalcontrastleft,thethermallosesitsidentity.Whatisthisultimatelevelisnotanobviousquestion.Indeed,itcanbeeasilyestablishedthatthethermalwillneverreachthelevelofitsinitialtemperature.Thereasonisitspartialdilutionbyentrainmentofsurrounding\ruid(whichchangesasthethermalcrossesisotherms)anditsconsequentdilution.Apracticalapplicationofthissituationisthedumpingofwasteinastrati edbodyofwater:Thedumpedwastesinksfromthesurface,graduallymixeswithsurroundingwaterduringitsfall,andeventuallysettlesdownatsomeintermediatedepth.ThedeterminationofthatdepthiscrucialinwaterqualitystudiesandpermittingThestrati edenvironmentischaracterizedbyitsstrati cationfrequencyNde nedfromN2= gdT dz(10.43) 10.4.THERMALSINSTRATIFICATION177 Figure10.11:Alaboratoryexperimentofathermalsinkinginastrati edenvi-ronment.Notetheultimatearrestandspreadingofthethermalonceitlosesitsbuoyancy.(FromScorer,1997)whereT(z)isthetemperaturepro leoftheambient\ruid.Totrackathermalinthisenvironment,thesamequantitiesareneededasbefore,namelythethermal'sradiusR,volumeV=mR3,verticalpositionz,verticalvelocityw=dz=dt,reducedgravityg0,andtotalbuoyancyB=g0V.Thedi erencewiththeprevioussectionisthatnowthetotalbuoyancyBisnolongeraconstantofthemotion.Threeequationsareatourdisposal:ThemassbudgetdV dt=Au=aR2w;(10.44)themomentumbudgetd dt(Vw)=2 3g0V;(10.45)whichremainunchanged,andtheheatbudget,whichisd dt(VTthermal)=AuT(z)=aR2wT(z)(10.46)Usingthereducedgravitylocallyexperiencedbythethermal,g0= ggTthermal T(z)],thelastequationcanberecastasd dt(Vg0)+d dttV gT(z)]=aR2w gT(z)Using(10.44)andthefactthatdT(z)=dt=(dT=dz)(dz=dt)=w(dT=dz),itreducestod dt(Vg0)= N2Vw:(10.47)EliminatingVfromEquations(10.44),(10.45)and(10.47)byusingV=mR3yieldsasetofthreeequationsforthethreeunknownsRwandg0 178CHAPTER10.PLUMESdR dt=a 3mw(10.48)d dt(R3w)=2 3R3g0(10.49)d dt(R3g0)= N2R3w:(10.50)Viewingthesethreequantitiesnolongerasfunctionsoftimetbutratherofelevationzandcallinguponw=dz=dt,wecantransformtheminto:dR dz=a 3m(10.51)d dz(R3w)=2R3g0 3w(10.52)d dz(R3g0)= N2R3(10.53)The rstoftheseequationsyieldsR=R0+a 3mz;(10.54)inwhichR0istheinitialradiusatthedeparturelevelz=0.AslongasN2isaconstant(linearstrati cation),thelastequationofthesetcan,too,beintegratedtoyield:R3g0=R30g00 3m 4aN2(R4 R40)(10.55)inwhichg00areistheinitialvalueofg0Thethermallosesallitsbuoyancywhenitsg0dropstozero,whichoccurswhenitsradiushasgrowntothevalueRendsuchthat3m 4aN2(R4end R40)=R30g00(10.56)thatisRend=R40+4a 3mR30g00 N214(10.57)Assumingthatthethermalstartedwithaninsigni cantradiusandhasvastlyexpandedduringitstravel,wecanapproximatetheprecedingexpressiontoRend4a 3mR30g00 N2144a 3m2B0 N214(10.58) 10.5.PLUMESINCROSS-FLOW179whereB0=V0g00=mR30g00isthethermal'sinitialbuoyancy.Translatingthisradiusintothecorrespondingelevationgivestheterminallevelwherethethermallosesitsidentity:zend=3m a(Rend R0)3m aRend108m2 a314B0 N214(10.59)Withtheparametervaluesa=190andm=254determinedattheendoftheprevioussection,wehavezend317B0 N214(10.60)Notethatatthelevelwhereg0=0,thethermalhassomeresidualverticalvelocityandwillovershootslightlyitslevelofneutralbuoyancy.ThisexplainsthebulgeonthefrontsideofthethermalseeninFigure10.11.10.5PlumesinaCross-FlowThelinethermalmodel.Problems10-1.Byusingablowerandsomepreheating,onecanadjustboththeupwardvelocityandbuoyancyoffumesexitingfromthetopofasmokestack.Speci -cally,twoscenariosarebeingconsidered,onewithmorevelocityandonewithmorebuoyancy,asfollows:Scenario1: Averageexitverticalvelocity=12m/sAverageexitbuoyancy=0.01m/s2Scenario2: Averageexitverticalvelocity=1m/sAverageexitbuoyancy=0.12m/s2Ineachcase,theexitdiameteris1.5mandtheentrainmentcoecientaistakenas0.115.Whichofthetwoscenariosgivesthehighestverticalvelocityatthecenteroftheplume20mabovethesmokestack?10-2.Younoticeabuzzardsoaringinacirclingfashionandguessthatitistak-ingadvantageoftheupwardmotionofathermal.Asyouhappentohave 180CHAPTER10.PLUMESmeteorologicalgearwithyou,includingaradarpro ler,youdeterminethatthebuzzardis\ryingatanaltitudeof80mandthatthetemperatureatthecenterofthebird'scircleis0.30Chigherthanoutsidethethermal,wherethetemperatureis25C.Whatistheradiusofthethermalanditscenterverticalvelocity?Also,howoldisthisthermal?10-3.10-4.10-5.Showthat,forathermalrisinginahomogeneousambient\ruid,w2isequaltog0z=2.Doesthisrelationhaveanyparticularsigni cance?10-6.Establishtheformofthetotalenergyofathermal(kineticpluspotential)forathermalrisinginauniformenvironmentanddetermineitsvariationalongthepathofthethermal.Resolveanyparadox.10-7.Itwasmentionedattheendofthesectiononthermalsinastrati eden-vironmentthat,onceitreachesitslevelofneutralbuoyancy,athermalstillpossessesaresidualverticalvelocity.Whatisthatvelocity?And,atwhatultimatelevelzdoestheverticalvelocity nallyvanish?Assumethattheinitialradiusofthethermalwasnegligiblecomparedtoitsradiusatthelevelofneutralbuoyancy.10-8.10-9.