e sinϑ dθ dt θ ˆ,!!!!!!!!!!!!(10.1)! dt! ˆk + x dt dt dt ˆk = ( u r , v r , w r )!!!!!!!!!!!!(10.5)!

Transcription

1 Chapter(10.((The(Effect(of(the(Earth s(rotation( Inthischapter,wewilldeveloptheequationofmotionandvorticity equationinarotatingframeofreference MotioninaRotatingFrameofReference Letusconsiderafixedvector e inareferenceframerotatingattherate of Ω.ToanobserveronanonFaccelerating(inertial)frame,thefixedvectorin therotatingframemayappeartorotate.thus, d e = lim Δt 0 e(t + Δt) e(t) Δt = e sinϑ dθ θ ˆ, (10.1) where θ ˆ istheunitvectorintheazimuthaldirection.therightfhandsideof (10.1)canbewrittenas e sinϑ dθ θ ˆ = Ω e. (10.2) Letusnowconsiderafluidparcelat r = xî + yĵ + z k = x, y, z ( ). (10.3) Then,thevelocityofthefluidparcelisgivenby where u = d r = d u r = d r $ # & " % ( xî + yĵ + z ˆk ) = dx î + dy ĵ + dz ˆk + x dî + y dĵ + z d ˆk = u r î + v r ĵ + w r ˆk + x Ω î + y Ω ĵ + z Ω ˆk = u r + Ω r, (10.4) r = dx î + dy ĵ + dz ˆk = ( u r, v r, w r ) (10.5) representsvelocitymeasuredintherotatingframeofreference.becauseofthe Figure10.1.Avector r fixedinarotatingcoordinatesystem. 237

2 accelerationoftherotatingcoordinatesystem,velocityobservedbysomeonein aninertialcoordinatesystemwillseetheeffectofrotation Ω r.nowtaking timederivativeof(10.4),wehave where a = d u = d u $ # & + Ω u = d u r $ # & + d Ω r $ # & " % r " % r " % a r = d u r $ # & " % r + Ω ( u r + Ω r ) = a r + 2 Ω u r + Ω Ω r, (10.6) r = du r î + dv r ĵ + dw r ˆk = du r, dv r, dw $ r # & (10.7) " % istheacceleratingintherotatingframeofreference.becauseoftherotationof theearth,weseetwoextratermsin(10.6):thecoriolisaccelerationand centrifugalacceleration.forthesakeofbrevity,letusnowdropthesubscript r in(10.6) TheFictitiousAccelerationinaRotatingFrame Onarotatingframeofreference,weseetwo fictitious mechanisms changingtheaccelerationoffluid.thecoriolisacceleration, 2 Ω u,produces forceperpendiculartoboth Ωand u.consideringthedirectionofrotation vectorinthenorthernhemisphere,windvectorwillbedeflecteowardits rightfhandsideduetotherotationeffectoftheearth(figure10.2a).asaresult, anystraightpathwillbecurveowarditsrightfhandside(figure10.2b).letus consideranarbitrarypoint, P(r,φ,θ),onthesurfaceoftheearth.Atthispoint wehavevelocityvector u = (u r, v φ, w θ ).Further,earth srotationvectoris describedas Ω = Ωsinθ, 0,Ωcosθ ( ), (10.8) where Ω = Ω.Wehaveonlyradialandlatitudinalcomponentsofearth s Figure10.2.VelocityvectoranhedirectionofCoriolisacceleration(left)and theeffectofcoriolisaccelerationonthetrackofaparcel. 238

3 rotationvector.then,theresultingcoriolisforceisgivenby $ 2 Ω u & = 2& & %& ˆr ˆφ ˆ θ Ωsinθ 0 Ωcosθ u r v φ w θ ) ) ) () = 2Ωcosθv φ ˆr 2Ω( cosθu r sinθw θ ) ˆφ 2Ωsinθv ˆ φθ. (10.9) Thecentrifugalacceleration, Ω Ω r,canberewrittenas Ω Ω r = ( ΩR)Ω = Ω 2 R ( ) ˆR, (10.10) where R istheunitvectorinthedirectionperpendiculartoandawayfromthe rotationaxis(seefigure10.2).equation(10,10)canberewrittenas Ω Ω r = Ω 2 R ( ) ˆR = Ω 2 R E, (10.11) where R E istheradiusoftheearth.thus,theearth srotationproduces accelerationofafluidparcelawayfromtheearth srotationaxis.thisforceis proportionaltothecosineoflatitudeasshouldbeexpected. Figure10.3.ThecentrifugalforceofmagnitudeΩ 2 R = Ω 2 R E cosθ isproducedin thedirectionperpendiculartoandawayfromtherotationaxis TheEquationofMotioninaRotatingFrameofReference. Withtheaidof(10.6),theequationofmotioninarotatingframeof referencecanbewrittenas D u Dt = 2 Ω u 1 ρ p + g 2 Ω Ω r +ν 2 u +ν u ( ), (10.12) 239

4 where u isvelocitymeasuredinarotatingframeofreference.becauseofthe rotationoftheearth,wehavetwo fictitious accelerationontherightfhandside of(10.12);theyarecoriolisaccelerationandcentrifugalacceleration.the centrifugalaccelerationiseasilyaddeothegravitationalaccelerationtoyield effectivegravitationalaccelerationasshowninfigure10.4.thecentrifugal accelerationismuchweakerthanthegravitationalacceleration;thus,the effectivegravitywillnotbemuchdifferentfromthetruegravity.thus,(10.12) canberewrittenas D u Dt = 2 Ω u 1 ρ p + g e +ν 2 u +ν u ( ). (10.13) Wewilloftendropthesubscripteineffectivegravity,sincethistermisnot significantlydifferentfromtruegravitationalacceleration.inoceanographic applications,wehavetoadhetidalforce.then,theequationofmotionis modifiedas D u Dt = 2 Ω u 1 ρ p Φ W +ν 2 u +ν u ( ), (10.14) wherethegravitationalaccelerationanidalaccelerationwerewritteninterms oftheirpotentialfunctions. Figure10.4.Effectivegravity, g e,atagivenpointdiffersfromthetruegravity, g, becauseofthecentrifugalacceleration, Ω Ω r. (Q1)Howisthemaximumcentrifugalaccelerationcompareswiththe gravitationalaccelerationinmagnitude? 10.4.VorticityEquationinaRotatingFrameofReference Vorticityequationcanbederivedbytakingcurlof(10.14).The derivationprocedureisexactlysameasdelineatedinchapter7.4.the 240

5 gravitationalaccelerationanidalaccelerationtermsvanishasyoutakecurlof (10.14).OnlyexceptionistheCoriolisterm.TakingcurloftheCoriolis acceleration,wehave ( 2 Ω u ) = 2ε ijk ( ε klm Ω l u m ) = 2( δ il δ jm δ im δ jl ) ( Ω l u m ) = 2 = 2Ω j Ω i + 2u i j 2Ω i Ω j 2u j i ( Ω i u j ) 2 ( Ω j u i ) = 2 Ω ( u ) 2 ( Ω ) u. (10.15) Byadding(10.15)tothevorticityequation(7.18),wehave D ω Dt = 2 Ω ( u ) + ( ω ) u 2 ( Ω ) u ω ( u ) + 1 ρ ρ p 2 +ν 2 ω = ( ω a ) u ω a ( u ) + 1 ρ ρ p +ν 2 ω, (10.16) wheretheabsolutevorticity ω a = 2 Ω+ ω (10.17) isdefinedasthesumofrelativevorticity, ω,andplanetaryvorticity, 2 Ω. Recognizingthat Ωisessentiallyaconstantvector,wecanrewrite(10.16)as D ω a Dt = ( ω a ) u ω a ( u ) + 1 ρ ρ p 2 +ν 2 ωa. (10.18) Equation(10.18)isidenticalinformwiththevorticityequation(7.18)except thatweusedabsolutevorticityinsteadofrelativelyvorticity. LetusnowconsideralocalCartesianplaneonthesurfaceoftheearthas showninfigure10.5.onthisplanethelocalverticalcomponentofabsolute vorticityis ω a = ( ξ a,η a,ζ a ) = ( 0, 2Ωcosθ, 2Ωsinθ). (10.19) Then,thelocalverticalcomponentofvorticityequationcanbewrittenas ζ t + u ζ x + v ζ y + w ζ z ( ) w = ξ +ξ a x ( )% +η w y ζ +ζ a # x + v & ( $ y + 1 # ρ p ρ 2 x y ρ p & # % (+ν 2 ζ $ y x x + 2 ζ 2 y + 2 ζ & % (. (10.20) $ 2 z 2 241

6 Figure10.5.AlocalCartesianplanecenteredatlatitudeθ.Thelocalplaneis tangentialto ˆφ and θ ˆ,andisperpendicularto ˆr.Thelocal x directioncoincides withthedirectionofφ whereasthelocal y directioncoincideswiththe directionofθ.thelocal z directionisidenticalwiththedirectionof r. Notethattheeffectofearth srotationinthefirsttwotermsontherightfhand sideof(10.20)isnaturallysmallnotbecauseξ a issmallbutbecausethevertical velocityissmall.thus,theimportanceoftheearth srotationonalocalplane appearsonlyinthethatis, ζ t + u ζ x + v ζ y + w ζ z ( )% w w = ξ +η x y ζ +ζ a # x + v & ( $ y + 1 # ρ p ρ 2 x y ρ p & # % (+ν 2 ζ $ y x x + 2 ζ 2 y + 2 ζ & % (. (10.21) $ 2 z 2 Theverticalcomponentofplanetaryvorticity, 2Ωsinθ,iscalleheCoriolis parameter,whichisanimportantparametermeasuringtheimportanceof earth srotation.basedon(10.21),theimportanceofplanetaryvorticity increaseswithlatitudeandiszeroattheequator. Figure10.6.Afluidparcellocatedatagivenlatitude. 242

7 Inordertounderstanhephysicalmeaningofabsolutevorticityinmore detail,letusconsiderafluidparcellocatedatagivenlatitudeasshowninfigure 10.6.Thisfluidparcelmoveswiththeenvironmentalfluid.Then,thenorthern andsouthernedgesofthefluidparcelrespectivelymoveatthelinearspeedof and u N = r cos( θ + dθ )Ω, (10.22) u S = r cos( θ dθ )Ω, (10.23) where Ωistheangularvelocityofearth srotation.undoubtedly,thesouthern edgeismovingfasterthanthenorthernedge,yieldingacounterclockwise rotationasshowninfigure10.6.insphericalcoordinatesystem, " $ 1 $ ω = $ r 2 cosθ $ $ # ˆr r cosθ ˆφ r ˆ θ r φ θ u r r cosθu φ ru θ Thus,thelocalcomponentofvorticityisgivenby %. (10.24) & 1 # ru ω r = θ r 2 cosθ φ r cosθu & φ % (, (10.25) $ θ 1 r cosθu = φ = Ω cos 2 θ = 2Ωsinθ, (10.26) r 2 cosθ θ cosθ θ whichisidenticalwiththeverticalcomponentofearth srotationderivedin (10.19).Thus,thelocalverticalcomponentofplanetaryvorticitymeasures vorticityoffluidonthelocalplane.thisvortexmotionisduetotherotationof theearthasexplainedin(10.22)and(10.23)and,henceforth,iscallehe planetaryvorticity GeostrophicEquation Foraninviscidfluid,equationofmotiononalocalCartesiancoordinates inarotatingframeofreferencecanbewritteninacomponentformas and ( ) t + u u = 2Ωcosθw + 2Ωsinθv 1 p ρ x Φ +W x ( ) v t + u v = 2Ωsinθu 1 p ρ y Φ +W y ( ) w t + u w = 2Ωcosθu 1 p ρ z Φ +W z, (10.27), (10.28). (10.29) 243

8 Foranincompressiblefluid, u = 0,verticalvelocityistypicallymuchsmaller inmagnitudethanhorizontalvelocitiesforgeophysicalmotions.then,the equationofmotioncanbefurthersimplifiedas t + u x + v y = fv 1 p ρ x v t + u v x + v v y = fu 1 p ρ y, (10.30), (10.31) and 0 = 1 p g, (10.32) ρ z where f = 2Ωsinθ isthecoriolisparameter. Arigorousderivationof(10.32) requiresthesofcalledscaleanalysisandisconsideredbeyonhescopeofthis book.letusfurtherassumethatthenonlineartermin(10.30)f(10.32),i.e., u u,aswellastimederivativeterm,i.e., u t,aresmallerthantheother terms,anhetimescaleofmotionisshortenoughtoignoretidalforcing.then, theequationofmotioninthe xy Fplanecanbewrittenas ρ f ˆk u = H p, (10.33) where H isthehorizontalcomponentofdeloperator.termsin(10.33) representthemajorcomponentsin(10.27)and(10.28),anddescribethesof calledgeostropicbalance.theverticalcomponentofmotionissimply H L Figure10.7.Geostrophicbalancebetweenpressuregradient(bluearrow)and Coriolisacceleration(redarrow)anheresultinggeostrophicwind(blue curvedarrow).inthepresenceoffriction(blackarrow),windspeedisreduced resultinginweakercoriolisacceleration.theresultingflowcrossestheisobars asshownbycurvedredarrows. Here,weassumehatgravitationalaccelerationisessentiallyvertical. 244

9 hydrostaticbalance.inthepresenceoffriction,thegeostrophicvelocity diminishessothatpressuregradientforcebecomesstrongerthancoriolisforce. Thus,theflowwillcurvetowarhelowerpressure(seeFigure10.7).Ifa motionoccursonaplanemuchsmallerthantheearth sdimension,local Cartesiancoordinatesareoftenconvenienttodescribethemotion.Further,the Coriolisparameter, f,canbeassumedasaconstantonthesmalllocalcartesian plane.suchanapproximationiscallehe f Fplaneapproximation. Ifamotionbecomeslargeenoughtoignorethelatitudinalvariationofthe Coriolisparameter,thelatitudinal( y )variationofthecoriolisparameterona localcartesianplanecanbeexpressed,byusingtaylor sexpansion,as where f (y) = f (y = 0)+ f y y=0 y, (10.34) f y = ( y 2Ωsinθ ) = 1 ( R E θ 2Ωsinθ ) = 2Ω cosθ = β. (10.35) R E Thus,wecanwrite(10.34)as f = f 0 + β 0 y, (10.36) where f 0 = f (y = 0) = f (θ 0 ), (10.37) β 0 = 2Ω R E cosθ 0, (10.38) andθ 0 isthelatitudeofthecenter( y = 0 )ofthelocalcartesianplane.equation (10.36)iscallehe β Fplaneapproximation.Notethatthemagnitudeof β is onlyoftheordero(10 6 ) FO(10 7 ) ofthemagnitudeof f inmidflatitudes. (Q2)Whatisatypicalmagnitudeof f inmidflatitudes?whatisatypical magnitudeof β inmidflatitudes? (Q3)Whatisagenerallatituderangeforwhichthevariationof f shouldbe consideredonalocalcartesiancoordinates VorticityConservationLawunder β FPlaneApproximation Underthe β planeapproximation,aninterestingformofvorticity equationcanbederivedfrom(10.30)and(10.31).theverticalcomponentof vorticityisderivedbysubtracting y derivativeof(10.30)from x derivativeof (10.31).Thatis, 245

10 # v t x & ) % (+ u $ y x + v,# +. v * y- x & # % (+ $ y x + v &# % ( v $ y x & % ( $ y # = f x + v & ) % ( u $ y x + v, +. f, (10.39) * y- whereweusehefactthat f isnolongeraconstant.withtheaidofthefact that f doesnotchangeintime,(10.39)canberewrittenas D Dt ζ + f ( ) = % # x + v & ( ζ + f $ y ( ), (10.40) whereζ + f istheverticalcomponentofabsolutevorticityonalocalcartesian plane.further,continuityequationforauniformdensityflowcanbewrittenas H 0 x + v H " % w $ dz = # y & z dz, (10.41) 0 whichforathinlayerwithuniformzonalflowcanberewrittenas " x + v % $ H = w(z = 0) w(z = H ). (10.42) # y & Iftheverticalvelocityvanishesat z = 0,thenwehave x + v y = w(z = H ) = 1 H H DH Dt Withtheaidof(10.43),wecanwrite(10.40)as D Dt ln ( ζ + f ) = Dln H Dt orequivalently,,. (10.43) D ln ζ + f $ * ) # &, = 0,orsimply, D ζ + f $ # & = 0. (10.44) Dt ( " H % + Dt " H % Theterminparenthesisin(10.44)iscalledpotentialvorticity.Thus,inashallow slaboffluid,potentialvorticityisconservedfollowingthemotion.bearinmind that(10.44)considersonlytheverticalcomponentofvorticityinanarrowslab withuniformhorizontalvelocity.thus,(10.44),atbest,isanapproximation. " Nevertheless,(10.44)isausefultoolforpredictingmotionoffluidinthe atmosphereanheocean.atleast,itisagoooolforunderstandingthe generalbehavioroffluidinarotatingframeofreference. LetusconsiderwindapproachingamountainasshowninFigure10.8.As theflowapproachesthemountain,thedepthofthelayer, H,decreasesthereby " Conservationofpotentialvorticityisderivedinamorerigorousmannerin termsoftheertel stheorem,whichisbeyonhescopeofthistextbook. 246

11 producingnegativevorticityaccordingto(10.44).thus,theflowturnsaround themountainasdepictedinfigure10.8. Figure10.8.Descriptionofaflowaboveamountain.Basedontheconservation ofpotentialvorticity,negativevorticityisproducedalongthefootofthe mountain.dependingonthedirection,therefore,theflowisdeflectedonthe differentsidesofthemountain. Inasimilarmanner,ifaflowisdeflectedmerdionallyarounhecentrallatitude θ =θ 0,planetaryvorticity f variesaccordingtothelatitude.then, dζ = df accordingto(10.44)if H isheldconstant.thus,asflowmovesnorthward,then negativerelativevorticity,ζ,producesasouthwarddeflection.ontheother hand,southwardshiftofaflowproducesapositiverelativevorticity,which deflectstheflownorthwardasshowninfigure10.9.theconservationof potentialvorticityisafundamentalphysicalmechanismofrossbywaves, althoughthedetailsaremuchmorecomplicatehandescribedhere. Figure10.9.PropagationofRossbywavesanhemeridionalexcursion accordingtotheconservationofpotentialvorticity

12 10.7.TheRossbyNumber Therelativeimportanceoftheearth srotationismeasuredbytherossby number.therossbynumberisexpressedastheratioofcentrifugalacceleration tocoriolisacceleration.thatis, R o = rω 2 fu = u fr = O U $ # &, (10.45) " fl % whereu and L arethescalesofhorizontalvelocityandmotion,respectively.if Coriolisaccelerationismuchstrongerthancentrifugalacceleration, R o should beasmallnumber.inotherwords,planetaryvorticityproducedbytheearth s rotationismoreimportantthancentrifugalaccelerationproducedbythe (relative)motionofthefluid.asaresult,effectoftheearth srotationcannotbe ignoredinexplainingtheparticularfluidmotion. ThisnonFdimensionalnumbercanbeinterpretedinaslightlydifferent way.equation(10.45)canberewrittenas R o = rω 2 fu = u fr = 1, (10.46) ft whichistheratioofthetimescaleofearth srotationtothatofthemotion;note thatthecoriolisparameterhasthedimensionoftime F1.Thus,ifthetimescaleof themotionislargerthanthetimescaleoftheearth srotation,therotationeffect oftheearthshouldbeimportant.inotherwords,thetimescaleofthemotionis longenoughto feel theeffectoftheearth srotation. Equation(10.45)canberewrittenas L = U f For R 0 <<1, 1 R 0. (10.47) L >> U f. (10.48) Thus,foragivenvelocityscale,theeffectoftheearth srotationincreasesasthe lengthscaleofamotionincreases.forabarotropicocean,phasespeedof gravitywaveisgivenby gh,where H isthedepthoftheocean.then,the Rossbyradiusofdeformationisgivenby L = gh f = gh ~ O(2000 km). (10.49) 2Ωsinθ Foramotionwiththespeedof~100msF1,whichisthetypicalspeedofsynoptic scalefeatures,thecorrespondingrossbyradiusofdeformationis L = U ~ O(1000 km). (10.50) 2Ωsinθ 248

13 TherearemanydifferentformsofRossbyradiusofdeformationaccordingto applications TheEkmanNumber Asmentionedearlier,themajorbalanceofforceinarotatingframeof referenceisbetweenthecoriolisforceanhepressuregradientforce(see (10.30)and(10.31)).Thatis, ρ fu = p L,or p = O( ρ ful). (10.51) Therelativeimportanceoftheviscoustermscanbemeasuredwithrespectto thecoriolisterm,i.e., E k = viscous force Coriolis force = µul 2 ρ fu = ν. (10.52) 2 fl ThisratioiscalleheEkmannumber.Notethat(10.52)canalsobewrittenas E k = ν fl 2 = U fl UL ν = R o R e, (10.53) whichistheratiooftherossbynumbertothereynoldsnumber.thus,alarge Reynoldsnumbermeansthattheeffectofviscositycannotbeignoredin comparisonwiththecorioliseffect. WhileEkmannumberistypicallyverysmallforoceanicandatmospheric motions,thereisaverynarrowregionnearasolidboundarywhereeffectof viscositycannotbeignored.equation(10.52)showsthatthelengthscaleofthe narrowregion,whereviscosityisimportant,shouldbeextremelysmall.this narrowregioniscalleheboundarylayer.foramotionwithaspeedof100m s F1,thelengthscaleoftheregionwhereviscouseffectiscomparabletoCoriolis accelerationis L = ν f ~ O(1 m). (10.54) Equation(10.54)givesanideaastothescaleoftheboundarylayerinthe atmosphereanheoceans.inthislayer,wehavetoconsidertheeffectofthe earth srotationaswellasthatofviscosity.anexcellentexampleistheekman layer,inwhichtheviscouseffectofwindstressanhatoftheearth srotation playanimportantroleindeterminingthedynamicsoftheflowinthelayer.( 249