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Diagnosing vertical mixing in a two‐layer exchange flow Pawlowicz, Rich; Farmer, David D. 1998

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JOURNAL OF GEOPHYSICAL  RESEARCH, VOL. 103, NO. C13, PAGES 30,695-30,711, DECEMBER  15, 1998  Diagnosingvertical mixing in a two-layer exchangeflow Rich Pawlowicz Departmentof Earth andOceanSciences,Universityof BritishColumbia,Vancouver,Canada  David M. Farmer Instituteof OceanSciences,Sidney,British Columbia,Canada  Abstract. Most classificationschemesand analysesof estuarineand exchangeflows use only salinity as a tracer. Temperatures are generallyignored. However,a proper understanding of the effectsof surfaceheatingcan explainobservedseawardchangesin the slopeof temperature-salinity correlations.A theoryis proposedrelatingchangesin temperatureand salinitywith transportand mixing parametersin a two-layerexchange flow. Resultsshowthatalong-channel changesin the slopeof T-Scorrelations arevirtually independentof vertical mixing but are directlyrelatedto horizontallayer transport. Changesin the layer salinitiescan be relatedto variousratiosof horizontaland vertical transports.Combiningthesetwo featuresof the theorypermitsa diagnosticdetermination of Lagrangiantransportand mixing from standardhydrographicobservations of layer temperatureand salinityand an estimateof the surfaceheatinput. The theoryis applied to observations madethroughthe spring/neap tidal cycleof mixing in Haro Strait,British Columbia.  1. Introduction  manet al., 1995]. Mixing at convergence frontsdrivenby tidal velocitieshas alsobeeninvestigated[Farmeret al.,  Haro Strait, alongwith the Straitof Georgiaand Juande 1995]. FucaStrait(Figure 1), form an estuarinesystemdominated Duringthesummerof 1996a fieldprogramtookplacein by twowatermasses, a freshlightsurfacewateroriginating an attemptto understand theseandotherissues.This proin the FraserRiver plumeanda deepsalinelayeroriginatgramincludeda large-scale conductivity-temperature-depth ing in the Pacific. Freshwater from the FraserRiver covers (CTD) surveythat was repeatedseveraltimesduringthe the Straitof Georgiain a thin buoyantplume,which then courseof a 2-week cruise, spanningthe period between flowsout to the Pacificvia Juande FucaStrait,entraining two successive neaptides. Preliminaryanalysisof the hywaterfromthelowerlayerduringitsjourney.A deepinflow drographicdatasetshowedthat althoughthe T-Scorrelation compensates for this lossof lower layer water. Transport at any particularstationwas usuallycloseto beinglinear, throughJohnstoneStrait at the northernend of the Strait of as would be expectedin situationswhere only two water Georgiais usuallyconsideredto be small. It is well-known massesare involved,the slopeof the correlation steepened [Herlinveauxand Tully, 1961;Thomson,1981]thatmixing as locationsprogressed seaward(Figure2). Not only did occursprimarilyin thenarrowpassages andcomplexgeomthe slopechange,but the T-S characteristics of the bottom etry of the Victoria Sill/Haro Strait/Boundary Passregion, water also changed in a complex way. A summary of T-S which therefore acts as a control of some kind on the overall correlations of the watersin this estuarinesystemappears circulation.Further,it hasbeeninferredfrom a crudepato fall withina downwardcurvingor banana-shaped region, rameterization basedon proxymeasurements thatthebreakratherthanalonga straightline. A similarseaward progresdownof stratificationis modulatedby the spring/neap cysionin the slopesof T-S correlations in this estuarybased cle in the magnitudeof tidal currents[GriffinandLeBlond, on observations overa greaterspatialdomainwasdescribed, 1990]. However,thereis little knowledgeaboutthe mechalthoughnotexplained,by Herlinveauxand Tully[ 1961]. anismsthroughwhich tidal currentsdrive mixing. PreviIt seemspossiblethatsurfaceheatingwasresponsible for ousworkhasconcentrated on identifyingregionsof mixing thisslopechange.The fact thatthiseffectalsoappearsin throughestimationof turbulentdissipation ratesfrom nu- historicaldata suggeststhat it is a robustfeatureof this remericalmodelsand/ormicrostructuremeasurements[Crawgion andhenceworthyof investigation.Most analysesof ford, 1991; Gargett,1988; Gargettand Mourn,1995;Foreestuarinedynamicsrely on observedsalinitiesalone[e.g., HansenandRattray,1965],temperatures beingignored,so Copyright1999 by the AmericanGeophysicalUnion. that the possibilitiesof exploitingand understanding this behavior have apparently not been considered before. The Papernumber 1998JC900024. 0148-0227/99/1998JC900024509.00 analysisdescribedhereinis dividedinto two parts. First, a 30,695  30,696  PAWLOWICZ ANDFARMER: VERTICAL MIXINGINEXCHANGE FLOWS 30'  20'  10'  123øW G  50'  Pass  40'  Haro Strait  48øN 30.00'  30'  Victoria Sill  20'  50øN  30'  49øN  30'  joen Pacific Ocean  48øN  126øW  125øW  124øW  123øW  Figure 1.Fraser River estuary, withinset showing details ofHaro Strait region (indicated byrectangle at bottom). Depths intheStrait ofGeorgia approach 400m,whereas Juan deFuca Strait isnodeeper than about 200m. Shallow sills(•-, 100m)offVictoria andin Boundary Pass areseparated by300m deep  waters inHaroStrait.Rosario Straittotheeasthasa depth ofonly• 50m.V,B, andG markthelocations  ofhydrographic stations discussed inthetext.Locations of1976 current meter moorings inHaroStrait are indicatedby "+".  kinematic theoryisdeveloped inwhichtheeffects ofheating 2. Theory andmixingareexplored inthecontext ofa two-layer model. willbedeveloped asfollows. Budget equations Flowstructure is specified through a setof nondimensional A theory for mass and for tracers in a two-layer system will be deratios.Thetheoryis general andnotrelatedto anyspecific andsolvedto relatetemperatures region. Toillustrate theutilityofthisanalysis, inthesecondrived,nondimensionalized, and salinities at either end of a channel(estuary,strait)segpart,weapplythetheory diagnostically to determine flow ment to one another. The result is a mappingbetweenobserstructure fromhydrographic observations in theHaroStrait vations of layer temperature and salinity in twogeographregionanddiscuss theimplications oftheresults.  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS 14  30,697  ,  13  12  •11 310  E 9  a) All stations i  6  i  26  i  28  30  !  32  34  Salinity 12  ,  2  ,  ,  1  11 '%  •s  o•  %ß  c) B,G stations  b) V stations 6 28  30  32  34  Salinity  28  30  32  34  Salinity  Figure2. T-Scharacteristics of watermasses atinshore andseaward endsof HaroStraitregion,showing (a) all profiles discussed in text,(b) seaward (Victoria Sill)stations, and(c)inshore (Boundary Passand Straitof Georgia) stations. Dashed lineshows trendof VictoriaSill T-Scharacteristics, whileflatterdotdashed lineshowstrendof Straitof Georgiastations. Thedifferentslopesleadto anapparent downward curvatureof the correlationin (a). The sourceof the intrudingwatermasswith a salinityof 29 is not known.  ical locationswith a set of four nondimensionalparame- nusof the estuaryoccursat somea:> L. This systemrep-  onlya partof thecomplete estuary, awayfromthe tersdescribing theratiosof varioustransports. Thisforms resents (in effect)a classification system for estuaries andexchange direct influence of the river and ocean. Layer transports flows. However,it will furtherbe shownthatknowledgeof one dimensionalvariable,the heatflux, will allow the determinationof almostall otherdimensional parameters.Thus theclassification schemecanbeuseddiagnostically to deter-  areui(x), in layersof depthDi(x) andwidthai(x). The transport hereis notnecessarily thatmeasured asa meanby Euleriantechniques.In nonlinearflows,nettransport determinedvia Lagrangian techniques, e.g.,traceradvection,  mineintegrated horizontal andverticaltransports (i.e.,mix- canbe somewhatdifferentfrom thatdeterminedby Eulerian techniques suchascurrentmeters.Thedifferentlayersare ing) in thissystem. characterized by a temperature Ti(x) anda salinity$i(x) 2.1. Governing Equations thatvarycontinuously alongthechannel.Forconciseness a tracervariable Oiis usedtorepresent eithersalinity Considera two-layersystem(Figure3). Upper(lower) generic in thefollowingdevelopment. Mixingbelevelvariableshavethe subscript i = 1(2). The inputof or temperature by entrainment velocfresh water occursfor some a: < 0, and the ocean termi- tweenthetwolayersis characterized  30,698  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS  The equations arenondimensionalized using  H  F=  $1'T1  w2  z = ui & =  D  Ul 1  0  X  x  Xz' Uu'i OsS  (8) (9)  whereX is thelengthof thepartof theestuarybeingconsidered,U is thelowerlayertransport at x = 0 (lowerlayer transportis usedsinceit is not directly dependenton the freshwater outflow),andOs,•' = ]0x(0) - 02(0)1is thedifferencein tracervaluesbetweenthe two layersat x - 0. In nondimensional terms,thesalinitydifference between upper andlowerlayersat z - O,AS' (0) - S• (0) - S•(0) - 1 at all times. The nondimensional temperaturedifference  AT'(O) = -1 in summer. In winter,however, AT'(O) = + 1. Althoughwe considerprimarilysummerconditions, the  Figure3. Geometry of thetwo-layer system. Thesurfaceanalysisfor a wintercaseis very similar. areais axX, butmasstransfer parameterized usingentrain-  mentvelocities w•,ebetween thetwolayers occurs through Verticalexchangewill be consideredconstantoverthedomain of the system,with  a (possibly smaller)areaaeX.  a2w2 -  W  (12)  a2wx RW (13) ities wi, which are not necessarilyequal. In the usualcase considered in, e.g., fjords,entrainmentis assumed to occur whereR = wx/w2 _> 0 is an entrainment ratio,describonly intothe upperlayer (wx = 0), butno suchrestrictionis ingthe asymmetry betweenmixingintotheupperandlower imposedhere.Layertransports alsovarycontinuously along layers. Valuesof R will reflectthe physicalmechanisms the channel.Finally,the upperlayeris subjectto an inputof  at work in the estuary.For example,turbulentmixingwith  heat,F = H/pcp, representing thenetheatflux intothe wx • w2 implies R • 1, whereasif the dominantmech-  surfaceH, dividedby a representative densityp anda heat anismis entrainmentinto the upperlayer, thenwx m 0 or capacity%. We assumea quasi-steady state. The relevantconservaOtherusefulnondimensional parameters, with definitions tion equationsfor volumeandtracersare chosenfor their physicalrelevance,are an aspectratio betweenthe upwardverticaltransportWX andU OUl Ox  Ou2 Ox  --  w2a2 -- wl a2  (1)  =  wxa2 - w2a2  (2)  w2a202 - Wla201 4- Fax  (3)  WX  Ox  0u202 Ox  wla201 -- w2a202  w2a2(02- 0x) + Fax  002  --U20X = wxa2(0x- 02)  > 0  -  (14)  axF  (15)  (4)  (5)  (6)  whereaxF is the surfaceheatinputrate, which will be consideredconstant,andW O•, is the mixinginputof heat.B is positivefor summerconditionswhenthereis a net inputof heatinto the waterandnegativefor winterconditionswhen thereis a net lossof heat. Finally,the excessoutflowin the  upperlayerdueto thefreshwater inputis characterized asa fractionof the exchangeflow,  f -Uf'  (16)  so that 1 + f' represents the ratio of horizontaltransports betweenthe upperandlowerlayersat x - 0.  The barotropicflow shallbe takenasconstant, -u2 - f _>0  U  anda heatinputratio B -  where it is understoodthat the heat flux term in (3) applies only in the equationfor temperature. It is convenientto rewritethelattertwo equationsby substituting thosefor volume conservationto get  •'llOX =  A=  (7)  Droppingprimesin nondimensional variables(andusing an asterisksuperscript whendimensional quantities arere-  sincein the steadystatea net input of freshriver watermust ferred to), we have be balancedby an equal seawardflow f. Implicit in this Ou2 = n(1 - R) assumption is a restrictionto timescaleslongerthantidal. Ox  (17)  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS  30,699  which canbe solvedimmediatelyfor lower layer transport:  u: - 1 + A(1 - R)x  (18)  Using (7), we haveupperlayer transport  711-- 1 + f + A(1 - R)x  •0) (19)  •1) _ ß -  Layertransport increases by a factorA(1 - R) between the inshore(x - 0) and seaward(x - 1) endpoints.It is sensible to restrict attention to cases where flow is unidirectional  AR  I+A(1-R)  within eachlayer, i.e., 1  < +  (20)  •(•)  However,in mostestuariesthe layer transportincreasessea-  wardowingtoentrainment, sothatexpected valuesareA(1R) _>0orR_< 1. Substituting(18) and (19) into the nondimensional tracer equations,we finally have (21)  O01 Ox  =  A(02 --01) I q- f q-A(1 - R)x  +  BA I q-f q-A(1 - R)x  002 = 1AR(02 --01) Ox + A(1 - R)x  Figure 4. NondimensionalT-S characteristicsof (left) inshore and (right) seawardends of the two-layer system. Layer characteristics are shownby opencircles,which are joined by solid lines representingthe typical appearanceof actualobservations in whichthe two layersare separatedby a regionof mixed water. Labeled offsetssummarizethe resultsof the analysisfor pure exchangeflows. The leftmost dashedline showsthe "mixing line" of expectedT-S characteristicsin caseswhen heat input is negligible. Hatching indicatesthe lowerextremityof the triangular-shaped region in which the T-S characteristics of the surfacelayer at the seawardend of the sectionmust appear. These exact solutionsfor pure exchangeflows are approximatelytrue for  (22)  whereagainthe terminvolvingB in (21) appliesonlyto the temperatureequation. Solutionsto this systemcan be approached in two ways.On physicalgroundsonemightconsiderspecifyinglayer inflows(i.e., conditionsat x = 0 for the upperlayer andx - 1 for the lowerlayer). However,it is mathematically moreconvenientto specifyall conditions at x -  NondimensionalSalinity  0 and solve for those at x = 1. This is also more con-  nonzero f with2•replacing A and/•replacing R (seetext).  sistentwith the way in which observations are interpreted. Temperature andsalinityprofilesmeasured at a seawardstation are to be related to those measured inshore. Solutions to  is a a smoothlyvarying,monotonicallydecreasing, positive 0 for r/--> •, hasa thissetof equations thereforedefinea nonlinearrelationship functionthat asymptoticallyapproaches  betweena setof parameters {A, B, R, f} definingtheflow removable singularity g(0) - 1/2, andhasa limitg(-1) anda setof along-channel changes {AS1, AS2,AT1,AT2} +1. in salinityand temperature(nondimensionalized by across-  The four pointsrepresenting thelayercharacteristics at ei-  layerchanges AS*(0) andAT* (0)) thatformtheobserva- ther end of the channelare collinearonly if B -- 0; that is, tions.HereAOi = Oi(1)--Oi(O)andA0(0) = 01(0)-02(0). heat input is negligible. If B > 0, then the seawardline, whoseslopeis -(1 + AB), will be steeper thantheinshore 2.2. Analysisfor Pure Exchange Flows linewhoseslopeis -1 (in thewintercase,sincebothAT(O) The caseof pure exchangeflows, f = 0, is analytically andB haveoppositesigns,the seawardline will havea slope straightforward,albeit tedious. The relevantrelationships 1-AB, alsosteeperthantheinshoreline of slope1, although between{A, R, B } andtheobservations aresummarized in in the oppositedirection).Sincethe slopeof the mixingline is someintermediate Figure 4. Pointsrepresentingseawardconditionsare to the joining the lower layer characteristics right of thosegiving inshoreconditions;salinity increases seaward,asexpected. Horizontal (salinity) distancesin Figure 4 are rational combinationsof the nondimensional parameters.However, vertical(temperature)distancesare morecomplicated.The effectsof surfaceheating(nonzeroB) appearin a term in-  value,1 < I+ABg < I+AB, thepoint{S2(0),T2(0)} must  volvingthefunction g[A(1 - R)], where  wardtheupperright,somewhere in thetriangular-shaped regionwhoselowerlimitsareindicatedby hatching.The general patternin whichT-Scorrelations on the right-handside appearsteepergivesrise to a vaguelybananashapedclus-  g(•/) _ (1+•/)log(1 r/2+•/)- •/  (23)  lie strictly abovethe mixing line joining the two "source" waters(the deep seawardwater massand the inshoresurface water mass).That is, the T-S characteristics of all water massesobservedin theestuarywill fall withina regionabove  thisline.Thepoint{ $• (1), T1(1)} mustlie evenfurtherto-  30,700  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS  a) aS 1'  '"  '....  ß •;;.:9; ;.;.;.:.:.:.:.: ........... •$..... •o:...... ß '..........'.'-'-'."'""$".':'::"• ...... :':: :•.•::,•. .........  ""•'.•  .......... T.if' .... ;.  •+ 1  ::1: o ITI  ............. -'.......  .................... '-' =•' ......4:................................  •'  '•...• '•"•''_._%" ...... ;1i.,li..•. ,•.-oi?• ?." •ili.l.l.... •ß i''""•i•-'J  0  •  •  0.5  1  '  1.5  •  2  •  2.5  • 3  A/(l+f) horiz(  transport vertical mixing  b)6S1  1.5  f:O f=0.1 f=0.5  --...... 0.5  I  0.5  " 1  i  i  1.5  2  2.5  A/(l+f)  Figure 5. (a) Salinitydifferences betweenthetwolayersat theseaward edgeof thesystem,asa function of aspect, entrainment, andtransport ratiosA, It, and1+ f. (b) Changein theupperlayersalinitythrough thesystem.Bothchanges areshownasa functionof entrainment andaspectratiosin naturalcoordinates. Salinitiesareindependent of the heatinput. Solutions areshownfor pureexchange flows(f=0, solid contours),aswell astwo differentestuarineflows(f=0.1, dashedcontours,andf=0.5, dottedcontours). Theblankareaat toprightcontains parameter combinations thatviolatetheconstraints of unidirectional flowswithineachlayer.Thisboundary of thisregionis constant in coordinates for A andIt butvaries with f in naturalcoordinates.  ter of points,concavedownward.However,it is important to understandthat otherrelationships shownin Figure4 are notnecessarily alwaystrue.For example,theseawardupper layertemperature (salinity)01(1) maybe greateror smaller thaninshorelowerlayertemperature (salinity)02(0). 2.3. Analysis for Estuarine Flows  001  A  A  I +f (02 --01) + I +f B (24) A  I q-fit('lq-f)(O 2--01)  (25) = Ait(02 - 01). In the generalcaseof estuarineflowsfor whichf > 0, analyticapproaches quicklybecomeintractable.Numerical coordinates arethus.• = A/(1 + .f), i.e., solutions overa gridof { A, It, B, f } valueswerecomputed The"natural" thetotaltransport in theupper layer, and1• usinga nonstiffRunge-Kuttaordinarydifferentialequation A using ofthevertical andhorizontal fluxratios. solver [Shampineand Reichelt,1997]. However,compar- R(1 + f), a product isonsbetweenthepureexchange andestuarine flow condi- Similar factors arise if we assumeinsteadR m 0 (i.e., that is primarilyintotheupperlayer),although the tionsaresimplifiedusinga systemof "natural"coordinates entrainment equations aredifferent. Thegeneral utilityof these basedon the existingnondimensional parameters.These resulting is somewhat empirical.However, aslongasf is naturalcoordinatescanbe derivedheuristicallyby assuming coordinates shown IA( << • (i.e.,thatlayertransport isnotsignificantlynot"toolarge,"it will beshownthattherelationships trueif wesimplyreplace the changedthroughthe estuary),so that the tracerequations in Figure4 areapproximately  (21) and (22) canbe simplified:  set{A,R,B} withtheset{.•,i•, B}. Notethatthisimplies  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS  30,701  a)bSl/bS 1.5  I .-' , .'  • .. i  i  i,  I•.:  I'  I,:  !..  h: ,i  .•.:  0.5  ,,,  •  -"-"-.-•.--• . .• ................. 1.25.............. - _ - - • - .-.."--,% .""-_•-=_-_-._-_ =.•.............. 1.5'-' ..-I'"%..... .:-.:•-:-"•- r?;-_•_'•._'•-__-...---•:  ....' ..' •  ..  •  .. .."  . -"L' .....  -,, , .i.' ,' .'  I ,  ..'/  I I :'  I,  .' /,  I,..  • I,..' /,..'  I': ':'  if: •  -  I,• .'." /, I,: '/  ,  ,  ,-  ..... •  • /  -" / / ."  "'" /' :  I, : ' :'  ...' • • .. •  .-' ,  T.T .....__- ----.;.=  '"';1.'  ...)/ ..-'• .- •  -• /  '' ."""' / '  I' ." "."  1  1.5  _ ..;• ...,  ."l '  ß  ..-• ..-/  I"' ,  "'/'"''  ' " / • '" /' ' "' ! ' ." ]•  I-" 0.5  .;.' ß  ." / • '"" /• "'  / 2  2.5  N(l+f)  b)AS2/AS 1 .....  • ..--..'-..•,._ _ ...I  ........-..•........ ,,.•.....• ........ • .'...,,,: ,,• •  ......... 1.5  I '•. •,,,,•  _12'.5.  ' ' t .......  '" '; "'1;'%"'%""'•'"'"'"' [ .' ''1  ..........  ' -' --_"'"':"• ...............................  ..... ,..,'.•.:.-:.:............... 'T..-....• .......  ....... ß 'k-5,_ .....  ••  '-.•. '" :': '•. :d""•.  I  ß......... ::::..::..':-?:...::•4•s,,, .  ' .............. ::"""':':':  5 ................................................... ..................  2 ............  ----1•'7•"---•-:-'-•:..-•-_•. .-......• ...........1.5....  ...................  ;.•?.:..--.•..;•::;.-.:.Ztt.•._--.  .................ß. _1._25 •,•'-'-' •:2'•.............-'-'"'2'"'-'-'"'-'- '-- • zb- - :1..25 .......................  •,'"  ............................  ................................ ¾.'•:..........  1 ................................................... - -1- ...... •. ..................  0.75  e..-,_•  1 ......... - -1- - - -  .......................................................  0.?• .........  o.7•- -  ......  f=O f=0.1 f=0.5  0.5  0  I 0.5  I 1  I 1.5  I 2  I 2.5  3  N(l+f)  Figure 6. (a)Estimates of.2.- A/(1 + f). (b)Estimates of/• - R(1+ f). Bothestimates areshown as a functionof true values.Solutionsare contouredfor pureexchangeflows(f=0, solidcontours),aswell astwo differentestuarineflows(f=0.1, dashedcontours,andf=0.5, dottedcontours).Note thatsolutions  areexactfor pureexchange flows(horizontal or verticallines)butonlyapproximate for estuarine flows. The blankregionin thetoprightcontains flowsthatviolatethe conditionof unidirectional flow within layers.  the top right cornerof Figure 5a. That area of the parameter spacedescribesflows in which layer transportbecomes very small owing to lossesfrom entrainment.Obviously,it is when layer transportbecomessmall at somepoint in the AS(I) - S2(1) - S1(1)asafunction of• and/•forthree systemthat the effectsof extrafreshwatertransportmay bedifferentvaluesof f (recallthatAS(O) - 1 by definition). comelarge. Figure 5b showsthe along-channelchangein upperlayer The seawardchangein this differenceis stronglydependent  thatonlythreeof the four nondimensional parameters canbe easilyestimated.Additionalinformationwill be requiredto specifyf. Considerdifferencesin layer salinities.Figure 5a shows  on the entrainment ratio R. If R > 1, i.e. the flow entrains  into the bottomlayer, then this differenceincreases.Drhwing analogieswith typical observations in which salinities tend to vary somewhatcontinuouslythroughthe layers, a verticalsalinitysectionalongthe lengthof an estuarywould showmore stratificationand more closelyspacedisolinesto seaward. Conversely,if entrainmentis predominantlyinto the surfacelayer (R < 1), the differencedecreases, the strat-  salinity, AS1-- S1(1) - S1(0),asa function of.•, •, and f. Therearetwocasesof interest.WhenRA (= RA) > 1, the estuaryis well mixed. That is, more salt is lost/gained owing to vertical mixing than to horizontaladvection.All salinitiesat the seawardend are greaterthan salinitiesin-  shore,i.e., $1(1) > $2(0). Isolinesin an along-channel  sectionin sucha systemwould intersectthe surfaceand/or bottom. For RA < 1, transportpredominates and section ification is at a maximum closest to the river mouth. If, isolineswould be closerto horizontal,intersectingthe left however,mixing occurs,but there is no strongpreference and/orright edgesof the section.This representsa partially towardupward or downwardentrainment(R m 1), the dif- mixedestuary.Again, the casesfor nonzeronet outflowcolferenceremainsroughly constant.Isolinesof salinity in a lapsequitewell usingthe naturalcoordinates. By combiningsomeof the analyticsolutions(Figure4), verticalsectionwould have the sameslopefor all x. The formulas for .• and/• alone.Forpureexnaturalcoordinatesappearto satisfactorilycollapsethe data wecanderive for nonzerof as longas R << 1 + I/A, i.e., awayfrom changeflows,the following are exact:  30,702  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS  a) -(1 +AT/AS), R=1/(1+f) ß  ..  ß •  ..  ;;  \  ,•,. '.,  ..  ß .  1.5  •...  ;  \'..  -..  • .., • ..  •..  'X;:-..'x,'...  .•  ,.  •  ,.  •,  '-' ....  "......... .  0.5  ...  •, ...  '•"'. \•".  \  •  .•  • •  '...  •,  ,•, .,.  ''. ....... ' ' '  .....  ......  .  ,•,  ..... ,,,,,,  ....  ........ T' -.  /.s-,,' ........•--,,  ß  -.....  '........  ß  •.s.,•...- ....  "  '  ß.....  ß  0  O.5  1  1.5  2  2.5  3  A/(l+f)  b) fixed A=(1+f)  ..... i / cxl... ..""% i.  t i/. i.,' / i i.•,. ß .- '- - , /  1.5  ,'  d'  I•. •  ,, t  ,..  ,•  , ..'  I'•  ,i  :  I:  0.2  ..";'  .'  i;  :'  I  0  /  / ..-'/ •...'  I' :  i.  0  / .." '  [..' • ..'1 •  ......" ..'"'""'"" ..', / ,,.% / .," ,.-,' /  t  .'  ,'  .j-  .,,7/  ..' i ...  I  0.4  I,  0.6  I o.8  , 1  2  B  Figure7. Estimates forAB (a)keeping k - R(1+ f) fixed, and(b)keeping 2i - A/(1 + f) fixed. Solutionsare shownfor pure exchangeflows (f=0, solid contours),as well as two differentestuarine flows(f=0.1, dashedcontours,andf=0.5, dottedcontours).Note that solutionsare exactandentirely independent of R for pureexchange flows(verticallinesin b), butonlyapproximate for estuarine flows.  ASx  AS2  A = AS(1)'R- AS•  in summer conditions whenAT(O) = -1. SinceB > 0 in  (26)summer,thisimpliesa steepeningof thecurvein theseaward direction, i.e.,negative slopes thatbecome morenegative. In  However, asFigures 6aand6bshow, thefollowing approx- winterconditionstheseawardslopeis equalto 1 - AB, with imatesolutions holdwellfor estuarine flows,aslongas B < 0. We againhavea steepeningin the seawarddirection  R << +  A$1  AS 2  2i m AS(1)'/•mAS1  but in the oppositesense;positiveT-S slopesbecomemore positive. Interestingly,the slopechangeis independent of the entrainmentratio R. In fact, sinceW appearsin the  (27)denominator of B and the numerator of A and hence cancels  In Figures 6a and6b theexactrelationships for f - 0 •e in their product,the slopechangeis independentof vertical indicated bythestraight lines.Although it mayappe•that mixingaltogether,dependingonly on theheatflux inputand  thecollapse ofsolutions for• (Figure 6a)isnotverygood the layertransport. fornonzero •, •specially toward thetopright, it should be notedthatfor R - 0.•, co•ection bythefactor1 + f decreasesbiasesin the estimationof A from a maximum of 60% down to about 6% of its true value.  The additionof a heatflux inputtermmakessolutions  forchanges in differences between layertemperatures more complexthanthosefor salinity.It will beusefulto consider  theslopeof theT-Sco,elation.FromFigure4, AT(1) = -1 - AB AS(1)  (28)  UOr_ -- [1+AS(1) ' (29) ABFa•X AT(1) or  .fi.BU(1 Fa, +f)©T X • --[1+AT(1)] AS(1)](30) This resultprovidesa methodby whichtransportestimates can be made usingestimatesof heat flux and observations of temperatureand salinityalone. Althoughexactlytrue in exchangeflows,this relationshipis alsoapproximately true  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS  a) i  i  .........................  i  ..............................  i  1.75 ................................  -1.-79.7c  .......•.5......."1'•--•' .......... i'.•' ....................... 1.5................. 1.5 --  •:• .................  1.5  R=1/(1+f)  i  • 7.•..... "E7,5;,"d ......  30,703  •."-" ................... •:-25•:•'• ..................... 1.25 .................. '-:1'•:•i i;,;-,' .............. • ......................."1:........ 1 ...............{.........................i......... ....................  0.75..•)7,]•5• .......................................... 0.75"•,•  ...............  cœ........ •E.,5" o.5................................ o:•......... -:(h',5:'o.s......  0.5 o.-,',c  ....  i  0.5  ..6•5c.2s ..........................  •2s- ...............  i  i  i  1  1.5  2  0:...250.25 ................... i  2.5  3  A/(l+f)  b) fixed A=(l+f)  1.5  0.5  0  0.4  0.6  2  Figure8. Estimates forB (a)keeping/•- /t(1 + f) fixed,and(b)keeping A - A/(1 + f) fixed. Solutionsare shownfor pure exchangeflows (f=0, solid contours),as well as two differentestuarine flows(f=0.1, dashedcontours,andf=0.5, dottedcontours).Estimatesareexactandentirelyindependent of R for pureexchangeflows,as shownby the horizontalor verticallinesin (a) and (b) respectively,and approximatefor estuarineflows.  in estuarineflows(Figure7). Sincequiteaccurateestimates of B are possible(see below), the accuracyof suchtransport estimatesis, in principle,limitedby the accuracywith which A can be computed(Figure6a). In practice,the relativecrudeness of estimatesof fluxesandthe geometricparameterssuchas surfaceareaal X will probablybe limiting factors. There is anotheradvantageto the use of slopeas a parameter.In attemptingto fit actualobservations into a two-layerparadigm,thereis alwaysthe problemof specifying layer depths.If the T-S correlationis linear,thenthe slopeis independent of this choice,which then entersthe equationsonly throughthe calculationof a (dimensional) differencein temperaturebetweenupperand lower layers, ©T. However,this temperaturedifferenceis only weakly dependenton the choiceof layer depth. For example,it is easyto demonstrate thatin the caseof a linearprofile,computedtemperature differences arecompletelyindependent of the specifiedinterfacedepth. One estimatefor B canbe formedby combining(27) and (30) and simplifying:  AS(I) av() 1_-  AS•  Figure 8a showsthat this estimateis quite accurateover a large rangeof A and f, and figure 8b showsthat the estimateis virtuallyindependent of/t overa wide rangeof values. Thus knowledgeof heatflux can be usedrathereasily to make accurate estimates of W and hence of vertical en-  trainmentrateswithin the region. A secondestimateof B canalsobe formedfrom the relationships in Figure4 using the last unusedpiece of information,AT2. It involvesthe  inverseof g(r/) andis lesssatisfactory in itsaccuracy.  Theaboverelations let usdetermine/•,A, andB from observations.However,no simpleapproximation hasbeen foundfor estimatingf. This mustbe specifiedusingother information.In thenextpartof thispaperwe applythistheory to a set of observations.  3. Mixing and Transport in Haro Strait A field programwascarriedoutin the summerof 1996to betterunderstandmixing and transportprocesses and their evolutionthroughthe spring/neap cycleof tidal amplitudes in theHaroStrait/Boundary Passarea,a complexregioncontrollingthe estuarinecirculationoff the eastandsouthcoasts of Vancouver Island(Figure1). As partof thisinvestigation,  30,704  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS Salinity  Salinity  Salinity  •5 27 29 31 33 3.5•5 2.7 29 31 33 35 •5 27 29 31 33 35 50  5O  5O  ,.  lOO  100 .....................  100  150  150  150  200  a) V stn  200  b) B stn  300  250  300'  Temperature  06  200  250 t  250  8  300  Temperature  10 12 14  06  8  10 12 14  Temperature  06  8  10 12 14  5O  50.t'"50  100  100  ..................  100  150[  150  150  2001  200  2501  250  300 •  300  200  f) G stn  250 300  Figure 9. Profilesof temperatureat (a) VictoriaSill (b) BoundaryPass,and (c) Straitof Georgia. Correspondingsalinityprofilesare shownin (d), (e), and(f). Estimateddepthof interfacebetweenlayersis shownby the dottedlines. Profilesgo to the bottom. Deepestwatersare in the middleof Haro Strait, separatedfrom the ,,• 200 m depthsoutsidethe straitby ,,• 100 m sills in BoundaryPassand southof Victoria.  a regularhydrographic surveywascarriedout. Stationswere occupiedduringnighttime,a periodwhen tideswere relatively weak. Thereis a strongasymmetrybetweendaytime andnighttimetidesin thisareaduringsummermonths.This regularschedulewas chosento reducetidal aliasingproblemsandalsoto prevent,asfar aspossible,"contamination" of the profilesby purelylocalturbulentphenomena. 3.1. Observationsand Layer Approximations Three setsof stationsare usedhere, representingconditions at the inshore, middle, and seaward ends of the com-  thanvalueschosenthisway, sincethe layerinterfacewould then lie within a region of near constanttemperatureand salinity. In the secondprocedurethe layer depthwas chosento coincidewith the zero crossingof the first baroclinic modeof horizontalvelocityat theM2 frequency.Thischoice hassomeattractionfrom a dynamicalviewpoint.However, sincethe conceptof modesimplicitlydeniesmixing(which is verystrongin thisregion),it is alsoof somewhat dubious merit.Happily,bothprocedures gaveverysimilarresults. Stations at the seaward or "exit" end ("V" stations)are  shown in Figures 9a and 9d. These stationsare located within a deep region west of the Victoria Sill and showa relativelythick andstratifiedsurfacelayer of depthapproximately70 m. Figures 9b and 9e show profilesobservedwest of the BoundaryPassSill ("B" stations),at the boundarybetween the BoundaryPassand Haro Straitregions.The deeplayer hasits upperboundarynear100m. This depthis greaterthan that usedin stationsinshoreand seaward. There are many possibleexplanationsfor this, includingchangesin crosschannelwidth and/orinterfacedepthsas well as aliaseddynamics(seealsosection3.5 andthediscussionof Figure 15). Note thatthe layerdepthsareimportantonly in sofar asthey affectlayer averages;theyarenotexplicitlyusedin the analysis. Althoughnot consideredin the errorestimatesbelow,  plete region(Figure 1). We can thereforeconsideralongchannelsegmentsthroughBoundaryPassalone, through Haro Strait alone, or throughthe completeregion. Applicationof the theorydevelopedaboverequiresthat profiles obtainedat thesestationsmustbe interpretedin the context of a two-layersystem. Sincethe flow parametersare thus at leastpartiallydependenton the choiceof layer interface depth,someconsideration mustbe givento the methodof choosing thisparameter.In theinshore/Strait of Georgiastations(Figures9c and9f), the interfacewaschosento lie at the baseof the thermoclinemarkingtheFraserRiverplume. For stationsoff Victoria and within BoundaryPass,two differentprocedures wereused.In thefirsta decisionwasmade to definethe lower layer as includingthatpart of the water columnwith roughlyconstantT-Scharacteristics, while the results are not too sensitive to reasonable variations in these upperlayerincludedthatpartwith a roughlylinearvariation depths. Stationsat the inshoreend are shownin Figures9c and in properties.It is unlikelythatlayerdepthswouldbegreater  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS b) Deep  a) Surface 34  ......  c) Deep-Surface  3z  33  3,•  32•  3; 3(  2•,  28  2[  27 26  2(  25 24 26 28 30  2! 2  4  6  June/July1996  e) Deep  d) Surface 12  11  olo  •  .  .  12  tures(salinities)in bothlayersdecrease(increase)seaward. Althoughgeographicallythe B stationslie muchcloserto the G stations than to the V stations, their characteristicslie  31• t 3'  .•'30 • 29  30,705  f) Deep-Surface  ......  approximately midwaybetweenthetwo. Thereis somedifficultyin establishing conditionsnearthe surfaceinflow(G stations),shownby the largevariabilityin meanlayer characteristicscomparedwith thoseat otherlocations. The observations form a time series,showingchangesin layercharacteristics as the tidesbuildfrom neapconditions on June25 to spring(maximumvelocity)conditionsnear July 3. Upper and lower layer characteristics at the B (and V) stationsconverge(Figures10c and 10f), indicatinga decreasein stratificationdue to a greateramountof mixing. Surfacecharacteristics at G stationsare highly variabledue to small-scalevariabilitywithin the Straitof Georgia,andit is difficult to discernany trend.  11  3.2. Freshwater Outflow, Surface Heat Flux, and Geometry  10  Accordingto preliminary figuresobtainedfrom the Water Surveyof Canada,the mean dischargerate of the Fraser  9  Riverat Hopewas6750rn3/sin June1996. Thisrepresentsabout50% of the totalrunoffinto the Straitof Georgia [Griffinand LeBlond,1990]. Net outflowthroughJohnstone Straitis assumedto be negligible(seealsosection3.5). The net outflow throughHaro Strait is thereforeassumedto be  7  June/July 1996  • 104 m3/s duringtheperiodin question. It is notpos-  sible to be more precisebecausethe effectsof wind in the Figure 10. Time seriesof temperatures and salinitiesin the Strait of Georgiacan resultin temporaryfluctuationsin the two layers for the V (*), B (o), and G (+) stations,show- freshwateroutflowthroughthe Haro Straitregion. ing (a) surfacelayersalinities,(b) bottomlayersalinities,(c) Themeanheatfluxwasestimated to be• 150W/rn2, salinitydifferencesbetweenlayers,(d) surfacelayertemperfromabout-50 W/rn2 at nightto peakvalues of atures,(e) bottom layer temperatures,and (f) temperature ranging some400 W/rn2 duringthedaytime.Sensible andlatent differencesbetweenlayers. fluxeswere estimatedfrom surfacebuoymeasurements and  werefoundto be verysmall(of order10 W/m2). The 9f. Stationsfrom two locations(hereafterreferred to as "G" stations)on either side of the entranceto BoundaryPassare  largestuncertaintyin this estimatearisesfrom the effects  used. The chosen locations lie well within the Strait of Geor-  thoughdaysweregenerallysunny,photographs takenduring the cruiseshow a surprisingamountof thin cloud, so this value is probably not unreasonable.Since the transittime for surfacewater throughHaro Strait is severaldays, it is  gia and are relatively unaffectedby processes local to the BoundaryPassentrance,henceare goodrepresentatives of the boundaryconditions.In the contextof a two-layer system theseprofilesshowa fresh,warm surfacelayer with a thicknessof approximately20 m abovea deeperlayer with more graduallychangingcharacteristics. There is evidence of the intrusionof a differentwatermassat a depthof about 40 m in sometemperatureprofiles. This is indicatedby a local temperatureminimum, as well as the finger-likesignatureof interleavingin the T-S diagram(Figure 2). The sourceof this intrusionin not known. Attemptsto trace it in the completedata set were inconclusive.Although the profilesof temperatureand salinityfor B andG stationsare quite different,the slopeof their T-S characteristics is very similar.We returnto this point later. Layer characteristicswere obtainedfor each profile by takingmeanvaluesof temperature andsalinityaboveandbelow the interfacedepthsdeterminedabove.Figure 10 summarizesthe layer characteristics at all locations.Tempera-  of cloud fraction, which was assumed to be about 0.6. Al-  reasonable to use the mean value in calculations.  This is not  a very goodapproximationwhenconsidering the Boundary Passregion alone, sincetransittime there is probablyless than 1 day. This is one causeof the poor resultsobtained over this particularsection(seebelow). The surface areas over which these fluxes affect the water column are estimated from the chart as follows:  stations  G to B (Boundary Passsection), 100krn2;stations B to V (HaroStraitsection), 1400krn2;stations G to V (complete region),1500krn2. We havemadeana prioriassumption that mosttransportoccursthroughthe BoundaryPass/Haro Straitpassage.The areacalculations thereforeignorewaters eastof San JuanIsland (includingRosarioStrait). However, the Haro Strait section includes the shallow waters on the western side of the strait as well as the considerable the Victoria  Sill.  area of  30,706  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS 5-  a) nondimensionalparametersfor completeregion  -1  ....  4-  A  - 0.8  3-  - 0.6  R - 0.4  B = 0.19  +0.09/-0.09  I  0  - 0.2  I  I  0.2  0.4  I  0.6  b) transportsin completeregion 25-  WX = 18 +14/-5.1 20-  •)15-  I+A(1-R)) (•10-  5-  I  0  I  I  0.2  0.4  I  0.6  f  Figure11. (a) Nondimensional parameters and(b) dimensional transports diagnosed for thecomplete region(usingstationsG andV).  Thetotalheatinputintothewatercolumn depends onthe of sucha procedure werenotveryuseful, having extremely product of themeanheatfluxandsurface area.Inspectionlargebiases anderrors. In ordertoforma consistent picture, of theformulas derived in thefirstpartof thepapershowsthefollowing procedure•is •applied here.First,estimates are thatdimensional estimates of transport will bedirectlyproportionalto the totalheatinputandhencedirectlyproportionalto uncertainties in areaandmeanflux. For example, a 25% overestimate in thesevalueswouldresultin a 25% overestimate in derivedtransports.However,estimatesof  formedof parameters A, /i•, andB fromthe observations discussed above.For differentvaluesof f, theproductAB is usedin combinationwith the surfaceheatinputto form an estimateof the dimensionallower layer transportat the inshoreendU, thebarotropicoutflowrateUf, andthelayer  nondimensional transport ratios{A,/•, B} areindependent transport at theseaward endU[1 + A(1 -/•)]. It is shown of thesevariables.  thatdifferentaspects of the flow havedifferentsensitivities to f, andhencethe choiceof this valuehasan importance  3.3. MeanTransport Estimates  thatvaries according totheflowparameter under considera-  Meanflowparameters aredetermined bysubstituting meantion. Figures1la and12ashowthenondimensional parameters layer characteristics into the formulasdeterminedabove. Uncertaintiesfor all estimatesare determinedusinga boot- estimatedfor the Haro Strait and completesections.Estistrapapproach(seeappendix).Theseintervalsresultfrom matesof A and/• vary with f, whereasthosefor B are inof f. However,A > 1 and/i• < 1 at all times; considerationof the variability in the layer averagesonly. dependent They do not take into accountunsuitabilityof the theory, thatis, thereis moreverticalexchangethanhorizontaltransthereis entrainment intoboththelower approximations madein the derivationof thetheory,inaccu- port,andalthough intotheupperlayerpredomiraciesin the surfaceheat fluxesand geometricalparameters andupperlayers,entrainment mustincrease (areas,etc.), or biasesin the sampling.Someof theseeffects nates.Also,A(1 -/•) > 0, solayertransport are irrelevantto the analysis,and othersare presumedto be seaward. Values for/i• are similar in both sections and are small.They are somewhatgenerous in thattheyincludethe similarto thosefoundfor the BoundaryPasssectionaswell effectsof spring/neap variability,aswell as spatialvariabil- (notshown).Valuesof A for the completesectionaremore than double those for the Haro Strait section. Values of A for ity at any one time. Only three of the four governingparameterscan be re- the BoundaryPasssection(not shown)are similarto those liably determinedfrom this data set. Externalinformation for the Haro Straitsection.This impliesthatthe increasein in eachsection, proportional toA(1 -/i•), is mustbe usedto determinethe fourth.In principle,it is pos- layertransport sibleto separately estimatef with datafromtwo adjacentor alsosimilar. However,B is greaterin the Haro Straitsecto theoverallregion(approximately 0.5 comoverlappingsections(suchas we havehere)by considering tioncompared conservation of massin eachlayer. In practice,the results paredwith 0.2). Sincethe surfaceheatflux anddifference  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS -1  5-  nondimensional parameters fol Haro Strait ----t ....-I....  - 0.8  4-  - 0.6  3-  2-  - 0.4  B = 0.5 +0.4/-0.3  0.2  1-  20-  30,707  i  i  i  i  0  0.2  0.4  0.6  b) transportsin Haro Strait WX = 8.2 +7.3/-3  --  15-  E10-  U(I+A(1-R))  5-  .........  i  0  .......  u  0.2  0.4  0.6  f  Figure12. (a)Nondimensional parameters and(b)dimensional transports diagnosed fortheHaroStrait region(usingstationsB andV).  betweenupperandlowerlayertemperatures is similarover the completesection. For the Haro Strait sectionf should the wholeregion(Figure1Of) andknowingthatthatthe area be somewhatsmallercomparedto the overall value in orof theBoundaryPassregionis negligiblecomparedwith the der to account for the effects of entrainment in the northwhole, ern section.We thereforeestimatethat the layer transport  (notincluding thefreshwater outflow of ,-, 1 x 104 ma/s) is ~ 5 X 104 m•/s (f ~ 0.2)attheentrance toBoundary Bcomplete 0.2 Pass, ~ 7 x 104ma/s atthenorthern endofHaroStrait,and Are&Haro(WXOT)complete (WX)complete ~ 10 x 104ma/s south of Victoria(summarized in Figure Areacomplete(WXOT)Haro (WX)Haro 13). The meanlayer velocitywithin Haro Strait,in a layer BHaro  =  0.5  • 2  (32)  Thusabouthalfof theupwardverticaltransport WX ormixingoccursin theHaroStraitregion.Theotherhalfmustoccurin theBoundaryPassregion.RA > I for thecomplete region,implyingthatthisis a well-mixed estuary overall. After substitution of the geometricalandheatflux parametersdiscussed above,transportestimates are madefor the Haroandcompletesections (Figures1lb and12b). Since the sections overlapandsharea commonseawardbound-  averaging 70 m thickand5 km wide,is about20 cm/s. This is impliesa displacement of about60 km or the approximate distancebetweenthe Straitof Georgiaandthe Straitof Juan de Fuca in about3 days, consistentwith earlier estimates.  The seaward increases in transport arecaused by entrain-  ment.About9 x 104 m3/sof lowerlayerwateris mixed intotheupper layer,andabout 6 x 104 ma/sofupper layer  thecomplete section (~ 18x 104 m3/s)  wateris lostto thelowerlayerin eachsection. Transportestimatesfor the BoundaryPasssectionhave beeninferredfrom differences betweenthe completeand Haro sections, ratherthancomputed directly.Attemptsat directestimatesprovedhighly unreliable.This is because the transportestimaterelieson divisionby a slopechange, proportional to theproductAB. The slopechangeacrossthe BoundaryPasssectionis very small,with AB • -0.05 40.5. Since this interval containszero, the range of values of 1lAB is quitelarge. However,this smallslopechange  waterinputtotheStraitof Georgia, i.e.,Uf ~ 104 m3/s,  the observedestimate.This valueis quite smallbecausethe  ary,computed valuesfor the barotropic transport Uf and totallayertransport at the seaward endshouldbe similar. Seaward transport in the lowerlayer,U[1 + A(1 - R)], appears relativelyindependent of f andis in therangeof 10 x 104 m3/sfor bothsections. Verticaltransport WX is independent of f (in the approximate formulasused). Roughlyspeaking, entrainment intothesurface layerin the Haro Strait section is about half the entrainment found for  thetheory.UsingU ~ 5 x 104 m3/s Conversely, theestimates of inshore layertransport U and doesnotinvalidate fluxesof 150W/m 2,theexpected valueof AB netflowUf arestrongly dependent onf (although theirsum andsurface isnot).Assuming thatthenetflowisequivalent tothefresh- is about 0.03, certainly consistentwithin the error bars of wehavef somewhere in therangeof 0.1 - 0.3 or ~ 0.2 for  surface area of the northern section is small.  30,708  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS  Strait of  BoundaryPass  Haro Strait  Juande Fuca  Georgia  time intervalbetweenbetweenobservations is only slightly greater than the transit time throughthe region), greater problemsarisewith otherunknowns.Lackingotherknowledge,the freshwaterfractionsf shall be takenas fixed and equivalentto the valuesdeterminedabovefor themean.This fraction,specifiedat theentrance,parameterizes to someextent the mixing processes in the Straitof Georgia.Henceit is plausibleto keepthis constant,at leastoverthe timescale of the field program.Transportestimatesaredirectlydepen-  Figure 13. Summary of transport estimates in Harodent onthedaily mean heat flux, which weshall also assume Strait/Boundary Pass system. Alltransports areinunits of tobeconstant throughout thisperiod. Although thisisa 104m3/s.  weakpointintheanalysis, there issome evidence tosupport  suchan assumption.The daily changesin sea-surface temperaturemeasuredby a surfacemeteorologybuoywhichcan 3.4. Spring/Neap Variations be attributedto diurnal cycling (not shown),were roughly Since the observationsform a time series,we can use the similaroverthe entireperiodof the experiment,suggesting two-layertheoryto investigate changes in mixingoverthe thattheheatcyclealsoremainedroughlyconstant.Weather spring/neap cycle. The datacanbe subdivided intothree wasgenerallygoodthroughoutthefieldprogram. No error estimates will be made since the subdivided data sections:observations takenonJune25-27,representative of theendof theneaporthebeginning of thespringperiod;ob- are, in general,too sparseto supportusefulbootstrapping. servations takenaroundJuly 1-2, representative of thepeak It is presumedthaterrorswill be of the sameorderasthose of thespringtides;andobservations takenaroundJuly4-6, found for the mean valuesabove(i.e., of order50% in transtrendsin theresultshere, representing theendof thespringperiod.Figure10 shows port).Sincewe shallbediscussing thattheretendsto bea generallineartrendin valuesthrough rather than focusingon absolutevalues,a more critical asthespringtides.Exceptions arethesurfaceobservations at sumptionis that biasesin the theoryremainroughlyconG stations,which showa fair degreeof variabilitywithout stant. a well-defined trend. We shall take the T-S characteristics Figure 14 showsresultsfor the completeregion. Tidal at this surfaceinflowto be constant(andequalto themean) elevationsare plottedto showthe spring/neapcycle. Note that total upperlayer transportU(1 + f) is roughlyconoverthe entireperiod. of Usefulapplication of thetheoryis a littlemoreuncertain stant. This providessomeconfidencein the assumption thanforinvestigations of themeanflow.Althoughthequasi- constantf. On the otherhand,the layer transportoff Vicfromsome9 x 104 m3/sto 13 x 104 m3/s. staticapproximation is probablynot strictlyvalid(sincethe toriaincreases  I  a)Tidal elevation  0  .........  -1  -2  ,  10  i  i  i  i  i  i  i  i  i  b) nondimensionalparametersfor completeregion  -1  - 0.8  --  R  - 0.6m 0.4  0.2  0  25-  c) transpodsin completeregion  20-  WX  •.15E  .......................  U(I+A(1-R))  50 24  i  i  i  i  i  i  i  i  i  i  i  i  i  25  26  27  28  29  30  1  2  3  4  5  6  7  June/July1996  Figure 14. (a) Tidal elevations,(b) nondimensional parameters, and(c) dimensionaltransports diagnosed for the Haro Straitregion(usingstationsG andV).  PAWLOWICZ AND FARMER:VERTICALMIXING IN EXCHANGEFLOWS In general,the mean of the valuesconsideredhere will not  30,709  flux,asbefore,is estimated at about1150 W/m 2 (alsocon-  be identicalto the meanvaluesdeterminedin the previous sistentwith theJune/July estimates fromWaldichuck [ 1957] section.This is partly a consequence of the nonlinearityof citedby HerlinveauxandTully). It is unlikelythat we can the formulas,providingan additionalsourceof uncertainty. get agreementto better than the correctorder with this data. The increasein transportis due to an increasein the verSubstituting, the surfacelayertransport tical mixing WX (whosedeterminationis independentof Fa•X  f) from14 x 104 m3/s to 20 x 104 m3/s,rather thanany  changes in the inflowU. The entrainment ratioR is roughly constantthroughoutthisperiod,althoughthe aspectratio A increases.Mixing processes in thisregiondo not appearto actasa barrierto volumetransportin andoutof the Straitof Georgia.This transportappears,at leastin thislimitedsegmentof data,to be roughlyindependent of the spring/neap cycle. However,mixing greatlyaffectsexchangewith the  S(1 + f) •  -(1 + AT/AS)O  • 10x 104m3/s (33)  The cross-sectional area of the upperlayer in the Strait of Juande Fuca is about100 m x 20 km; this impliesa mean  velocityof about5 cm/s.  Again, this cannot be consideredanythingmore than a rough"orderof magnitude"calculation,since(amongother problems)theseestimatesindicateno increasein layertransStrait of Juan de Fuca. portthroughHaro Strait.However,it is pleasinglyconsistent with the detailedanalysisconsideredabove. 3.5. ComparisonsWith Historical Observations A number of attemptshave been made to directly estiThe along-channelchangein the slopeof T-S charac- mate layer transportin this region usingarraysof current teristicswas first notedby Herlinveauxand Tully [1961]. meters. Observations in mid Juan de Fuca Strait obtained From their figure 31, T-S slopesin July are approximately in the summerof 1975 showeda layer transportof about  etal., 1994].Observations atvir-0.15øC/pptin theStraitof Georgia; -0.7s,øC/pptoffVic- 25X104m3/s [Labrecque toria(line5); and-1.2øC/ppt at themouthof theStraitof tually the samelocationin April-June 1973 showeda layer JuandeFuca(line 2). It is notpossibleto estimatelayertem- transport ofabout10 x 104m3/s [Godin etal.,1981]with peratures(or their difference)from their presentation.We a netoutflow of order4-2x 104m3/s.Layertransports in Johnstone Strait at the northern  dividethe flow (roughly)into two sections,eachwith a sur-  end of the Strait of Geor-  faceareaa•X m 2000km2. Thenormalized slopeincrease giawereabout3 x 104mS/s,witha netinflowof about netoutflow, beingtherelatively for eachsectionAT/AS m 1.15.The temperature differ- 1 x 104m3/s. In general, encebetweenupperand lower layerswill be assumedto be  smalldifferencebetweentwo largelayer transports,wasimpossibleto determinereliably. Spatialsamplingwas sparse  similar to that found above, O m -1.15øC, and the surface West  East  50  lOO  150  Haro Strait  200  July-August 1976 currents in cm/s contour interval 10 cm/s 25O  flooddirectionpositive  300  0  I  I  I  I  .1  I  2  3  4  5  ,  I  I  I  I  I  I  6  7  8  9  10  11  distance(km)  Figure 15. Mean residualflow structureacrossHaro Strait in summer1976. Currentmeter locations  (rn)arelabeledwiththemeanflowperpendicular to thesection.Contour linesrepresent themeanfield  determined through a fittingprocedure thattookintoaccount theeffects of (large)vertical excursions by  the moorings(seetext).  30,710  PAWLOWICZAND FARMER:VERTICAL MIXING IN EXCHANGE FLOWS  enoughthat considerable uncertainties existedin the distri- This providessome confidencethat computedvertical enbutionof currentsin spiteof the useof 42 currentmeters. trainmentratesare equallyreasonable. However,the layer transports themselves areprobablyreasonable.Note that the layer transportfor Johnstone Strait, 4. Discussion and Conclusions althoughrelativelysmallcompared withthatfoundfor Juan A quasi-steady kinematictheoryhasbeendeveloped to de Fuca Strait, is of similar magnitudeto the layer flow at  theentrance toBoundary Pass(about 5 x 104 m3/s)found relateflowparameters in two-layer estuarine or exchange in thiswork.  flowsto along-channel variationsin the observational vari-  Therearenodirectobservations of transport between the ablestemperature andsalinity.Thedifference between the StraitofGeorgia andBoundary Pass. Thelowerlayerinflow twoclasses offlowliesintherespective presence orabsence atthispointwillacttorenewthedeepandintermediate wa- ofanetmeanflow.Analytic solutions arefound forthecase tersin theStraitof Georgia.LeBlond et al. [1991]found of exchange flows.Theseforma mapping between a setof thatcurrent records fromthedeeppartoftheStrait,some50 nondimensional ratiosof flowparameters andobservational kmnorthofBoundary Pass, hadapulsed character. Theyin- variables. Although analytic solutions arenotpossible for ferredthatthese pulses werethesignature ofgravity currentsestuarine flows,it is shownthatif thenetflowis known, emanating fromBoundary Passduringneaptides.A mean a simplecorrection factorcanbecomputed suchthatthese inflowrateof about4.6 x 104m3/s wascomputed fora 3 solutions collapse ontothose found fortheexchange flow.  month period inthesummer of1984.Thisisvirtually identi- Previous workofthisnature hasconcentrated onsalinity caltotheestimate madehere.However, thepresent analysischanges alone.However, thepresence ofanadditional tracer expands thepossibilities. Forexample, theefshowslittle evidenceof thespring/neap modulationin trans- (temperature) portthatformedthebasisof thereasoning of LeBlondet al. fectsof summersurfaceheatingresultin a seawardincrease in the slopeof the T-S correlationof verticalprofiles. Upper layer watersbecomewarmer as they flow seaward,and Therearenopublishedestimates of transport withinHaro (depending on the detailsof mixing) lower layer watersbeStrait itself. However,a planar array of 18 currentmeters come warmer astheyflow inshore.Surfacecoolingin winter was moored in Haro Strait in the summerof 1976 [Webster, [1991].  1977]. Mooringlocations areshownin Figure1. Figure alsoincreases theslopebutwithopposite sign.In bothcases 15 showsthe mean currentsnormalto the section. Mean this slopechangeis independentof the amountand details  layertransports wereestimated in severalways:(1), area- of verticalmixingbutis inversely proportional to thelayer weightedaverages werecomputed usinga varietyof rea- transport.Verticalmixingaffectsonlythe horizontaland sonablychosenareas,(2), meancurrents werefittedto ei- verticaldisplacement of T-Scurves.Theincrease in slope ther a planeor parabolicspatialfunction,whichwasthen meansthatsummaryT-Sdiagrams for all watersin anestuspatiallyintegrated, and(3), in orderto takeinto account arywouldhavea slightlycurvedshape,concave downward theeffectof (sometimes large)verticalexcursions thatoc- in summer.Suchfeatures havebeennotedbefore,although curredwhenstrongcurrents dragged themooringdown,raw notexplained, in theestuarine systemoff theeastandsouth currentswere fittedto a spatialfunction,whichwasthen coastsof Vancouver Island.Usingthetheory,relationships averaged in time to find a meancurrentfield andspatially canbe foundbetweenhydrographic observations andflow integrated.Resultswere fairly consistent no matterwhich parameters. It is theconsideration of T-Sslopechanges that technique wasused,showingan outflowin theupperlayer leadsto a relationship betweenlayertransport andnetsur-  of about10 x 104 m3/s, anda lowerlayerinflowof about faceheatflux.Knowledge of thelatter,alongwithotherre14 x 104m3/s. Although thisimplies a netinflowof wa- lationships between salinities, canthenbeusedtodiagnose ter throughHaro Strait,it is likelythattheseestimates suf- layertransport in boththehorizontalandvertical. fer fromthesameproblems asthoseof Godinet al. [1981] The theoryis appliedto a setof observations from the discussed above.In particular,thestrongdeepcurrents are HaroStrait/Boundary Passareaof coastalBritishColumbia. basedmostlyon measurements at a singlemeter. Another Althoughthegeometryof thisregionis greatlycomplicated possibilityis thata netinflowdoesexistin HaroStrait.This by numerousislandsand channels,the theoryappearsto wouldbe balancedby a net outflowin the channelsfur- providereasonable estimates of bothhorizontal andvertical ther east,primarily RosarioStrait. However,it is gener- transports, summarized in Figure 13. Althoughthe mechallythought[e.g.,Thomson, 1981,Figure10.23,Waldichuck,anismsdrivingthe mixing are not part of this theory,the 1957,p. 414] thata netinflowexiststhere.Althoughuncer- resultsobtained canbeusedto differentiate between various taintyexistsaboutthenetflow,layertransport valuescom- possibilities.Two differentregionsof vastlydifferentsur-  parewellwiththe8 - 11 x 104m3/s foundhere.Notethat faceareawerecompared. Verticalmixingtransports were unlikelythatmixing the interfacebetweenthe layersis at the surfaceat the east- similarin bothregions.Thusit appem's is caused by a geographically diffuse process. If that were ern end of this sectionbut reachesdepthsof 140 m at its western end. This is another indication that not much imthe case, then the amountsof mixing would be somewhat portanceshouldbe placedon the layerdepthschosenin the proportionalto surfacearea. Instead,most of the mixing probablyoccursin a few specificregions,probablythe sill analysisabove. Overall, it appearsthat horizontaltransportestimatesde- regionsin BoundaryPassand off Victoria. However,there rived usingthe theorydevelopedhereare quitereasonable. arelimitationsto thiswork. The transportin RosarioStraitis  PAWLOWICZ AND FARMER: VERTICAL MIXING IN EXCHANGE FLOWS  30,711  neglected.Layerdepthswerenot knowndirectly.Although percentiles,matchingthe intervalcontainedin the standard the theoryis not too sensitiveto the exactdepthschosen,it errorof an estimatewith a Gaussiandistribution. still representsa possiblesourceof error.  Changes overthespring/neap cyclewerealsoconsidered. Acknowledgments. Work discussed here waspartially funded  These results have ahigh degree ofuncertainty. However, al- by theU.S. Office ofNaval Research under contract N00014-95-10497duringpostdoctoral workat IOS by R.P.,andpartiallyby the though vertical mixing does increase during spring tides, the National Sciences and Engineering Research Council ofCanada relative change is somewhat smaller thana factorof 2. The under grant OGPO194270. ratio of upwardto downwardmixing appearsto be remain aboutthe same.Inflow from the Straitof Georgiadoesnot appearto vary,remainingroughlyconstant.This may be an indicationthata hydrauliccontrolof somekind is important in governingthedynamicsin thisregion.The changesin vertical mixing, however,causethe transportinto Juande Fuca Strait to change. Griffin and LeBlond [1990] and LeBlond et al. [1994] proposeda model of the exchangein which mixing variedthroughthe spring/neapcycleaccordingto a  References Crawford, W. R., Tidal mixing and nutrient flux in the watersof southwestBritishColumbia,in Tidal Hydrodynamics, editedby B. B. Parker,pp. 855-869, JohnWiley, New York, 1991. Elton, B., and R. J. Tibshirani,An Introductionto the Bootstrap, Monogr.on Stat. andAppl. Probab.,vol. 57, ChapmanandHall, New York, 1993. Farmer, D. M., E. A. D' Asaro, M. V. Trevorrow, and G. T. Dairiki,  Three-dimensionalstructurein a tidal convergencefront, Cont. Shelf Res., 15, 1649-1673, 1995. speed,which was to parameterizethe presenceof hydraulic Foreman, M. G. G., R. A. Walters, R. F. Henry, C. P. Keller, and jumps with enhancedmixing. Their tunedmodel success- A. G. Dolling, A tide model for easternJuande Fuca Strait and fully reproducedobservedvariabilityin surfacesalinity. the southernStrait of Georgia,J. Geophys.Res., 100, 721-740, Froude number based on the cube root-mean  cube current  Finally, the idea that measurements of surfaceheatfluxes canbe combinedwith thoseof watertemperatureandsalinity to form estimatesof horizontaland verticaltransportsis potentially a powerful tool in efforts to understandestuarine and exchangeflows. The theorydevelopedhere allows the studyof geographically complexregionsfor whichdirect measurements are notpossible.It is alsousefulin providing bulk estimatesof vertical mixing, a parameterthat is difficult, if not impossibleto determineusingtraditionalmeans but is crucialin understanding the flow of waterandnutrients in coastal areas.  1995.  Gargett,A. E., A "largeeddy" approachto acousticremotesensing of turbulentkineticenergydissipationrate•, Atmos.Ocean,26, 483-508, 1988.  Gargett,A. E., and J. N. Moum, Mixing efficienciesin turbulent tidal fronts:  Results from direct and indirect  measurements  of  densityflux, J. Phys.Oceanogr.,25, 2583-2608, 1995. Godin, G., J. Candela,and R. de la Paz-Vela, On the feasibilityof detectingnet tranportsin and out of GeorgiaStrait with an array of current meters,Atmos. Ocean, 19, 148-157, 1981.  Griffin, D. A., and P. H. LeBlond, Estuary/oceanexchangecontrolledby spring/neaptidal mixing,EstuarineCoastalShelfSci., 30, 275-297, 1990.  Hansen, D. V., and M. Rattray, Jr., Gravitationalcirculationin straits and estuaries,J. Mar. Res., 23, 104-122, 1965.  Appendix' Error Estimates  Herlinveaux,R. H., and J.P. Tully, Someoceanographic features of Juan de Fuca Strait, J. Fish. Res. Board Can., 18, 1027-1071,  The bootstrapis a computer-based techniquefor assign1961. ing measures of accuracyto statisticalestimates[Efronand Labrecque,A. J. M., R. E. Thomson,M. W. Stacey,andJ. R. Buckley, Residualcurrentsin Juande Fuca Strait,Atmos.Ocean, 32, 7•bshirani, 1993]. Only the availabledata are used, with 375-394, 1994. no extra assumptions aboutprobabilitydistributions.This LeBlond,P. H., D. A. Griffin, andR. E. Thomson,Surfacesalinity techniqueis mostpowerfulwhendealingwith nonlineares-  variationsin the Juande Fuca Strait: testof a predictivemodel, Cont. Shelf Res., 14, 37-56, 1994. relieson resamplingspecifiedpopulation(s)of data. In this LeBlond, P. H., H. Ma, F. Dougherty,and S. Pond, Deep and incase the theoretical framework assumes that characteristics termediatewater replacementin the Strait of Georgia,Atmos. timates derived from data sets of an intermediate  size.  It  at four locationsare known (upper and lower layers at inshoreand seawardpositions),thus there are four populationsof temperature/salinity pairs. The resamplinginvolved in bootstrapreplicationis carriedout by stationwithineach population(temperature andsalinityarenotconsidered to be independent). The usualbootstrapestimateof standarderroris basedon the sum-of-squares formulafor the standarddeviation.However, applicationof the estimationformulasfor the nondimensionalparameterssometimesinvolvesdivisionof numbersvery closeto zero. The resultingsetof resampledestimatescontainvery large values(i.e., form a "heavy-tailed" distribution),resultingin standarderror estimatesthat are also quite large and do not accuratelyreflect the spreadof the bootstrapreplicates.Error estimatesare basedinstead on a percentileinterval of the bootstraphistogram;that is, the error barsspanthe rangebetweenthe 15.87 and 84.13  Ocean, 29, 288-312, 1991.  Shampine,L. F., and M. W. Reichelt, The MATLAB ODE Suite, SIAM J. Sci. Comput.,18, 1-22, 1997. Thomson,R. E., Oceanography of theBritishColumbiacoast,Can. Spec.Publ. Fish.Aquat.Sci., vol. 56, Can. Dept. of Fisheriesand Oceans, Ottawa, Ont., 1981.  Waldichuck,M., Physicaloceanographyof the Strait of Georgia, British Columbia, J. Fish. Res. Board Can., 14, 321-486, 1957.  Webster,I., A physicaloceanographicstudy of Haro Strait: A data summaryand preliminaryanalysis,Contract.Rep. Ser. 773, Inst. of OceanSci., Sidney,B.C., 1977. D. M. Farmer, Institute of Ocean Sciences,P.O. Box 6000, Sidney, B.C., CanadaV8L 4B2. (dmf@ios.bc.ca) R. Pawlowicz,Departmentof Earth andOceanSciences,University of British Columbia,6270 UniversityBlvd., Vancouver,B.C., CanadaV6T 1Z4. (rich@ocgy.ubc.ca)  (ReceivedDecember18, 1997; revisedMay 19, 1998; acceptedJune15, 1998.)  


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