JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 97, NO. C7, PAGES 11,299-11,321, JULY 15, 1992

Laboratory Simulation of ExchangeThrough Fram Strait

KENNETH HUNKINS

Lamont-Dohert• Geological Observator• of Columbia University, Palisades, New York

J. A. WHITEHEAD

Woods Hole Oceanographic Institution, Woods Hole, Massachusetts

Laboratory experiments and theory were conducted to observe the flow patterns and transport in both buoyancy-drivenand wind-driven rotating fluids. In "lock-exchange"experiments, water with one density flows into a second basin after a sliding gate is removed. Water of a second density flows back into the first basin. The size and location of the currents for various values of density difference,rotation rate, and assortedsidewall geometrieswas recorded. Volume flux of the fluid was also measuredand comparedwith a theory for lock-exchangeflow of a rotating fluid. In a separategroup of experimentswith a passiveupper layer, easterly winds (like those in the Arctic Ocean) drive the upper level water into the Arctic Ocean and thereforeoppose the buoyant exchange. Westerly winds would drive the water out of the Arctic Ocean. This indicates that the exchangebetween the Arctic Ocean and the Greenland-Norwegian Sea is likely to be driven by buoyancy rather than by driven by wind. Crude estimates of the volumetric and fresh water exchange rate from the lock-exchange formulas are compared with observed ocean fluxes, and approximate agreement is found.

INTRODUCTION ated with the term strait and cannot be expected to exert the same kind of control on exchange that, for example, The seasin high northern latitudes have a special interest the Strait of Gibraltar exerts on flow between the Atlantic for oceanographybecause they provide a major heat sink for Ocean and the Mediterranean Sea. Ice and Arctic waters the world ocean and are an important component of global flow southward through Fram Strait while Atlantic waters convectionand climate. In recognitionof this important role flow northward. Much of this transfer takes place in two there has been increasing attention in recent years to both narrow boundary currents. The East Greenland Current observationaland modeling programsfor the Arctic Ocean on the western side of the strait exports cold water of low and the Greenland Sea. The severe climate and the pres- salinity from the Arctic Ocean to the Greenland Sea and ence of sea ice, particularly in the Arctic Ocean, severely the West Spitsbergen Current on the eastern side carries restrict field operations, so oceanographicdata have accu- warm saline waters northward. Strong fronts in the strait mulated slowly and are still quite limited. It is necessaryto separate the two principal contrasting water masseswhich extract as much information as possiblefrom the data avail- have their origins, respectively, in high polar latitudes and able and to interpret them with theoretical models designed in the subtropics. Within these seas there is a wide range to test ideas about the physical processes. Simple labora- of mesoscale motion superimposed on the mean flow. In tory models of the Arctic Ocean and Greenland Sea will be Fram Strait, transient propagating eddies with diameters described here which have been developed with the object ranging from several kilometers to 100 km have frequently of clarifying ideas about their circulation and the exchange been observedalong the marginal ice zone both in remotely between them. These models explore idealized situations sensedimages and in hydrographicsurveys [Johannessen et based on relevant physical concepts but are not designedto al., 1987; ManIcy, et al., 1987a; Ginsbergand Fedorov,1989]. realistically mimic currents and hydrography. They are ex- Meanders of both the ice edge and the front along the East pected to contribute toward agreementon the nature of the Greenland Current are commonly seen, and one subice me- important physical processesin these oceans, to stimulate ander was seento developinto a cycloniceddy [ManIcy et more detailed modeling, and to aid in planning future field al., 1987b].Another type of feature is the nearly stationary exploration. eddy which has often been observed over the Molloy Deep in Before describing the model experiments we will review the northern central part of the strait. This cyclonic eddy is briefly the oceanography of the seas on which these stud- about 60 km acrossand seemsto be topographically trapped ies focus. The deep basins of the Arctic Ocean and the [Wadhamsand Squire,1983; Bourke et al., 1987]. Greenland Sea are connected by the gap between Green- Despite its lack of a sill or narrow constriction, Fram land and Spitsbergen known as Fram Strait. This opening Strait does appear to act as a barrier to exchange since is 450 km wide acrossits narrowest section with depths ex- strong contrastsexist between the water massesof the Arc- ceeding 2000 m for a distance of 100 km along that same tic Ocean and Greenland Sea. The Arctic Ocean is covered section(Figure 1). Thus, despiteits name, Fram Strait in its deep regionswith perennial drifting sea ice and an up- lacks the narrow width and shallow depth usually associ- per water layer about 100 m thick with salinity of only 31 to 32 ppt (Figure2). This upperlayer, freshened by river runoff Copyright 1992 by the American GeophysicalUnion. from surrounding continents, forms a cold, low-salinity lens which overlies a transition layer of increasing salinity and Paper number 92JC00735. density between 100 and 300 m. The lid of low-salinity 0148-0227]92 ]92 JC-00735$05.00 water on the Arctic Ocean prevents convective circulation

11,299 11,300 HUNKINS AND WHITEHEAD: SIMULATED EXCHANGETHROUGH FRAM STRAIT

180'

- . CNUKCHI EAST . ß . SEA SIBERIAN ' .. - . SEA

BEAUFORT SEA

LAPTEV

ARCTIC OCEAN

90* EURASIAN BASIN KARA SEA

BAY

BARENTS SEA

GREEN LANi SEA

ICELAND

NORWEGIAN SEA

o

Fig. 1. Bathymetry of the Arctic Ocean and Greenland Sea.

I

I0O0

2000

3000 -05•

4000 -(a) I I • i • I I I I I ! I I I I I I I I I I I I I I ! I I !

I0O0

2OOO

•000

4000

Fig. 2. Potential temperature and salinity sectionsacross the Arctic Ocean and the Greenland and Norwegian Seas(adapted from Aagaard,[1989]. in the upper layers. Weak stability prevails below 300 m cooled in winter to becomedenser than the underlying wa- where temperature and salinity are relatively uniform. In ter and convective overturn has been observed to depths the Greenland Sea, on the other hand, vertical stability is greater than 1000 m and may even penetrate to the bottom weak throughout the water column with pronouncedthin- during particularlysevere winters [GSP Group,1990]. The ning and even vanishing of the surface layer near the center ventilation of the Greenland Sea leads to formation of North of the cyclonicgyre (Figure 3). There surfacewaters are Atlantic Deep Water which contributes significantly to the HUNKINS AND WHITI•.HI•.AD:SIMULATI•.D EXCHANGI•. THROUGH FRAM STRAIT 11,301

24 21 20 õ0 48 47 winds and ice motion, Colony and Thorndike[1984] esti- mated currents as a residual of unexplained variance, while Coachmanand Aagaard [1974] used the densityfield with the assumptionof negligible currents at great depth. These results show that wind maintains the ice circulation, which in turn drives the upper ocean. The mean ice drift pattern may be interrupted in late summer when the polar anticy- clonic wind system weakens and even reverses. Ice drift in ...?.... both the gyre and the Transpolar Drift Stream may reverse 2000- for a period of severalweeks each summer [McLaren et ai., 1987; Serrezeet ai., 1989]. Althoughthere are no data yet on ocean currents during these reversals,theory indicates that . . the baroclinic spindown time for the upper Arctic Ocean is 3000- ß

. too long for any current changeto occurduring theseevents. The cyclonicgyre in the Greenland Sea is alsodriven mainly by winds, since calculations of Sverdrup transport based on POTENTIAL TEMPERATURE •C observed wind data bear a quantitative resemblance to the geostrophicsurface circulation determined from the density 24 21 20 õ0 48 47 field [Aagaard,1970]. The northwardSverdrup transport o I I • • I j -- :-•-- • -- 34.90.,.• must be balanced by a western boundary current which con- stitutes part of the East Greenland Current. The Greenland gyre did not reverseitself even when the wind stresscurl re- versed sign for a period of 5 months. I000• Within Fram Strait, wind is not the principal driving force for currents beneath the ice since it has been found that the S< 34.95 x10 -3 mean geostrophiccurrent, determined as a residual from ice and wind measurements, accounts for 2 to 5 times as 2000 much mean ice motion as local wind [Moritz and Colony, 1988]. Flow in the strait must thus be drivenby densitydif- ferences between the Arctic Ocean and the Greenland Sea, and an explanation for it in these terms was provided by 3000 . ß Wadhamset ai. [1979] using a rotating tank experiment. When a radial barrier, was introduced into a circular zonal front, the rotational constraint against meridional flow was SALINITY(xib -3) ß ' ' broken and a narrow boundary current rapidly transported fresh water toward the rim of the tank in an analog of the Fig. 3. Potential temperature and salinity sections extending East Greenland Current. The laboratory boundary current north-northwestacross the GreenlandSea [Carmack 1986]. closelyresembled the uniform potential vorticity model cal- culatedby Manley et ai. [1987a]on the basisof the section globalmeridional transport of heat. The physicaloceanog- shownin Figure 5. In that model the exponentialprofiles for raphy of theseseas has been reviewedby Carmack[1986, interfacedepth and velocity have e-foldingwidths scaledby 1990]and Aagaard[1989] and the physicaloceanography of the internal radiusof deformation(9.4 km for their data). Fram Strait by Hunkins[1990]. Details of the origin,distri- Using this and appropriate valuesfor other parametersthey bution, and transformation of the water masses in this re- found a volume transport for the East Greenland Current gion havebeen discussed by Coachmanand Aagaard[1974], of 1.1x10• ms s-•, which compareswell with the valueof Aagaardet al. [1985],and Rudeis[1986, 1987]. 0.9x10• ms s-• foundby Rudeis[1987] on thebasis of bud- The circulation of ice and upper layers in the Arctic Ocean get calculations although it is much less than the result of is driven predominately by winds. Surface circulation in the 3x10 • m s s-• arrived at from moored current measurements Arctic Ocean is anticyclonic, while in the Greenland Sea it by Foidviket ai. [1988]. On the easternside of the strait is cyclonic(Figure 4). The motion of sea ice is knownes- the ice-free West SpitsbergenCurrent carries warmer, more pecially well, since it has been tracked in recent years with salinewater north into the Arctic Oceanalong the surfaceto the aid of automatic data buoys and manned ice stations a latitude of about 81øN, where it descendsto becomea sub- [Colonyand Thorndike,1984; Colonyand Rigor, 1991]. In surface current. Mean surface winds and hence ice stresses the Beaufort Sea, that part of the Arctic Ocean between on the water are directed toward the southwest across the Alaska and the north pole, mean ice motion describes a entire strait, so the West SpitsbergenCurrent flowsagainst clockwisegyre, while in the regionbetween Eurasia and the the wind and thus must also be driven by buoyancyforces. north pole the Transpolar Drift Stream is directed toward Note that wind stresses at coastal stations are not a reli- Fram Strait and then continues into the Greenland Sea with able indicator of stresses over the strait. Recent data on the East Greenland Current. More than 70% of the vari- winds in the Fram Strait region are summarized by Jonsson ance in ice motion within the Arctic Ocean is explained by [1989], and pressuredata are summarizedby Colony and geostrophicwinds. To determine this fact, surface currents Rigor [1991). Theseobservations and ideasagree with the in the Arctic Ocean below the ice have been observedby two concept of the Arctic Ocean as an estuary of the Atlantic independent methods, both of which show a clockwisegyre Ocean with saline inflow at depth and fresh outflow at the similar to that of the ice. Using observationsof geostrophic surface. Since Fram Strait is very wide, rotational effects 11,302 HUNKINSAND WHITEHEAD: SIMULATED EXCHANGE THROUGH FRAM STRAIT

180 ø

ß

ß ß

ß

90 ø

o

Fig. 4. Schematicsurface circulation of the Arctic Oceanand GreenlandSea, Redrawn from Tchernia [1980]. are important and the front betweenthe two water masses dict ice coverageusing observed mean monthly atmospheric is inclined across the strait. This is most evident in the East forcingand a prescribedinflow through the Faeroe-Shetland Greenland Current where the front slopesupward from west Channel. This model evolvesinto a two-layer exchangeafter to east. In the sectionshown in Figure 5 the interfaceas de- a 20-year integration, but publishedresults focus more on fined by the 0øC isotherm slopesupward from 180 m on ice coveragethan on currents.This modelwas alsoused to the westernedge of the sectionto intersectthe surfaceat a studyinterannual ocean forcing of the ice pack[Fleming and location 80 km to the east. Semtner,1991]. Another numerical model with extensiveice A number of analytical and numericalmodels have been dynamicsis that of Hibler and Bryan [1987],which differs presentedto explainvarious aspects of the circulationin the from the previousone in beingdiagnostically constrained to Arctic Ocean, the Greenland Sea and Fram Strait. A two- remain closeto observedvalues of temperature and salinity layergeostrophic model of Arctic Oceancirculation based on on a 3-year time scale. A third three-dimensionalnumerical estuarineresearch has been developed by Sti!lebrandt[1981] model of the Arctic Ocean and Greenland Sea which repro- in which ice and an upper layer freshenedby runoff and ducesmany of the observedfeatures is that of Piacsek ct precipitationflow out while a deepAtlantic layer of higher al. [1991].A simplernumerical model has been developed salinity flowsin through Fram Strait. Entrainmentof the by Woodand Mysak[1989] which covers only the Greenland lowerlayer into the upper producesa balancedexchange. A Sea. In their model, Sverdruptransport toward the north modified version of this model has been compared by RudeIs in the central and eastern ice-free areas is matched on the [1989]with an alternativemodel of hisin whichthe mixing westernside with a southward coastal boundary current, the of the lowerlayer into the upperis accomplishedby •helf East Greenland Current, which is basedon an analytical so- processesrather than by entrainment. He finds that shelf lution given by Gill [1982]to the problemof the flow in a processesappear to dominateat the presenttime. Con- channel after removal of a dam. siderable effort has been devoted to numerical modeling of In this paperwe approachthe questionof the flowthrough polar and subpolarseas and approacheshave involved one-, Fram Strait by observingflows in laboratorymodels and dis- two-, and three-dimensionalconfigurations with various cussingtheoretical implications of theseflows. Experiments formulations for ice behavior. A numerical model of air- in buoyancy-drivenexchange (section 2) producea qualita- sea-ice circulation which includes ice dynamics as well as tive picture that resemblesthe flow in Fram Strait. These time dependencehas been used by Semtner[1987] to pre- suggestiveobservations do not answerthe questionas to HUNKINSAND WHITEHEAD: SIMULATED EXCHANGE THP•OUGH FP•AM STP•AIT 11,303

MX-I MX-2 MZ-6 190 189 187 186 194 4 5 6 7 8

----I.5

TEMPERATURE (*C)

'""'"•4.90

SALINITY (ppl)

2OO ß

/ I •28 / " \

SIGMA-!

20i 410 60i 80I I00I 120I 140i 160i 180i 200i 220i 240 KILOMETERS

Fig.5. Temperature,salinity and density ((rt) sections across Fraxn Strait at 79ønorthextending from 60øW to 40øE[Manlc• ctal., 1987a]. why Fram Strait happensto be the location wheredifferent location of the currents for various values of density differ- water massesare separated. Wind-driven experimentsand ence,rotation rate, and assortedsidewall geometries with theory(section 3) showthat wind(f- planedynamics) will varyingdegrees of resemblanceto FramStrait. The hopeis drive fresh water into the Arctic Ocean rather than allow it that comparisonof the flow patternswith the flowsknown to leave.A quantitativelock-exchange theory is producedin to exist in Fram Strait will enable one to more clearly under- section4 that predictsthe magnitudeof flux as a function standthe dynamicalorigin of someof the currentsin Fram of depth and densityof the low-salinitylayer in the Arc- Strait. tic Ocean. This theory is tested by laboratory experiments In all experiments,there was a layer of deep salty wa- in section5, and implicationsto the freshwaterbalance of ter with the dynamicallyactive shallow layers above them. the Arctic Ocean are calculated in section 6. Concluding This deep layer representsthe deeperwater in the Fram remarks are made in section 7. Strait regions,below approximately 300 m depth.The shal- low layersrepresent surface waters in the Arctic Oceanfor BUOYANCY-DRIVEN EXCHANGE the first basin and the Norwegian-GreenlandSea for the Laboratoryexperiments were conductedto observethe other basin. Theselayers serve as the sourcewaters for the flowpatterns in density-drivenrotating "lock-exchange"ex- East Greenland Current and the West SpitsbergenCurrent, periments,where water with onedensity flows from its orig- respectively.In orderto makethe situationas simpleas inal basinthrough an openinginto a secondbasin after a possible,no attemptis madeto incorporatethe important slidinggate separating the twofluids is removed.Likewise, deepercurrents in Fram Strait. Thesedo haveimportant the water of differentdensity in the secondbasin flows back oceanicvolume and salinity fluxesbut they are slowerand into the first basin. The intent was to observe the size and associatedwith smaller pressuresthan the surfacecurrents. 11,304 HUNKINSAND WHITEHEAD:SIMULATED EXCHANGE THROUGH FRAM STRAIT

Four experimental configurations were employed as sulas were constructed so that the cylinder had two basins sketched in Figure 6. The first two shared the same con- joined by a strait 40.5 cm wide. The configuration was tainer geometry. A 1.06-m-diameter cylindrical container intended to. geometrically model the Arctic Ocean, Fram was mounted on a rotating turntable, and two false penin- Strait, and the Greenland-Norwegian Sea. In the first con- figuration, there was a layer of fresh water in one basin and TOP VIEW, (1 & 2)) none in the other. In the second there were surface lay- ers in each basin of differing density. The third and fourth configurationsemployed tanks of different geometries. In the third, a rectangular tank 183 cm long, 19.2 cm wide, and 24.8 cm deep was used so that some of the possibleef- fects from curvature of the sidewalls could be eliminated. 108 The sliding gate was inserted perpendicular to and midway along the long axis of the tank. This resulted in a strait as wide as the two equally sized upstream basins. In the fourth configuration,a 2-m basin was divided along a central cord by a sliding gate that was removedto start the experiment. This last configuration was expresslyintended to produce SIDE VIEWS the largest possiblehydrostatic number in order to resolve (1) (2) whether eddies and slow velocities observed in the earlier

, runs were produced by a small hydrostatic number. Scalingconsiderations produce seven dimensionless num- bers- There are ten independentvariables in the experi- ments(except for the first that had nine): gravity g = 980 cm s-•' depthof the shallowlayers h• 5 cm and of the deep layer h•. = 20 cm (in a few caseswe had to resortto 18 cm becauseof geometricalconstraints), Cori- olis parameter f = 4•r/T where T is period of rotation of TOP VIEW, 3 thetank, (f wasvaried from 0.2 s -• to 1.7s-•), kinematic viscosityof the waterv = 0.01 cm2 s-•, densityof the two t surfacelayers p• = 0.998 gcm-3 andp• = 1.000gcm -3 24.8 (notpresent in the firstconfiguration) and of the deeplayer ps = 1.002gcm -• the widthof the gateL• and finally the width of the basins L2. Since there are three dimen- Mirror (for side view) sionalunits of mass,length, and time, seven(six for the first) dimensionlessnumbers result. Eachof thesewill now 19.2 be discussed. Dimensionless number 1' Depth of the bottomfluid. This is given by the ratio of shallowlayer to deep depth h•/h2 I_ J which is 0.25. This being less than 1 implies that dynam- ics of the bottom fluid should be less important than the dynamics of the surface layers, but it is not clear that the TOP VIEW, (4) deep layer can be completely ignored. Some experiments TOP V IEW,(4) will show that the surface jets possessthe dynamics that jets are believed to have when they lie over infinitely deep fluid. Dimensionless number œ: Friction. The Ekman number of the bottomfluid. E = v/fh• is 10-4 sothe usualEkman layers will be found on the bottom. We note that Ekman number of the shallow layers is going to be expressedas the above Ekman number divided by the square of the depth ratio. This dimensionlessnumber is 16 times larger than the Ekman number of the deep fluid so there should be slight viscous coupling between the layers. Unfortunately, 2OO the physicsof that coupling is poorly understood. Spin-up time estimates will be given later to indicate that viscous forces are smaller than inertial forces in the experiments. Dimensionless number 3: The Boussinesq approxima- tion. The density difference between the fluids is fixed at 0.002gcm -s and0.004 gcm -s for all experimentsand is small compared to the absolute values of the density. There- fore, the Boussinesqapproximation should be valid. Dimensionless numbor J: Width of the gate. The ratio of Fig. 6. Sketch and dimensionsof the four experimental appara- tuses. All photographs are oriented so the equivalent of the Arctic the width of the gate to the Rossby radius of deformation Ocean is at the top. based upon density difference between the two top fluids. HUNKINS AND WHITEHEAD: SIMULATEDEXCHANGE THROUGH FRAM STRAIT 11,305

Thisis writtenas W = f L•/[gh•p•- p2)/p3]•/2 andis in- density current with a "nose" similar to that observed by versely proportioned to the square root of the Burger num- Stern, et al. [1982],and Griffithsand Hopfinger[1983] was ber. It may be as large as 20 for Fram Strait and is varied observedto emergefrom the front and move along the right in the experiments. hand wall away from the layer of fresh water. The sequence Dimensionless number 5: Steepnesso• the •ront between of photographsin Figure 7 showsthe front between the two the two upper fluids. This is not necessaryfor the first con- waters after 22 s. The density current extending into the figuration. It is given by the ratio of the Rossby radius clear basin is the principal mechanism for transporting fresh based upon density difference between the two top fluids to water far into the second basin. the Rossbyradius based upon density difference between the Back at the gate region, something else of interest was freshand bottom fluids. In all experimentsthis is 2•/2. happeningat the other end of the front, where it intersected Dimensionless number 6: Width o/ the Rossby radius the left-hand wall. Fresh water was fed into the front by a basedupon density difference betweenthe two top fluids com- current in the upstream basin that flowed along the left-hand pared with the depth o[ the upper layer. We call this a hy- wall. At the point where this water fed into the front, there drostaticnumber H•, - [g(p•- To derive was some inertial "overshoot" of the fresh water similar to shallow water equations, this must be large, and it is almost that calculatedby Hermann et al. [1989]. The overshoot always large in the ocean. H• is varied in the experiment. produced a meander that gradually caused the left-hand In experiments 1 through 3, H• spans the range 4 to 0.5. point where the front touched the wall to move downstream, Experimental configuration4 was constructedto make both and in fact the entire meander migrated slowly into the sec- H• and W large together. ond basin. As time progressed,the meander became larger, Dimensionless number 7: Ratio betweengate width and a and by the time the nose had circled the downstream basin, dimensionoI the basin,L•/L2. This numberis varied in the the meander was quite sizeable. The migration of this me- experiments. In order to attain a flow that is representative ander was slowerthan the propagation of the nose along the of a steady flow, we try to make this as large as possible. other wall, but the volume of the fluid within the meander Dimensionless numbers 1, 3, and 5 are fixed for all the was larger. Therefore although the density current on the experiments described in the next section. The others will right-hand wall is the principal mechanismfor moving fluid be given where appropriate. far into the secondbasin, the meander can move greater vol- umes of fluid small distances into the second basin. The me- Qualitative Results ander processcan potentially confusemeasurements within the strait that attempt to measure volume flux from the In the first configuration, 20 cm of salt water was added fresh water basin into the other basin. This process was to the tank and spun up to a counterclockwiserotation with observedfor all three valuesof W and H•. a periodof 15 s (f = 0.84s-•). A watertightsliding gate The first configurationdemonstrated the existenceof the was then inserted acrossthe strait down to a depth of 8 cm, bore and the development of a meander at the mouth that is and 5 cm of fresh water was slowly added to the top of associatedwith overshoot of the fluid flowing along the left- one basin, resulting in a water depth below the layer ha of hand wall. Both processeshave been studied previously. 17.5 cm. The run commenced when the gate was raised. This also provides a comparison with the second configu- Three runs were conducted with increasing f and therefore ration, which was felt to be a closer approximation to the increasingW and decreasingH•. situation in Fram Strait. In this configuration, 5.2-cm-deep The front adjusted gravitationally in approximately the layers of water with differing densitieswere placed in both first rotation period to develop into a rotational density cur- basins. The lightest water was fresh water and the interme- rent. As is shown in Figure 7, the fresh water began to flow diate water had a density midway between the fresh water toward the other basin by turning to the right looking from and the deep water (whosedepth was again 20 cm). Af- the fresh water basin toward the other one. It flowed along ter the tank was spun up, the gate was removed. A front the front until it hit the right hand wall. At that point, a formed within one rotation period as it had in the first run, but significant qualitative differencesfrom the flow in the first configuration were soon apparent: The greatestdifference (Figure 8) was that two density currents moved in opposite directions from the points where the front intersected the sidewalls. Each density current was found in its respectiveright-hand wall looking from its basin to the other one. The freshwater current flowed out over the top of the intermediate water. The nose appeared to be similar to the laminar nosesin the first configuration. The intermediate water current had a nosethat wedgedbetween the fresh water and the deep water. The noseof this density current looked more laminar and smaller than the nose of the one layer problem. Second, there was a region of pronounced overshoot of both of the currents coming out of the basins on their re- spective left-hand walls. That region was bigger than previ- Fig. 7. Top view of dye spreadingin the first configurationwhere ously seen and essentially resembled overshoot observed in there is a layer of fresh water dyed black 5 cm deep in one basin over 17.2-cm-deep salt water. There is 22.5-cm-deep clear salt numericalexperiments [Hermann et al., 1989]. Here the re- water in the second basin . Time of the photograph is 22 s after gion of overshoot was clearly found to be more unstable than removal of the gate. in the numerical experiments. The overshoot sometimes de- 11,306 HUNKINSAND WHITEHEAD: SIMULATED EXCHANGE THROUGH FRAM STRAIT

Fig.8. Topviews of dyespreading in thesecond configuration after approximately 60 s. Photographsshow dyed freshwaterexperiments on the left and dyed saltwater experiments on the right. Values of W foreach pair are 5.3, 10, and 22 from top to bottom. veloped a dipole eddy structure shown in the photographs are shown in Figures 10 and 11. As before, the wall jets in Figure 8. The dipoleindicates that there is considerably appearedto be approximatelythe sameshape and strength more dynamic (baroclinic)coupling between the two layers as in case 2. The eddies in the front between the two lateral than there was between the one layer and the deep fluid in walls have much more spaceto develop,but they develop run 1. Finally, more eddies were produced along the front much less vigorouslythan in the precedingcases. This can and these seemed to also have a dipole character. be attributed to a jet that is apparently less throttled by For the third configuration, parameters were similar to a narrow opening. The overshoot on the left-hand walls the correspondingruns in the second,so a closecomparison is about the same as in the third case. There is a clear of the effects of curving versusstraight sidewallscould be impressionthat the surfacewall jet speedstill scaleswith made. Photographsat roughly the same times as Figure 8 the gravitywave speed but the internalwall jet propagated are shown in Figure 9. The wall jets appeared to be ap- more slowly. proximately the same shape and strength as in case 2. The Quantitative Results overshoot on the left-hand walls is much less, but the eddies are more pronounced,especially those coming off the den- Velocityof the nosesof the intruding currentswas esti- sity current that movesout of the freshwater basin. There matedfrom the experimentalresults by measuringthe dis- is a clear impressionthat the curvature of the walls is very tance traversedduring one time interval. In the secondcon-- important to both overshoot and instability processesfor figurationphotographs were not takenfrequently enough to these currents. determinein detail whetherthe noseswere changing their The purposeof the fourthconfiguration was to observe speed during the experiment, so efforts were made to de- experimentswith the same Hy as before but to have larger termine the velocity as early in the run as possible. The

W than.. before. For this configuration, the rotation rate and velocitymeasurements should be regardedas havingrela- density difference were similar to the correspondingruns in tively large errors, of the order of tens of percentor more. the second and third, so a close comparison could be made. In the third configuration,the trajectories of the nosesare Photographsat roughly the same times as Figures 8 and 9 shownin Figure 12 and 13 and velocitieswere obtained from }IUNKINS AND WHITEHEAD: SIMULATEDEXCHANGE THROUGH FRAM STRAIT I 1,307

Fig. 9. Top views of dye spreading in the third configuration after approximately 60 s. Staxting from the left the photographs show successivelydyed freshwater and then dyed saltwater experiments that have the same value of W. Pairs of photographs have values of W of 1.2, 2.5, 5, and 10 from left to right.

the calculated slope of the rms best fit lines shown. All r = d/(•,f)'/2ß If wetake d = 5 cm,•, = 0.01cm 2 s2, , and velocitiesare given in Table I along with rotation periods f = 1.7 (the largestvalue), then r = 38 s. But the top layer T, strait and basin widths œ• and /,2, and Rossby radius could probably not spin down this rapidly. There is no rigid based on the density difference between the two upper flu- boundary at the top of the top layer, and force would have to idsR = [g (pa- p2)ha/p3]a12/f. Averages of thevelocity be transmitted to the bottom fluid. Although the transmis- data were taken down the columns as well as across the sion of drag through density interfaces is poorly understood, rows. They reveal that velocity decreasedsystematically it is believed that spin down time between the top fluid and with increasingrotation rate and that the freshwater nose the bottom fluid, i.e., between two fluids of different salinity, went faster than the intermediate water noseß Dimensionless wouldbe muchlarger than this value. It is known[McDon- numbers are shown in Table 2 where those velocities were ald and Dicke, 1967]that stratificationgreatly inhibits spin- dividedby wavespeed (g'h,) •12, where down. Also, the spindown time of the bottom layer would We note that as rotation rate is increased, the Rossby ra- be 4 times this value. In addition velocity showed little de- dius is systematicallyreduced, the Ekman numberis alsode- crease with time in Figures 12 and 13. Sidewall friction is creased,the hydrostatic number Hy decreasesfrom greater not well understood, and its role in the density currents is than to less than 1, and the ratio W of width of the tank to less certain. Its effect was ignored in previous studies by Rossbyradius becomesprogressively greaterß From the ex- Stern et al. [19821and Gri.(fithsand Hopfinger[19831. perimental measurementin Table 1 of a systematicdecrease In spite of the small friction, the noses were observed to in nosevelocity alone it is not clear whether the decreasein slow down significantly with increasing rotation, and there velocity and the tendencyfor more eddiesis determined by were also more eddies observed. We wondered whether the a decreasein E, a decreasein Hy, or an increasein W. slowing down of the nose speed and the increased eddy ac- The followingconsiderations indicate that Ekman friction tivity is determined by a decreasein H• or by an increasein is probably negligiblefor these experiments. The Ekman W. The fourth experiment had different relative values of spindowntime for fluid next to a rigid boundary is equal to W and H• comparedwith the third, so one could get some 11,308 HUNKINSAND WHITEHEAD: SIMULATED EXCHANGE THROUGH FRAM STRAIT

Fig. 10. Top view of dye spreadingin the fourth configurationfor W = 10. (Left), Freshwaterdyed, t = (top) 45 s, and (bottom) 121 s. (Right), Salt water dyed, t =(top) 92 s, and (bottom) 210s. indication of the answer by comparing the results. Unfortu- Rhines-Johnsonovershoot [Hermann et al., 1989]on the left nately the comparison was not conclusive. For 60 s the inter- side of the strait looking downstream was widely observed, mediate nose had significantly slower nose speed for larger but it did not apparently lead to a large transport far from W (in configuration4 as comparedto configuration3), but the opening. The density current on the right wall remains the same was not true for 30 s. Conversely,for 30 s the top as the location of greatest transport far from the opening. nose had faster speed with larger W. Eddy activity seemed to be less with larger W, but this result is subjective. WIND-DRIVEN EXCHANGE In summary, the density-driven flows can get set up as a steady feature in spite of the transient nature of the ex- The experiments on buoyancy-drivenexchange previously periment. The nose velocities are very steady, especially described were motivated by the observation that circula- when H• and W are large. Surface nose velocity scalesas tion of the upper layers within Fram Strait is evidently not (g•h)•/2 for W andHy large. Intermediatenose velocity forced by wind and ice stress. However since large-scalecir- exhibitsa lessdistinct correlation with (g*h)•/•. In some culation of the upper layers in both the Arctic Ocean and casesit exhibited a significantly smaller velocity than the the GreenlandSea is observedto be strongly(even primar- top nose; this may be the result of some dead water effect. ily) wind-driven it is possiblethat exchangethrough the Unfortunately scatter for the intermediate nose velocity was strait may be forced by pressure gradients arising from re- very large, and the resolution of the dependenceof it on the mote winds. Experiments were run to study the motion of a external parameters is beyond the scope of this study. No surface layer in a divided basin with wind forcing. The tank stationary front is found at the mouth. Eddies are plentiful used in these runs had an inside diameter of 45 cm and a (baroclinicinstability?) on the basisof theseobservations height of 30 cm and was divided along a diameter into two the front would be expected to twist, undulate, and shed semi circular basins by a wall with a gap of 10 cm in the eddies perhaps forever. Significant eddies are produced for middle (Figure 14). A salinelayer 18 cm deepwas first in- small H• but less so for larger Hy and W. The Hermann- troduced into the tank and spun up to solid rotation. Next, HUNKINS AND WHITEHEAD: SIMULATED EXCHANGE THROUGH FRAM STRAIT 11,309

Fig.11. Topview of dyespreading in thefourth config•rration for W = 20. (Left),Freshwater dyed is shownat a timeof (top)46 s and(bottom) 157 s. (Right),Salt water dyed is shownat a timeof (top)60 s and(bottom) 240 s. a freshwaterlayer wascarefully floated onto the surfaceof Sea,on the otherhand, the windstress curl is positive[Jon- both basins. No gate was used,and there was no initial sson,1991], sincethis sea is locatedin the shearbetween pressuredifference between the two basins.In the absence the polar easterliesand the zonal westerlies.Wind stress of wind there would have been no exchange. The tank was curl vanishedin the model at the gap, thus simulatingFram rotated at a rate of 0.25 revolutionsper secondfor all runs. Strait, wherethe curl of the mean wind stressis nearly zero. The upperlayer was1.2 cm deep, muchless than the lower Dimensionalparameters for this experimentare presentedin layerdepth, so theseruns corresponded to reducedgravity Table 3 wherethey are comparedwith correspondingvalues dynamicson an f-plane. for the Arctic Ocean and Greenland Sea. Dimensionless pa- A small turbine blower mounted on the tank directed an rameters derived from these values are shown in Table 4. As air streamparallel to the wall in a well-definedjet with a in the earlierexperiments there is fairly goodcorrespondence width of about 10 cm. The axis of the jet correspondedwith betweenthe oceanand laboratoryvalues for Rossbyand Ek- the wall diameter,so wind shearwas of oppositesign in the man numbers. The laboratory value for Burger number S is two basins.Using an idealizedglobal surface wind system largerthan the oceanicvalue but still lessthan 1 asdesired. as a conceptualmodel, this setup bearssome resemblance The hydrostaticnumber H is only slightlylarger than 1 in to the stresson the Arctic Oceanand GreenlandSea (Fig- the model,and thereforehydrostatic conditions may not al- ure 15) althoughthe actualcenter of the polargyre is dis- waysprevail. The verticalgeometric ratios are similarfor placedfrom the northpole to the centerof the Arctic Ocean ocean and model, but the horizontal ratio is much larger [Uolonyand Rigor,1991]. It is throughthe ice- packthat in the model than it is in the ocean. A much larger tank wind stress is transmitted to the Arctic Ocean, and as we than was practically possiblewould have been necessaryto noted in the introduction,pack ice describesan anticyclonic obtain better agreementfor this ratio. gyrein this ocean.Mean ice vorticityin the Arctic Ocean, Soon after the blower was started the dyed upper layer and hence ice stress curl exerted on the water, is negative beganto flow throughthe gap from the "subpolar"to the (R. Colony,private communication, 1990). In theGreenland "polar"basin, where after a few rotationsa circularanti- 11,310 HUNKINSAND WHITEHEAD:SIMULATED EXCHANGE THROUGH FRAM STRAIT

160 160 140t D=90.17 +0.30t R=0.95 150 D= 76.47 +0.76t R = 0.98 e// E E 140

120 130

120

1 oo 110

80 ' I ' I ' I ' I ' 100 0 20 40 60 80 100 20 40 60 80 100 time, s (a) time, s (b)

140 160 D = 79.77 + 1.44t R = 1.00

130 D = 71.47 + 0.96t R = 0.99 140

E 120

110 120

lOO lOO D = 82.70 + 1.73t R = 0.99

9o D = 88.48 + 0.80t R = 0.93

80 0 10 20 30 40 50 60 70 o 10 20 30 40 50 60 time, s (c) time, s (d)

Fig. 12. Trajectoriesof the nosesfor theconfiguration 3 runs. Periods of rotationare, (•) 7.5, (6) 30, (e) 15 and (d) 60 s. The squaresare for the freshwaternose and the diamondsare for the intermediatewater nose.

300 300 / D = 14.48 + 2.13t R -- 0.98 E 200 200

D= -8.19+1.83t R=1.00

ß

100 100

D = - 13.37 + 0.92t R = 0.96

0 ß , • , 0 100 200 300 0 100 200 300 400 time, s (a) time, s (b)

Fig. 13. Trajectoriesof the nosesfor the configuration4 runs. Periodsof rotationare, (•) 30 and (b) 60 s. The squaresare for the freshwaternose and the diamondsare for the intermediatewater nose. HUNKINS AND WHITEIIEAD: SIMULATED ]•XCHAN•E THROUGH FRAM STRAIT ! !,3 ! !

TABLE 1. Velocity Measurementsof Buoyancy-DrivenExperiments

Config. T R L, La V! Vi VA• --1 --1 --1 s cm cm cm cm s cm s cm s

15 5.0 40.5 107 2.3 none

30 7.8 40.5 107 1.6 1.1 1.35

15 4.1 40.5 107 1.2 0.9 1.05

7.5 1.9 40.5 107 0.8 0.3 0.55

60 16 19.2 91.5 1.44 1.73 1.59

30 7.8 19.2 91.5 0.96 0.80 0.89

15 4.1 19.2 91.5 0.75 0.76 0.76

3 7.5 1.9 19.2 91.5 0.67 0.30 0.49

60 16 200 100 1.83 0.39 1.14

30 7.8 200 100 2.13 0.92 1.52

Average 1.27 0.80

TABLE 2. DimensionlessNumbers for the Buoyancy-Driven Experiments

Config.

3x10 -5 0.1 1 0.6 none 0.38

6x10 -5 5.3 1.6 0.49 0.34 0.38

3x10 -5 10 0.8 0.37 0.27 0.38

1.5)<10--5 22 0.4 0.25 0.09 0.38

1.2)<10 -4 1.2 3.2 0.44 0.53 0.21

6)<10 -5 2.5 1.6 0.30 0.24 0.21

3)<10 -5 5.0 0.8 0.23 0.25 0.21

1.5xlO -5 10 0.5 0.20 0.10 0.21

1.2)<10-4 10 3.2 0.57 0.12 2.0

6)<10 -5 20 1.6 0.65 0.28 2.0

Average 0.39 0.24 11,312 ItUNKINSAND WHITEHEAD: SIMULATED EXCHANGE THROUGH FRAM STRAIT

BLOWER an external frame of reference. Rotation in the %ubpolar• basin is cyclonic with an angular velocity nearly the same as that of the tank and remains nearly motionless relative to the tank frame of reference. TOP Several features of these experiments can be analyzed and VIEW explained by simple vorticity dynamics. Since the lower layer in the model was much deeper than the upper layer, a single-layer model was used in which gravity is reduced by the ratio of the density difference to the mean density. Depth of the Ekman layers was only about 1 mm, much less than upper layer depth, so the layer is taken to be inviscid 45crn with forcing by Ekman pumping on the top and dissipation through Ekman suction at the bottom. Friction on sidewalls will be ignoredhere, sincetheir area wassmall relative to the area of the top and bottom surfaces. The Coriolis parameter is taken to be constant, since it can be demonstrated that in a rotating two-layer system,the depth of the upper layer in the basic state is constant and there is no vertical vor- tex stretching in the radial direction to simulate variation of the Coriolis parameter with latitude. The vertical vorticity equation derived from the inviscid shallow- water equations SIDE [ Gill, 1982]is VIEW dt -t-(f -t-co) -t- - 0 (1) 50cm where the Coriolis parameter is given by f = 2f• in a tank with angularvelocity f•, the relative vorticityis Ou/Oy, the total derivativeis d/dr = and the other symbolshave their usual meanings. Conservation of mass, which requires that changesin height of a water column be equated with the divergence of flow, which is composedof a horizontal flux through the sidesof the column and a vertical flux through the top and Fig. 14. Sketchof rotating tank used for wind-drivenruns. bottom induced by Ekman pumping, is given by dt - of co (2) POLAR EASTERLIES d(h+r/)_I curlr+ 60 ø where h is the depth of the interface below the undisturbed ZONAL water surface, r/ is the elevation of the surface above the same reference level, v is kinematic viscosity, and curl r -- 30 ø [0%/0z - 0r•/0y] with r• and % the componentsof surface WESTERLIES wind stress. Elimination of horizontal divergencebetween TRADE (1) and (2) yieldsan equationfor potentiaJvorticity with forcing and damping terms, O dt +•1 dt •curlr- co=0 (3) Since the Rossby number is small in these experiments, Fig. 15. Idealized surfacewind circulationon a longitudinally nearly geostrophicand hydrostatic balanceis assumed.Hy- uniform Earth. drostatic balance requires the surface elevation r/ = •h, is a small number in these experiments aJlowingr/to be ne- glectedin (3). The densitydefect is definedas • = 2 [pl - cyclonicgyre beganto develop(Figure 16). After a period p2/pi + p2]. The velocitiesfor approximategeostrophic bal- of about 30 s almost all of the freshwater layer had been ance are transferred to the "polar" basin, leaving only a thin layer around the periphery of the "subpolar" basin and exposing the deeplayer at the surfacein the centralarea (Figures17 and 18). The fresh water is trapped in a circularlens in the "polar"basin which is shownclearly in sideprofile (Fig- - + -- ure 19). The resultsare sketchedin Figure20. This polar . gyrerotates rapidly in a clockwisedirection with an angular This allowsco to be eliminatedfrom (•), yieldinga quasi- velocitywhich is nearly the sameas that of the tank but of geostrophicvorticity equation with layer thicknessh as the oppositesign and thus remainsnearly stationaryrelative to dependent variable HUNKINSAND WHITEHEAD:SIMULATED EXCHANGE THROUGH FRAM STRAIT 11,313

TABLE 3. Typical DimensionalParameters for Wind-Driven Experiments

Parameter Notation Ocean Laboratory

Depth, upper layer hi 100 m 1.2 cm

Depth, lower layer h2 2000 m 17.5 cm

Gap width L• 4.5x105 m 10.2 cm

Basin width L2 10? m 43.8 cm

Density,upper pl 1026kg m-s 998 kg m-s

Density,lower p2 1028kg m-s 1025.3kg m-s

Coriolisparameter f 1.4x10-4 s-• 3.142s -•

Accelerationd.ue to g 9.83 m s-2 9.80 m s-2 Gravity

Viscosity,vertical v• 10-2 m2 s-• 10-6 m2 s-•

Viscosity, vh 10-4 m2 s-• 10-6 m2 s-• horizontal

Current velocity U 0.25 m s-• 10-2 m s-•

Densitydefect /• = 2[p•- p2/p•+ p2] 2x10-s 25x10-s Internalwave speed c = (lghi)l12 1 m s-1 0.054m s-1

Internal radiusof R = c/f 7.1 km 1.7 cm deformation

TABLE 4. Dimensionless Parameters for Wind-Driven Experiments

Parameter Notation Ocean Laboratory

Rossbynumber R0 = U/(fϥ) 4x10-s 3.1x10-2

Burgernumber S = R/L 1.6x10-2 0.17

Hydrostatic number H• = R/hi 71 1.4

Ekmannumber E = •/(fh22) 1.8x10-5 1.0x10-s

Depth ratio hi/h2 0.05 0.068

Lengthratio L•/L2 4.5x 10-2 0.23

Here the time-dependent terms have been linearized and we f 0t • + • curlr have assumed• • f to be consistentwith geostrophy. •g0 V2h_• [Oh 1 This quasi-geostrophicequation will be applicableonly dur- (4) ing early stages of spinup, since in the later stages of the Vh]=0 tank runs it was observedthat • = f and that surfacefronts 11,314 HUNKINSAND WHITEHEAD:SIMULATED EXCHANGE THnOUGH FRAM STRAIT appear.In orderto evaluatethe relativeimportance of the inertial, time term in (5) may be ignored. The third and variousterms in (4) the equationis scaledwith the follow- fourth terms represent forcing and damping, respectively, i.s characteristic e= y') = h'= and are retained in all cases as essential to the problem. h]ho, r' -- r]r0. Scaledequation (4) is then With this approximation,(4), whichis a dimensionalequa- tion, reducesto a linear equationfor vorticity diffusion, h-gV at Oh0'-•+ •1 curlr- • •7• h= 0 (6) where• = tig(•/2fa)•1• is the vorticitydiffusivity. With pfL curl' va h' = 0 (5) this simplifiedequation it is possibleto interpret a number of the features of wind-driven flow in the tank. In the early whereR -' (tigho)•/a/fis the Rossby radius of deformation. stages of developmentwhen velocities are still small, a near Sincein theseexperiments (R2/L 2) '- 0.028,the first, or balancewill exist betweenthe first two terms of (6) as Ek- man pumpingforces water downwardat the top to deepen the upper layer. At this early stage the growth rate of the layer, Oh 1 0--•-----p--]curlr (7) is the same as the vertical Ekman pumping velocity. In order to check the observeddeepening of the layer against thissimple relation, an estimateof windstress curl is needed. Wind stressin this experimentmay be calculatedfrom the Ekman mass transport relation My = r,/f (8) whichgives the vertically-integratedtransport normal to the wind stress vector. This transport will be established within one or two rotations of the tank after the wind com- mences.Since essentially all of the upper layer in one basin was transferred to the other basin in a period of about 60 s, the masstransport can be estimatedfrom the knownvol- ume,gap width, and time. The valuefound by this method is My -- 0.64kg's -• m-•. Thenmultiplication of thisvalue by f givesthe maximumstress, r0 -- 2.0 Pa, at the gate region of the tank. Fig. 16. Wind-drivenexperiment. Top view,one revolution (4 s) after blower was started.

Fig. 18. Wind-driven experiment. Top view, eight revolutions (32 s) after blower was started. The dyed freshwaterlayer has Fig. 17. Wind-drivenexperiment. Top view, threerevolutions been almost completely transfered to the "polar" basin where it (12 s) after blowerwas started. is trapped in an anticyclonic gyre. HUNKINSAND WHITEHEAD: SIMULATED EXCHANGE THROUGH FRAM STRAIT

Fig. 19. Wind-drivenexperiment. Side view showing deep anticyclonic gyre of freshwaterafter 180 s.

1 V2hp]-• curl r = 0 (9) Solutionsof thisequation may be compared with qualitative observations,after manytank revolutions,of the thickness asshown in Figures18 and 19. With the parabolicform of the wind stress,(9) becomes 02 h 02 h axe_+ • =-Fy (10) whereF = 2 ro/pf•L2. Boundaryconditions are h(x, y) = 0 on the circlex 2 + y2 _ L2. SIDE TOP VIEW VIEW The solutionof (10) whichsatisfies the boundarycondi- tions is Fig. 20. Schematicdiagram of resultsof wind-drivenexperiment after most of the freshwater layer has been transferred to the h(x,y)=-•-Fy(x2 -t-y2 -L 2) (11) "polar"basin. Abbreviations are PAG for polaranticyclonic gyre and SCG for subpolar cyclonic gyre. Contoursof surfaceheight departuresshow an elevated domein the uppersemicircle and a depressionin the lower The curl of the wind stressmay be estimatedif the stress semicirclecorresponding with anticyclonic and cyclonic flow fieldis known.Details of the stressfield were not measured, respectively(Figure 21). Thisquasi-geostrophic flow gener- but a parabolicform of the profileis a reasonableassump- ally resemblesthe tank results,but limitationsof the math- tion, ematicalmodel do not allowit to describethe completere- movalof the upperlayer from the lowersemicircle, since r•--ro --1 r•--O the linearmodel requires the upperlayer to be presentev- ['½] erywhere.Also the contoursare not so perfectlycircular in this calculationas they are in late stagesof the actual Thecurl of the wind stress isthen -Or, lay = -2r0y/L 2, experiment,andthis also is likely due to the limitations of whichvaries from zero at y = 0 to a maximumvalue of linearity. -2rollat x = 0,y = L. Tocalculate therate of deepening A furthertest of the calculations against observation is of the layer,take the meanvalue of the windstress curl, the maximumthickness of the upperlayer. Maximumthick- -roll = 2.0 Pa/0.45 m = 4.4 Pa m-• ness,as calculated from (11) usingthe windstress arrived at earlierand the values in Table3, occur• at x = 0, y = L/3•1• From (7) the rate of deepeningfor the upperlayer is then witha value,h(0, L/3 •/2) = 0.88m. Thiscalculated maxi- calculatedtobe 1.4 mm s -•, a valuewhich agrees fairly mum depth isseveral times that of the layer observed inthe wellwith the observation that the layer was 10 cm deep center of the anticyclonic gyreafter many minutes when an afterabout 60 s. After many revolutions asteady state will approximatesteady state had been reached. The discrely- bereached which may be described byomitting the time ancyindicates that linear Ekman theory isnot adequate to termin (6), predictthe final steady state. This was anticipated earlier 11,316 HUNKINSAND WHITEHEAD: SIMULATED EXCHANGE THROUGH FRAM STRAIT

Ov• fh• + f = (13) Oz H• Ova fha o• + f = //• (•4) and thermal wind between the two layers f(• - v•)= g, Oh,o= (•5) One can derive differential equationsfor h•, ha, v• and va asfollows: Take a/a• of (15) and use(13) and (14) to get

(16) a•h•Oza f•g' ( • 1) fallsg'Ha -0.1 We define z as being zero in the middle of the channel. The solution of ha is H3 Ha z z -0.3 h•= m + n• + • •osh• + Z sinh• (•7) and from (12), the solutionof h• is h•= H•HaH•-F-'---• - B cosh •-z Asinh •z (lS) Fig. 21. Contoursof surfaceheight departure based on a linear where vorticitymodel with a parabolicwind stressprofile representing the polar easterlles.Contour heights are in dimensionlessunits, [8/FL 3] h(x, y)]. Arrowheadsrepresent flow direction. I• = • p(H•g'HaHa+ H•) To solvefor velocity,use (13) and integrate: when the relative vorticity and the Coriolisparameter were found to have nearly the same magnitudeafter many rev- olutions. There must be more friction than than has been v•= f {Ha H•- H• + Ha- H2 } z- fH•RB sinh •-z accountedfor here. It may be that someeddy viscosityde- velopsin theselater stagesto augmentmolecular viscosity. fRAH• cosh•--FCz (•9) Sidewall friction has been ignored and may also represent someof the missingfriction. or use(14) and integrate H3-H•-Ha LOCK-EXCHANGE THEORY va= f H•-F Ha z+ A theory for velocityof the flow and the shapeof the interface is formulated here without the third underlying fRB sinhz fRA coshz layer. Becauseof dynamicalcomplexities involved with the H• •+ H• •+D (20) twofluid layers lying over a third layer,it wasfelt that three and (15) dictatesthat (7 = D. layerswas too complicated for a firstattempt. In addition,it Three constants remain to be determined, presumably was anticipatedthat a quantitativelaboratory experiment by the inclusionof additionalgeometrical or dynamiccon- would be desirableto check the predictions,and a three- siderations. Here, only the solution for the fiat bottom layerexperiment did not appearto be practical.It is hoped (H• = Ha = Ha = H) will beinvestigated further, and then that the formulation can be extended to the three-layer case onlywith the assumptionthat the heightprofile is antisym- in the future. metric, so that B = 0. Under theseconditions, equations Considertwo basins,basin 1 of depth H• with water of (17)-(20) reduceto densityp• andbasin 2 of depthHa with waterof density The two basinsare separatedby a straightsmooth channel h;= H2 Asinh Rz (21) of depth Ha with a closedgate. The surfaceof the water is initiallythe sameevery,where, but uponremoval of the gatethe verticalinterface between the twowaters of differ- ha= '•-H '•'A sinh•z (22) ent densitieswill experiencea complicatedcollapse due to buoyancyforces. After sometime we expectthe lightfluid fz fRA coshz to travelalong the top of the channel out of basin 1 into 2 n • -i-(7 (23) basin 2 and the denserfluid to travel along the bottom of the channelout of basin2 into basin1. Duringthe co.!lapse, fz fRA coshz the seasurface between the twobasins will haveadjusted so va=-•--F H •-FC (24) that the volume flux of th• two'currents is equal. Becauseof the symmetry of the profile, the assumption The governingequations for the flowin the laterstage are of equal and opposite volume flux through the lock region conservation of depth, requiresthat vz -- -va at z = 0. This requiresthat C = 0, so only the constantA remainsto be determined. h, (•) + h•(•) = m (•) Gill [1977]showed that the constantpotential vorticity conservationof potential vorticity current has a Bernoulli function that is easily determined ex- HUNKINSAND WHITEHEAD:SIMULATED EXCHANGE THROUCH FRAM STRAIT 11,317

cept for a constant. Since there is no dissipation in the cur- center of gravity of eax:hcolumn of width dx. The product rent, this constant is conservedalong streamlines. Therefore of these is integrated across the currents and summed for the symmetric solution for the current extends throughout the two nosesto give the entire region from behind one nosethrough the passage to behind the other nose. The symmetric solution exists -.long the entire extent of the flow field. 9• wAp•= •.•_•/2Q2f•1• 9Ap h•_ dx- Either energy considerationsor Bernoulli's law has been used to solve for the final constant in hydraulics problems. Here we utilize the time-dependent energy equation by tak- Q1/x/2 h2)dz ing the dot product of the Euler equationswith velocity for •l-lj_x[2 9Aph, (h-• + (28) a two layer rotating fluid Equations(27) and (28) are to be set equal, and since Q2, h l '-' H- h2, and A2 -- A2, they simplify to =0 and integrate over the entire volume including the region containing the transient nosesof the currents as they move Using equations(21)-(24) equationis (29)is found after intothe otherbasins. The volumeintegral of u-V < • 6 > some manipulation to reduce to - < 6. V p > along with continuity V. 6 = 0 can be convertedinto a surfaceintegral by Green's theorem for flux of kinetic energyinto or out of the volume plus pressurework -•--•-F•-•24R • 1-cosh +4sinh • = on the volume surface. We assume that the boundaries of the volume are sufficiently far from the noses that velocity is zero and therefore the surface integrals are zero. Hence we find a simple balance between changein potential energy where and the increasein internal kinetic energy in the currents. • H A sinh2-• = •- (31) • Ot + g < wAp>= 0 (26) These two equations are satisfied for the values This energy balance resemblesthat used by Yih [1980, • A p. 205]. 2-•'- 2.5940 • -- 0.07514 (32) We do not know the detailed flow in the nose region [Stern, 1980; Stern, et al., 1982; Gri.ffiths and Hopfinger, SinceA was the last remaining unknown, volumeflux can 1983]but we can assumethat the noseis fully developed. now be determined from the integral Hence it will be self-similar between a time t and a time t + •t. The similarity assumption requires that the volume Q• =-Q2 = h• v• dx (33) of the moving nose region be unchanging,in which case we J-•,12 can set ci = Qi/Ai, where Qi is the volumeflux of the ith current behind the nose and Ai is the cross section area of the current(see Figure 22). Q1=fAR (•cosh• •-• • -2Rsinh •-•• ) (34) Thus the increasein internal kinetic energy in time equals For the values of A and • given above, the formula reduces ci times the area averageof kinetic energy acrossthe current. to These are summed for the two on the left and the two on g'H a the right to give Q•=0.156 f (35) The prediction of the width of the current and the volume •

LOCK-EXCHANGE EXPERIMENTS

Laboratory experiments were conducted to measure the transport by lock exchangeflow with constantpotential vor- ticity. The apparatus was a cylindrical tank 2 m in diame- ter with a 10-cm-high wall. The bottom was fiat and level to better than 1 mm. Artificial walls were constructed as sketchedin Figure 23 sothat two basinswere separated by a smooth 10 cm wide gap. Volume of each basin was measured Fig. 22. Sketchof the changein energyand the extensionof the by filling with water up to 5 cm deepand was23500 cm -z. noseregion with velocityci = Qi/Ai. The apparatuswas placedon the 2 m rotating turntable at 11,318 HUNKINS AND WHITEHEAD: SIMULATED EXCHANGETHROUGH FRAM STRAIT

tracting the density before the run from the density after the run for each basin and then averagingthe absolutevalue of the change for the two basins. Density transport was then calculatedfrom this result by dividing Ap by the time r the gate was left open. The results are shownin Figure 24, where density trans- port was plotted as a function of density differencebetween the two basins times depth of the water. The data for 15 s were expectedto be well within the rotating limit and were the most numerous observations. The data are compared with a line from a prediction of lock- exchangederived in section 4. The line was produced using the prediction of volumetransport Q• = 0.156g•h2/f from lock-exchange constantpotential vorticity theory equation(35). Volume flux is multiplied by p• - p2 to predict a density flux. This density flux is divided by the volume of the basin to predict density change per unit time. Using the formula and the valuesfor basin flux and rotation given above and a value of g = 980 cm s-2, the predictionis that transportis propor- tionalto 0.0077[(p•- p2)H]2 asshown by the solid line. The data from experiments with a period of 15 s have an empiri- calbest fit of 0.0039[(p• -p2)H] TM asshown by the dashed line. This formula seemsat first glance to not to agree very well with the rapidly rotating prediction. However, satis- factory agreement between data and theory is found to the P! P2 left where[(p• - p2)H]is of theorder 10 -2 to 10-•. In that region some of the data lie above the theoretical curve but bow. Wo isht, closerto 1, the transport seemsto curve over as though low rotation effectswere beginning to be felt. Figure 24 has two Fig. 23. Top view of the cylindrical basin used to test the constant potential vorticity theory. additional features that give a clearer perspectiveof the ob- servations. First the 60 s data are added as open squares. The data lie slightly abovethe fifteen seconddata. Second,a the Coastal Research Center at Woods Hole Oceanographic predictionfor transport for nonrotating lock exchangetrans- Institution. port was included as the top thick line. This was produced In a typical run, one basin was filled with fresh water and using a prediction for the velocity of nonrotating flow from the other with slightly dyed water of known salinity. Sam- Yih[1980] V-207], U = 0.71(gH(p, -p2)/(p, + p2))'12. pies of water were taken from each basin and the density To use this formula to predict density flux, U is multiplied measured. A removablewatertight gate would separatethe two basins. The turntable was then set into rotation and the period of rotation recorded. The depth of the water in -2 the vicinity of the gate was also directly measured. After a motionlessstate was achieved in a rotating frame, the gate -3 was removed and the density current would start. Within a few seconds the nose was seen to start into the other basin, curve to the right, and lead to a continuousflow into the -4 other basin. When the nose was seen to have moved two thirds around the perimeter of the basin, the gate was re- placed, the time of gate opening was noted, water in both basins was thoroughly mixed, and samplestaken. The ex- periment could then be regarded as the initial conditionsfor -6. * a second experimental run with smaller initial density dif- ference. The difference in dye intensity never got so small -7 . that it was impossibleto see the nose, even though the color -2 -i 0 of the two waters gradually becamesimilar. In all, a total of 49 runs were conducted, 40 with a ro- tation period of approximately 15 s and 9 with a period of approximately a minute. Each run took between 20 and Fig. 24. Experimentalmeasurements of densitytransport Ap/'r 40 rain to conduct. The density of the samples was mea- versusupstream parameter I = (p2 - p• )H. The data with a ro- suredwith a densiometeraccurate to 0.0001gcm -3. The tation period of approximately 15 secondsare given as diamonds. Data with a sixty-second period are open squares. The solid line initial difference in density between the two basins p2 - p• is from a rotating lock exchange formula equation 35. Dashed was calculated and multiplied by depth H. After each run line is a best fit line 0.0039((p2 - p•)H) TM- The bold line is the density changeduring the run Ap was calculatedby sub- from nonrotating lock exchange theory. HUNKINS AND WHITEHEAD: SIMULATED EXCHANGE THROUGH FRAM STRAIT 11,319 by HL/2 to predict a volumeflux which is again multi- could range from over 400 m to approximately 100 m in the plied by p• - p2 to predict a density flux. This density flux center of the basin. The most sensiblenumber appears to is divided by the volume of the basin to predict a density be H - 200 m that is representativeof the 34.6 ø]oo haline transport. The prediction is that transport is proportional depth near Fram Strait. The primary balance for fresh wa- to 0.0035[(p2- p•) HIs/•. In comparingthe data with both ter in the Arctic Ocean is fresh water added by river runoff lines for rotating and nonrotating flow, one seesthat the the- and freshwater removal by rafting of ice out of the ocean. ory for nonrotating flow predicted far too large a transport We take Q! - 104m s s-• as the valueof riverrunoff mi- to the left, where rotational effects were evidently impor- nus ice rafted south by currents from Aagaard and Carmack tant, but did come close to the data on the right. On the [1989].These numbers in (40) give other hand, the rotating theory lies within the data on the left but lies above it on the right. The data with a rotation AS = 1.06 o/oo period of 60 s clearly lie above both the 15 s data and the Aagaardand Carmackcite Paquetteet al. [1985]as assign- rotating theory to the left, but below the nonrotating the- ing the East Greenland Current an averagesalinity of 33.7. ory The empirical best fit for both the 60- and 15 s data is They also use for the Norwegian Sea a salinity of 34.6 and 0.0036[(p•- p•) HI •'7•. Thishas a powerlaw almostex- thus use valuesof infiowingand outflowingwater differing actly halfway between the rotating and nonrotating theory. by approximately1%o, a differencein agreementwith this To the right all the data lie slightly below both theories,but prediction. Thus it appears that the lock-exchangeflow is they appear to approach the nonrotating theory. Since the consistentwith the magnitudeof freshwaterrenewal actually nonrotating theory is expected to be valid when it predicts found in the Fram Strait region. lessflow than the rotating theory, the data seemto confirm the nonrotating theory in this region. CONCLUDING REMARKS

LOCK-EXCHANGE CALCULATION In this seriesof studies,assorted features in laboratory ex- perimentsthat resemblethe currentsand transportsin Fram A simple salt and volume balance for the Arctic Ocean- Strait have been observed. In the first dam-break experi- Norwegian Sea through Fram Strait will be made assuming ments, the transport of fluid from one basin to another by that the Arctic Ocean is a "mediterranean basin" in a man- narrow boundary currents was clear. The width of the cur- ner similarto that doneby Bye and Whitehead[1975]. The rents appeared to be scaledby the Rossbyradius of deforma- model will incorporate conservation of volume tion, although the photographswere not judged to be unam- Qi - Qo -]-Q! (36) bignonsenough to produce quantitative data. The current flowingfrom the fresh water side (which correspondsto the where Qi is volume flux of water into the Arctic Ocean Arctic) to the intermediateside (which correspondsto the through Fram Strait, Qo is volume flux out of the Arctic GreenlandSea) is a modelof the East GreenlandCurrent. Ocean through Fram Strait, and Q! is the volume of fresh The complexity of the real coastline and the bathymetry is water added to the Arctic Ocean through flows into it from lacking in the model. The laboratory current had a velocity other regions. The model will also incorporate conservation thatscales closely with the velocity scale Vs/(g'h) •/a = 0.4. of salt in the simple form Usingtypical values of g = 9.8 m s-a, Ap/p = 1.8x10-3 SiQi = SoQo (37) (from Figure 5), and h =200 m, this givesan expectedve- locity of the East Greenlandcurrent of 0.77 m s-•. This By writing the conservationof salt in this way, one assumes is roughly the velocity obtained by subtracting the wind- that there is a simple correlation between the salinity of the inducedmotion from observedice motion [Hunkins,1990]. currents into and out of the Arctic Ocean through Fram The velocity of the current from the intermediate side to Strait. Although such a correlation is obeyed closely by the freshwater side which correspondsto the West Spitsber- the two-fluid laboratory modelsinvestigated here, the actual gen Current was considerably slower and its velocity scale oceanicsituation is more complicated.Equations (36) and is not clear. The extensionof the West SpitsbergenCurrent (37) are now combinedto give into the Arctic Ocean is poorly studied and its speed seems QoAS- S,Q! (38) to be not documented,although observationsof the tongue of highest temperature in the Arctic Ocean suggestthat the We assumethat Si and Q! are fixed by nature and that Qo current remains on the right-hand side of the Arctic Ocean and AS are unknown. The dynamic lock-exchangeformula for hundredsif not thousandsof kilometers[Hunkins, 1990, Figure 7]. Thereforeit is unclearwhether the slownessof Qo- 0.156g'H2f = 0.156 g/•ASH pf • = SiQ! AS (39)the laboratory current duphcates the ocean. The extensive eddies in the laboratory experiments seem is then incorporated to reduce the system to one unknown. to duplicate the numerous eddies observed in the vicinity This gives the formula of Fram Strait. The eddies in the density-driven labora- tory experiments roughly appeared to be a Rossby radius AS = v/pfSiQl/O.156g/•H • (40) of deformation in size, and this appears true for the ocean To evaluatethis, we take valuesof p = 1027kg m-s, /3 = eddies also. The eddiesdo not appear to be of only one sign 0.71 kg m-s (ø/oo)-•, g = 9.8 m s-• Si =35ø/00as a (seeespecially Figures 8-11). The density-drivenflows did representative number for oceanic salinity, and f = 1.36 x not initiate a large cyclonicgyre south of the strait of much 10-4 s-•. The valueof H to usewould be that appropriate larger size, and one may suspect that the gyre in the Green- for the upper halocline water of the Arctic Ocean. Depend- land Sea may arise from wind driving interacting with the ing upon where one looked and what criterion one used,this density-driven flow. Combining buoyancy and wind driving 11,320 HUNKINS AND WHITEHEAD: SIMULATEDEXCHANGE THI•OUGH FP•AMSTP•AIT is beyondthe scopeof the presentinvestigation and must Given that there is abundent evidence that the Fram be left for further work. Strait flow is buoyancydriven [Hunkins,1990], the apparent If we assume the value of A in section 4 is not fixed, as successof the lock-exchangeformula is.not really surpris- we have assertedthere, the volume flux predicted by the ing, since it could be derived from a thermal wind relation lock-exchangetheory reveals novel results when plotted as alone. However, the use of that simple formula would require a function of A as shown in Figure 25. For small A, the numerous assumptions. It is hoped that the lock-exchange interface between the two fluids extends across the entire derivation given and its being accompaniedby quantitative openingand there are localcurrents at the edge.There is laboratory data are more convincing. a barotropicshear across the entiresection (from equations Without doubt numerous real features have been omitted (23) and (24)) that transportsfluid in the "wrong"direc- from this study. A list of these includes a shelfiikebathymet- tion, i.e., low-densityfluid towardthe low-densityend, and tic profile for the eastern wall, sidewall slope for all sides,the denser fluid toward the dense end. This barotropic trans- real coastal shape, continuousstratification, sphericalgeom- port is greaterthan the baroclinictransport within the side etry including the beta effect, sea ice, turbulence, and the boundarylayers. Conversely,the barocliniccomponent of actual temporal forcing(if it were known). Most of these, the flow transportsfluid in the "correct"direction i.e., low- except for the challenging• plane, could be incorporated in densityfluid awayfrom the low-densityend and dense fluid future studies, but it is hoped that this study provides the awayfrom the denseend. The net resultis densitytransport first approximation. the "wrong"way for small A. ForA = HI2 sinh(),/2R), the In closing,it must be remarked that the surfacewater of boundary ]ayersintersect the bottom and top and the cur- the Arctic Ocean is greatly undermixed. If the low-salinity rent gets progressivelynarrower as A increasesabove this water were mixed downward more vigorously, the exchange value. The barotropic transport decreasesand the total would be greater, and salinity differencebetween the surface transport becomesmore positive with increasingA. The water and deep water in the Arctic would be less. Equation predictedvalue of A/H that satisfiesthe energyconstraint (40) indicatesthat salinitydifference would be inverselypro- lies above the point where the transport crosseszero. And portional to H, that was taken as the depth of the bottom finally as A approachesinfinity, the transport approaches of the halocline. But velocity would stay constant, since it 0.5. This reverseflow phenomenonhas also been noted by is proportional to the square root of the product of den- Pratt and Armi [1990]. sity and H. Rafting of the Arctic ice, which removesmore fresh water than the lock-exchangeflow, is proportional to velocity of the East Greenland Current times the width of RIGHT the current. The width is proportional to the Rossbyradius of deformation, which is also proportional to velocity. So width also stays constant. Thus rafting is expected to be WAY[o.sl 9- only weakly dependent on the mixing strength in the Arctic Ocean. It seems therefore that the processesthat remove Qf fresh water from the Arctic Ocean are independent of the g,Hz vertical mixing coefficient there.

Acknowledgments.Support by the Office of Naval Research,Arctic Sciences,under contractN00014.87-K-0204 MH is acknowledged.The photographicwork and valuable assistancewith the laboratory experimentswas providedby RobertE. Frazel, whom we thank.This WRONOI• q P-•• I ? ! work representsa collaborationbelween the two authors;however, W'^Y' section3 wasprimarily an effort of the first authorwhile sections4 and 5 were primarily an effort of the secondauthor, who was alsosupported Fig. 25. Nommlized density transport as a functionof the last by the Office of Naval Research,Coastal Science section, under grant N0001'4-89-1-1037for studiesof lock-exchangeflows. Lamont-Doherty freep•rameter A. The sketchedsections below the curveillustrate GeologicalObservatory contribution 4933; WoodsHole Oceanographic the position of the interfacefor the three v•lues of A. Institution contribution 7985.

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