Localized Convection in Rotating Stratified Fluid

JOURNAL OF GEOPHYSICAL RESEARCH, VOL. 101, NO. C10, PAGES 25,705-25,721, NOVEMBER 15, 1996

Localized convectionin rotating stratified fluid

J. A. Whitehead Department of Physical Oceanography,Woods Hole OceanographicInstitution, Woods Hole, Massachussetts

J. Marshall Department of Earth, Atmospheric and Planetary Science,Massachusetts Institute of Technology, Cambridge, Massachusetts

G. E. Hufford Woods Hole OceanographicInstitution, MassachusettsInstitute of Technology Joint Program, Cambridge, Massachusetts

Abstract. We study the convectiveoverturning of a rotating stratified fluid in the laboratory. Convection is induced from the surface of a salt-stratified fluid by the introduction of salty fluid over a circular area. The external parameters are buoyancy forcing of strength, B0, applied over a circular area of radius R•, the rotation rate as measuredby f, ambient stratification N, and the depth H. The experimentsare motivated by physical scaling argumentswhich attempt to predict the length and velocity scalesof the convectivechimney as it adjusts under gravity and rotation and breaks up through baroclinic instability. The scalesof interest include the number, size, and typical speedsof the fragmentsof the brokenchimney, the final depth of penetrationof the convectivemixed layer, and the total volume of convectivelyproduced water. These scalesare tested against the laboratory experiments and found to be appropriate. In this idealized problem we have found the depth of penetration dependsonly on the size and strength of the forcing and the ambient stratification encountered by the convection event; it does not depend explicitly on rotation. The implications of the work to deep water formation in the Labrador Sea and elsewhere are discussed. Finally, the study has relevance to the role and representation of baroclinic eddies in large-scale circulation of the ocean.

Introduction Chapman,1995]; or, in the openocean, by deepconvection events which are manifested as a very deep mixed That deep water in the oceansis cold, and the deep- layerinduced by wintertimebuoyancy loss [Storereel, el est water is the coldest, has been known since the al., 1971]. The latter processis the focusof this study. 18th century [Ellis, 1751], althoughsalinity can com- Deep ocean convectionis apparently only found in re- plicate this simplepicture [Veronis,1972]. In general, gionsthat are preconditionedregions, where gyre-scale the oceans are stably stratified everywhere, except in and mesoscale circulation have weakened the ambient top and bottom mixed layers; even polar oceans are stratificationsufficiently for rapid cooling/evaporation/ stably stratified in spite of the fact that the coldest freezingto producevery deep mixed layersduring pe- water is created there and sinks downward to spread riods of maximum coolingin winter. During the for- out to temperate and tropical latitudes. Thus surface mation of the deep mixed layer, water moves vertically cooling/evaporation/iceformation do not frequentlydi- with velocitiesof 2-10 cm s-1 within the deepmixed rectly produce water that is denser than the deep- layer [Storereelel al., 1971;Schott and Leaman,1991]. est water in the region. Instead, in polar oceans the The vertical movement is both upward and downward dense water "makes its way" downward by two differ- and is recorded episodicallyby current meters; these in- ent routes: by flowing downward as density currents dividual convectioncells have becomeknown as plumes, along the continentalslope from areasof origin in shal- and they act to efficiently "mix" the layer. The mixed low seas[see e.g., Nansen,1906], [Gawarkiewiczand layer depth increasesfrom h,undredsof metersto possi- bly thousandsof meters over the period of a few days. After and possiblyduring the latter stagesof the deep Copyright1996 by the AmericanGeophysical Uifion. mixed layer formation, the entire volume of mixed-layer Papernumber 96JC02322. fluid sinks by spreading laterally at its base and con- 0148-0227/96/96JC-02322509.00 tracting laterally at the surface. Simultaneously, the

25,705 25,706 WHITEHEAD ET AL.: CONVECTION IN ROTATING STRATIFIED FLUID sides of the deep mixed layer develop geostrophiced- Numericalstudies by Jonesand Marshall [1993]have dies which augment this sinking. also supported the scalingstested by Maxworthy and Many theoretical, numerical, and laboratory model- Narimousa[1991, 1994] in the laboratory.Using nonhy- ing studies have clarified the dynamics of such deep drostatic models,they focusedon both the production ocean convectionevents. Killworth [1976]; Crdponel of the small-scaleconvection cells and the emergenceof al. [1989],and Madecel al., [1991],for example,focused the larger aggregatescale in which lumps of cold wa- upon generation of a large-scaleflow while parameter- ter cluster together. Legg and Marshall [1993]found izing the convectivescale. The dynamicsof adjustment that the lumps of densewater tended to pair up with with the surroundingstratified fluid and the production correspondinggyres of cyclonicsurface fluid to produce of eddies along the region dividing the cooled and un- "beton" pairs, which movethe densefluid laterally from cooled fluid were topics of interest. The cooling was the place of origin. regarded as a distinct event with the lateral adjust- Helfrich and Battisti [1991]studied the flow created ment processeshappening concurrently. Linear theory by a steady point buoyancysource in a rotating strat- of rotating Rayleigh-Bdnard convectionsummarized in ified flow. Helfrich [1994] conducteda similar study the work by Chandrasekhar[1961] was employed to ad- of flow createdby impulsivesources (called thermals). dress the dynamics of the plumes. Davey and White- Scaling considerationswere developedfor velocity and head[1981], for example,showed that convectioncells length scales and successfullycompared with labora- wider than an appropriately defined Rossby radius of tory experiments. An important result in both studies deformation(gApH/po)•/2/f would not grow, where g is that fluid would accumulate at a terminal depth until is accelerationfrom gravity, Ap/p is the proportionof the beton mechanismproduced eddy pairs that would vertical densityvariation (but an adversedensity gra- remove the fluid. dient, with densestwater on the surface),H is depth The present study considersflow created by an ex- of the fluid layer, and f is the Coriolis parameter. Fer- tended but confinedbuoyancy source in rotating, strat- nando et al. [1991] and Bubnovand Golitsyn[1990], ified fluid. It can be considered to be an extension of used scalingarguments to explore the nonlinear regime, Maxworthyand Narimousa[1991, 1994]with stratifica- determined the region of parameter space where rotation added. Or, it can be consideredto be an extension tion was dominant, and produced information about the of Helfrich and Battisti [1991]with finite buoyancysize length and velocity scaleswithin that region. Fernando, added. Laboratoryexperiments (in section3) will test et al., [1991]identified the length scale scalings(outlined in section2). trot- (Bolf3)•/•' (1) DynamicalConsiderations suchthat the critical depth, found experimentally to be Nondimensional Numbers 12.7/rot, is the boundary layer thicknessbeyond which rotation becomes important in steady convection. The When an unstratified fluid of total depth H is forced significanceof/rot is that for distancesgreater than/rot to convect by uniform surface cooling in the presence the effectsof rotation are important whereasfor smaller of rotation f, there is only one nondimensional com- distancesrotation is not. Typically, in the ocean, /rot bination of the main external parameters B0, f, and is about 100 m. Here B0 is buoyancy flux defined for H heatingor coolingas B0 = gc,H!/pcp, whereg is ac- , X/Bo/f3 trot celerationof gravity, c• is the coefficientof thermal ex- R0= H = H' (4) pansion,H! is the heat flux per unit area, p is den- which will be called the natural Rossbynumber R3 sity, and cp is specificheat, respectively.For salt flux, (its physicalsignificance is reviewedin Marshallet al., Bo = g/•Fs, where/• is the salt coefficientof expansion [1994]).Here/rot = v/Boll 3 canbe consideredto be a and Fs is the volume flux of salt. Maxworthy and Na- measure of the vertical scale to which convectionpene- rimousa[1991, 1994] presentedevidence in supportof trates in an unstratified fluid during an inertial period. the importance of/rot and introduced the associateddi- If R• is large, then the convectionwill be limited by the mensionless "natural" Rossby number defined in terms oceandepth beforeit feelsthe effectsof rotation;if R• is of external parameters small, then the convectionwill come under geostrophic controlbefore encountering the bottom. Briefly,R• is = (2) large in the atmosphereand small in the ocean. Only in oceanicregions which experiencedeep convection can In addition, a new length scale,a radius of deformation, R3 get small enoughfor rotation to be important. In the ocean,typical values for f, B0, and H are f • 10-4 s-•, B0 • 10-7 m2 s-3 (correspondingto a heatloss of was found to determine the diameter of groups of • 2000W m-2), andH • 1000-4000m, givingan R• individual thermals that aggregatedon the bottom in in the range.0.08-0.4 so rotation cannot be neglected closed cells. in the ocean. In the laboratory experiments presented WHITEHEAD ET AL.: CONVECTION IN ROTATING STRATIFIED FLUID 25,707 here R• rangesfrom 0.04 to 0.41, and in our numerical made up of many aggregates.Some parts of the subse- experimentsR• variesfrom 0.05 to 0.58, both spanning quentdiscussion are due to Visbecket al. [1996],where the oceanic regime. more details of the argumentsmay be found. In addition to R•, two additional nondimensional We consider first the vertical evolution of the mixed numbers are needed to fully describe the system when layer at a position in the center of the well-mixed chim- stratification and the size of a forcing region are in- ney. For a sufficiently wide chimney a water parcel in cluded. The first is N/f, comparingthe relative sig- this position will be unawareof any spatial inhomogene- nificance of stratification as measured by the frequency ity in either coolingor overturning. Before rotation be- N = [(-g/p) (Op/Oz)]1/2 to rotationmeasured by comes important, the evolution of the mixed layer can f. The secondis Rs/H, an aspect ratio comparing be described by a one-dimensional model. If there is R, the radius of a circular forcing area, to the depth no entrainment of fluid acrossthe base of the chimney, of the water column. In sites of deep ocean conver- then [seeTurner, 1973], tion, N rangesfrom 2 x 10-4 to 10-3 s-1; thusN/f varies from i to 10. In the laboratory experiments /2Bot presentedhere, 0.86 < N/f • 6.64. They are thus h- V in the lower limit of thermocline ocean values, mak- As rotation becomesimportant, the chimney adjusts ing them particularly suited to deep ocean convection toward geostrophicbalance. Simple analytical expres- sites. Typical length scalesfor Rs are set by meteoro- sionsare found in the work by Gill e! al., [19741and logical forcing and preconditioningdynamics and can Cv@on½! al. [1989]. From geostrophicadjustment the- range from a few to many hundredsof kilometers. The ory we expect the wall of the chimneyto have a width depth H variesfrom 1000 to 4000 m. This givesa range Ra = Nh/f, the Rossbyradius of deformation. The of 1.25 • Rs/H • 100. In our experiments,Rs/H surfaceconvergence has an associatedcyclonic velocity spanned 0.25-0.75. They are thus at the very lowest urim, which may be inferred from thermal wind thus limit of real ocean values; nonhydrostatic effects may thus be exaggerated. If viscosity and thermal conduction are of impor- dz- pf •r ' (6) tance,the Taylornumber Ta = f2H4/•2 and Rayleigh numberRa = BoH4/k2• may alsobe important.The implying critical value of Rayleigh number, which must be ex- _ ghApl,. (7) ceeded for convection to be important, has been related uri,•- Pof RD ' to the Taylor number[Chandrasekhar, 1953; Nakagawa whereAplr is the densityjump acrossthe zoneassumed and Frenzen,1955] accordingto the relationship to be of width RD, r is the radius, and p0 is a constant reference density. Ra!(cr) -- kTa2/3, We define the chimney geometry such that the den- where k is a constant dependenton the boundary condi- sity differenceradially outward from the center is the tions. Bubnovand Senatorsky[1988] determined k ex- same as the vertical density difference from the spread- perimentally and found it to lie between 2.39 and 8.72, ing levelto the surface.Then by definition(gApl,/po)= dependingstrongly on the boundary conditionsused. (gAplz/po)=N2h.Hence, from (7) In the laboratory it is impossibleto achieve a value of H similar to that of the ocean. Using molecular val- = (8) uesof v and n, typical laboratoryvalues of Ra! range sinceRa = (Nh/f). The rim currentwill continueto from 1013to 1015,if f rangesfrom 0.01 to 0.1 s-1, grow • h evolvesin time until the onset of baroclinic H • 0.1 m, and Ta • 105,giving a criticalRayleigh instability. At that time eddies will form and begin to numberof Ra!(cr) • 106. We thusexceed this criti- transportwater (at a speed(7)) awayfrom the chimney cal value by several orders of magnitude. In the ocean and also bring surroundingwater in. f • 10-4 s-1, H ,•. 103 m, • • 10-6 m2 s-1, and The instability timescale may be inferred from bar• n ,.• 10-7 m2 s-1 (for thermaldiffusion in water),giv- clinic instabilitytheory [Eady, 1949]. The growthrate ingRa! • 1027and Ta • 1022vastly exceeding critical values. w of the f•test growing mode in the rim current is

Convection in a Stratified Rotating Ocean Considerbuoyancy forcing applied over a finite circular area only, so the parametersto be consideredare B0, whereRi- N2/(Ou/Oz)• is the large-scale Richardson N, f, H, R, and time t. Previouswork [Leggand Mar- number. If the rim current is in thermal wind balance shall, 1993; Jonesand Marshall, 1993]has focusedon with speed Nh, then • = I and the evolution of both individual and collections of con- . (10) vection cells. In the presentstudy we concernourselves with the developmentof the entire chimney which is At this time, t • f-1 our scalesbecome 25,708 WHITEHEAD ET AL.' CONVECTION IN ROTATING STRATIFIED FLUID

Urim ~ v/Bo/f - Urot (11) whichreduces to h~ ds (BoRs)l/a t•d •,o /rot -- V/Bo/f, (12) = N ' (19) and The quantity ds is the scalingfor the maximum depth f achievedby the convectivemixed layer, assumingall h ~ • /rotß (13) availablepotential energy is convertedto eddy kinetic The above scales are interesting inasmuch as the in- energyby baroclinicinstability. The result(19) is de- stability timescaleand the lengthscaleof the preferred rivedin moredetail in the workof Visbeck,et al. [1996]. mode are seemingly independent of the stratification This scalingfor penetrationdepth has no f dependence. N. It is also fascinating to see the unstratified scales Now let us investigatethe scalesof the baroclinicin- Urot and /rot appearing in this stratified problem; but stability in the context of the evolutionof the mixed see$peer and Marshall[1995] where they appearas key layer. We knowfrom the solutionof Eady's[1949] baro- parameters in the hydrothermal plume problem also. clinic instability problem that Equilibrium State Nh /eddy~ /•D ~ f --ls . (20) So the initial stagesof the vertical evolution of the mixed layer are governedby the one-dimensionalre- As h increases,then so must ls in accordancewith the sult, which is valid until baroclinicinstability setsin on expressionsderived abovefor h. The numberof eddies a timescalet ~ f-i. At that time the baroclinicin- present is dependent on the integral number of wave- stability facilitates an exchangeof fluid and properties lengthsthat may be containedabout the circumference between the chimney and the surrounding region. This of the chimney, that is, lateral exchangeinhibits the further vertical evolution 27r./•s of the mixed layer, so that the layer deepensless rapidly M ~ (21) than the t 1/2curve. Eventually, an equilibriummust be reachedbetween the densityforcing through the surface Again, this numberwill changewith time as/eddygrows. and the lateral removal of densefluid by the baroclinic This prediction is borne out by the data presentedbe- eddies, as detailed in the work of Legg and Marshall low; initially, h is small, and the chimney begins to [1993].This equilibriumnecessarily halts further deep- break up into eddies. As time goeson, h increases,and ening of the mixed layer; the depth at this time will be one sees the eddies coalesce. Ultimately, the number the maximum depth achievedand may be derivedas fol- of eddies actually shed is smaller than the number of lows. The balance between the buoyancy flux through wavesinitially visible at the onset of instability. At the the surfaceand the lateral transportis [Visbeck,et al., equilibrium depth we expect, setting h = ds, 1996] R•/af Ms z i/a ß / Bøg dA-1] p0 / v'p'dldz(14) o where v is the horizontal cross-stream velocity, the To summarize, we now have scalingsfor the velocity prime denotesdeviation from the average,and the bar of the rim current in the baroclinic zone, the number of denotes a time average. From energy analysis of the baroclinic eddies formed, their associatedlength scale, baroclinicregion we canwrite [Eady,1949; Green, 1970; and the equilibrium depth of the convectivelyproduced seealso Visbeck,et al., 1996] densewater. These scaling predictionsare henceforth v'p'- g p'2, (15) denoted by the additional subscript s Npo u• - Nd• -(Bo R•)•/a (22) where pt is the perturbation density anomaly achieved when moving a water parcel in the baroclinic region. = : Since(as discussedabove) f f gP'~N2h , (16) MsR•2/a f Po = nl/a (24)

vtpt scalesas ds= (RsBø)l/a. v'p'~ P---2-øNah2 (17) g Note that the first three above are independent of N, Then the equilibriumbalance (14) becomes and the last one is independent of f. Finally, it is interesting to considerthe final volume 1B0•rR•~ 12•rRsNaha , (18)of densewater produced during a convectionevent. An g g upper limit is given by the fastest rate at which dense WHITEHEAD ET AL.: CONVECTION IN ROTATING STRATIFIED FLUID 25,709

water may be produced. The shortest time in which Source Water Reservoir the system can reach ds is given by the time it would take the one-dimensional model to evolve to ds. Once the one-dimensional model ceasesto be valid, vertical Spray Nozzel evolution is slowed under the stiffnessimposed by rotation. So the shortest time required by the system to reachds is givenby equatingh in (5) with ds in (25) which gives the timescale Removable Shroud to contain mist ' Then the rate Q at which densewater is producedscales SourceRadius R s as the volume of the cylinder beneath the sourcedivided by ts so

1/3 /•5/3/•2/3 = :v , p Stratified water Many of the abovescalings, but particularly(22)-(25), will now be tested against the laboratory results.

Laboratory Observations of Deep Water Convection in Rotating • ..•• TurntableRotating Stratified Fluid Figure 1. Sketchof the apparatusfor producinga Althoughmany theorieshave been developed and ob- buoyancyflux with a paint sprayeronto the surfaceof servationstaken of rotating convection,little is known a rotating stratified fluid. about the convectiveoverturning of a rotating stratified fluid. Here attempts havebeen made to systemati- callydevelop experiments and techniquesto investigate iment. Lateral circulation of water in the tank from convectionin stratified fluid. We are particularlyinter- small unevenfilling is quite small and is measuredfrom estedin convectionwhich is producedby the addition of the drift of columnsof potassiumpermanganate to be density over a limited surface area since it has obvious less than 0.1 mm s- 1. implications for the ocean. To generateconvection, the additionof salty water Density Increase by Spraying at the surfacewas preferred to surfacecooling. For 60 s a fine mist of almostcompletely salt-saturated The first laboratory experimentwas exploratoryin nature, although some quantitative information about eddy size and velocity magnitudeswas obtained. The apparatus(Figure 1) consistsof a linearlystratified rotating fluid producedby the Oster methodin a square glassrotating tank 113x114 cm 2 whichis 50 cm deep. In the Oster methodthe tank is filled usingtwo identical cylindricalsource tanks, one with fresh water and the other with salty water. Both have free surfacesex- posedto the air. Water from the fresh tank is slowly ':introducedto the rotatingtank througha rotatingcon- • -20 nection and into a bed of fine stones on the bottom of the tank. Meanwhile, water from the salty tank is fed into, and mixed within the freshwatertank, so that the levelsof the two tanks slowlydecrease together. The densityof the introducedwater increaseslinearly with -30 , . time, and eachelement of water injectedspins up in the 1.00 1.01 1.02 stonebed so that the net effectis a rotating stratified bodyof water. Stratificationin a tank filled carefullyby Density(gcm '3) this method can be relatively constant;Figure 2 shows Figure 2. A typicaldistribution of densitywith depth density versusdepth at the start of a typical exper- in the test tank after it is filled by the Oster method. 25,710 WHITEHEAD ET AL.' CONVECTION IN ROTATING STRATIFIED FLUID

Figure 3. Views of eddiesproduced in experimentswith the paint sprayerforcing after the shroudhas beenremoved at (a) 715s after start and (b) 180s after start. The top parts of the figuresview ofcontain the water.a side In thisview experimentalof the water run,through f - 0.15a 415ømirror.s-1, N - 0.43Thes -z,bottom and B0parts - 0.32show cmthe2 s-to•p . bluefood coloring(p closeto 1.2 g cm-a ) is sprayed finished and the shroud was removed. The side view onto the top surface of the stratified water that was in the mirror was visible for the duration of the run. 30 cm deep through a 30-cm-diameter hole. The mist A second, black and white camera recorded a close-up was produced by a paint sprayer, which was located of the side view in order to record the advance of the about 120 cm above the tank. The sprayer was en- mixed layer. The depth of the dyed mixed layer was closed in a lightweight shroud to prevent salt spray occasionallyrecorded by eye with a meter stick during from contaminating the laboratory equipment. The the experiment for additional calibration. sprayer and shroud were quickly removed after 60 s As time progressed,the mixed layer advanceddown- of spraying. The total volume of salt-saturated solu- ward during the first 60 s and stopped advancingwhen tion introduced through the hole was measuredto be the spray was turned off. After the shroud was removed, 61 cma to within 10% error, givinga buoyancyflux it was clear that the patch of densedyed fluid had, more Bo = #ApV/•rr2t of roughly0.3 cm2 s-3. This pro- oftenthan not, brokeninto a numberof eddiesas shown duced greater values of buoyancyfluxes than could be in Figure 3. These eddieswere correlated with cyclonic obtained from surface cooling. For comparisonthe surfacecirculation. However,as the eddy field slowly buoyancyflux gc•H!/•rr2pcp from H!: 500-Wcooling broke up, a deeper patch of dyed fluid was invariably overthe samearea is about0.03 cm 2 s-3. (Forthis cal- seento lie to the right (lookingradially outwardfrom culationwe useda specificheat of 4.1 J g-1 m-1 oc-1 ' the centerof the densitysource) of eachcyclonic eddy. density= 1 g cm-a, g = 980 cm s-2, and a = We now recognizesuch eddy pairs as a common labo- 2 x 10-4øC-1.) ratory exampleof the heton arrangement[Helfrich and Two cameras rotating with the turntable produced Battisti, 1991],in which the anticycloniccomponent of video imagesof the movementof the dyed fluid and of the circulation within such a heton is concentrated at pellets scattered on the top surface. The experiment depth rather than at the surface. Photographs from w•s recordedon videotapefor at least 115rain after the four runs are shownin Figure 4. The variation in eddy sprayer was turned off. A color camera obtained a top sizes and depth of penetration is obvious. Each row has view together with a side view through a 4150mirror. the same value of f with the top row being larger by a The top view was only obtainedafter the sprayinghad factor of 4. Each column has the same value of N with WHITEHEAD ET AL.' CONVECTION IN ROTATING STRATIFIED FLUID 25,711

the left column being larger by a factor of about 4. The 5. The centrifugal accelerationhad to be limited to a correlation of eddy size with row and depth of penetra- small fraction of gravity so that the surfaceof the fluid tion with N is clear. These are the end-members of nine remained almost perfectly flat. This prohibits use of suchruns which showedan unmistakablerelationship of valuesof f > 0.5 s-•. eddy size with f and depth with N. Selectionof values The aboveconsiderations resulted in the followingex- of buoyancy flux B0, fluid depth H, stratification N, perimentalparameter ranges: 0.2 < N < 0.9 s-•, 0.125 and rotation rate f dependedon the followingfactors: < f < 0.5, s-•, and 0.17 < B0 < 0.3 cm2 s-3. Since the scalinglaws predict low sensitivityto Bo, somead- 1. For a set of experimentsin which Rs, N, and f are ditional runs were conductedby varying B0 through systematically varied, B0 is set at the maximum value 2 ordersof magnitude(with N being appropriatelyad- we can produce. This ensures that convection cells are justedto resultin the samepenetration depth). In these as vigorousas possible.The presentvalue of B0 ranged runs, N/f was mostlyless than 1 and so not in the op- from 0.17 to 0.3 cm 2 s-3 in most of the runs. This timal parameter space. produced values of /rot between 1 and 13 cm so that rotational effects were experiencedwithin distancesless Measurements taken and associated errors. than the depth of the tank. The use of larger rather Estimates of eddy size were made by measuringthe di- than smaller valuesof B0 avoidsa problem that arisesin ameter of every clearly visible cycloniceddy recordedon thermal coolingexperiments, in whichcooling is limited the video after the patch of densiftedfluid had spread by heat flux through the top and which typically lead out sufficiently to clearly identify each eddy. Speeds to B0 values which are an order of magnitude smaller were found by measuring the distance that small pa- than used here. If as is typical, rotation rates in the per discs moved in 3 or 4 s. These measurementswere range0.1 > f > 1.0 s-• are usedin suchexperiments, taken during the same time interval as those of radial values of/rot are so small that molecular processesbe- distance. The depth of the mixed layer was identified come important. The only way to removethis problem as the bottom of the dyed layer, and measurementsof in thermal experiments is to use smaller values of f. this depth were taken directly by eye as the experiment However, these experiments require smaller values of progressed(d,), fromthe sideview tape (ds•), andfrom N if the deformationradius is to be small enoughto the top view throughthe mirror (dry). allow baroclinicinstability- (seecomment 3 below). Error in the velocity measurements is difficult to de- This low stratification invariably producesvery shallow fine since the most rapid velocity of the surface pellets mixed layers with baroclinic eddies limited to the top was selected for measurement. The actual measurement centimeter or two near the surface. It is thus difficult to of the position of the particle and the time is very pre- observe. Moreover, the Rayleigh number of this shal- cise(to a coupleof millimetersand to 0.03 s, respec- low layer is small. The present experiment, with its tively), but there is no way to guaranteethat this is supporting scaling arguments and larger values of B0, the true peak velocity in the entire velocity field. We overcomesthese shortcomings. suspectthere is no other velocity twice as great as the 2. Although the largest possiblevalue of B0 is de- values we report, for instance, but the span shown is as sirable, only a small amount of saturated salt solution good as can be obtained with our present facility. should be added to increase the density of the mixed Error for size and wavenumber of the eddies depends layer. Given the salinity of saturated salt solution this as much on identification of the eddies as on experi- fact not only limits buoyancy flux but also was found mental precision. The dye technique appears to allow to limit N < 1.0 s-•, sincefor greatervalues of B0 the every eddy to be filled with dye. In observingthe ed- mixed layer penetrated to the bottom of the tank. dies in real time or by viewing the videotape, we found 3. It was also necessaryto ensure that the baroclinic no evidence that eddies without dye were present. It is eddy size be of the order of or less than the radius of important to keep in mind that the eddiesand currents the source. More importantly, no experiments were con- generate their own mesoscalevariability. ducted in which the eddiesreached the outer wall during Error in measurement of convective depth from the the run. The eddy-sizeconstraint dictates roughly that side view camera was estimated to be about 2 cm for Nd/f < 40 cm. With N equal to its greatest value the most extreme cases. This error was mostly from of ~ 1.0 s-• and d of ~ 10 cm this implies,roughly, uncertainty of parallex corrections, since the side view f > 0.25 s-•. In our experiments,runs with f- 0.125 contained no information about whether a particular gave the largest permissibleeddies, and since eddy size deeply penetrating blob lay closeto the camera or fur- was found to be inverselyproportional to f, smaller val- ther away. Error in the measured depth from the video ues were not used. Indeed, smaller values would result camera that showed both a top view and the side view in the eddiesreaching the outer wall of the tank before mirror was limited by resolution of the video camera. the mixed layer had reached its maximum depth. Each individual video line resolved about 0.5 cm, but 4. It was important to try and conduct the experi- the estimate of depth could be off by a total of four ments in the range with N/f- i or greater, the pre- or five lines depending on lighting, lumps in the pene- dominant range of interest in the ocean. trating eddies, and other visual interpretation aspects. 25,712 WHITEHEAD ET AL.' CONVECTION IN ROTATING STRATIFIED FLUID

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Table 1. ResultsFrom the SprayingExperiments With Rs= 15 cm.

f N Bo d, ds,, dm Eddy Size Velocity (s-x) (s-x) (cm2s-a) (cm)(cm)(cm) (cm) s

0.250 0.83 0.29 - - 7.5 12,18,20 1.0-1.2 0.250 0.41 0.27 - 12 12.5 16,16,18 0.94-1.5 0.250 0.22 0.27 - > 25 b 20,20,21 0.8-1.5 0.125 0.83 0.29 10.2 27,28 1.2-2.1 0.125 0.41 0.27 12 12 14.3 32 1.5-1.6 0.125 0.24 0.29 - 25 b 27,32 1.6-2.1 0.500 0.82 0.29 6 - 7.5 7,8,8,8,9 0.47-0.9 0.500 0.45 0.28 - 12 11.6 6,7,8,8,9 0.6-0.8 0.500 0.20 0.28 - 25 22.5 7,7,8,8,8 0.6-1.3

Thus the error of the two camerasis comparable. Depth shownwith logarithmic axes. The least squareslog-log measurementsranged from 4 to 27 cm, but depth er- regressionformula is shown,and the exponentof-0.97 rorsdo not rangefrom plusor minus8 to 50%, sincethe is very closeto -1. In Figure 5c the sensitivity to vari- small depthsexhibited smaller uncertaintiesin parallex ation of N is shown. There is little variation with N and variation in individual blobs. Thus a more realistic compared to the sensitivity to f. The sensitivity to N rangeof uncertaintiesis from plusor minus8 to 30%. is clearly very weak. The data for eddy size are plotted It should be emphasized, however, that the biggest against(BoR•)•/a/f suggestedby (23)in Figure5d. source of scatter in the data is due to natural variabil- The regressionline has a slope of 2.3 and a correlation ity associatedwith eddies. This scatter is visible in the coefficient of 0.96. All the experiments in Table 1 had rangesshown in the tables and figuresand is not a con- roughly the same value of B0 and so should have the sequenceof measurementerror. The scatter could only samevelocities according to (22). The nine valuesof be reducedsignificantly by a much greater statistical maximum velocity were averaged and then divided by study, which is prohibitivelyexpensive, rather than by the averagevalue of (BoR•)1/3. The resultwas 0.9. improvementof measurementtechnique. Data for mode number were taken as the number of eddiesin Table 1. That number plotted against the for- Paint sprayer results. After somepreliminary mula ,•,•2/3 ½/'-'0•,/•/3 , as suggestedby (24), givesa line of runs, nine experimentswere conductedwith this paint slope 0.95 with a correlation coefficient of 0.98. The sprayer arrangement. The results of these measure- lengths and velocitiesdivided by their scalesare also ments are shown in Table 1 and in Figures 5 and 6. In shown plotted against the two dimensionlessnumbers two casesthe dye was closeto the bottom. The letter b denotes this in Table 1. N/f and R• in Figure 6. Although there is scatter, there is only weak dependenceon these dimensionless Equations(12) and (23) for /rot and l, predictthat numbers. Moreover, the coefficientsare in the range of the size of the eddies is sensitive to rotation but not 0.9 for velocity and 2.3 for length scale. to stratification. This is borne out by the data shown in Figures5a and 5b wherethe eddy size for five ex- Experiments With Steady Buoyancy Flux perimentsis plotted againstthe Coriolisparameter. It is clear that the data are not only relatively insensi- Although the paint sprayer method had the advan- tive to N but that they follow a trend like the inverse tage of producingan unobstructedplan view of the ed- of f. In Figure 5a the data on eddy size exhibit clear dies, its most serious limitation was that the sprayer sensitivitywith f to within a few percent.The average had to be stopped to enable the convection to be ob- eddy sizesand standarddeviation for the three progres- served. This means that, unfortunately, there is no sivelygreater values of f are (in centimeters)28.2-t-3.4, guarantee that paint-sprayer velocities, eddy sizes, or 16.9+ 2.2, and 7.7 + 0.7, respectively.The standardde- penetration depth lie at their final values. The second viations are 13% or lessof the mean values,yet N varies set of laboratory experimentswas designedto overcome by a factor of 4. In Figure 5b the data on eddysize are this problem. To generate a buoyancy flux at the top

Figure 4. Four photographs(directly from videotape)of dyedfluid after the paint sprayeris removed.is a side viewEach throughwastaken an downward angled mirror. from aboveThe twothe toprotating photostank.have In athe value top ofof }ach - 0.5photo•;raph s -• and twobottom have a valueof f - 0.125s -•. The left columnhas N of about0.8 s-•, whereasthe righthas N of about0.2 s-•. 25,714 WHITEHEAD ET AL' CONVECTION IN ROTATING STRATIFIED FLUID

35 4O X N=0.8 + f=.5 + f=.5 30- A N=0.4 + f=.5 3O + f=.5 4' N=0.2 + f=.5 25- n f=.25 n f=.25 2O a f=.25 21)- ß f=.125 ß f=.125

15-

X . ß i ß i i 10- 0.00 0.20 0.40 0.60 0.80

N•/s) (c) 5 ! 0.1 012 0.4 0.6 4O (a) y=l.2+2.3x R=0.96

lOO 30'

X N=0.8 0 ß i ß i ß i ß i ß i ß A N=0.4 2 4 6 8 10 12 14

4' N=0.2

0.1 1 y (d) f(s-1) (b)

Figure 5. a) Eddysize as a functionof Coriolisparameter f for assortedvalues of N. b) Same as in (5a but on log-logcoordinates. A leastsquares fit line and formulaare given.The slopeis very closeto -1. (c) Eddy sizeas a functionof stratificationN. d) Eddysizeas a functionof (BoRs)l/a/f. A leastsquares fit lineand formula are shown.

surface, a cylinder of either 15-, 30-, or the 45-cm diam- was of the desired rate if the standing water depth in eter with Whitman filter paper attached to the bottom the cylinder was 2 or 3 cm. This ensuredthat the flow was placed in contact with the free surface of the water into the tank was spatially uniform. (Figure 7). Althoughthis had the limitation that the In practice, two values of porosity were employed. overheadview of the experimentwas partially obscured, In the 15-cm-diameter cylinder, Whitman 5 paper was the buoyancy flux could be continuedindefinitely, al- used, since it was readily availablefrom the stockroom. lowing the depth of penetration to be monitored over To start the experiment, the fluid depth in the cylinder a long period of time. As in the first experiment, the wasbrought to 1.7 cmby manuallypouring in 300cm a ambient fluid consistedof a linearly stratified rotating of brine. A pump was immediately started and sup- fluid producedby the Oster method in the sameglass pliedfluid at a volumeflux of Q = 0.25-ma s-1. This tank (113x114cm2). The porosityof the filter paper was the rate that water was estimated to seepthrough was selected so that the flux through the filter paper the paper from the supplier's specificationsof porosity WHITEHEAD ET AL' CONVECTION IN ROTATING STRATIFIED FLUID 25,715

-1

y = 0.76 + 0.06x R = 0.49 y = 0.66 + 1.23x R = 0.83 ß i ß i ß i ß i ß ! ! ! ß

2 -7

, y = 2.5 -0.58x R = 0.42 y= 2.5-,0.068x R= 0.58 ß i ß i , i ß ! ß 0

0 2 4 6 8 0.1 0.2 0.3 0.4 0.5 N/f Ro Figure 6. Velocity and size of eddiesdivided by their scalesas a function of 2 dimensionless numbersthat varied in the runs shownin Table 1. Least squaresfits to the data are also shown along with their formulaeand regressioncoefficient.

with a 1.7-cm headß In this manner a constantdepth penetrationcould be measuredas accuratelyas possi- of fluid in the cylinder was maintained. This depth was ble, to within some2 cm. The depth of the dyedmixed measured periodically and found to remain to within layer was occasionallyrecorded by eye with a meter 0.1 cm of the desired depth. Thus the flow of brine stick duringthe experiment.The greatestuncertainty into the stratified tank is estimatedto producea buoy- is from parallex, since some of the fluid is closerto the ancyflux B0 = QgAp/15•'•r= 0.28 cm•' s-a. Paper camera and other fluid is further away. A color camera of different porosity, Whitman 542, was used for the wasused to obtaina top viewplus a sideview through larger-diameterexperiments, 30 and 45 cm. This pro- a 45o mirror. The cylinder obscuredthe central area in ducedvalues of B0 = 0.17and 0.3 cm•' s-a for standing the top view,so only intrusive fluid that spreadlaterally depths in the cylinder of 1.7 and 3 cm, respectively. could be seen from above. The side view in the mirror Smaller depths were used for some experiments,since was visiblefor the entire duration of the experiment. larger heads producedsignificant sag of the paper in As time progressed,the mixed layer advanceddown- these larger cylinders. ward. The principalmeasurement was of the depth of In all experiments the procedure was the same. A the dye. The time for the dye to reach each centime- fixed volumeof brine dyedwith blue food coloringwas added to the cylinder. A pump was then activated to supply identical fluid continuouslyat the rate of vol- • camera ume flux estimated to flow through the filter paper.

The depth of fluid abovethe filter paper was monitored VCR s and monitors and observedto vary by lessthan 30% for the duration of the experimentof either 15 or 20 min. By varying source diffuser the salinityof the fluid in the cylinder,buoyancy fluxes couldbe achievedequal to that of the paint sprayerof source water

0.3 cm•' s-a, downto a value100 timessmaller. rotating table As before,two camerasrotating with the turntable recordedthe movementof the dyedfluid andof pellets scattered on the top surface. The data were recorded i sliprings on videotape for a duration of about 20 min. The cam- Figure 7. Sketchof the apparatusfor producingcon- era recordingthe sideview wasset sothat the depthof vection continuouslywith time. 25,716 WHITEHEAD ET AL.- CONVECTION IN ROTATING STRATIFIED FLUID

: . :• .... . - ;;•,•-•.,:.:i'-!:i!•i;.;?•;;:!i:'":'-'"";?c•;',*•'•?•-•.•?:?;.;-'• --:•' :':;.;.-.:.:.".-':'i•::i•i--• i.-,••-- .-,• 5" "'-:?:'.-?":-":--.' ....;:;::. . --:,-•

...... • ...... :':-..'.•. ....• .....:-....-...?.: ...'...... : •::. '...... :::..:-]: ... . .:• ,-.....

......

.:. •-:•:•;;•:••:,'•.: * •..•,,,:•,:.•.:.. -,•.... .:..:'•....::::::::::::::::::::::::::...:.?.:•.:..•. :..-':•.... •.}..? :.4:'::'- ...... ': '%'•'":':::i. . .:,. .'::..:;-::.

,-?::E• ...... :•...... •:: , •-•.• ?•:;•.. ----,• . •.;•' -..-•..•,•.•/•,.•::.:..: '%"•":'•:*:'?½:.:• . :.-:½:",:•,,...,,•...... •-.• ...... '- .-:-:]•;•...... •'

:• :.• :.•f'•".zX:•:::.:•w:•]•}:...•.•&'...... '"'•:•...... •..•:•:•:-,• ' ..( ':'d '...... :':';":'':":' '"""; :':"::' :''::":""" '"'"";' :':':;:g•}:½.•::i-•:::::-'z::. •.;.•:...... ;•-:,:.,,::½-,-...:•: .:...... -.;..•,•-.•....:....-i;?• ...... •?...;: ..•:..... • :...•: .;..:•-;.•:•i:;'•;:• ....;...:..----• ....

• ...... **:•. " .----•,.•c•..•-.."""-...... --::-f•':•"•re"'--'--•m::;:'i•:...•:•::•:•::

.•. •,.$:::: *'*"• - - ;•:a•...... :;•',,:,.•..• .... "' '...• .s;.-...•'•**•:--•? ....•';½::•:-•"• ...... -":...... -...... "...'.'"" Figure 8. Views of eddiesproduced in experimentswith continuousforcing. The tops of the panels contain a side view of the water through a 45ømirror. The bottoms of the panels show the topview of the apparatus.a) After 150s with f = 0.25s -•, N = 0.22s -•, andBo = 0.32cm2/s 3, the downwardintruding water is splitting. The radius of the sourceis 7.5 cm, and the Rossbyra- diusl• is 5.4 cm. (b) After 300 s, two eddypairs are well formedand are conveyingthe fluid away from the center. The two surface eddies are cyclonicand clearly indicated by the circular spirals that go counterclockwiseinward. The two deeper eddies are not very visible in this picture, but they are locatedroughly under the armsextending from the spiralsto the centerof the tank. c) An experimentwith a largersource radius of 22.5 cm after 150s with f = 0.5 s-•, N - 0.43 s-•, and B0 = 0.17 cm2/sa. A numberof wavesare seen around the rim of the intrudingwater. The Rossbyradius is smaller(l• = 3.1 cm) than for Figure8a and Figure8b. (d) After 360 s a number of eddies transport the dense water sideways. ter depth level was taken visually from the videotape. ally when the advance slowed, the side view revealed The depth levels were calibrated by an image of a ver- considerablelateral slumping of the sides of the dyed tical meter stick that was placed in the middle of the region, but unstable eddies had not yet developed. As tank, then moved to a number of locations closer and time progressed,however, the patch of densedyed fluid further away from the camera. This calibration was broke into a number of eddies as observed in the ear- also checkedagainst other geometricalfeatures such as lier experiments. The mixed layer depth at the longest a background grid and calibration tapes on the tank permissible time was recorded as the final depth in Ta- edges. A correction was also made for bulging of the ble 2. This longest time was either the duration of the filter paper from the hydrostatic load of the dyed fluid. ßrecording or the time when an eddy hit the sidewallsof This was a little over a centimeter in the experiments the tank. with the 45-cm-diameter source and less for the smaller Some aspectsof the circulationare revealedin top source. This sag was visible through the side view mir- and side photographsdisplayed in Figure 8. The dye ror and was directly subtracted from depth data. The would typically collect under the sourceas a rapidly estimate of plus or minus 2 cm includes this correction. deepeningmixed regionfor early times. At somestage, As in the experiments driven by the paint sprayer, the waveswould appear around the rim of the dyed fluid. mixed layer penetrated rapidly downward in the early The dye would quicklyaccumulate at lobeswhich were part of the run. Its rate of advance slowednoticeably af- offsetby a cyclonicsurface circulation. The lobeswould ter a time which was relatively easy to discriminate but pair with such cyclonesand propagate away from the generally never ceasedfor the duration of a run. Usu- source region. WHITEHEAD ET AL' CONVECTION IN ROTATING STRATIFIED FLUID 25,717

.:.

.?:?, ..:;';::::"i'":::::•::. '." '. •:•: *:ii;:.'•::".,,".;11' .el;:: ;;:::-::'"', !;:..;...:. ::-.....:..:.:;: ..:......

;:"-/ ....:!{t;?-':.- ½•:---:-:-:::'S•-•-,::•..

::::. .•,•.:.•:•a• ...... ,•,'.,..."•'•%!, X2'** ?

::**::x.::..:-:.:-...... •..•....

Figure 8. (continued)

Our theory for the final depth suggeststhat there is test this, five runswere conducted (July 14-21), where no dependenceon f. This is well borneout by the data B0 was varied over a range of 2 orders of magnitude. shownin Figure9a, wherethe progressionof the depth This Wasaccomplished by varyingboth the salinity of with time is shown for a number of runs with roughly the injected fluid and the value of N to keep the ratio thesame value of N (about0.4 s- •). Valuesof f range B•/•/'Nconstant. Figure 11 shows the depth of the dye over a factor of 4. Despite a fourfold variation in f, versus time for the five runs. The depth is clearly ap- the depth for large time rangesbetween 15 and 20 cm proaching~ 16 cm for most cases. Three trajectories or about 30%, less than one tenth the variation in f. track one another fairly closely.They correspondto the For experimentsin which N varied by almost a factor three data points that lie in the upper left in Figure 10 of 4, there is a lot of sensitivity,however, as shownin and may correspondto a dynamicbalance (not neces- Figure 9b. The depth varieswith N by about a fac- sarily coveredin our scalingdiscussion) valid for small tor of 4. The scatterin Figure 9a of about 0.3 and N/f. The trajectory for the run denoted by crosses the scatterin Figure 9b of about 4 differby morethan had the largestvalue of N/f, and it actuallyagrees a factor of 10. This is significantlygreater than any most closelywith the data shown by solid squaresin possiblemeasurement error or scatter from mesoscale Figure 10. With sucha small numberof runs one can- variability. Within the range that is thought to apply not be definitive about the value of the exponents of to the theory, for N/f > I and R[ < 0.6, the final the variousterms, but the followingestimate is useful: depth of the convectionas a functionof the predicted Averagingall five final depthsfrom Table 2, we obtain depthdp - (BoR)•/a/Nseems to clusterabout a lin- 13 cm with a standard deviation of 2.1 cm, 16% of the ear regressionline (Figure10). Data from experiments averagevalue. If this deviationis dueentirely to scatter whoseparameters span a factorof 5800are shownin the of n in the formula B '•IN, then the rangeof n around Figure 10. The squaresrepresent the datajudged to be 1/3 is estimatedas In 1.16/ln 10 = 0.06,suggesting that in the most relevant parameter spaceand these have a 0.27 > n < 0.39. rangeof 4 in f, 2.6 in N, and 3 in R• for a total range of 31. The data show reasonablygood scatter about Discussion the least squaresfit line with a regressioncoefficient of 0.81. The constantof proportionalitybetween h and d, The laboratory experiments clearly show that the in (19) is 4.6. downward advance of the mixed layer ceasesafter a A final test of the theory was made from the predic- given time. Then dense water formed by the surface tion that depth should be proportional to •l/3/N.•0 To buoyancy flux spreadslaterally within baroclinic eddies 25,718 WHITEHEAD ET AL.' CONVECTION IN ROTATING STRATIFIED FLUID

l0 25 x f=0.500 N=0.43 + f=0.500 N=0.36 ß f=0.250 N=0.40 20- ß f=0.250 N=0.34

DA l•A r-, f=0.125 N=0.44 l'lA ß f=0.125 N=0.38 l'laa E •5- -10 IX ,r• +% m/ ß X Aa•-i • Xx o 10- ß X + [] ß ß X A ß + x ß X & + ß X ß x -2O ß x _

-30 ' i ' i ß I I ! ß i ß 0 100 200 300 400 500 600

Time (s) (a) (BoRs)1/3 [] N/f>l, Ro*<0.6 N (cm) Ipl q- N/f

X Ro*--0.6 ale + a•X 4- -10 Figure 10. Final mixed layer depth as a function of ß the depth predictedby (19). The datum with R• - 0.6 Aß is shown by a cross;for this caseone might expect little Aß ß ß baroclinicinstability. Data with N/f < 1 for which ß f=0.500 N=0.29 one would expect relatively weak influence of rotation, ß -2O & f=0.250 N=0.22 are shown by pluses. Data with both N/f > i and x f=0.500 N=0.38 /• - 0.6, which are most likely to be in the range of validity of the theoretical assumptions behind the + f=0.250 N=0.40 scaling arguments, are shown by squares. The least ß f=0.500 N=0.80 squares fit has a slope of 4.6 and an offset of- 1.9. [] f=0.250 N=0.86 -30 ' ß , ß i ß i 0 100 20O 300 400 eddy size exhibited significant correlation with formu- Time (s) (b) las for these scales from the theory of Visbecket al., Figure 9. a) The advanceof depthof the mixedlayer [19961. with time for runs with similar values of N, 2 values In contrast to the paint sprayer experiments, the ex- of Rs, and more strongly varied values of f. The plot perimentsdriven through the diffuserwere operatedfor illustrates the relative insensitivity of depth with f and a full 15 min. In all the experiments,eddies had clearly to a lesserextent with Rs. See runs May 11-18 on started to transport density laterally, suggestingthat b]e 2. (b) The advanceof depthof the mixedlayer with the duration of the experimentswas long enoughfor the time for runs with assorted values of N, one value of scalingto be appropriate. Indeed,the observeddepth of Rs, and two values of f. This illustrates the sensitivity of depth with N. See runs May 3-10 on Table 2 the mixed layer had good agreementwith the formula in the following ways: depth showedlittle variation when f was varied with all other parameters held constant; and ambient fluid is drawn in to undergo convection. there was dependenceon N as predicted; five experi- The depth of the mixed layer at this time is consistent mentswith the parameter group Bo1/3/N held constant withthe formul• (BoR)•/•/N in thatit isnot sensitive exhibited similar depths even with B0 varied by 2 or- to f and is proportional to B 0•/•/N . ders of magnitude;and finally, experimentswith N/f The mechanismof lateral spreading by the baroclinic and Ro* > 1 displayed a reasonably linear correlation eddiesstrongly resemblesthe heton mechanism.In the between depth and our prediction of it. experiment we observeda cycloniceddy lying over, but Perhaps the most significantresult is that although laterally offset from, a deeper mass of dyed fluid as the mixed layer deepeningis arrested by baroclinic in- foundanalytically by Crdponet al. [1989].In the nu- stability, a phenomenonthat relies on energy stored in merical experiments we observed cyclonic eddies near the "thermal wind" and hence reliance on f, the final the surfacesoffset from anticyclonic eddies at greater depth does not depend explicitly on rotation rate. In- depths. stead, the final depth of the convectivelyproduced wa- In the experiment driven by the paint sprayer, mea- ter dependsonly on the size and intensity of the forcing surements of the velocity, the mode number, and the and the ambient stratification. Implicit in the deriva- WHITEHEAD ET AL.' CONVECTION IN ROTATING STRATIFIED FLUID 25,719

tween d, and ds is a gauge of the efficiencyof that Bo=0.3 [] process. This balance, then, is important not only in Bo=0.17 X the context of convectionbut also anywhere that the efficiency of baroclinic eddiesis of interest, for example, Bo=0.03 q- in the parameterization of baroclinic eddiesfor use in Bo=0.01 A coarse-scale numerical models. Bo=0.003 ß In the case of convection,where forcing persistslong •o enoughfor the formation of barocliniceddies, we pro- •-• o ß•:+a o posethat the scalessummarized below (denoted by "0" -10- •1- [] [] in anticipationof their applicationsto the ocean) will [] governthe evolutionof a convectivechimney. The con- •x X• Aß A stants of proportionality are deducedfrom our labora- -15 tory experimentsin the preceedingsection. 0 500 1000 1500 R•/3f Time (s) Mo-- 1.0 '•/a (28) Bo Figure 11. Depth versus time in five experiments where Bo was varied through 2 ordersof magnitude'and in which N was adjusted for each run so that depth as vo- . predictedby (25) remainsconstant. l0- 2.3(BøRs)x/a tion of ds is the balance between buoyancy forcing at f , (30) the surface and the lateral transport of dense fluid by baroclinic eddies by the heron mechanism. Again, re- do- 4.6(BoRs call that one of the most important ramifications of the N ' (31) calculated constant of proportionality for depth is an To understand when such scaling would be relevant indication of the efficiency of the baroclinic transport. to the ocean we need to take a closerlook at the length Energy arguments were used in our scaling, assuming of time neededfor mixed layer deepeningto be arrested. that all available potential energy was converted to ki- Recall that the shortest time necessaryfor the chimney netic energy. The constant of proportionality, 4.6, be- to evolve to dfinalis given by

Table 2. Parametersof the ExperimentsWith SteadyForcing.

Final Date Diameter f N Bo Depth N/f (BoRs)x/a/N, cm s- 1 S- 1 cm 2 s- 3 cm cm

May3 15 0.250 0.40 0.30 8 1.6 0.15 3.3 May 4 15 0.250 0.22 0.30 18 0.9 0.15 6.0 May5 15 0.250 0.86 0.30 4 3.5 0.15 1.5 May6 15 0.500 0.80 0.30 4.5 1.6 0.05 1.6 May9 15 0.500 0.38 0.30 9 0.8 0.05 3.4 May 10 15 0.500 0.29 0.30 17 0.6 0.05 4.5 May 11 30 0.250 0.40 0.17 15 1.6 0.11 3.4 May 12 30 0.500 0.36 0.17 16 0.7 0.04 3.8 May 13 30 0.125 0.38 0.17 18 3.0 0.31 3.6 May 16 45 0.500 0.43 0.17 17 0.9 0.04 3.6 May 17 45 0.250 0.34 0.17 17 1.4 0.11 4.6 May 18 45 0.125 0.44 0.17 17 3.5 0.31 3.6 May 20 45 0.250 0.44 0.30 18 1.8 0.15 4.3 May 24 30 0.250 0.37 0.30 21 1.5 0.15 4.5 July 14 45 0.500 0.84 0.30 12 1.7 0.05 2.2 July 15 45 0.500 0.42 0.03 14 0.8 0.02 2.1 July 18 45 0.500 0.22 0.00 14 0.4 0.01 1.9 July 20 45 0.500 0.28 0.01 13 0.6 0.01 2.2 July 21 45 0.500 0.56 0.17 14 1.1 0.04 2.8 July 22 45 0.050 0.19 0.03 11 3.8 0.60 5.1 July 28 45 0.250 0.85 0.27 8 3.4 0.15 2.2 45 0.125 0.90 0.30 6 7.2 0.41 2.1 July 29 , 25,720 WHITEHEAD ET AL.: CONVECTION IN ROTATING STRATIFIED FLUID

The depths predicted above are roughly consistent with recordeddepths of mixingat Bravo [Lazier,1980]. t - •4 B0J ' (32) For example,the estimateof 100 W m-2 is appropri- usingour empiricalconstant of proportionalityfor dfinal ate to winter 1968, where the recordeddepth of mixing and the x/• from the one dimensionalderivation. Us- was 800 m and we predicted 695 m. Winter 1972 was ing the following values typical of a Mediterranean extremelycold with heat fluxesreaching 800 W m-2. convectiveevent, heat flux m 800 W m-2 which, if That winter's recorded depth of mixing was 1500 m, a = 10-qøC-1,implies B0 = 2 x 10-7 m2 s-3 (theac- where we predicted 1262 m. tual in situ valueis sensitiveto temperatureand salinity Given these estimates it seems likely that Labrador and it is probablyless) over an area Rs = 20-45 km, we Sea overturning is affectedby lateral exchangein baro- find using(32) thatt=4-7x l0 ss, orm 4-7 days. clinic eddies. In that case, baroclinic eddies formed at Althoughthe Mistral blowsfor up to a week,forcing of the edgesof the convectionsite provide mechanismsfor this magnituderarely persistsfor more than 1-2 days. the transfer of convectivelyproduced water into the sur- We concludethen that it is unlikely that barocliniced- roundingwater (includingthe DeepWestern Boundary dies are impacting on the formation of Mediterranean Current (DWBC)). deep water (althoughthey are undoubtedlyimportant One might also expect that the final volume of con- in its dispersalonce formed). However,the discussionof vectively produced water manufactured by one of these the duration of the coolingevent is subjectto so many events would be governed by the above scaling theo- local factorsthat further refinementis necessarybefore ries. LabradorSea Water (LSW) is associatedwith the being applied to any particular ocean event. (DWBC) as it leavesthe Labrador Basin, and trans- Our estimates can be used to indicate what condi- ports have been calculated from hydrographic sections tions are neededfor deep water formation. Let us take through the current. However, very little is known as an example a preconditionedregion with a radius of about the frequency, duration, or position of convective 50 km, N = 10-a s-1, with B0 = !.25 x 10-7 m2 s-3 events in the Labrador Sea. It is not clear that water (correspondingto a relatively large heat flux of 500 with the characteristicsof LSW is produced every year; W m-2). We find that dois about850 m. Howcould but the signal of LSW associatedwith the DWBC is rel- deeper penetration be attained? It is unlikely that B0 atively steady. In unravelingthese questionsit will be could be significantlylarger. However, Rs and N de- useful to understand how much water is modified in a pend on the preconditioningprocess, so let us examine given convectionevent. Such estimates can be made us- their effects. To obtain a large increasein predicted ing the ideas here given a hydrographicor survey of the depth, for instance by a factor of 2, R• would need to ambient stratification of the region and meteorological be 8 times greater, whereasN would only have to be surveys of typical storms. 2 times smaller to get a similar changein depth. Thus the formula indicates that deep convectionis sensitive Conclusion to preconditioning, since gyres of moderate size with lower N are preferredfor deepconvection compared to Without doubt, the independenceof convectiondepth larger gyreswith larger N. from f is interesting and important. It may be sur- Now let us considerthe Labrador Basin, assuming prising to many, and the thorough proof of the scal- weather station Bravo to be typical of the entire basin. ing theory that leads to this result is not completely There is some evidencethat the buoyancyflux mea- straightforward. Perhapsthe easiestway to summarize sured here is representative of the basin based on an- the result is that the velocity of the eddies scaleswith alyzedfields from EuropeanCentre for Medium-Range Nd and that buoyancyscales with N2d, so buoyancy Weather Forec•ting. Then, choosingaverage winter flux integratedover depth scales as Nada and that must valuesfrom the Bravorecords over several years, we find be integrated around a circumference2•rRs. In steady for warm wintersa basin-wideheat lossof 100 W m-2, state, that flux times circumferentialarea, must be pro- and for extreme cold winters values of 600 W m -2. portional to buoyancyflux B0 times surface area, and These heat lossespersist for a month or more. this leads to the scaling for depth as presented. The Usingan N m 9 x 10-4 s-1 calculatedfrom CTD data entire argument dependson the selectionof Nd for the stratificationencountered by the convectiveoverturning velocity scale and the conceptthat eddy flux equals and settingRs • 100 km with a heat flux = 100 W m -2 velocityscale times buoyancyscale. If the velocityis (implyinga B0 = 2.5x 10-8 m2 s-a), (32)gives a depth ultimatelydriven by the conversionof potentialenergy d = 695 m with a minimum formation time of t = into kinetic energywith little dissipation,the aboveve- 33.5 days,long comparedwith f0 instability time series. locity scalewould prevail. However,it is possibleto The caseof B0 = 600 W m-2 correspondsto a final argue that other velocityscales could be producedby depth d = 1262 m and a minimum formation time of surfacecooling. t = 18.5 days. Note that the strongerthe forcingis, the However, a variety of data consistentwith these scal- f•ter the convectivepenetration and the deeperthe ing relation have been found. Thirty-one laboratory mking is. experimentswith penetrationof a cooledlayer into a WHITEHEAD ET AL.: CONVECTION IN ROTATING STRATIFIED FLUID 25,721 rotating stratified fluid produce data that are consistent Killworth, P. D., The mixing and spreadingphase of MEDOC with the proposedscaling. Nothing has been found at 1969, in Progress in Oceanography,vol. 7, pp. 59-90, variance. Although the experiments are by no means Pergamon, Tarrytown, N.Y. 1976. completely definitive becauseof scatter from mesoscale Lazier, J. R. N., Oceanographicconditions at oceanweather variability, we feel that the results provide strong sup- ship Bravo, 1964-1974, Atmos. Ocean, 18,227-238, 1980. port for the scaling. Legg, S., and J. Marshall, A Heron model of the spreading phase of open-oceandeep convection, J. Phys. Oceanogr., Acknowledgments. Support from the Office of 23, 1040-1056, 1993. Naval Researchis gratefully acknowledged.G. H. was Madec, G., M. Chattier, P. Delecluse, and M. Crdpon, A supported by an AASERT grant. Support for labora- three-dimensional numerical study of Deep-Water forma- tory experiments were funded by the Ocean SciencesDi- tion in the northwestern Mediterranean Sea, J. Phys. vision of the National ScienceFoundation under grant Oceanogr.,21, 1349-1371, 1991 OCE92-01464. This is Woods Hole OceanographicIn- Marshall, J., J. A. Whitehead, and T. Yates, Laboratory and stitution contribution 9104. numerical experiments in oceanic convection, in Ocean Processesin climate Dynamics: Global and Mediterranean References Examples, edited by P. Malanotte-Rizzoli and A. R. Robin- son, p. 173-201, Kluwer Acad., Norwell, Mass., 1994. Bubnov, B. M., and G. S. Golitsyn, Temperature and veloc- Maxworthy, T., and S. Narimousa, Vortex generation by ity field rdgimesof convectivemotions in a rotating plane convection in a rotating fluid, Ocean Modell., 92, pp. fluid layer, J. Fluid Mech., 219,215-239, 1990. 1007-1008, Hooke Inst., Oxford Univ. 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