Proc. Natl. Acad. Sci. USAVol. 77, No. 9, pp. 5043-5047, September 1980Geophysics

Meridional heat flux of the North Atlantic Ocean(ocean heat budget/climate)

CARL WUNSCHDepartment of Earth and Planetary Sciences, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139

Contributed by Carl Wunsch, June 26, 1980

ABSTRACT Using self-consistent geostrophic general cir-culation estimates for the North Atlantic Ocean as determinedby an inverse method, I calculate the geostrophic and totalmeridional heat flux. The result is to a large extent dictated bythe assumption of a fixed volume flux through the FloridaStraits. There is no fundamental discrepancy between oceanicheat fluxes deduced from atmospheric residuals and those es-timated directly in the ocean, including the estimates madehere, simply because the error bars on all calculations are verylarge.

The partition of the global flux of heat from equator to polebetween the atmosphere and ocean is one of the critical prob-lems in understanding present climate and its potential changes(1), and a considerable literature has collected on this question(2-8). Using radiation budget calculations, largely inferred fromsatellite data, plus direct observations of the atmosphere, Oortand Vonder Haar (3) found, from their residual, that the oceanshould be carrying about 50% of the poleward heat flux at about200N i.e., about 2.9 PW (1 PW = 1 petawatt = 1015 watts).It is not always remembered that the formal error bars on thiscalculation are quite large (making the residual not significantlydifferent from zero) and that the systematic errors may becomparable (8).From bulk transfer formulas, Bunker (9) found that poleward

of 200N in the Atlantic, total air/sea transfer could account for1.1 PW (6). This is less than 50% of the value formally requiredby Oort and Vonder Haar. From the results of Stommel andCsanady (5) one has the impression that the Pacific Oceanprobably carries only a fraction of the heat of the Atlantic, andin fact may be carrying it the wrong way [i.e., equatorward, asBryan (2) found at 320N]. Hastenrath (8) applied bulk formulacalculations similar to those in ref. 9 to all three oceans. Hisvalue at 20'N plus his substantial poleward flux in the Pacificis quite similar to the Oort and Vonder Haar values.

Bryden and Hall (6), using a hydrographic section across theAtlantic at 240N (10), estimated the meridional heat flux as 1.1PW. Their method is similar to that in ref. 2. Note that theBryden and Hall value is in close agreement with Bunker'snumbers. Using the same data, in conjunction with other In-ternational Geophysical Year hydrography, Roemmich (7)estimated the heat flux across this same section as 1.2 PW andas 0.8 PW at 360N. His results are based upon an inversemethod (11). Roemmich (7) attempted to force the heat fluxacross this latitude to be as large as that required by Hastenrath(8); he found that this high heat flux was incompatible with therequirement of mass conservation across the line (to be moreprecise, it is incompatible with present estimates of how largethe deep water velocities in the Atlantic are likely to be, a pointdiscussed further below.)

Here I will make another attempt to estimate the North At-lantic heat flux by direct calculation from the oceanographic

data. The result will then be compared to some of the previousheat flux estimates in order to determine whether there is somereal incompatibility of the results.

MODELSThe discussion will be pivoted on a group of model circulationscalled North Atlantic-i (NA-1) and also based upon an inversemethod (11). The circulation per se will be discussed elsewhere.For our present purposes only an outline is necessary. The hy-drography used is displayed in Fig. 1; almost all of it can befound in refs. 10 and 12. The underlying assumption of theNA-1 models is that in the subareas displayed in Fig. 1 the flowis geostrophic and mass is conserved (although not all possiblesubareas were used to generate constraints). If a particular watertype (described as lying between two potential density surfaces)does not intersect the sea surface within a subarea, then its massis conserved to within the observational errors in that region.Total mass (top to bottom) is also conserved. Notice that mod-ification of water mass types does occur in the circulationscheme, but substantially so only when the water mass contactsthe atmosphere and no conservation constraints are imposed.In addition, it is required that the total salt in any given subareabe conserved [expressed as anomaly relative to a standardtemperature-salinity (T-S) curve]. The data used in ref. 7 area subset of those employed here, consisting of the three zonallines at 24°, 36°, and 480N.

Let vR(xi,z) = vRz be the geostrophic velocity relative to somearbitrary reference level in station pair i at depth z (the thermalwind). Let bi be the unknown reference level velocity. Then,following ref. 11, we write the various conservation constraintsas a set of conditions upon the b1 in the form

A b = - F, [1]in which the elements A,1 are normally the area occupied instation pair i by water type j (this will be the full area of the pairif the particular conservation statement applies to total mass)and rF is the mass flux of water type j that is not conserved withthe initial reference level.

Because the Atlantic is essentially closed to the north (13), thetotal flux of mass and salt across any complete zonal section mustvanish. Constraints with this effect were part of Eqs. 1. In ad-dition, the geostrophic heat flux across a zonal section may bewritten

HF = cP, P(VR(Xi,Z) + b1)T(xi,z)dz Axi [2]

or

F, Axibi fi dz

= HF- cp 2 Axi

dIdz, [3]

Abbreviation: NA, North Atlantic.

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-60 -50 -40West longitude, °

FIG. 1. Hydrographic station pairs (dots) used to compute generalcirculation models. The total transport (top-to-bottom integratedmass flux) is displayed for one of the models, NA-1A. The contourinterval is 10 metric megatons/sec. Numerals are local transports inmegatons/sec.

in which di is the water depth and Axi is the station spacing ofpair i, cp is the heat capacity of water, p is density, T is tem-perature, PVRi = p(Xi,Z)VR(xi,z), etc. If we define

H = c Axi d pT(xtz)dz}

(the vector of station pair heat contents), we have then

E Hibi = HF - cP F Axi pVRiT(xi,z)dz.i i~~~~~~/

HEAT FLUXThe heat fluxes as a function of latitude for NA-1A and NA-1Bare displayed in Fig. 2; despite the considerable difference ininitial reference level, the two results are very similar, and at-tention therefore will be focussed primarily upon NA-1A.We display in Fig. 3 the determination of the North Atlantic

heat flux by Bunker (9) (summed by M. M. Hall and H. L.Bryden, personal communication) the annual mean global valuerequired by Oort and Vonder Haar (3), and the value computedat 240N by Bryden and Hall (6) as well as Roemmich's (7)preferred values at 240 and 360N.The geostrophic heat flux computed in NA-1 is not the total

oceanic heat flux. We must add the Ekman (wind-driven) heatflux. The Ekman contribution may be written approximatelyas

HFE = Cp -TE,Pfin which r is the zonal component of wind stress and TE is themean temperature of the water moving in the surface Ekmanlayer. At 240N Bryden and Hall (6) took the Ekman depth tobe 50 m, TE = 260C, and a value of VE = Tlf = 5 X 106 m3/secfrom ref. 15. This 5 X 106 m3/sec flowing poleward must bereturned at depth. If this return flow is assumed to occur at theaverage temperature of the section (5.40C and probably too lowa temperature), the net flux of heat due to the Ekman com-ponent is then 0.43 PW. This value has been added to theNA-1A values in Figs. 2 and 3; a similar procedure was used toestimate the Ekman flux at the other latitudes in the models.At latitudes poleward of 300N, the Ekman flux is equatorward(i.e., negative).

+101

[4]

Because this equation is in the same form as those in Eq. 1, itwas carried as part of the inversion to find the solution b. Onecan impose a heat flux HF on the solution, or allow it to be de-termined from the rest of the system. We chose the latter pro-cedure by multiplying Eq. 4 by a very small number, and de-termining HF from the system residual.

Eqs. 1 were suitably normalized and scaled (14). They werefound to be underdetermined, and rank deficient as well. Therewas nominally N = 128 equations in M = 411 unknowns bi, butthe system rank, k, was determined to be about 75. The systemwas solved by the singular value decomposition in the form

k U7T(- F)b = L T(- ) v k =75,

i=1 Xi

in which the X4 are the nonzero singular values, and

A = U A VT,

34

[5] l

,

vz

in which the U vectors are the columns of matrix U and similarlyfor V; A is the diagonal matrix such that A1i = Xi. The choiceof rank 75 in Eq. 5 leaves a residual in Eq. 1. [For present pur-poses I am neglecting the difference between the system (Eqs.1) and the actual scaled and weighted system that wassolved. ]

Models NA-1A and NA-lB differ only in the initial referencelevel assumed [2000 decibars (20 megapascals), and the bottom,respectively]. The vertically integrated mass transport fromNA-IA is shown in Fig. 1.

10 20 30 40 50 60North latitude,

FIG. 2. Meridional heat flux as a function of latitude for modelsNA-1A and NA-IB; the values are geostrophic contribution only. Thetotal, geostrophic plus Ekman, heat flux is shown for NA-1A. TheBryden and Hall (6) geostrophic component at 24°N is also shown(denoted BH). The time of year in which the individual hydrographicsections were obtained is noted along the 1013-W coordinate. Noticechange in sign from top to bottom of figure.

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'0 ~~~~~~~mean

i+1015 rath0 ~~~BH

rtange OV SeptI\ N

Roemmich \L

+lol4NA-1A-0<^KBunker;} ~~~~~~~total mean

°l-1014 _ OV annuanal-

co mean~~~~V ea

_101S -10 20 30 40 50 60

North latitude, °

FIG. 3. Total heat flux in NA-1A compared to Bunker's (9) an-nual mean (M. M. Hall and H. L. Bryden, personal communication),the Oort and Vonder Haar (3) annual mean (denoted OV), Hasten-rath's (8) calculation from bulk formulas (similar to Bunker's),Roemmich's (7) preferred values from another inverse calculation,and the total from ref. 6 (BH) at 241N. Annual extreme values forsome of the Oort and Vonder Haar estimates are also displayed, asis an estimate of the annual cycle in the ocean at 24°N obtained fromthe transport cycle of the Florida Current. Estimates of the standarderrors for the NA-1 calculation are shown at a few latitudes. Noticethat NA-1 curves cross the OV curve at high latitude.

The present estimates of total oceanic heat flux are plottedin Fig. 3 together with several previous attempts. Bryden andHall (6) pointed out the nearly exact agreement of their 24°Nvalue with that of Bunker (9). Given the large uncertainties inthe bulk formulas, and the error field for the oceanic values thatwill be discussed below, this agreement is undoubtedly fortui-tous. But it is important to notice that the direct oceanic cal-culations, even though differing in detail, all yield essentiallythe same value at 24ON, a point to which we will also return.

At middle latitudes, Fig. 3 shows that all the oceanic etMateslie below those of Oort and Vonder Haar; if the Pacific Oceandoes not make a major contribution toward the difference (wehave already noted the possibility that the North Pacific heatflux is actually equatorward), then we need to explore thequestion of whether there is a real discrepancy between theoceanic and meteorologically determined values. Noticehowever, that the NA-1A values actually cross those of VonderHaar and Oort at about 530N and the only possible discrepancyis at middle latitudes.

Bounds on the oceanic heat fluxThe first clue as to how representative these estimates may becomes from the close similarity of the heat fluxes for bothNA-IA and NA-1B and for the Bryden and Hall value at 24°N.The convergence of the values is, despite some considerabledifferences in actual geostrophic flow patterns, essentially in-evitable.

There are a number of ways to quantify the possible rangeof heat flux values given that the inverse solutions (and all otherknown schemes) are essentially underdetermined.

First, notice that the models were constrained to conservetotal mass and total salt across any zonal line. These two con-straints may be written as

EbiAxi fd p(xz)dz

-E Axi fd p(xlz)vRjdz [6a]

or

E biRi = -g, [6b]

in which Rj is the mass per unit length of station pair i and glis the mass flux due to vR, and

E bi Axi X pjS(xjz)dzadi

=- F, AX, p(xjz)S(xi,z)vRidz [7a]ioor

E bjCj = - g2, [7b]in which C, is the salt content of pair i and g2 is the salt flux dueto VR. Now, generally speaking, temperature, salinity, anddensity are highly correlated in the ocean. If, for example, theT-S relationship were purely linear, then the requirement thatsalt be conserved (Eq. 7) also requires that there be no heat fluxeither-i.e., compare Eq. 4 and Eq. 7 with T = aS, a a constant(in the absence of a significant atmospheric transport of watervapor). Stommel and Csanady (5) point out the intimate rela-tionship between the shape of the T-S curve and the ability ofthe ocean to transport heat. The capacity of the ocean totransport heat and at the same time dispose of the highly cor-related flux of salt is restricted.We can write

T = aS + flp + e,

in which a and : are constants and e is a residual. At 360N thecorrelations between T. p, and S are such that only 4% of thetemperature variance is not accounted for linearly by S and p.Thus heat transport across this line occurs in two ways; via thesmall fraction of the temperature variability that is not linearlydependent upon S and p, and the fact that there are small re-siduals in Eqs. 6 and 7. These residuals, in turn, may be inter-preted in several ways. First (5, 15), there is a real water vaporflux in the atmosphere, which must be balanced by a returnflow in the ocean. Stommel (15) has estimated that in this regionthe water vapor flux in the Atlantic is of order 0.08 X 106 m3/secsouthward; in addition, there may be as much as 1 X 106 m3/secflow from the Bering Sea into the Atlantic (16). Both of thesesouthward flows are a small fraction of the Ekman flow esti-mated above. If they enter at 25'C and exit at 00C (an extremerange and impossible for Bering Sea water) the maximumchange in our estimated heat flux is about 0.1 PW. Thus theexistence of small residuals in the conservation equations(whether from atmospheric water vapor fluxes, net inflows fromthe Bering Sea, or-as is more likely to be the dominant portionhere-errors in the data) is unlikely to modify the heat fluxestimates in any significant way.

Given, then, the basic validity of the constraint equations,we can attempt now to quantify the differences between theheat fluxes of the two models and any other acceptable circu-lation model. Because the system of Eq. 1 is both underdeter-

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mined and rank deficient, the matrix has a null space spannedby vectors Ni such that

ANi = 0,i=k + 1,M (76,411)and we can add to b any combination of Nj we wish withoutviolating the conservation requirements-i.e., we can make thesubstitution

Mbo-b+ Z oyjNp [9]

p=k+ i

in which the yj are completely arbitrary because there is noinformation about them in the conservation laws. Thus to withinthe small residuals

EZRiNji = RTNj = 0 [10]

and

CNj = CTNj = 0,

i.e., both the mass content and the salt content of all the stationpairs are nearly orthogonal to the null space of A (Ci and Ri aredefined in Eqs. 6 and 7, and N1p is element i of vector N1).The "true" b differs from any model-determined b only by

a sum of vectors of the form of Eq. 9, and we could find the trueb if we had enough independent constraints. Let us thus writethe "true" geostrophic heat flux as

HF = cp E Axi fd Pi(VRi + bi + E yjNji)T(xiz)dz

AxE jCP f piVRiTidZ + E HibiJ

+ E yjHTNj. [11]I

The first two terms are the heat flux determined from any givenmodel upon choosing the correct rank. The third term is thatpart of the heat flux governed by the components of b that wehave not been able to determine. The total velocities in the twomodels are vRA + bA and VRB + bB, in which these are therelative and reference velocities of the two solutions A and B.The difference between these two velocity fields, and the dif-ference from any other acceptable solutions for the circulationpattem,'can differ only by a projection onto the null space. Thusthe difference between any model determination of the heatflux and the true heat flux depends upon the projection of thevector H onto the null space corresponding to the last term ofEq. 11. But'because T, S. and p, and hence H, C, and R, are sonearly linearly dependent, the projection of H onto the nullspace is very slight. For example, the fraction of the length Hin the 240N section accounted for by the 75 Vi is in the rangeof 98%, leaving only 2% that can produce heat flux differencesin different models (i.e., HTVVTH/HTH - 0.98) and is simplyanother way of demonstrating the strong linear dependenceof T upon S and p. But because the projection onto the nullspace is nonzero, one could produce heat fluxes of arbitrary sizeand sign by introducing arbitrarily large yj into expressions 9and 11. The price paid would be a rapid growth in the totalvariance of b, which quickly becomes unreasonable. This is theexplanation of why Roemmich (7) could not force an arbitrarilylarge heat flux across 24ON; it would be better to say that sucha heat flux could be found but would require oceanic velocitieson very small spatial scales far beyond any ever observed.

This latter argument implicitly assumes that the Nj showmore small-scale structure than do the Vj. Generally speakingthis conclusion is correct; the velocity field lying in the nullspace is of higher wavenumber-i.e., is "wigglier"-than that

in the range of A. But one might anticipate the overall oceancirculation could be simply multiplied by an arbitrary scalarso as to make the entire system run much more rapidly orslowly, thus greatly increasing or decreasing the heat fluxes.Such a large scale arbitrariness is prevented in the present cal-culation, as well as in those of refs. 6 and 7, by fixing the totalwater flux through the Florida Straits at around 30 X 106 m3/sec(there are slight deviations from this value in the calculations,which are of no significance). This number is based upon thedirect measurements of Niiler and Richardson (17) and it servesthe purpose of setting an overall strength to the circulation toa large (but incomplete) degree. If much smaller numbers areused for the Florida Straits transports, then much smaller overallheat fluxes result. The entire set of solutions is thus dependentupon the validity of this number, which is suggested (17, 18)to be accurate to roughly + 3 X 106 m3/sec. Compared to mostoceanographic parameters, this is a well-determined value, andits availability tends to focus attention on the 240N section.The annual cycleUsing the estimate (17) of the annual cycle in the FloridaCurrent from a summer maximum of 33.6 X 106 m3/sec to awinter low of 25.4 X 106 m3/sec, one can make a crude estimateof the seasonal range of oceanic heat flux. This is displayed inFig. 3 at 240N; it was computed by assuming the seasonalvariation in transport can be multiplied by the average tem-perature of the Florida Straits, and that the changed flow re-turns at the average temperature of the Bahamas to Africasection [essentially the basis of the Bryden and Hall estimateof the mean heat flux (6)]. A more sophisticated estimate couldbe made by deducing the entire change in the general circu-lation under a variety of imposed fluxes at the Florida Straits;but, because we do not know the complete seasonal change inbaroclinicity, such a calculation does not seem worthwhilenow.The implied change in seasonal heat flux at 240N is, as seen

on Fig. 3, fairly small. In addition, there should be a contribu-tion from the seasonal change in Ekman flux, but this is unlikelyto enlarge the range very much.The eddy fluxIt is, then, unlikely that one can find much higher heat fluxesin the ocean if the data being used are at all representative. Theonly remaining possibility lies in eddy heat fluxes. But these tooseem insufficient to change the picture much. In the interiorocean, away from the western boundary currents, directmeasurements of v'T' give values too small to contribute sig-nificantly. Of course an eddy heat flux in a region of strong T-Scorrelations has the same problem of disposing through mixingof the corresponding eddy salt flux as does the large-scaletime-average flow.To some extent the eddy field is accounted for in the model

calculations already described. Along any section, we maywrite

v = vo + v', T = To + T',

etc., in which

(v) =vo, (T) =Toetc., in which the bracket denotes a spatial average and the vsare total velocities (including b). The heat flux may thus bewritten

HF =cpvoTO X area + cp (v'T') X areaorHF = cpvoTo X area + CpCVT (T'2)1/2 (v'2)'/2 X area,

in which

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CVT = (v'T')/((T'2)1/2 (V'2)1/2)is the spatial correlation function. In general, this correlationis quite small (at 240N it is -0.06, at 40'N it is 0.04). The totalheat fluxes plotted in Figs. 2 and 3 of course include the con-tributions from these correlation terms even though they tendnot to be statistically significant and like all correlations tendto be biased away from 0. Taken at face value, they can be re-garded as eddy fluxes of heat, although we cannot distinguishbetween "standing eddy" contributions and purely temporaleddy contributions and there are problems of spatial resolutionof the data.One can define (2) zonal and vertical average temperature

fields in each section and rewrite the heat flux as

HF = cp fTj(z)pvj(z)dA + cpf Tj'(x,z)pv1'(x,z)dA [12a]

orHF = cpfT2(x)pv2(x)dA + cpf T2'(x,z)pv2'(x,z)dA, [12b]

in which the primed variables are the corresponding deviationsfrom the averages. The heat flux contribution from the firstterm on the right in 12a tends to dominate that of 12b and ac-counts for most of the total heat flux. This result suggests, asprevious investigators have found, that the heat flux is governedby an overturning cell rather than by the more intense verticallyaveraged flows characterized by the Gulf Stream and itslarge-scale returns. Evidently, mixing of water with the largetop-to-bottom temperature contrast provides the dominant heatflux.

At 240N much of the warm salty water that is going north-ward in the Florida Straits is returning southward as a deep flowof cold fresher water. Of the 31 X 106 m3/sec total, about 16X 106 m3/sec is returning equatorward, fresher, at a temper-ature of less than 4.60C. The conversion of one water type toanother occurs at high latitudes in the model, where the watermass is exposed to the atmosphere.

If T' and v' are referenced to the average temperature andvelocity of each section, they tend to -be highly nonnormallydistributed. In order to gain a crude, but probably not unreal-istic, error bar for the heat flux, the standard errors for corre-lation coefficients for variates T1' and v1' of Eq. 12a werecomputed (19). The number of degrees of freedom was chosensomewhat arbitrarily to be 3 times the number of station pairsin the section (3 vertical degrees of freedom). The computationof a more rigorous error bar is complicated because the bi arelinearly dependent upon the vRi and both fields contribute afraction of the heat flux. The resulting standard errors of theheat flux for this oversimplified estimate are shown on Fig. 3at some latitudes.

Generally speaking, the error bars grow with increasinglatitude, reflecting the increasing complexity of the baroclinicstructure. The error at 240N, is surprisingly small. Inspectionof the section in ref. 10 suggests that it is almost pathologicalin the featurelessness of the baroclinicity

DISCUSSIONThere is a pleasing gross similarity between the Oort andVonder Haar (OV) values and those deduced from NA-1, andthere is no quantitative discrepancy between the oceanographicvalues and the meteorologically derived ones. Oort and VonderHaar (3) suggest that the standard error of their results is 2-3PW at low latitudes and 1-2 PW at mid-latitudes. Within theseerror bars one cannot distinguish the oceanographic and at-mospheric estimates (at some latitudes one cannot distinguishthe Oort and Vonder Haar values from 0). In addition, Has-tenrath (8) notes that reasonable estimates of systematic erroralso remove any discrepancy of the size displayed in Fig. 3.

The NA-1 models, and all similar large-scale circulationschemes for the ocean, combine hydrographic data from dif-ferent seasons of different years. The months when the datawere obtained are denoted in Fig. 2. Thus the October 1957240N section is forced by the circulation scheme to be com-patible with the April-May 1959 section at 360N. I have plottedon Fig. 3 the Oort and Vonder Haar estimates of the range ofthe seasonal cycle in the ocean transport of heat at several lat-itudes. At 20'N, for example, they estimate 0.3 PW is goingequatorward in July, whereas 5.4 PW is headed poleward inMay.

It is a problem to understand how the ocean could producesuch a large variability in the meridional heat flux. There aretwo basic mechanisms available. If, because of the seasonalwarming, the southward return flow near surface occurs atmuch higher temperatures in the summer than in the winter,without a corresponding increase in the temperature of thepoleward flow, then much more heat can be carried equator-ward. The second mechanism would correspond to a changein the barotropic flow-i.e., in b. This latter mechanism, asnoted above, would make little difference, at least at 24ON. Butcoupled with possible, but at the moment almost completelyunknown, changes in the Pacific Ocean heat flux with season,it appears that, within the overall error bars, such changes mightbe accomodated. The enormous size of the annual range sug-gested in ref. 3, and the shortness of the temporal record, impliesa great uncertainty inall the averages.Our basic conclusion then is that there is no fundamental

discrepancy between the meridional heat flux estimates com-puted from atmospheric residuals and those computed directlyfrom oceanic observations. Plausible estimates of the uncer-tainties in both types of calculation yield overlapping values.The oceanic data base needs to be improved to permit the cal-culation of true average values, and an understanding of theannual cycle of heat flux and storage.

This work was supported in part by the National Science Foundationunder Grant OCE78-19833. This is Mid-Ocean Dynamics Experimentcontribution 143 (Polymode). This paper was written while I was onleave at the Center for Earth and Planetary Physics, Division of AppliedSciences, Harvard University.

1. National Research Council (1979) Carbon Dioxide and Climate:A Scientific Assessment (Natl. Acad. Sci., Washington, D.C.).

2. Bryan, K. (1962) J. Geophys. Res. 67,3403-3414.3. Oort, A. H. & Vonder Haar, T. H. (1976) J. Phys. Oceanogr. 6,

781-800.4. Bennett, A. F. (1978) J. Phys. Oceanogr. 8,785-798.5. Stommel, H. & Csanady, G. T. (1980) J. Geophys. Res. 85,

495-501.6. Bryden, H. L. & Hall, M. M. (1980) Science 207,884-886.7. Roemmich, D. (1980) J. Phys. Oceanogr., in press.8. Hastenrath, S. (1980) J. Phys. Oceanogr. 10, 159-172.9. Bunker, A. F. (1976) Mon. Weather Rev. 104, 1122-1140.

10. Fuglister, F. C. (1960) Atlantic Ocean Atlas of Temperature andSatinity Profiles and Data From the International GeophysicalYear of 1957-1958, Woods Hole Oceanographic Institution Atlas(Woods Hole, MA), Ser. 1.

11. Wunsch, C. (1978) Rev. Geophys. Space Phys. 16,583-620.12. Worthington, L. V. & Wright, W. R. (1970) North Atlantic

Ocean Atlas of Potential Temperature and Salinity in the DeepWater, Including Temperature, Salinity and Oxygen ProfilesFrom the Erika Dan Cruise of 1962, Woods Hole OceanographicInstitution Atlas (Woods Hole, MA), Ser. 2.

13. Leetmaa, A. & Bunker, A. F. (1978) J. Mar. Res. 36, 311-322.14. Wiggins, R. A. (1972) Rev. Geophys. Space Phys. 10,251-285.15. Stommel, H. (1980) Proc. Natl. Acad. Sci. USA 77,2377-2381.16. Coachman, L. K. & Barnes, C. A. (1961) Arctic 14, 147-161.17. Niiler, P. P. & Richardson, W. S. (1973)J. Mar. Res. 31,144-147.18. Richardson, W. S., Schmitz, W. J., Jr. & Niiler, P. P. (1969)

Deep-Sea Res. Suppl. 16,225-231.19. Kendall, M. G. & Stuart, A. (1969) Distribution Theory, The

Advanced Theory of Statistics (Griffin, London), Vol. 1.

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