Downloaded 10/02/21 06:02 AM UTC 2342 JOURNAL of PHYSICAL OCEANOGRAPHY VOLUME 33

NOVEMBER 2003 MARSHALL AND RADKO 2341

Residual-Mean Solutions for the Antarctic Circumpolar Current and Its Associated Overturning Circulation

JOHN MARSHALL AND TIMOUR RADKO Massachusetts Institute of Technology, Cambridge, Massachusetts

(Manuscript received 21 May 2002, in ®nal form 25 March 2003)

ABSTRACT Residual-mean theory is applied to the streamwise-averaged Antarctic Circumpolar Current to arrive at a concise description of the processes that set up its strati®cation and meridional overturning circulation on an f plane. Simple solutions are found in which transfer by geostrophic eddies colludes with applied winds and buoyancy ¯uxes to determine the depth and strati®cation of the thermocline and the pattern of associated (residual) meridional overturning circulation.

1. Introduction the ACC eastward and, through associated Ekman currents, induce an Eulerian meridional circulation⌿ (the Thermocline theory offers plausible explanations of Deacon cell, see Doos and Webb 1994) that acts to the structure of midlatitude ocean gyres in linear vor- overturn isopycnals, enhancing the strong frontal region ticity and Sverdrup balance as reviewed, for example, maintained by air±sea buoyancy forcing B. The potential in Rhines (1993) and Pedlosky (1996). Theories of energy stored in the front is released, we imagine, ocean currents in zonally unblocked geometries such as through baroclinic instability, and the ensuing eddies the Antarctic Circumpolar Current (ACC) are much less induce an overturning circulation ⌿* that tends to re- well developed, however. In the absence of meridional store the isopycnals to the horizontal (see Fig. 2). In boundaries, Sverdrup balance no longer applies and it the theory presented here, it is the interplay of the ad- is much less obvious how a meridional circulation is vection of buoyancy in the meridional plane by⌿ and maintained. Furthermore, just as in zonal-average theory ⌿* that sets the structure of the ACC. of the atmosphere, geostrophic eddy transfer in the ACC We suppose the following. plays a central role in its integral balances of heat, momentum, and vorticityÐsee, for example, McWilliams 1) Transfer by geostrophic eddies, balancing momen- et al. (1978), Bryden (1979), Marshall (1981), de Szoe- tum and buoyancy input at the surface, sets the strat- ke and Levine (1981), Johnson and Bryden (1989), Gille i®cation and vertical extent of the ACC. This as- (1997), Marshall (1997), Phillips and Rintoul (2000), sumption yields predictions for the depth of pene- Karsten et al. (2002), Bryden and Cunningham (2003). tration and strati®cation of the ACC and its baro- For a recent review of observations and theories of the clinic transport as a function of wind and buoyancy ACC, see Rintoul et al. (2001). forcing and eddy transfer. In this paper, we put forward a simple theory of the 2) There is an approximate balance between⌿ and ⌿*; ACC and its associated meridional overturning circu- the MOC of the ACC is the ``residual'' circulation lation that makes use of zonal average residual-mean ⌿res ϭϩ⌿⌿ * that advects buoyancy (and other theory. Key observations of the Southern Ocean are tracers) in the meridional plane to offset sources and summarized in Fig. 1 in which we show the time-mean sinks. Here we will solve for the pattern of ⌿res given surface elevation, a schematic of the meridional over- the pattern of wind and buoyancy forcing at the sur- turning circulation (MOC), the streamwise-averaged face and assuming a closure for ⌿*. buoyancy distribution, and the thermal wind. We suppose, as sketched in Fig. 2, that westerly winds ␶ drive Before going on, we emphasize that here we develop an `` f plane'' theory of the ACCÐthere is no account taken of the ␤ effect. The relation of this study to pre- Corresponding author address: John Marshall, Department of vious ␤-plane investigations is discussed as we proceed Earth, Atmosphere and Planetary Sciences, Massachusetts Institute of Technology, Rm. 54-1526, 77 Massachusetts Ave., Cambridge, and in the conclusions. MA 02139. Our paper is set out as follows. In section 2 we for- E-mail: [email protected] mulate the problem by developing and applying resid-

᭧ 2003 American Meteorological Society

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC 2342 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33

FIG. 1. Key observations of the ACC: (a) the time-mean surface elevation measured from altimetry. Contour interval is 0.8 ϫ 104 m2 sϪ1. The bold lines mark the boundaries of circumpolar ¯ow. (b) A schematic of the currents and overturning circulation in the Antarctic region modi®ed from Fig. 164 of Sverdrup et al. (1942). (c) The streamwise-average buoyancy distribution computed from gridded hydrography; contour interval is 10Ϫ3 msϪ2. The vertical dotted lines denote the average latitude of circumpolar ¯ow marked in (a). (d) The thermal wind velocity computed from gridded hydrography by integration of the thermal wind assuming zero current at the bottom; contour interval is 10Ϫ2 msϪ1. Modi®ed from Karsten and Marshall (2002b). ual-mean theory to the ACC. In section 3 we ®nd so- a latitude circle. Eddy ¯uxes normal to these contours lutions for the vertical structure of the ACC and its are then by construction due to transient rather than associated MOC. In section 4 we discuss and conclude. standing eddiesÐsee Marshall et al. (1993).

2. Residual-mean theory applied to the ACC a. Residual-mean balances of momentum and We assume at the outset that ``zonal average'' theory buoyancy has relevance to the ACC. However, rather than aver- aging along latitude circles, we reference our along- 1) BUOYANCY stream coordinate to a mean surface geostrophic contour x (see Fig. 1a). Thus our zonal average,() , ``follows the The time-mean steady buoyancy equation can be writ- stream,'' as in de Szoeke and Levine (1981), rather than ten in the familiar form:

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC NOVEMBER 2003 MARSHALL AND RADKO 2343

⌿res, we eliminate (␷ ,w ) using Eqs. (2) and (3) to ob- tain1 ץ Bץ J (⌿ , b) ϭϪ[(1 Ϫ ␮)␷ЈbЈ], (4) yץ zץ y,z res

١´*where Jy,z(⌿res,)bbbbϭ (⌿res)yzϪ (⌿res)zyϭ v and ␮ is given by

wЈbЈ 1 ␮ ϭ . (5) ΂΃΂΃␷ЈbЈ s␳ Here

s␳ ϭϪb yz/b (6)

FIG. 2. Schematic diagram of the Eulerian mean (⌿ ) and eddy- is the slope of mean buoyancy surfaces. The parameter induced transport (⌿*) components of the Southern Ocean meridional ␮ controls the magnitude of the diapycnal eddy ¯ux: if overturning circulation driven by wind (␶) and buoyancy (B) ¯uxes. The associated velocity is computed from the streamfunction as (u, ␮ ϭ 1, then the eddy ¯ux is solely alongb surfaces, y), where y is a coordinate pointing northward the diapycnal horizontal component vanishes, and theץ/⌿ץ ,zץ/⌿ץ␷) ϭ (Ϫ and z points upward. The sloping lines mark mean buoyancy surfaces advective transport captures the entire eddy ¯ux; if ␮ b. The eddy buoyancy ¯ux␷ЈbЈ is resolved into a component in the ϭ 0, horizontal diapycnal eddy transport makes a con- b surface and a horizontal (diapycnal) component. tribution to the buoyancy budget. Diagnosis of the eddy- resolving ``polar cap'' calculations presented in Karsten et al. (2002) shows that the interior eddy ¯ux is indeed ,B closely adiabatic but that, as the surface is approachedץץץ bץ bץ ␷ ϩ w ϩ (␷ЈbЈ) ϩ (wЈbЈ) ϭ , (1) w b tends to zero, leaving a horizontal eddy ¯ux di- z Ј Јץ zץ yץ zץ yץ rected acrossb surfaces. The implications of these dia- where (␷ ,w ) is the Eulerian mean velocity in the me- batic eddy ¯uxes are studied in section 3f. Elsewhere ridional plane,b is the mean buoyancy, and variables in our study we assume that all diabatic eddy ¯uxes are have been separated into mean (zonal and time) quan- zero. tities and perturbations from this mean caused by tran- Note the following. sient eddies. Here, for simplicity, we have adopted a 1) Streamfunction ⌿*, Eq. (3), is de®ned so that, in the Cartesian coordinate system (see Fig. 2). Note that we limit of adiabatic eddies, vanishing small-scale mix- are in the Southern Hemisphere: x increases eastward, ing, and air±sea buoyancy ¯uxes (␮ ϭ B ϭ 0), Eq. y increases equatorward, z increases upward, and the (4) reduces to J(⌿ ,)bbϭ 0. Then is advected by Coriolis parameter f Ͻ 0. In Eq. (1) the buoyancy forc- res ⌿res, suggesting that classic inferences of overturning from air±sea interaction and small-scale mixing pro- ing in the Southern Ocean based on tracer distri- cesses has been written as the divergence of a buoyancy butions (see Fig. 1b) are sketches of the residual, ¯ux B. rather than of the Eulerian mean ¯ow. Our goal now is to express Eq. (1) in terms of the 2) Streamfunctions ⌿*,⌿ , and hence ⌿res unequivo- residual circulation ⌿res: cally vanish at the surface because w ϭ wЈϭ0 there.

⌿ϭ⌿ϩ⌿res *, (2) where⌿ is the overturning streamfunction for the Eu- 2) MOMENTUM lerian mean ¯ow and ⌿* is the streamfunction for the We now wish to express the momentum balance in overturning circulation associated with eddies (see Fig. terms of residual, rather than Eulerian-mean velocities. 2 and caption). The key step is to note that if the eddy This is desirable because the buoyancy equation [Eq. vЈbЈ can be written´١ uxvЈbЈ lies in theb surface, then¯ -١b, where, fol´*entirely as an advective transport, v 1 lowing Held and Schneider (1999), v* is de®ned in To arrive at Eq. (4) from Eq. (1), decompose the eddy ¯uxes (␷ЈbЈ ,wЈbЈ ) into an along-bw component (ЈbЈ /s ,wЈbЈ ) and the re- terms of a streamfunction ⌿* given by ␳ maining horizontal component (␷ЈbЈ Ϫ wЈbЈ/s␳, 0) (see Fig. 2). The divergence of the along-b component is then written as an advective wЈbЈ transport ⌿* ϭϪ . (3) ,(wЈbЈ/s␳, wЈbЈ) ϭ ␷ *b yzϩ w*b ϭ J(⌿*, b)´ ١ b y where ⌿* is given by Eq. (3). This is combined with mean ¯ow HerewЈbЈ is the vertical eddy buoyancy ¯ux and b y is advection in Eq. (1) to yield the lhs of Eq. (4). The divergence of the mean meridional buoyancy gradient. the diapycnal (horizontal) eddy ¯ux leads to the last term on the rhs In more precise terms, to express Eq. (1) in terms of of Eq. (4).

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC 2344 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33

(4)], to which the momentum equation is intimately linked, is most succinctly expressed in terms of residual velocities; the momentum and buoyancy equations must be discussed together. The statement of Eulerian-mean zonal-average zonal momentum balance is, in the steady state (remembering that px ϭ 0, etc.), Ϫ f ␷ ϭ F, (7) where f is the Coriolis parameter,␷ is the Eulerian- mean meridional velocity, andF combines together mo- FIG. 3. The residual ¯ow ⌿res ϭϩ⌿⌿ * is assumed to be directed mentum sources and sinks and momentum ¯uxesu␷ , along mean buoyancy surfacesb in the interior but to have a diapycnal vu ) terms. To express Eq. (7) in terms of)´such as Ϫ١ component in the mixed layer of depth hm (denoted by the horizontal the residual meridional velocity dotted line).

␷res ϭ ␷ ϩ ␷*, where With these assumptions, the steady-state mixed layer ⌿* buoyancy budget can be written asץ ⌿ץ ␷ ϭϪ and ␷* ϭϪ , (8) ץ Bץ boץ ⌿resץ zץ zץ ϪϭϪ(1 Ϫ ␮) ␷ЈbЈ, yץ zץ yץ zץ we add Ϫ f␷* to both sides of Eq. (7). The resulting residual momentum balance can be written as where bo(y) is the mixed layer buoyancy. Integration ⌿* over the depth of the mixed layer hm, noting that ⌿resץ ⌿ץ f res ϭ F ϩ f , (9) ϭ 0 at the surface, gives zץ zץ bץ where we have used Eq. (2) and ⌿* is chosen as in Eq. o Ä ⌿ϭresͦzϭϪhom B , (11) yץ .to ensure that the residual buoyancy balance, Eq (3) (4), takes on a simple form. where To make further progress, we now make some sim- 0 ץ .plifying assumptions BÄ ϭ B Ϫ (1 Ϫ ␮) ␷ЈbЈ dz (12) yץ ͵ oo Ϫhm b. Simpli®ed system is the net buoyancy supplied to the mixed layer by air± In the interior of the ACC we suppose that 1) buoy- sea buoyancy ¯uxes and by lateral diabatic eddy ¯uxes. y in the localץ/␷ЈbЈץ ancy forcing (due both to convection and mixing pro- The relative importance of Bo and cesses) vanishes, that is, B ϭ 0 in Eq. (4), and that 2) buoyancy budget of the diabatic surface layer is not yet the eddy ¯ux is directed entirely alongb surfaces, that clear. Speer et al. (2000) argue (and show supporting is, ␮ ϭ 1 in Eq. (4). Thus observational evidence) that air±sea ¯uxes can provide the necessary warming to allow a surface ¯ow directed J(⌿res, b) ϭ 0 (10) away from Antarctica as suggested in Fig. 1b: indeed in the interior, implying that there is a functional rela- they diagnose the sense of ⌿res from Eq. (11) using Ä Ä observations of Bo (assuming that Bo is dominated by tionship between ⌿res andbb : ⌿res ϭ⌿res( ). This functional relationship will be set, we suppose, in the surface surface heat ¯uxes). In theoretical calculations, Marshall mixed layer. (1997) also assumes that the eddy contribution in Eq. We suppose the following about the mixed layer: 1) (12) is negligible. Calculations presented in section 3f below, however, suggest that diapycnal eddy ¯uxes It is vertically homogeneous and of constant depth hm, as sketched in Fig. 3. Furthermore, we set entrainment could play an important role in the buoyancy budget of the surface layer in the ACC. ¯uxes at the base of the mixed layer to zero (BϪhm ϭ 0) and neglect the seasonal cycle. 2) Eddy ¯uxes have Equation (11) sets the functional relationship between Ä a diabatic component; ␮ in Eq. (4) varies from 0 at the ⌿res and bo.IfB Ͼ 0 (corresponding to local buoyancy ,y Ͼ 0ץ/boץ surface to 1 at the base of the mixed layerÐsee Treguier gain by the mixed layer), then, because 2 ⌿Ͼ0 and so (noting that ⌿ ϭ 0) the et al. (1997) and Fig. 3. res | zϭϪhm res | surface ¯ow in the mixed layer is directed equatorward. If the mixed-layer buoyancy gradient is constant, then, ac- 2 Note that it is likely that the depth hm of the ``diabatic'' layer is cording to Eq. (11), the strength and sense of the residual not, in general, the mixed layer depth (A.-M. Tregueir 2001, personal circulation will be directly proportional to BÄ at each communication). It is more likely that hm at a given location is the depth of the deepest isopycnal that occasionally grazes the surface latitude. because of eddy dynamics or the seasonal cycle. In the momentum equation [Eq. (9)], we suppose that

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC NOVEMBER 2003 MARSHALL AND RADKO 2345

3 (y terms can be neglected, allowingF to be set ⌿* ϭ k | s␳␳| s . (17ץ u␷/()ץ .zץ/␶ץequal to the vertical divergence of a stress, F ϭ In the ACC, * is negative, that is, in the sense to Then Eq. (9) can be written as ⌿ return the isopycnals to the horizontal, balancing⌿ as .(⌿* sketched in Fig. (2ץ ␶ץ ⌿ץ f res ϭϩf . zץ zץ zץ 3. Solutions for the ACC and its overturning Integrating from the surface where ⌿* ϭ⌿res ϭ 0 and circulation ␶ ϭ ␶o (the surface wind stress) through the surface Ekman layer to depth z in the interior, where ␶ ϭ 0, we Let us now draw together our key relationsÐEqs. obtain (11), (13), and (17)Ðand seek solutions. They are ⌿res(z) ϭϪ␶ o/ f ϩ⌿*(z). (13) ␶ o ϩ⌿ ϭ⌿* and ⌿* ϭ k|s |s , (18) Equation (13) is one of our key relations and equates f res ␳␳ the residual ¯ow to the sum of an Eulerian circulation, where the directly Ekman-driven current Ϫ␶o/ f, and a ¯ow associated with eddies, Eq. (3). The physical content of ⌿ϭ⌿res res(b) (19) Eq. (13) is a statement of steady-state integral momentum balance: multiplying through by f, we see that the set in the mixed layer by Eq. (11) and s␳ is given by net Coriolis torque on the residual ¯ow over the water Eq. (6). column is balanced by wind stress applied at the surface Rearranging the above, we can write (for s␳ Ͻ 0, the and eddy form drag at the bottom. Note that in the ACC, case considered here) f Ͻ 0 and ␶ Ͼ 0, and so Ekman ¯ow is directed equa- 1/2 ␶ ⌿ (b) torward at the surface: ⌿ Ͼ 0, as sketched in Fig. (2). s ϭϪϪo Ϫ res . (20) To proceed further, we must express ⌿* in terms of ␳ []fk k b. In other words, we must ``close'' for the eddy ¯ux. In that which follows, the system of Eqs. (18)±(20) will Ä be solved for a given surface pattern of ␶o(y), Bo(y), and c. Closure for ⌿* bo(y). If the eddies are ``adiabatic'' in the interior, then ␮ ϭ 1 in Eq. (5), and Eq. (3) can be written in terms of a. Solution technique horizontal buoyancy ¯uxes as follows: Any physically meaningful model of the ACC should wЈbЈ ␷ЈbЈ be characterized by the poleward shallowing of the ther- ⌿* ϭϪ ϭ , (14) mocline in which s Ͻ 0 (see Fig. 1c), and therefore bbyz ␳ we consider only the solution branch corresponding to yielding the more ``conventional'' de®nition of ⌿*Ð the negative sign in Eq. (20). The equation for the slope see, for example, Andrews et al. (1987) and Gent and [Eq. (20)] is rewritten as a ®rst-order partial differential McWilliams (1990). We adopt the following simple clo- equation inb using Eq. (6): sure for the interior eddy buoyancy ¯ux: 1/2 ␶ o(y) ⌿res(b) ␷ЈbЈϭϪKby, b ϪϪ Ϫ b ϭ 0. (21) yzfk k where K is an eddy transfer coef®cient that is assumed [] to be positive. Then ⌿* can be written as Despite the seemingly complicated form of its coef®- cients, this equation can be easily integrated along char- ␷ЈbЈ b y acteristics, since ␶ (y) is prescribed and ⌿ (b ) can be ⌿* ϭϭϪK ϭ Ks␳, (15) o res bbzz evaluated at the base of the mixed layer using Eq. (11).

The characteristic velocities ␷ c and wc are where S␳ is given by Eq. (6). Furthermore, if we suppose that K is itself proportional to the isopycnal slope [this dy assumption and its relation to the prescription of the ϭ ␷ c ϭ 1 dl horizontal variation of the Ks, suggested by Visbeck et 1/2 al. (1997), is discussed in the appendix]: dz ␶ (y) ⌿ (b) ϭ w ϭϪϪo Ϫ res , (22) K ϭ k | s |, (16) c ␳ dl[] fk k where k is a positive scaling constant, then ⌿* can be and, of course, ⌿res andb do not change at a point written as moving along the characteristic

d⌿res/dl ϭ 0 and db/dl ϭ 0, 3 This is a very good approximation in the ACCÐsee the estimates ofuЈ␷Ј based on ¯oat data by Gille (2003). where l is the distance along a characteristic.

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC 2346 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33

TABLE 1. Numerical constants used in solutions for circumpolar ¯ow.

Ϫ4 2 Ϫ2 Surface (speci®c) wind stress ␶ o 2 ϫ 10 m s

Meridional scale of ACC Ly 2000 km

Zonal scale of ACC at 55ЊN Lx 21 000 km Vertical scale H 1km

Mixed layer depth hm 200 m Coriolis parameter f Ϫ10Ϫ4 sϪ1 Planetary vorticity gradient ␤ 10Ϫ11 sϪ1 mϪ1 Ϫ3 Ϫ2 Buoyancy change across ACC ⌬bo 7 ϫ 10 ms Ä Ϫ9 2 Ϫ3 Net buoyancy forcing Bo 2 ϫ 10 m s Eddy transfer coef®cient K 500 m2 sϪ1 LK Eddy parameter k ϭ y 106 m2 sϪ1 H Thermal expansion coef®cient ␣ 10Ϫ4 K Ϫ1 Ϫ1 Ϫ1 Speci®c heat of water Cp 4000 J kg K Density of water ␳ 103 kg mϪ3

y FIG. 4. The assumed forcing functions: (a) wind ®elds [␶o(y)]/␶o ϭ b ϭ⌬b , (25) Ä Ä oo y and {0.3 ϩ sin[(␲y)/Ly]} and (b) buoyancy ¯ux [B(y)]/Bo ϭ Ly sin[(2␲y)/Ly], plotted as a function of y/Ly. where ⌬bo is the buoyancy drop across the ACC at the surface. It is worthy of note that f appears in Eq. (22) in The patterns of net buoyancy forcing over the ACC combination with the wind stress, and so ␶o and f have are uncertain. We experiment with the following form: the same status and could both be allowed to vary: solutions to Eq. (21) can readily be found with variable 2␲y BÄÄ(y) ϭϮB sin . (26) f. However, the closure assumption, Eq. (16), is based o ΂΃Ly on f-plane ideas and may require modi®cation in the presence of ␤ (see the appendix). For this reason, we If the positive sign is chosen, buoyancy is gained on con®ne ourselves here to f-plane solutions. the polar ¯ank of the ACC and is lost on the equatorial Reduction of the problem to a system of ordinary ¯ank, as suggested by the observationsÐsee Speer et differential equations, as in Eq. (22), enables analytical al. (2000). Note that the form Eq. (26) ensures that the expressions for the thermocline depth to be found for net buoyancy ¯ux vanishes, when integrated across the simple forcing functions (see section 3c). For more com- jet. plicated and realistic patterns of forcing, the system of Eq. (22) requires an elementary numerical calculation. 3) NUMERICAL CONSTANTS Some solutions are presented below in section 3d. Table 1 de®nes and gives typical values of the parameters used in our calculations. Note that if the ther- b. Forcing ®elds Ϫ4 Ϫ1 Ä mal expansion coef®cient ␣ ϭ 10 K then a Bo of Ϫ9 2 Ϫ3 1) WIND 2 ϫ 10 m s corresponds to a heat ¯ux of 10 W Ϫ2 Ϫ3 Ϫ2 m and a buoyancy jump ⌬bo ϭ 7 ϫ 10 ms We consider two idealized wind pro®les, one that corresponds to a temperature jump of 10ЊC. increases linearly from Antarctica:

␶ooy(y) ϭ ␶ (y/L ), (23) c. Depth of the thermocline and a second pro®le with more structure and perhaps Integration of the system in Eq. (22) is particularly more realism: simple for those characteristics that emanate from the

base of the mixed layer at which ⌿res ϭ 0, corresponding ␲y Ä ␶ (y) ϭ ␶ 0.3 ϩ sin . (24) to B ϭ 0Ðno net heating of the surface layers. Then oo L []΂΃y Eq. (22) reduces to They are plotted in Fig. 4a. Numerical constants that dy ϭ ␷ ϭ 1 are assumed are set out in Table 1. dl c

1/2 dz ␶ (y) 2) SURFACE BUOYANCY AND BUOYANCY FLUX ϭ w ϭϪϪ o . (27) dlc fk We suppose that the surface buoyancy ®eld increases [] linearly away from Antarctica: One such characteristic corresponds to the base of the

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC NOVEMBER 2003 MARSHALL AND RADKO 2347

FIG. 5. The strati®cationb obtained for the linear wind pro®le, Eq. FIG. 6. The strati®cationb (contour interval is 10 Ϫ3 msϪ2) obtained Ϫ3 Ϫ2 (23), in the case of vanishing ⌿res (contour interval is 10 ms ). for the wind pro®le, Eq. (24), in the case of vanishing ⌿res. A linear A linear buoyancy pro®le is assumed at the sea surface, Eq. (25). buoyancy pro®le is assumed at the sea surface, Eq. (25). The assumed The depth of the zerob surface is given by the formula Eq. (29). numerical constants are set out in the table. Numerical constants used are set out in Table 1. thermocline z ϭϪh(y), which, in our case, is best de- d. The strati®cation and overturning circulation ®ned by the isopycnal outcropping at y ϭ 0 (where b o 1) BÄ ϭ 0: THE LIMIT OF ⌿ ϭ 0 ϭ 0). Fluid below the thermocline [z ϽϪh(y)] is as- res sumed to be homogeneous with vanishing residual cir- Figure 5 shows theb ®eld obtained for the wind Eq. Ä culation: ⌿res ϭ 0. The consistency of our assumption (23) when B ϭ 0 and so ⌿res ϭ 0. The depth of each that ⌿res ϭ 0aty ϭ 0 at the base of the mixed layer isopycnal has been computed from Eq. (27); the deepest is ensured by using the buoyancy forcing function Eq. isopycnal has the form given by Eq. (29). Indeed all Ä (26) with B | yϭ0 ϭ 0, as required by Eq. (11). isopycnals in Fig. 5 are parallel to one another. This can The depth of the thermocline under these assumptions readily be understood by inspection of Eq. (20): because reduces to a ®rst integral of Eq. (27): ⌿res ϭ 0, s␳ only depends on ␶o(y) and so is independent

1/2 of depth. Although we keep f constant in our theory, y ␶ (l) it is helpful to phrase the discussion in terms of potential h(y) ϭϪ Ϫ o dl. (28) ͵ fk vorticity (PV). The isentropic PV gradient (IPVG) is 0 [] 2 N f(ds␳/dz) ϭ 0 because theb surfaces are parallel to The fact that the thermocline depth depends on͙␶o is one another with f constant. a consequence of our closure assumption Eq. (17)Ðalso The strati®cation is given by see the appendix where key parametric dependencies 2 are summarized. N ϭ b zoϭ b /h, (30) For the linear variation of the wind stress ␶(y) ϭ ␶ y, o where b is given by Eq. (25) and h by Eq. (29). Eq. (28) yields o As discussed in the appendix, our expressions for 2 ␶ 1/2y 3/2 thermocline depth and strati®cation are consistent with h(y) ϭϪ o L (29) the results of Marshall et al. (2002), Karsten et al. 3΂΃fk ΂΃ L (2002), and Cenedese et al. (2004) derived from studies and de®nes the depth of the deepestb surface plotted of laboratory and numerical lenses. The relation of these in Fig. 5. The vertical scale is set by results to those of Bryden and Cunningham (2003) are also discussed. 1/2 2 ␶ o Figure 6 showsb for the wind ®eld Eq. (24): in this L ϭ 1886 m 3΂΃fk y case Eq. (27) had to be integrated numerically. The strati®cation has more structure because of the more for the typical parameters set out in Table 1. We see complex form of the driving wind ®eld, but again, with that the depth of the thermocline increases moving away PV uniform, all isopycnals are parallel to one another from Antarctica to reach some 1800 m on the northern because f is constant. ¯ank of the jet, not unlike the observations discussed In these solutions ⌿res ϭ 0 and the momentum budget in Karsten and Marshall (2002b); see their Fig. 4. Eq. (18) reduces to Solutions for the more realistic wind ®eld in Eq. (24) can be written down but do not take on a simple form. ␶O/f ϭ⌿* (31)

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC 2348 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33

FIG. 8. Isentropic PV gradient normalized by the nominal ␤-induced contribution (see text) for the solution shown in Fig. 7. Regions of positive IPVG are marked (ϩ); negative regions are marked (Ϫ).

2) BÄ ± 0: OVERTURNING CIRCULATION

Figures 7a,b showb and ⌿res for the wind ®eld Eq. (24) and the buoyancy forcing given by Eq. (26), choos- Ä Ϫ9 2 Ϫ3 ing the positive sign, and Bo ϭ 2 ϫ 10 m s , that is, we warm south of the ACC and cool north of it, as suggested by air±sea ¯ux observations. The sense of the residual circulation corresponds to upwelling of ¯uid in the deepest layers that outcrop at the poleward ¯ank of

the ACC (0 Ͻ y Ͻ Ly/4) and the shallow layers that outcrop on its equatorward ¯ank (3Ly/4 Ͻ y Ͻ Ly). At the surface, residual ¯ow converges in to intermediate

layers (outcropping at the latitudes Ly/4 Ͻ y Ͻ 3Ly/4), from the south on the poleward ¯ank and from the north FIG. 7. (a) The buoyancyb (contour interval is 10 Ϫ3 msϪ2) and Ä Ä on the equatorial ¯ank. Here ¯uid is subducted down- (b) residual circulation ⌿res (Sv) for nonzero thermal forcing B ϭϩBo sin[(2␲y)/Ly]. The wind stress is given by Eq. (24). ward and equatorward: we associate this ¯ow with Ant- arctic Intermediate Water. Eddies carry the momentum imparted by the wind ver- Note that now isopycnal surfaces are not parallel to tically through the column by interfacial form drag. The one another: the subtle variation in the thickness of is- relationship Eq. (31) was ®rst derived heuristically by opycnal layers implies the presence of interior PV gra- Johnson and Bryden (1989)Ðsee the discussion in Kar- dients. The spatial distribution of these PV gradients z) is shown in Fig. 8, where it isץ/ sץ)y) ϭ N 2fץ/qץ) sten et al. (2002) and Olbers and Ivchenko (2002). We b ␳ see it now as a special case of Eq. (13) applicable only normalized by the nominal value of PV gradient due to , not used in the theory. We see that the PV gradient in the limit that ⌿ ϭ 0. ␤ res reverses sign with : ¯uid upwells as it approaches In the homogeneous layer below the main thermo- ␷res Antarctica in regions of positive PV gradient and ¯uid cline z ϽϪh(y) we assume that ⌿* remains ®nite (ex- is subducted and ¯ows equatorward in regions of neg- actly balancing ) until interaction with topography can ⌿ ative PV gradient (we return to this point in the con- balance the surface stress through topographic form drag clusions). as described in Munk and Palmen (1951). In this ho- The magnitude of the overturning streamfunction mogeneous layer we set u ϭ u and suppose that abyss plotted in Fig. 7b is set by the assumed buoyancy gra-

⑀uabyss ϭ ␶o. (32) dient and surface buoyancy ¯ux at the sea surface. For the parameters chosen in Table 1 we ®nd Here ⑀ is a parameterization of bottom Ekman layer/ Ä topographic form-drag effects. BLLoyx We now go on to consider effects caused by the ®nite ⌿ϭres ϭ12 Sv ⌬b residual circulation by including nonzero buoyancy o forcing BÄ of the form in Eq. (26). (where 1 Sv ϵ 106 m3 sϪ1), broadly consistent with the

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC NOVEMBER 2003 MARSHALL AND RADKO 2349

FIG. 9. Vertical cross section of the zonal velocityu (y, z) that goes FIG. 10. Contours of the eddy-transfer coef®cient K (m2 sϪ1). The along with theb ®eld shown in Fig. 7. The contour interval is 10 Ϫ2 largest values of K, about 1600 m2 sϪ1, occur near the center of the msϪ1. ACC front.

integrated vertically and across the stream to obtain the analysis of Marshall (1997) who found residual circu- transport. We ®nd that the baroclinic transport depends lations of 10±15 Sv, but somewhat smaller than the 20 on external parameters as follows: Sv or so of overturning implied by air±sea ¯uxesÐsee Karsten and Marshall (2002a). In comparison with the ␶ L2 ⌬b baroclinic transport ϳ o . (33) zonal transport (ϳ100 Sv; see below), this ¯ow is weak. fk2 Ä We will discuss the processes responsible for Bo below in section 3f. The transfer coef®cient K(y, z) ϭ k | s␳ | plotted in Fig. 10 indicates strong spatial variation in the eddy activity, which, in our model, is mostly limited to the 3) ZONAL TRANSPORT regions of large vertical shear. For the assumed k ϭ 106 m2 sϪ1, the maximum value of K reaches 1600 m2 sϪ1, The large-scale zonal velocityu in Fig. 9 was com- not inconsistent with observational and numerical es- puted from the thermal wind balance assuming no mo- timates (see, e.g., Visbeck et al. 1997; Stammer 1998, tion below the thermocline [z ϽϪh(y)]. The general Karsten and Marshall 2002). Note that the former two structure of the thermal wind velocity ®eld has a peak papers attempt to estimate a vertically averaged eddy of7cmsϪ1, broadly consistent with the observations diffusivity over the main thermocline of the ocean, (see Fig. 1c). The zonal baroclinic transport directly whereas Karsten and Marshall estimate the near-surface computed from Fig. 9 is 75 Sv, however, considerable diffusivity (which is likely to be elevated above interior smaller than the observed 130-Sv transport of the ACC. values). The difference is perhaps due to our uncertainty in the choice of eddy transfer coef®cient, the prescribed linear variation in the surface buoyancy, Eq. (25), which does e. Meridional eddy heat and buoyancy ¯ux not impose a frontal structure, or simply the idealized The meridional eddy buoyancy ¯ux in the limit of nature of the model. Note, however, that to obtain the vanishing ⌿res is given by Eq. (31): absolute velocity we should add the depth-independent u ϭ [␶ (y)]/⑀ to this pro®le. Gille (2003) estimates ␶ o abyss o ␷ЈbЈϭb . a barotropic component to the stream of about 1 cm z f Ϫ1 Ϫ1 s .Ifuabyss has a peak value of 1 cm s , then the depth-integrated component contributes a transport of If b z is given by Eq. (30) and the wind®eld by Eq. (23), 40 Sv to the transport if the current is 3 km deep and then the integrated buoyancy ¯ux at midchannel is L ϭ 2000 km. 0 y ⌬boo␶ In the limit that ⌿ → 0, the baroclinic transport of ␷ЈbЈ dz ϭ . res ͵ 4 f the ACC depends linearly on the wind: this can be Ϫh shown by integrating up the thermal wind equation, If the buoyancy ¯ux is dominated by the heat ¯ux (note, f ubz ϭ y, from depth z ϭϪh, Eq. (29), to the surface, however, that freshwater ¯uxes may make an important using Eq. (21) (with ⌿res ϭ 0). The resultingu is then contribution in the ACC), then

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC 2350 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33

␳ Cp and so the total meridional heat ¯ux at midchannel in- H ϭ B, g ␣ tegrated around the ACC is given by

1 ␳␶C meridional heat flux ϭ⌬p bLo ϭ 0.3 ϫ 1015 W (34) 4 g ␣ oxf

for the parameters given in Table 1. ␮ ϭ 0), we obtain the following ordinary differential

This value is of the same order as estimated by, for equation for so: example, de Szoeke and Levine (1981), Marshall et al. b ␶ץ sץ bץ (1993), and, most recently, Gille (2003). Note that the h ooϭ B ϩ oks2 ϩ o, (38) yf΂΃ץ yץ yץabove estimate is independent of details of the eddy ¯ux moo closure because it pertains in the limit that 0 and ⌿res ϭ which can be solved given a distribution of B , ␶ , b , so Eq. (31) holds. Of course, in the limit that 0, o o o ⌿res ϭ and f. Integration of the ®rst-order differential Eq. (38) the meridional heat transport by eddies is exactly bal- requires a single boundary condition; if ⌿res ϭ 0 at the anced by transport of heat by the mean meridional over- base of the mixed layer at y ϭ 0, then Eq. (20) tells us turning⌿ . that The total heat transport across the ACC in our model Ä 1/2 is a function of the imposed Bo, Eq. (12). If there is ␶ (0) s (0) ϭϪ o . (39) warming south of the stream and cooling north of the o []fk streamÐas in the calculations presented hereÐthen ⌿ carries slightly more heat north than ⌿* does south. Figure 11a presents the numerical solution of Eqs. (38) We now discuss the role of diapycnal eddy heat ¯uxes and (39) in the case that the surface buoyancy forcing Ä vanishes (B ϭ 0) and for the wind pro®le given by Eq. in setting the pattern of Bo. o (24); other parameters are the same as were used to obtain Fig. 7. The solution is expressed in Fig. 11a in f. The role of diapycnal eddy heat ¯uxes Ä terms of the total buoyancy ¯ux Bo, which has been In the calculations shown thus far, a pattern of buoy- computed from so using Eq. (37). Thus Fig. 11a shows Ä that, as is to be expected from downgradient diffusion ancy forcing (Bo) was prescribed. It includes contri- in Eq. (35), eddy heat transfer in the mixed layer warms butions from air±sea ¯uxes (Bo) and from the lateral eddy heat transfer in the mixed layer [see Eq. (12)]. In the poleward ¯ank of the ACC and cools the equatorial this section we compute, as part of the solution, the ¯ank. This pattern is consistent with the one assumed Ä contribution due to mixed layer eddy buoyancy ¯ux for Bo a priori in section 3d. Addition of a ®nite surface ¯ux in Fig. 11b modi®es BÄ but, for a realistic parameter given a prescribed surface buoyancy forcing (Bo). Because the mixed layer is assumed to be vertically range, does not change the general structure of the total homogeneous, the mixed layer eddy ¯uxes in Eq. (12) buoyancy ¯ux.4 can be expressed as a function of the mean quantities Figures 12a,b show the buoyancy ®eld and residual immediately below the mixed layer using the interior circulation computed using the method of characteristics closure (see section 3a) for the buoyancy forcing in Fig. 11a. The overturning circulation plotted in Fig. 12b, reaching bץ bץ ␷ЈbЈϭϪK ooϭ ks , (35) a magnitude of some 6 Sv, is entirely driven by diabatic .y eddy processesÐthe air±sea ¯ux is assumed to be zeroץ y oץ We see, then, that diabatic eddy effects may make a where so(y) ϭ s␳(y, Ϫhm). For the linear surface buoyancy pro®le Eq. (25), convergence of the eddy ¯ux in nontrivial contribution to the residual-mean overturning Eq. (12) reduces to in the ACC. bץ sץ␷ЈbЈץ ϭ k oo. (36) 4. Discussion and conclusions yץ yץ yץ We have applied zonal average residual mean theory Using Eq. (17) evaluated at the mixed layer base, Eq. to develop a model of the ACC that combines the effects Ä (11) yields Bo as a function of so: 4 There is no reason to assume a priori that the combined effect of b ␶ץ BÄ ϭϪo ks2 Ϫ . (37) surface buoyancy forcing and eddies in Fig. 11b is a linear super- [(yf΂΃ position of these two contributions. The governing equation [Eq. (38ץoo is signi®cantly nonlinear. Nevertheless, in the parameter range under Using Eqs. (37) and (36) to simplify Eq. (12) (setting consideration, the two contributions add linearly.

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC NOVEMBER 2003 MARSHALL AND RADKO 2351

Ä FIG. 11. Total buoyancy ¯ux Bo (solid line) computed for a given Ä air±sea ¯ux Bo (dashed line). (a) The Bo when Bo ϭ 0: in this case diabatic eddy ¯uxes redistribute buoyancy within the mixed layer. Ä Ϫ9 2 Ϫ3 (b) The Bo when Bo ϭ Bo sin[(2␲y)/Ly], with Bo ϭ 1 ϫ 10 m s .

of buoyancy and mechanical forcing in a transparent FIG. 12. (a) The buoyancy ®eld and (b) the residual circulation for way. Simple solutions have been derived on the f plane the buoyancy forcing shown in Fig. 11a. The wind stress is given by Ϫ3 Ϫ2 that capture its mean buoyancy distribution, zonal cur- Eq. (24). Contour intervals are b ϭ 10 ms and ⌿res ϭ 2Sv, rent, and pattern of meridional overturning circulation. respectively. Here the residual overturning streamfunction, Fig. 11b, is driven entirely by diabatic eddy ¯uxes in the mixed layer. In our theory, transfer by baroclinic eddies balances imposed patterns of wind and buoyancy forcing and 1) controls the depth to which the thermocline of the component parts, then the depth of the thermocline de- ACC penetrates; and pends on the square root of the wind stress [see Eq. 2) plays, through diabatic ¯uxes directed horizontally (28)] and the strati®cation is given by Eq. (30). This through the mixed layer, an important role in redis- result ®nds support in laboratory (Marshall et al. 2002; tributing buoyancy in the mixed layer, supporting a Cenedese et al. 2004) and numerical studies (Karsten et meridional overturning circulation, surface conver- al. 2002) of the ACC. A corollary of this result is that gence at the axis of the ACC, and subduction. the zonal transport depends linearly on the applied wind We ®nd that the depth of the thermocline and baro- [see Eq. (33)]. This dependence should be compared clinic transport is intimately connected to the assumed with the predictions of other authors who argue that the balance between the applied wind stress and vertical baroclinic transport varies like͙␶ (Johnson and Bryden momentum transfer by baroclinic eddies [see Eq. (28)]. 1989), is independent of the wind (Straub 1993), or The overturning circulation ⌿res, by contrast, is asso- varies like ␶ (Gnanadesikan and Hallberg 2000; Karsten ciated with buoyancy supply to and from the mixed layer et al. 2002, and this study). Only in the latter two studies Ä B [see Eq. (11)] in which both air±sea ¯ux and diabatic is the strati®cation not set a priori. eddy ¯uxes play a role. We note the dependence of our solutions on the form assumed for our eddy closure and that of the eddy trans- a. Thermocline depth, strati®cation, and zonal fer coef®cient, Eqs. (15) and (16). The eddy closure y is conventional and ®ndsץ/bץtransport assumption ␷ЈbЈ ϭϪK If ⌿* depends on the square of the isopycnal slope, support in eddy-resolving numerical models of the as assumed in Eq. (17), and ⌿res is smaller than its ACCÐsee, for example, the recent study of Karsten et

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC 2352 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33

FIG. 13. The meridional IPVG deduced from a hydrographic data base by Marshall et al. (1993). Regions of positive IPVG are shaded gray; regions of negative IPVG are white. The arrows mark the sense of the sense of the implied residual ¯ow; poleward where IPVG Ͼ 0, equatorward where IPVG Ͻ 0. Note that the downward arrow around Antarctica is hinted at by the sign of the (very weak) IPV gradient but is not represented in the simple solutions discussed here. al. (2002). The assumed dependence of K on the local is ¯uxed down the isentropic PV (IPV) gradient, then y). Thus the sense of ␷res can beץ/IPVץ)( baroclinity yields a simple but important connection be- ␷res ϭϪ(K/ f tween K and mean ¯ow quantities, as described in the gleaned from inspection of observations of the merid- appendix. It is, however, an f-plane closure that may ional IPV gradient. The arrows in Fig. 13 [modi®ed from be modi®ed by the presence of ␤, see below. Marshall et al. (1993)] indicate the sense of the implied residual ¯ow, poleward where IPV increases poleward b. Meridional overturning circulation and equatorward where IPV increases equatorward. The shallow regions of reversed IPVG seen in the obser- The sense of the residual circulation shown in Fig. vations are very likely to be associated with low-PV 7b corresponds to upwelling of ¯uid in the deepest lay- ¯uid created by convection in mode-water formation ers that outcrop at the poleward ¯ank of the ACC and just equatorward of the ACC. These convected waters the shallow layers that outcrop on its equatorward ¯ank. are subducted in to the interior, carrying with them their At the surface, ¯uid converges in to intermediate layers, low-PV signal and reversing the sign of IPV. from the south on the poleward ¯ank and from the north on the equatorward ¯ank. It is tempting to associate the ¯uid upwelling around Antarctica with upper North At- c. The role of ␤ lantic Deep Water and the ¯uid subducted downward As discussed in the appendix, our eddy closure, Eqs. from the convergence zone farther north with Antarctic (15) and (16), is explicitly an f-plane parameterization, Intermediate Water. On the equatorward ¯ank of the with no sense of a critical shear for instability or eddy ACC, the residual ¯ow is to the south (for our chosen saturation. This closure may break down in the ACC pattern of BÄ in Fig. 4b), and so the eddy-induced mixed where ␤ constraints are likely to be importantÐsee, for layer transport here is larger than, and in the opposite example, Tansley and Marshall (2001) or Hallberg and direction to, the Ekman transport. This situation is not Gnanadesikan (2001). Further study is required to draw implausibleÐsee, for example, the inference from ob- such effects in to our theory. servations of these components by Karsten and Marshall Last, we emphasize that the ideas and formulation set (2002). The strength (and sense) of the overturning cir- out here have no direct counterpart in thermocline the- culation is of approximately 10 Sv and depends on our ory. Our solutions are not in Sverdrup balance because choice for the magnitude of BÄ . Note that our solution there is no ␤ effect. Because of the absence of merid- does not contain a representation of Antarctic Bottom ional boundaries, pathways for mean heat transport Water because of the absence of cooling around Ant- across the ACC are limited, and so eddies must be large- arctica in the assumed BÄ . ly responsible for poleward heat transport across the The large-scale potential vorticity ®eld is a primary stream (see, e.g., de Szoeke and Levine 1981). The ed- contact point between our theory and the observations. dies also balance the input of momentum by the winds.

Interior residual ¯ow ␷res is only possible if there is a The southward heat ¯ux is directly related to a down- meridional eddy PV ¯ux,␷ЈqЈ : Ϫ f␷res ϭ ␷ЈqЈ.If␷ЈqЈ ward momentum ¯ux through interfacial drag (see, e.g.,

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC NOVEMBER 2003 MARSHALL AND RADKO 2353

Johnson and Bryden 1989). This drag allows the surface Further, from Eq. (16) and supposing that | s␳ | ϭ h/Ly, momentum to be transferred to depth where it can be we can identify k as dissipated by mountain drag as the ACC ¯ows over the ⌬bL high ridges (Munk and Palmen 1951). Such a balance oy k ϭ ce . (A4) has been established as the dominant balance of the f ACC both in observations (Phillips and Rintoul 2000) Using Eq. (A4) our predicted depth of the thermo- and in numerical models (Ivchenko et al. 1996; Gille cline, Eq. (29) can then be written entirely in terms of 1997) and is at the heart of the simple model presented external parameters as here. 1/2 Tasks for the future are (i) to incorporate the effects ␶ oyL h ϳ . (A5) of the ␤ effect, (ii) to unravel the zonal-average 2D ΂΃⌬bo solutions discussed here to study the effect of regional topographic and forcing detail, and (iii) to study the role Furthermore, if we assert that Bo ϳ wEk⌬bo with wEk ϭ of interior mixing. ␶o/( fLy) being the scale of the Ekman pumping, Eq. (A5) can be expressed in terms of buoyancy forcing Bo, rather than the surface buoyancy ®eld b ; thus Acknowledgments. This work was greatly helped by ⌬ o numerous conversations about residual-mean theory and f 1/2 the ACC with our colleague Alan Plumb. Two anony- h ϳ wLEk y. (A6) mous referees and Richard Karsten made very helpful ΂΃Bo comments on the manuscript. We acknowledge the sup- This equation is the scaling for thermocline depth dis- port of the physical oceanography program of NSF. cussed in Marshall et al. (2002) and Karsten et al. (2002) and derived from their studies of labaratory and nu- APPENDIX merical lenses forced by buoyancy ¯ux rather than a prescribed buoyancy ®eld. It is interesting to note that

Eddy Closure and Scaling Results the baroclinic transport, Eq. (33), scales like ϳ(␶oL)/ f, independent of ⌬bo, if Eq. (A4) is subsituted for k. To place the eddy closure adopted in this study in Last we relate the above expressions to those dis- context [Eq. (16)] and to relate our results to previous cussed in Bryden and Cunningham (2003). The strati- work, following Visbeck et al. (1997) [using ideas that ®cation implied by the Marshall et al. (2002) scaling, go back to Green (1970)] we write the eddy transfer Eq. (A6), is coef®cient as follows: ⌬bB2/3 fB 2 N 2 ϭϳoo ϳ oK, f 21/2 2 K ϭ ␷ЈlЈϭcLLe ␳ y, (A1) hwfLEk yo␶ ͙Ri where we have set ⌬bo ϭ Bo/wEk and used Eqs. (A3) 2 2 2 1/2 where ce is a constant, Ri ϭ (N h )/u is the Richardson and (A6) to express K as K ϭ ce(Bo/ f ) Ly. The relation 2 2 2 number, u is the mean zonal current at the surface, f/ N ϭ [( f Bo)/␶o ]K is the key result of Bryden and Cun- ͙Ri is the Eady growth rate, L␳ ϭ (Nh)/ f is the de- ningham (2003), their Eq. (9). formation radius, and Ly is the meridional scale of the baroclinic zone. As discussed in, for example, Marshall et al. (2002), the eddy velocity implied by Eq. (A1) is REFERENCES

␷Јϳ( f/)͙Ri L␳ ϭ u, appropriate if the eddies garner Andrews, D. G., J. Holton, and C. Leovy, 1987: Middle Atmosphere energy over a deformation scale L␳ (see, e.g., the dis- Dynamics. Academic Press, 489 pp. cussion in Held 1999). The eddy transfer scale is as- Bryden, H. L., 1979: Poleward heat ¯ux and conversion of available potential energy in Drake Passage. J. Mar. Res., 37, 1±22. sumed to be Ly, the scale of the baroclinic zone. Note ÐÐ, and S. A. Cunningham, 2003: How wind-forcing and air±sea that this may be a problematical assumption in the heat exchange determine the meridional temperature gradient ACCÐthe eddy transfer scale could be set, for example, and strati®cation for the Antarctic Circumpolar Current. J. Geo- by the Rhines scale (u/␤)1/2, an obvious avenue for fu- phys. Res., 108, 3275, doi:10.1029/2001JC001296. ture enquiry. Cenedese, J., J. Marshall, and J. A. Whitehead, 2004: A laboratory model of thermocline depth and exchange ¯uxes across circum- The thermal wind equation tells us that polar fronts. J. Phys. Oceanogr., in press. de Szoeke, R. A., and M. D. Levine, 1981: The advective ¯ux of h ⌬b u ϭ o , (A2) heat by mean geostrophic motions in the Southern Ocean. Deep- Sea Res., 28, 1057±1084. fLy Doos, K., and D. Webb, 1994: The Deacon cell and other meridional cells of the Southern Ocean. J. Phys. Oceanogr., 24, 429±442. where ⌬bo is the buoyancy change at the surface over the scale L . Hence K, Eq. (A1), can be written as Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean y circulation models. J. Phys. Oceanogr., 20, 150±155. h⌬b Gille, S., 1997: The Southern Ocean momentum balance: Evidence o for topographic effects from numerical model output and altim- K ϭ cuLeyϭ c e . (A3) f eter data. J. Phys. Oceanogr., 27, 2219±2231.

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC 2354 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 33

ÐÐ, 2003: Float observations of the Southern Ocean. Part II: Eddy McWilliams, J. C., W. R. Holland, and J. H. S. Chow, 1978: A de- ¯uxes. J. Phys. Oceanogr., 33, 1167±1181. scription of numerical Antarctic Circumpolar Currents. Dyn. At- Gnanadesikan, A., and R. Hallberg, 2000: On the relationship of the mos. Oceans, 2, 213±291. Circumpolar Current to Southern Hemisphere winds in coarse- Munk, W. H., and E. Palmen, 1951: Note on the dynamics of the resolution ocean models. J. Phys. Oceanogr., 30, 2013±2034. Antarctic Circumpolar Current. Tellus, 3, 53±55. Green, J. S. A., 1970: Transfer properties of the large-scale eddies Olbers, D., and V. O. Ivchenko, 2001: Diagnosis of the residual cir- and the general circulation of the atmosphere. Quart. J. Roy. culation in the Southern Ocean as simulated in POP. Ocean Dyn., Meteor. Soc., 96, 157±185. 52, 79±93. Hallberg, R., and A. Gnanadesikan, 2001: An exploration of the role Pedlosky, J., 1996: Ocean Circulation Theory. Springer-Verlag, 453 of transient eddies in determining the transport of a zonally pp. reentrant current. J. Phys. Oceanogr., 31, 3312±3330. Phillips, H. E., and S. R. Rintoul, 2000: Eddy variability and ener- Held, I., 1999: The macroturbulence of the troposphere. Tellus, 51A- getics from direct current measurements of the Antarctic Cir- B, 59±70. cumpolar Current south of Australia. J. Phys. Oceanogr., 30, ÐÐ, and T. Schneider, 1999: The surface branch of the zonally 3050±3076. averaged mass transport circulation of the troposphere. J. Atmos. Rhines, P. B., 1993: Ocean general circulation: Wave and advection Sci., 56, 1688±1697. dynamics. Modelling Oceanic Climate Interactions, J. Wille- Ivchenko, V. O., K. J. Richards, and D. P. Stevens, 1996: The dy- brand and D. L. T. Anderson, Eds., Springer-Verlag, 67±149. namics of the Antarctic Circumpolar Current. J. Phys. Ocean- Rintoul, S., C. Hughes, and D. Olbers, 2001: The Antarctic Circum- ogr., 26, 753±774. polar Current systemÐChapter 4.6. Ocean Circulation and Cli- Johnson, G. C., and H. L. Bryden, 1989: On the size of the Antarctic mate, G. Siedler, J. Church, and J. Gould, Eds., International Circumpolar Current. Deep-Sea Res., 36, 39±53. Geophysics Series, Vol. 77, Academic Press, 271±302. Karsten, R., and J. Marshall, 2002a: Constructing the residual cir- Speer, K., S. R. Rintoul, and B. Sloyan, 2000: The diabatic Deacon culation of the Antarctic Circulation Current from observations. cell. J. Phys. Oceanogr., 30, 3212±3222. J. Phys. Oceanogr., 32, 3315±3327. Stammer, D., 1998: On eddy characteristics, eddy mixing and mean ÐÐ, and ÐÐ, 2002b: Testing theories of the vertical strati®cation ¯ow properties. J. Phys. Oceanogr., 28, 727±739. of the ACC against observations. Dyn. Atmos. Oceans, 36, 233± Straub, D. N., 1993: On the transport and angular momentum balance 246. of channel models of the Antarctic Circumpolar Current. J. Phys. ÐÐ, H. Jones, and J. Marshall, 2002: The role of eddy transfer in Oceanogr., 23, 776±782. setting the strati®cation and transport of a circumpolar current. Sverdrup, H. U., M. W. Johnson, and R. H. Fleming, 1942: The J. Phys. Oceanogr., 32, 39±54. Oceans, Their Physics, Chemistry, and General Biology. Pren- Marshall, D., 1997: Subduction of water masses in an eddying ocean. tice Hall, 1087 pp. J. Mar. Res., 55, 201±222. Tansley, C. E., and D. P. Marshall, 2001: On the dynamics of wind- Marshall, J. C., 1981: On the parameterization of geostrophic eddies driven circumpolar currents. J. Phys. Oceanogr., 31, 3258±3273. in the ocean. J. Phys. Oceanogr., 11, 1257±1271. Treguier, A. M., I. M. Held, and V. D. Larichev, 1997: On the pa- ÐÐ, D. Olbers, H. Ross, and D. Wolf-Gladrow, 1993: Potential rameterization of quasi-geostrophic eddies in primitive equation vorticity constraints on the dynamics and hydrography of the models. J. Phys. Oceanogr., 27, 567±580. Southern Ocean. J. Phys. Oceanogr., 23, 465±487. Visbeck, M., J. Marshall, T. Haine, and M. Spall, 1997: Speci®cation ÐÐ, H. Jones, R. Karsten, and R. Wardle, 2002: Can eddies set of eddy transfer coef®cients in coarse-resolution ocean circu- ocean strati®cation? J. Phys. Oceanogr., 32, 26±38. lation models. J. Phys. Oceanogr., 27, 381±402.

Unauthenticated | Downloaded 10/02/21 06:02 AM UTC