The Antarctic Circumpolar Current in three dimensions

Transcription

1 1 The Antarctic Circumpolar Current in three dimensions Timour Radko and John Marshall Department of Earth, Atmospheric and Planetary Sciences, Massachusetts Institute of Technology.

2 2 Abstract. A simple theory is developed for the large scale three-dimensional structure of the Antarctic Circumpolar Current and the upper cell of its overturning circulation. The model is based on a perturbation expansion about the zonal-average residual mean model developed in Marshall and Radko (23). The problem is solved using the method of characteristics for idealized patterns of wind and buoyancy forcing constructed from observations. The equilibrium solutions found represent a balance between the Eulerian meridional overturning, eddy-induced circulation and downstream advection by the mean flow. Depth and stratification of the model thermocline increases significantly in the Atlantic-Indian sector where the mean wind-stress is large. Residual circulation in the model is characterized by intensification of the overturning circulation in the Atlantic-Indian sector and reduction in strength in the Pacific region. Predicted three-dimensional patterns of stratification and residual circulation in the interior of the ACC are compared with observations.

3 3 1. Introduction The absence of land barriers in the Antarctic Circumpolar Current (ACC) results in distinct dynamical features that have no direct counterpart in the theory of midlatitude ocean gyres. Sverdrup dynamics, the cornerstone of subtropical thermocline theory (Rhines and Young, 1982, Lyuten et al., 1983), do not apply here. Both wind and buoyancy forcing play a role, as do geostrophic eddies which appear to be crucial in determining the stratification and transport of the ACC (see the review by Rintoul et al. 21). The difficulty of incorporating eddy transfer led to a rather slow development of conceptual models of the ACC (see, e.g. Johnson and Bryden, 1989; Marshall et al., 1993). Recently, however, residual-mean theories have been applied see Karsten et al. (22), Marshall and Radko (23, MR hereafter) which, we believe, capture the essence of the zonally averaged circulation and stratification of the Southern Ocean and fully embrace the central role of eddies. It is assumed that the Eulerian meridional circulation driven by the westerly winds (the Deacon cell), tending to overturn isopycnals, is largely balanced by the geostrophic eddies which act in the opposite sense. The Transformed Eulerian Mean formalism (Andrews and McIntyre, 1976) is used to represent the combined effect of eddy-induced and mean flow advection. While the simplified two-dimensional zonal-average view attempts to explain the integral characteristics of the ACC, such as the overall strength of the overturning circulation and eastward baroclinic transport, zonal averaging masks important three-dimensional effects. For example, inspection of the pattern of surface heat flux in the Southern Ocean (Fig. 1) reveals its strikingly nonuniform distribution along the path of the ACC, with air-sea heat fluxes warming the ocean at rates reaching 6W/m 2 into the ocean in the Atlantic and Indian ocean and becoming smaller and changing sign in the Pacific region. The downstream variation in the surface forcing may be a consequence of the asymmetry in the trajectory of the ACC; the current is partially steered by bottom topography (Marshall, 1995) and therefore does not exactly follow

4 4 latitude circles. As a result, the surface heat flux in to the ocean increases (relative to the streamline average) where the ACC meanders equatorwards in to warmer regions downstream of Drake Passage (see Fig. 1) and reduces when the ACC drifts polewards in the Pacific sector. Wind stress (Fig. 2a) also varies along the path of the ACC being significantly larger in the Atlantic-Indian sectors. In this paper we will argue that the spatially nonuniform forcing leads to downstream variation of the thermocline depth and stratification of the ACC (see observations of Sun and Watts, 22), and modulates the strength of the upper cell of the meridional overturning circulation. Figure 1. This study attempts to explain the three-dimensional time mean structure of the ACC and the associated pattern of the overturning circulation by extending the zonal average model of MR. Exploiting the asymptotic limit in which the downstream variation in buoyancy is assumed to be weak relative to its variation in the meridional plane, the problem is reduced to a system of equations which can be readily solved using the method of characteristics. The theory is analogous to linear models of forced stationary waves in the earth s troposphere (see the review by Held, 1983). The paper is set out as follows. After presenting key observations of the ACC in section 2, we develop a theoretical framework (Sec. 3) by introducing the quasi-zonal jet approximation. Residual-mean theory is used to incorporate the effects of mesoscale eddies. Simplifications introduced by the analytical model are then used to determine the interior structure of the ACC for various surface and boundary conditions. In Sec. 4 we briefly consider the simplest diagnostic model which involves prescribing the idealized distribution of the surface buoyancy (b m ), buoyancy flux (B) andthewind stress (τ) and computing the resulting interior fields. In Sec. 5 we study a different, and perhaps more physical, prognostic model in which the surface buoyancy distribution and fluxes are considered unknown and computed as a part of the problem. Model solutions are compared with the oceanographic observations. We summarize and conclude in Sec. 6.

5 5 2. Observational background Figs. 2-4 present key observations of surface properties and forcing of the Antarctic Circumpolar Current (to be used in our theoretical model). The pattern of zonal wind stress (τ) is shown in Fig. 2a, the surface buoyancy distribution (b m ) in Fig. 3a, and the air-sea buoyancy flux (B) in Fig. 4a. 1 To analyze the variation of these fields along the path of the ACC, it is convenient to reference our along-stream coordinate to mean surface geostrophic contours; these are indicated by the heavy solid lines in Figs. 2a,3a,4a (deduced from satellite altimetry), which mark the boundaries and the axis of the ACC. The width of the region bounded by these streamfunction contours is 1-2km (depending on the longitude). Projection of the zonal wind stress, surface buoyancy, and buoyancy flux on to this coordinate system is shown in Figs. 2b,3b,4b respectively, where the ordinate is now a geostrophic streamline rather than latitude. Figure 2. Several comments on the structure of the forcing fields are in order. Downstream (x) variation in (τ,b,b m ) contains a large signal in the fundamental (in x) harmonic whose wavelength equals the zonal extent of the ACC. The two lowest Fourier components in x the along stream average (n= mode) mode and the fundamental harmonic (n=1 mode) capture much of the large scale spatial variability of the ACC. Downstream variation in the wind stress (Fig. 2b) is rather moderate, about 2% of the mean. Winds over the ACC intensify in the Indian and Western Pacific sectors and weaken over Eastern Pacific and Western Atlantic. Surface buoyancy (Fig. 3b) has a similar pattern, characterized by weak downstream variation, whereas the air-sea flux (see Fig. 4b) is highly inhomogeneous. Buoyancy flux over the ACC is mostly positive (i.e. into the ocean) but changes sign in the eastern Pacific (12 W 6 W ). Figure 3. 1 It should be noted that there is a significant uncertainty in the observations, particularly with regard to the air-sea fluxes; the estimates of the surface buoyancy flux from various datasets (e.g. NCEP, COADS, SOC) differ by as much as 5%.

6 6 The following discussion will focus on the dynamics of the long wave components and therefore we also show (in panels 2c,3c,4c) the filtered data for (τ,b,b m ) consisting of the two lowest Fourier components in x: the streamline average mode and the fundamental harmonic. Fourier analysis indicates that the large-scale variation in surface buoyancy and air-sea buoyancy flux are very much in phase with each other but shifted somewhat to the west relative to the phase of the wind stress. Our objective is to study how the surface forcing fields drive and interact with large scale motions in the interior of the ACC and thereby shape the three-dimensional patterns of buoyancy and meridional overturning in the Southern Ocean. Figure Formulation 3.1. Elements of residual-mean theory Our starting point is the three-dimensional time mean equations of motion. We ignore the inertial terms in the time mean momentum equations, and write them as fv = P + τx x z (1) fu = P + τy, y z where (u, v) are the horizontal components of the Eulerian mean velocity v, (τ x,τ y ) are the wind stress components which we assume to be significant only in a surface boundary layer, and P is the dynamical pressure. The time-mean buoyancy equation is v b = (v b )+ B z, (2) where b is the time mean buoyancy and primes denote the perturbations from this mean due to transient eddies; B represents the vertical buoyancy flux due to small-scale processes and air-sea fluxes. In application to the ACC we interpret x as the along stream coordinate; y is the coordinate normal to the stream, referenced to a mean surface geostrophic contour.

7 7 Andrews and McIntyre (1976) showed that it is possible to incorporate the eddy-flux terms in the Eulerian equations by introducing residual velocities given by: v res = v + v, (3) where v =(u,v,w ) is the eddy induced velocity of residual mean theory, which is assumed to be non-divergent: v = v =. (4) We also require that the vertical velocity (w ) is zero at the surface (z =),and therefore integration of (4) from surface to depth z results in w = x Ψ u + y Ψ v,where Ψ u = z u dz, Ψ v = z v dz are components of a vector streamfunction. The eddy velocities can be compactly written thus: (u,v )= z Ψ, (5) w = h Ψ where Ψ =(Ψ u, Ψ v). In Appendix A we relate the vector streamfunction Ψ to the eddy buoyancy fluxes (v b ). The resulting model provides a physical basis for a parameterization scheme in which the eddy streamfunction (below the vertically homogeneous mixed layer) is determined by the local isopycnal slope as follows: Ψ u = k s x s x, u = k (s z x s x ), Ψ v = k s y s y, v = k z (s y s y ). (6) where k is a constant which sets the magnitude of the eddy transfer process. This parameterization is a direct extension of a two-dimensional closure introduced in MR and supported by the laboratory experiments in Cenedese et al. (24).

8 8 The buoyancy equation written in terms of the residual velocities (see Appendix A) reduces to v res b = B z, (7) where B = B + B includes the explicit buoyancy forcing (B) and a contribution from the diabatic component of eddy fluxes (B ). Here, however, we assume that eddies are adiabatic 2 and therefore B = Quasi-zonal jet approximation Our study is focused on a regime in which departure of the solution from its along stream average is asymptotically small. As shown below, this approximation makes it possible to theoretically analyze three-dimensional effects, while retaining a direct connection with the two-dimensional ACC model in MR. To develop a linear theory for the quasi-zonal current, we search for a solution by expanding in a parameter ɛ = b 1 << 1, b which measures the variation in buoyancy along the streamlines of the ACC relative to the cross-stream variation. Inspection of the surface buoyancy field in Fig. 3c indicates that ɛ.1, which justifies the asymptotic expansion in ɛ. Next, we separate the mean variables into the dominant two dimensional part [a function of (y,z) only] and a weak 2 This assumption is violated in the near surface regions, where the presence of the surface tends to suppress the vertical component of the eddy fluxes, which become directed across the isopycnals. However, the estimates in MR suggest that in the near surface mixed layer the explicit buoyancy forcing B significantly exceeds B and therefore diabatic component of the eddy fluxes can be neglected in this simple model of the ACC.

9 9 ( ɛ) three dimensional component. The linear analysis is focused on the dynamics of the fundamental harmonic in x: v = v (y,z)+v 1 (x, y, z)+o(ɛ 2 ), v = v (y,z)+v 1(x, y, z)+o(ɛ 2 ), b = b (y,z)+b 1 (x, y, z)+o(ɛ 2 ), P = P (y,z)+p 1 (x, y, z)+o(ɛ 2 ), B = B (y,z)+b 1 (x, y, z)+o(ɛ 2 ), τ x = τ x (y)+τ 1x (x, y)+o(ɛ 2 ), v 1 = Re[ˆv 1 (y, z)exp(ikx)] v1 = Re[ˆv 1(y,z)exp(ikx)] b 1 = Re[ˆb 1 (y,z)exp(ikx)] P 1 = Re[ ˆP 1 (y,z)exp(ikx)] B 1 = Re[ ˆB 1 (y,z)exp(ikx)] τ 1 = Re[ˆτ 1x (y)exp(ikx)] (8) where k is a wavenumber. B is the buoyancy forcing in Eq.(7), and τ x is the zonal component of the wind stress. The subscript (, 1...) in (8) pertains to (zero,first...) order quantities in ɛ. follows: To illustrate the expansion procedure, we write the buoyancy equation in (7) as (v + v ) (b + b )+(v + v ) (b + b ) = B z + B 1 z The leading (zero order) balance in ɛ is: whereas the first order balance requires that v res b = B z, (9) v 1res b + v res b 1 = B 1 z (1) The momentum equations (1), continuity equation (4), and the eddy closure (6) are similarly expanded in powers of ɛ. Not surprisingly, at the zero order in ɛ we recover a set of equations identical to those used in the two-dimensional model (MR), which is briefly reviewed below. The first order correction will give us information about longitudinal variations in the stream, the information we seek.

10 Zero order solution: the ACC in two dimensions The zero order component of the (y, z)-velocities can be expressed in terms of the scalar streamfunctions (time-mean and eddy induced) as follows: v = Ψ w z = Ψ, y v = Ψ w z = Ψ and the residual circulation is described by the streamfunction Ψ res (y,z) =Ψ +Ψ. y, The eddy parameterization (6) at the leading order reduces to Ψ = k s s = k s 2, v = Ψ z = k z s2, s = b y b z, (11) and the zero-order y-momentum equation is fv = τ z. (12) The problem is solved separately in a thin, vertically homogeneous mixed layer ( h m <z<) and in the stratified interior (z < h m ). For the interior it is assumed that eddies are adiabatic, and the forcing, mechanical and thermodynamical (B,τ), vanishes. Hence, the buoyancy equation (9) reduces to J(Ψ res,b)=, (13) where J(A, B) A y B z A z B y. Eq.(13) implies that the residual streamfunction and buoyancy are functionally related: Ψ res =Ψ res (b ). Integrating the buoyancy equation (2) over the depth of the vertically homogeneous mixed layer ( h m <z<), for the (two-dimensional) zero order component we obtain: Ψ res z= hm b m y = B, (14) where b m is the surface buoyancy. Finally, integrating the momentum equation (12) vertically from the surface to depth z (noting that Ψ = at the surface and τ is zero

11 11 at depth) we find that fψ = τ. Because Ψ res (y,z) =Ψ +Ψ,wearriveat,using Eq.(11), the key relationship of the MR model: ( ) τ Ψ res (b )= f + k s 2. (15) This equation can be easily integrated (see MR) along isopycnal surfaces: an algebraic transformation shows that the isopycnals represent the characteristics of Eq. (15), and the characteristic velocities v c and w c are: dy = v dl c =1 dz = w dl c = s = b y b z = τ fk Ψ res k. (16) Of course, Ψ res and b do not change at a point moving along the characteristics, and therefore for any given distribution of surface buoyancy and fluxes, the interior solution can be obtained by integrating the characteristic equations (16) downward from the surface. If, however, the surface Ψ res or b are unknown, as in the problem solved in Sec. 5, the method of characteristics can be readily modified to include an iterative adjustment to specified surface and boundary conditions First order correction To proceed to the next order, we rewrite the first order buoyancy budget (1) in terms of the complex amplitudes of the residual velocity and buoyancy (ˆv 1 res, ˆb 1 ). Since the x-component of the eddy induced velocity does not appear at this order the parameterization in (6) implies that u = O(ɛ 2 ) Eq.(1) reduces to b ˆv 1 res y +ŵ b 1 res z + u ˆb 1 ikˆb 1 + v res y + w ˆb 1 res z = ˆB 1 z. (17) The continuity equation (4) at O(ɛ) is:

12 12 ikû 1 + ˆv 1 res y + ŵ 1 res z =. (18) To simplify the algebra, it is also convenient to introduce the following O(ɛ) variables: ψ1 z ˆv 1 dz ψ 1 res (19) z ˆv 1 resdz, which represent the first order terms in the ɛ-expansion of the eddy streamfunction Ψ v in (6) and the corresponding residual streamfunction. The integral of the continuity equation (18) in z yields: ŵ 1 res = ψ 1 res y z ik û 1 dz. (2) Solution of the perturbation equations in (17)-(2) is greatly simplified by introducing a coordinate system associated with the zero-order buoyancy surfaces. As shown in Appendix B, an algebraic transformation reduces the system of partial differential equations in (y,z) in (17)-(2) to a system of ordinary differential equations in variable l which measures the distance along the old zero-oder isopycnals: dˆb 1 dl (ψ 1 res+ˆτ 1x /f)b z 2k s dψ 1 res dl = ik b R z z 2k s f ( ) dψ res dˆb 1 db + ik u ˆb 1 dl b z ˆP 1 dz = ik z û1dz. (21) where d dl is defined in (16). Dynamical pressure and the zonal velocity in (21) is related to the buoyancy by the hydrostatic and geostrophic relations: ˆP 1 = z h ˆb1 dz, û 1 = 1 f where h is the depth of the model thermocline. ˆP 1 y, (22) In that which follows, we solve the linearized problem by integrating the system of coupled ordinary differential equations for (ˆb 1,ψ 1 res ) in (21) downward from the base of

13 13 the mixed layer into the interior along the zero order isopycnals. Essentially, the method of characteristics introduced in Sec. 3.3 for the two-dimensional problem is extended to three dimensions. Implementation of this method depends on a particular choice of surface and boundary conditions, and specific details are given in Secs. 4,5. However in every case integration of Eqs. (21) requires knowledge of ψ 1 res at the base of the mixed layer. In Appendix B, we derive the (order ɛ) perturbation equations for the mixed layer and show that ψ 1 res can be expressed as ψ 1 res ( h m )= ˆB 1 Ψ res ( h m ) yˆb 1m iku ˆb1m h m. (23) y b m Equations in (21) and (23) are the key elements of our model, which, as we show next, make it possible to readily obtain steady linear three dimensional solutions for the ACC and its overturning circulation. 4. Diagnostic model: prescribing the surface buoyancy and the air-sea fluxes The simplest problem that can be treated by the method of characteristics, and the one which was solved in two dimensions by MR, involves prescribing the surface buoyancy distribution (b m ), wind stress (τ x ), net buoyancy forcing (B), and computing the resulting interior distribution of buoyancy and residual circulation. In order to relate our theoretical results to the observations, it is convenient to set the origin of coordinate system at 18 W. This projects the longitude range 18 W 18 E in Figs. 2,3,4 on to the computational interval <x<l x = 2π k in our model; the zero longitude therefore corresponds to x =.5L x. The two lowest spectral components of (τ x,b m,b) in Figs.2c,3c,4c, are closely approximated by the following analytical expressions:

14 14 Wind stress: τ x = τ x (y)+re[ˆτ 1x (y)exp(ikx)]; τ x = τ [.6+sin(π y L y )], τ =.1m 2 s 2 (24) ˆτ 1x =.3exp(.5πi)τ x. Buoyancy: b m = b m + Re[ˆb 1m exp(ikx)]; b m =(y/l y ) b, b =.15ms 2, (25) ˆb1m =.7 exp(.3πi) b sin(π y L y ). Air-sea flux: B = B + Re[ ˆB 1 exp(ikx)]; B = B sin(π y L y ), B =7 1 9 m 2 s 3 (26) ˆB 1 =.5(1 i)b. Note that we work in Cartesian coordinates and do not attempt to represent details of spherical geometry. The mixed layer depth is set at h m = 1m, and the constant that sets the magnitude of the eddy transfer coefficients K (see the Appendix A) is k =1 6 as estimated and used in MR. The zonal extent of the channel is L x = 2km, and the width of the model current is L y = 2km. As discussed earlier (Secs. 1,2), the patterns of buoyancy and buoyancy flux anomalies represent the enhanced warming of the ACC in the Atlantic and Indian ocean regions and cooling east of Australia (see Fig. 4). The complex coefficients in the ˆb 1,ˆτ 1x,and ˆB 1 terms reflect the phase of the fundamental (in x) harmonic relative to the 18 W 18 E interval. The zero order interior fields of buoyancy and residual circulation are obtained by integrating Eqs. (16) as in MR. The resulting solution, for surface conditions in

15 15 Eqs. (24)-(26), is shown in Fig. 5. The depth of the model thermocline reaches 2km on the equatorward flank of the ACC, not unlike the observations discussed in Karsten and Marshall (22a). The sense of the residual circulation corresponds to upwelling of fluid in the deep layers that outcrop at the poleward flank of the ACC; fluid is subducted downward and equatorward in the upper layers. The net strength of the overturning circulation is 18Sv, which is comparable to the earlier observational estimates (e.g. Karsten and Marshall, 22b) and the zonal-average theoretical model of MR. Figure 5. The next step is to compute the O(ɛ) quantities. For a prescribed buoyancy distribution in the mixed layer (ˆb 1m ), ψ 1 res ( h m ) is obtained from Eq.(23). Knowing (ψ 1 res, ˆb 1 ) at the bottom of the mixed layer, we integrate the interior equations (21) along the zero order (hence already known) characteristics. This integration, however, requires knowledge of the terms on the right hand side of (21), which, in our case, have to be obtained as a part of the solution. These terms are computed using an iterative procedure. First we solve (21) ignoring the RHS terms, and then use the resulting solution to evaluate them; pressure and zonal velocity are obtained from (22). The integration along the characteristics is repeated consecutively, and on each iteration we use the estimate of the RHS terms from the previous step. Typically, it takes less than 1 such iterations for the model to converge, within a negligible error of 1 8,toa sought after solution. The total buoyancy and velocity fields are obtained by adding the two-dimensional zero order components and the O(ɛ) correction (8). Fig. 6 shows the resulting horizontal residual velocities (u res,v res ) immediately below the mixed layer, superimposed on the surface buoyancy field. The surface buoyancy is higher than the zonal average for 3 L 8 x <x< 7L 8 x (the region in between 45 E and 135 W ) and lower outside of this interval. The horizontal residual velocity vectors (u res,v res ) are generally well aligned along the buoyancy contours; small, although visible, difference in the orientation of the two is a consequence of advection by the vertical component of the residual circulation

16 16 (w res ). Figure 6. Fig. 7a shows the distribution of w res at z = h m ; this quantity indicates the location and strength of the flux of buoyancy and passive tracers into (from) the diabatic mixed layer. The overturning circulation, as measured by w res, is characterized by a strong comparable to the mean downstream variation 3. The amplitude of w res is relatively weak for <x<l x /3 (an interval corresponding to 18 6 W ) but greatly intensifies after x = L x /3, reaching the values of ms 1 6m/year. Thus,our model predicts that the meridional overturning cell (MOC) in the Southern Ocean is greatly reduced in strength in the Pacific sector, but enhanced in the Atlantic/Indian sector. Note the significant difference in the structure and amplitude of w res in Fig. 7a and the Ekman pumping in Fig. 7b, which indicates the significance of eddy advection for the overturning circulation in the Southern Ocean. Figure Solving for the surface buoyancy and buoyancy flux While the foregoing setup is internally consistent and provides an important diagnostic tool for the dynamics of the ACC, one can readily question some of the implicit assumptions in the formulation of the model. In particular, the surface buoyancy distribution and/or the air-sea fluxes, which have been prescribed in Sec. 4, in reality may be a consequence of internal dynamics of the ACC and its interaction with the adjacent subtropical gyres. Although the issue of causality is beyond the scope of the present study, we now consider alternative, and perhaps more physically motivated, upper boundary conditions. In the following example we suppose that the surface buoyancy distribution and the air-sea fluxes are not known and should be computed as a part of the problem. 3 Recall that our linear theory makes no assumptions as to the relative strength of the zonally averaged and 3D components of the residual circulation.

17 17 Instead, the mixed layer buoyancy b m (x, y) is relaxed to a target buoyancy distribution b b (y)+re[ˆb 1 (y)exp(ikx)], which, we assume, is set by the atmosphere and therefore is to be regarded as known. The surface buoyancy flux is parameterized accordingly: B = λ(b m b ). (27) In this calculation we use λ =9 1 6 ms 1, which, for h m 1m, corresponds to a relaxation timescale of about two months; the resulting solutions show little sensitivity to the specific value chosen for this parameter. Inspection of the near surface atmospheric temperature (not shown) suggests that b can be approximated by the following simple analytical function: b = b + Re[ˆb 1 exp(ikx)]; b =(y/l y) b, b =.15ms 2 (28) ˆb 1 =.1exp(.3πi) b sin(π y L y ), which only slightly differs from the surface buoyancy distribution (25) used in our previous model (Sec. 4). At the same time it is sensible to consider the vertical distribution of buoyancy on the equatorial flank of the ACC as given. If the ACC (y <L y ) merges continuously with a subtropical gyre on the equatorial flank (y >L y ), then the stratification at y = L y may be affected, or even determined, by the dynamics of the subtropical thermocline. According to Karsten and Marshall (22a), the stratification on the equatorial flank of the ACC can be represented by a decaying exponential with an e-folding depth of h e 1m. Wethereforeuse b y=ly = b N (z) =A 1 exp(z/h e )+A 2. (29) The constants A 1 and A 2 are determined by requiring b N (z = h m )= b and b N (z = h) =,whereh is the depth of the model thermocline at y = L y.

18 18 Prescribing the density distribution at y = L y requires modification of the foregoing method of characteristics; we implement the following iterative procedure. First we compute the two-dimensional zero order solution. An initial guess is made for the surface buoyancy (we start with a linear b m (y)), the surface buoyancy flux is computed from (27), and the model is integrated using the method of characteristics in (16). The resulting buoyancy distribution at y = L y is compared with our target buoyancy distribution (29), and the difference between the two is used to adjust the surface buoyancy at the origin of each characteristic. The procedure is repeated until the buoyancy at y = L y converges, within an acceptable error ( 1 8 ), to the target distribution at the Northern flank (29). Figure 8. The zero order (two dimensional) components of buoyancy and residual circulation are shown in Fig. 8; this solution is similar to the one obtained with the model in Sec. 4 (see Fig. 5). In our case, however, the model is interactive, and does not rely on the a-priori knowledge of the air-sea fluxes. The implied streamline averaged buoyancy flux and surface buoyancy in the model are shown in Fig. 9. Buoyancy flux is positive (in to the ocean) with a maximum amplitude of m 2 s 3, broadly consistent with the observations in Fig. 4c. Figure 9. Next, the same iterative technique is applied to the three-dimensional component of buoyancy (ˆb 1 ). Eqs. (21) are consecutively integrated along the zero order characteristics; on each iteration the value of surface buoyancy at the origin of each characteristic (b 1m ) is adjusted to reduce the amplitude of b 1 at the point where this characteristic crosses the Northern boundary (y = L y ). The procedure is repeated until b 1 converges to zero at the equatorward flank of the ACC: b 1 y=ly =. The resulting distribution of buoyancy for surface and boundary conditions prescribed in (27)-(29), is shown in Fig. 1. The zonal buoyancy section along the center (y =.5L y ) of the model current in Fig. 1a is characterized by a pronounced warming, increase in stratification and upper thermocline depth in the eastern region.5l x <x<l x ( 18 E). To compare our results with

19 19 observations, particularly with regard to the downstream variation in hydrography, we turn to the recent data analysis by Sun and Watts (22). These authors present streamfunction projections of the historical hydrographic data the frame of reference directly associated with our (x, y)-coordinates. The pattern of buoyancy in Fig. 1a is suggestive of the observed temperature field along the stream function contours in the ACC (see Fig. 11 in Sun and Watts, 22). Likewise, the meridional section at 14 E in Fig. 3 of Sun and Watts (22) is qualitatively similar to the model prediction in Fig. 1b. Figure 1. Fig. 11 reveals the sense of the downstream variation in the structure of meridional cross-sections. In Fig. 11a (left) we show the difference in buoyancy between sections at 9 Eand3 E in our model. The corresponding potential temperature variation from the observations in Sun and Watts (22) is plotted in Fig.11b (right). The meridional coordinate in Fig.11b is the baroclinic mass transport streamfunction (Ψ 3 = 3 pδ dp). The range Ψ g 3 = Jm 2 in Fig.11b corresponds to the ACC width of 1 2km (depending on the longitude) and thus is broadly consistent with the zonal extent of the theoretical solution in Fig. 11a. Comparison of the two plots indicates that our model, despite being highly idealized, is able to capture the major features of the downstream buoyancy variation. Most of the spatial variability, in observations and in the model, occurs at the intermediate depth ( 5m) on the equatorward side of the ACC. Although the amplitude of the model buoyancy in Fig. 11a is significantly less than one might expect from the scale of the observed (Fig. 11b) temperature variation ( b gα T ), the discrepancy may in part be a consequence of not including high x-wavenumber spectral components that are present in Sun and Watts (22) observations but not in our model. Figure 11. The implied patterns of the surface buoyancy and the air-sea flux is shown in Fig. 12. While the downstream buoyancy variation is very gentle, less than 1% of the cross-stream buoyancy difference, the air-sea flux in Fig. 12a is highly non-uniform.

20 2 The amplitudes of the zero and first spectral harmonics are comparable, leading to their approximate cancellation in the eastern Pacific ocean and intensification in the Atlantic-Indian sector (L x /3 <x<2l x /3 in the model). Compelling evidence in support of the theory comes from comparing our solutions with the corresponding low modes the observed air-sea flux and buoyancy in Figs. 3c,4c. Both the phase and amplitude of the observed buoyancy and the air-sea flux are captured by the model predictions in Fig. 12a,b. Figure 12. Finally, it is of interest to examine the distribution of the large scale potential vorticity PV = fb z along the isopycnal surfaces in our model. In addition to being a useful dynamical tracer, the isentropic PV gradients measure direction and strength of the residual circulation (see the discussion in MR). Fig. 13 (top plot) shows a three-dimensional view of the isopycnal with b = 2 b. This surface is located on the 3 equatorward flank of the ACC, in the region where the residual circulation (see Fig. 7) is directed downward from the mixed layer. The distribution of the corresponding isentropic potential vorticity is shown in bottom plot of Fig. 13. An intriguing feature of this PV pattern is related to a region of low PV in the western part of the domain (x <.25L x ) the counterpart of south-eastern Pacific. This low PV fluid apparently originates near the surface in the region of surface cooling (see Fig. 12a), and then spreads, along the isopycnal surface, downward and eastward into the main thermocline. It is tempting to relate this low PV fluid with the Subantarctic Mode Water. Of course, mode waters are known to form convectively, within the deep winter mixed layers, physics which is beyond the scope of our idealized linear model. However, the dynamics considered herein may be essential in creating favorable conditions (surface cooling and subduction) for the formation of the low PV water, and thus a key player in determining its distribution. Figure 13.

21 21 6. Conclusions A model for the large scale three-dimensional structure of the Antarctic Circumpolar Current has been presented which assumes that the eddy transport in the interior of the ACC is entirely adiabatic; the diapycnal flux by small-scale turbulence is limited to the upper mixed layer. Tractability is achieved by expressing the buoyancy field in terms of the Fourier series in the zonal direction and drastically truncating the expansion. Our representation consists of only two modes: the dominant streamline average component and the first (fundamental) Fourier harmonic. Accordingly, the model is driven by the two lowest spectral components (n=,1 harmonics) of the wind stress and buoyancy forcing. The new theoretical framework makes it possible to determine the three-dimensional distribution of buoyancy and residual circulation for various surface and boundary conditions. In one of the examples (Sec. 4) we prescribe the surface buoyancy distribution and the air-sea flux and compute the interior fields. The prognostic model in Sec. 5, on the other hand, computes the buoyancy fluxes as a part of the problem, assuming that the stratification at the equatorward flank of the ACC is controlled by the circulation in the adjacent gyres and therefore is regarded as known; results of the two models are mutually consistent. The solutions represent a balance between the Eulerian meridional overturning, parameterized eddies, and the downstream advection of buoyancy by the mean flow. The latter component is absent in the zonal-average theories of the ACC (e.g. MR), and is shown here to play a significant role in determining the three-dimensional structure of the model thermocline. Residual circulation in the model is characterized by intensification of the overturning circulation in the Atlantic-Indian ocean sectors and dramatic reduction in its strength in the Pacific region, a pattern which is readily understood by considering the phases of the surface heat flux and the wind stress. Likewise, the model thermocline is relatively warm, stratified and deep in the region corresponding to 6 W 12 E.

22 22 These model predictions are consistent with the hydrography of the Southern Ocean. Ability of the model to reproduce the major features of the ACC indicates that this current can be viewed, at the leading order, as a superposition of the two-dimensional streamline-averaged flow and a linear forced long wave perturbation, analogous to the stationary waves in the earth s troposphere (Held, 1983). Acknowledgments. Support of the National Science Foundation is gratefully acknowledged.

23 23 APPENDIX A Eddy induced velocities of the residual-mean theory. Following Andrews and McIntyre (1976) (see also Ferrari and Plumb, 24, and Ferreira and Marshall, 24), we write the time-mean buoyancy budget (2) as: v b = F + B z, (A1) where b is the mean buoyancy and the eddy buoyancy flux F = v b = v b b b 2 b v b b b 2 b (A2) is separated in to components parallel and normal to the mean b surfaces. We are mostly concerned here with the second term in (A2), the skew flux which is equivalent to advection of b by the following eddy-induced velocity field: v = φ, (A3) where φ = v b b b 2. (A4) The first term in (A2) is zero for adiabatic eddies in which the eddy buoyancy fluxes are directed along mean buoyancy surfaces. If, however, eddies have a component normal to b contours, it is convenient (see MR) to combine the diabatic term with the direct diapycnal buoyancy flux (B) by rewriting (A1) as: v res b = B z, (A5) where B = B + B includes a contribution from the diapycnal component of eddy fluxes (B ). Residual velocities in (A5) are given by v res = v + v. Next, we note that the expression for the eddy velocities in (A3), (A4) greatly simplifies if the lateral scales significantly exceed the vertical scale ( z >> x, y ), a condition that is

24 24 generally well satisfied in the ocean (except in the relatively thin surface and bottom mixed layers). In this case, the horizontal eddy velocities reduce to the familiar forms ( ) u = u b z b z, v = z ( v b b z ). (A6) We adopt a conventional downgradient closure for the eddy buoyancy fluxes in (A6): u b = Kb x, (A7) where the eddy transfer coefficient K is not constant, but rather determined by the local properties of the flow field. MR used a model in which K is proportional to the isopycnal slope, a closure suggested by Visbeck et al. (1997) and supported by the laboratory experiments in Cenedese et al. (24). Extending this model for three-dimensional flows, we assume that u b b = k s x b x = k x bz bx (A8) v b b = k s y b y = k y b by z, where k is a positive constant. Parameterization in (A8) reduces (A6) to u = k z (s x s x ), v = k z (s y s y ). (A9) Eddy induced velocities can be expressed in terms of the vector streamfunction Ψ =(Ψ u, Ψ v) such that z (Ψ u, Ψ v )= (u,v ), and using (A9) we identify Ψ u = k s x s x, Ψ v = k s y s y, (A1) for the interior of the ocean below the mixed layer (z < h m ). In the mixed layer ( h m <z<), where isopycnals become vertical, (A1) does not apply. Instead, we require there that Ψ varies continuously from a finite value at z = h m to zero at the surface (z = ). Therefore, for both interior and mixed layer, we can relate the

25 25 eddy streamfunction and the eddy induced velocities as follows: z Ψ u = u dz, z Ψ v = v dz, w = h Ψ, providing a consistent model for the non-divergent eddy induced velocity field whose vertical component (w )vanishesatthesurface.

26 26 APPENDIX B Reduction of the first order problem to a system of ordinary differential equations. Following MR, we consider separately the adiabatic interior (z < h m ) and a thin, vertically homogeneous mixed layer ( h m <z<) see Fig.3 of MR. Interior dynamics Since the buoyancy forcing B is assumed to be negligible in the interior, the O(ɛ) buoyancy equation (17), written in terms of ψ 1 res, reduces to: ( z J(ψ 1 res,b )+J(Ψ res, ˆb 1 )=ik where (19) and (2) were used to eliminate (ŵ 1 res, ˆv 1 res ). z û 1 dz b z u ˆb 1 ), z < h m (B1) The explicit expression for ψ1 is obtained by expanding v in powers of ɛ: ( ) 2 v dz = k s 2 y b [ y = k ( z b) 2 = k b 2 ( z b 2 + Re 2 b ) y y b 3 (ˆb 1y b z ˆb 1z b y ) exp(ikx) z Therefore, picking out the first order ( ɛ) term, we find: ] + O(ɛ 2 ). (B2) ψ1 b y = 2k b 3 J(ˆb 1,b ). z (B3) The first order y-momentum equation written in terms of ψ 1 res is: f ψ 1 res z = ik ˆP 1 + f ψ 1 z + ˆτ 1x z. (B4) Integrating (B4) once in z and recalling that ψ 1 z= = ψ 1 res z=, we arrive at fψ 1 res = ik z ˆP 1 dz + fψ1 ˆτ 1x z=. (B5) Despite the seemingly complicated form of the governing equations (B1), (B5), they can be efficiently integrated along the old, zero order characteristics. For the characteristic velocities (v c,w c ) given by (16) d dl ( ) v c y ( )+w c z ( ) = y ( )+s z ( ) = 1 J(,b ), b z (B6)

27 27 dψ and therefore the first term in the buoyancy equation (B1) reduces to b 1 res z dl. Likewise, using (B6) we simplify the second term on the left hand side of Eq.(B1) to and thus Eq. (B1) reduces to ( ) ( ) J(Ψ res, ˆb dψ res dψ res dˆb1 1 )= J(ˆb 1,b )= b z db db dl, (B7) dψ 1 res dl ( dψ res db ) dˆb1 dl + ik u ˆb 1 b z z = ik û 1 dz. (B8) The momentum equation (B5) can also be greatly simplified by expressing it in terms of the (zero order) isentropic gradients. Using (B6), we rewrite (B3) as ψ 1 =2k s b z dˆb 1 dl. (B9) Substituting ψ 1 given by (B9) in (B5), we arrive at dˆb 1 dl (ψ 1 res + τ 1x /f)b z 2k s z = ik b z ˆP 1 dz. (B1) 2k s f Mixed layer The steady, vertically homogeneous, mixed layer buoyancy budget in 3D is b m u res x + v b m res y = B z. (B11) To proceed with the truncated spectral model, Eq. (B11) is integrated in z over the depth of the mixed layer h m, and the result is expanded in powers of ɛ using (8). The first order ( ɛ) component is given by: iku ˆb1m h m + ψ 1 res b m y +Ψ ˆb 1m res y = ˆB 1, (B12) where Ψ res,ψ 1 res are evaluated at the bottom of the mixed layer, resulting in an explicit expression for ψ 1 res z= hm : ψ 1 res = ˆB 1 Ψ res z= hm yˆb 1m iku ˆb1m h m. (B13) y b m

28 28 Equations in (B8), (B1), and the surface boundary condition (B13) are used in the model (Secs. 4,5) to efficiently compute the steady three dimensional components of buoyancy and residual circulation of the ACC.

29 29 References Andrews, D.G., and M.E. McIntyre, 1976: Planetary waves in horizontal and vertical shear: the generalized Eliassen-Palm flux and the mean zonal acceleration. J. Atmos. Sci., 33, Cenedese, J., J. Marshall, and J. A. Whitehead, 24: A laboratory model of thermocline depth and exchange fluxes across circumpolar fronts. J. Phys. Oceanogr., 34, Ferreira D., and J. Marshall, 24: Estimating eddy stresses by fitting dynamics to observations using a residual-mean ocean circulation model and its adjoint. J. Phys. Oceanogr., submitted. Ferrari R., and A. R. Plumb, 24: Residual circulation theory for the ocean. J. Phys. Oceanogr., submitted. Held, I.M., 1983: Stationary and quasi-stationary eddies in the extratropical troposphere: Theory. Large-scale Dynamical Processes in the Atmosphere. B.J. Hoskins and R.P. Pearce, Eds., Academic Press, Held, I.M., and T. Schneider, 1999: The surface branch of the zonally averaged mass transport circulation of the troposphere. J. Atmos. Sci., 56, Johnson, G.C., and H.L. Bryden, 1989: On the size of the Antarctic Circumpolar Current. Deep-Sea Res., 36, Karsten, R., and J. Marshall, 22a. Testing theories of the vertical stratification of the ACC against observations. Dyn. Atmos. Oceans., 36, Karsten, R., and J. Marshall, 22b: Constructing the residual circulation of the ACC from the observations. J. Phys. Oceanogr., 32, Karsten, R., H. Jones, J. Marshall 22: The role of eddy transfer in setting the stratification and transport of a circumpolar current. J. Phys. Oceanogr., 32, Luyten, J., J. Pedlosky and H. Stommel 1983: The ventilated thermocline. J. Phys. Oceanogr., 13,

30 3 Marshall, D., 1995: Topographic steering of the Antarctic Circumpolar Current. J. Phys. Oceanogr., 25, Marshall, J., and T. Radko. 23: Residual-mean solutions for the Antarctic Circumpolar Current and its associated overturning circulation. J. Phys. Oceanogr., 33, Rintoul, S.R., C.W. Hughes, and D. Olbers, 21: The Antarctic Circumpolar Current system. Ocean Circulation and Climate, G. Siedler, J. Church and J. Gould, Eds., International Geophysics Series, 77. Rhines, P.B. and W.R. Young, 1982: A theory of the wind-driven circulation I: Mid-ocean gyres.j. Mar. Res., 4 (Supplement), Sun, C., and D.R. Watts, 22: Heat flux carried by the Antarctic Circumpolar Current mean flow. J. Geophys. Res., 17, Visbeck, M., J. Marshall, T. Haine, and M. Spall, 1997: Specification of eddy transfer coefficients in coarse-resolution ocean circulation models, J. Phys. Oceanogr., 27, Department of Earth, Atmospheric, and Planetary Sciences, Massachusetts Institute of Technology, 77 Massachusetts Ave., Cambridge, MA, timour@ocean.mit.edu; marshall@gulf.mit.edu. With the extension package AGU ++, version 1.2 from 1995/1/12

31 31 Figure Captions 6 o E 9 o E 12 o E o 3 o E 3 o W 8 o S 7 o S 6 o S 5 o S 15 o E 4 o S 15 o W 18 o W o W 9 o W 12 o W Figure 1. Annual mean net surface heating of the ocean in Wm 2 estimated from the NCEP- NCAR reanalysis over (Czaja, 24, personal communication). Superimposed are the sea surface height contours (CI = 1 cm) which indicate the trajectory of the ACC.

32 32 3 (a) wind stress [1 1 Nm 2] (b) wind stress [1 1 Nm 2] (c) n =, 1 : wind stress [1 1 Nm 2] Figure 2. Fourier analysis of the observed (NCEP) surface wind stress (Ito, 24, private communication). (a)zonal component of the wind stress. Heavy solid lines are the streamfunction contours obtained from the sea surface height which mark the boundaries and the axis of the ACC (Ψ g = 5 1 4,, m 2 s 1 ). (b) Projection of the data on to the frame of reference associated with the streamfunction; the curved streamlines in the top plot are now straight and horizontal. (c) The filtered data consisting of the two lowest Fourier components in x: the streamline average mode and the fundamental harmonic.

33 33 3 (a) buoyancy [1 2 ms 2] (b) buoyancy [1 2 ms 2] (c) n =, 1 : buoyancy [1 2 ms 2] Figure 3. The same diagnostics as in 2 is now applied to the surface buoyancy field (from Levitus climatology).

34 34 3 (a) buoyancy flux [1 7 m2s 3] (b) buoyancy flux [1 7 m2s 3] (c) n =, 1 : buoyancy flux [1 7 m2s 3] Figure 4. The same diagnostics as in Figs. 2, 3 but for the air-sea buoyancy flux (based on the NCEP reanalysis).

35 35 z b(y,z) 2km y x z ψ res (y,z)l x km y Figure 5. The buoyancy (top) and the residual circulation (bottom) for the zero order (2D) component. The residual streamfunction (measured in Sv) represents the net strength of the overturning circulation.

36 36 L y x L x Figure 6. The horizontal components of the residual velocity (u res,v res ) immediately below the mixed layer, superimposed on the surface buoyancy field. Maximum horizontal velocity at the surface is v h = u 2 + v 2 =.14m/s.

37 37 a) w res ( h m ) x 1 6 L y 2 1 y 1 x L x 2 b) L y w Ek x y 1 2 x L x 18W 9W 9E 18E Figure 7. a) The vertical component of the residual velocity at the bottom of the mixed layer; w res represents the flux of buoyancy and the passive tracers from the adiabatic interior into (from) the diabatic mixed layer. The strength of the overturning circulation increases in the eastern part of the domain (Atlantic-Indian sectors) and decreases in the Pacific region (x <L x /3). b) Ekman velocity. Note the significant difference between w res and w Ek. Zero contours are represented by the dashed curves.

38 38 z b(y,z) 2km y x ψ res (y,z)l x z km y Figure 8. The streamline averaged buoyancy (top) and residual circulation (bottom) for a model in which we prescribe the stratification at the Northern flank of the ACC, and relax the surface buoyancy to a target distribution b.

39 39 b m (y) [1 2 ms 2 ] 1.5 2km y B (y) [1 7 m 2 s 3 ].6 2km y Figure 9. Streamline average surface buoyancy (top) and the air-sea buoyancy flux (bottom) for the model in Fig. 8

40 4 z a) b(y=.5l y ) 1m 2km x x W 9W 9E 18E b) b(14 E) x 1 3 z 1 5 2m 2km y Figure 1. Patterns of buoyancy in the model ACC. a) Zonal buoyancy cross-section along y =.5L y. Note the warming, increase in stratification and the upper thermocline depth in the eastern part of the section. b) Meridional section at x =.9L x, which corresponds to 14 E.

41 41 b(9e) b(3e) a) b) x z m 2km y Figure 11. Patterns of the downstream variation in buoyancy and temperature along the streamfunction contours of the ACC. a) Buoyancy difference between 9 Eand3 Efromthe theoretical model. b) Observed difference in the potential temperature fields from Fig. 9 in Sun and Watts (22).

42 42 a) L y B(1 7 m 2 /s 3 ).15.1 y.5 b m (1 2 m/s 2 ) b) L y 1 y.5 x L x 18W 9W 9E 18E Figure 12. a) Implied air-sea flux (zero contour is marked by the dashed line). b) The mixed layer buoyancy.

43 43 Figure 13. a) Three-dimensional structure of the isopycnal with b = 2 3 b. b) Distribution of the potential vorticity on the surface shown in (a). Note the region of low PV fluid in the upper left corner of the plot (counterpart of the southeastern Pacific).