## 1. Introduction

Oceanographers debate how dense waters, formed at high latitudes, are returned as lighter waters to the ocean surface, completing the meridional overturning circulation. Many argue that dense waters are upwelled through density layers across the abyssal ocean, requiring small-scale mixing processes such as energy dissipation over rough topography. Others argue that dense waters are transported along sloping isopycnals to the outcropping regions of the Antarctic Circumpolar Current (ACC) where vigorous winds of the Southern Hemisphere provide the energy required to convert dense water to light [see Kuhlbrodt et al. (2007), for a comprehensive review].

The Southern Ocean links the three major ocean basins and it is there that many water masses are either formed or modified (Sverdrup et al. 1942). The ACC is a zonal current, circulating around Antarctica, with a transport of 134 ± 13 Sv (Sv = 10^{6} m^{3} s^{−1}) as measured through the Drake Passage (Whitworth 1983; Whitworth and Peterson 1985). The ACC is the dominant dynamical feature of the Southern Ocean. In contrast, the Southern Hemisphere meridional overturning circulation (SMOC) is estimated to involve between 20 and 50 Sv of exchange between density classes over the entire circumpolar extent of the Southern Ocean. This exchange is thought to involve an upper and lower branch. In the upper branch, Upper Circumpolar Deep Water (UCDW) is converted to northward flowing Subantarctic mode and Antarctic Intermediate Waters. In the lower branch, UCDW and Lower Circumpolar Deep Water (LCDW) are converted to northward flowing bottom waters (Sloyan and Rintoul 2001).

Although it is not integral to the analysis, we choose to equate the overturning SMOC to the southward transport of UCDW. It is UCDW that feeds both the upper and lower limbs of the SMOC. We provide evidence that a southward transport of 20–50 Sv of UCDW into ACC (Sloyan and Rintoul 2001; Lumpkin and Speer 2007) is consistent with observed mixing rates. Both transformations above the mixed layer and vertical mixing in the ocean interior play important roles in determining the strength of the UCDW transport, and hence the SMOC.

The absence of land barrier(s) at latitudes and depths of Drake Passage (around 55°–60°S, and 0–1800 m respectively) denies the possibility of a mean geostrophic velocity across the ACC in depth or pressure coordinates. The SMOC in this region consists of a northward Ekman transport at the surface, as a result of the strong eastward wind stress and an eddy flux resulting from correlations between the thickness of isopycnal layers and the geostrophic flow. Only below 1800 m can the mean geostrophic flow contribute to the SMOC. Considering the overturning in density space, geostrophic flow across the ACC can contribute through both temporal and spatial correlations of the geostrophic velocity with the thickness of isopycnal layers—that is, in density space, both transient and standing eddies contribute to the overturning circulation. It is shown in section 4 that the effect of standing eddies can be neglected only in a small range of densities where a contour of constant potential vorticity (PV), on an isopycnal, runs along the entirety of the ACC (PV = *f*/*h*, where *f* is Coriolis frequency and *h* is thickness).

Unlike the ACC transport, which can be measured directly, the transport of UCDW can only be estimated using inverse methods and other indirect approaches. Inverse modeling has been used to estimate the southward transport of UCDW across 30°–40°S by Lumpkin and Speer (2007) and Sloyan and Rintoul (2001). They infer 20 and 52 Sv of UCDW, respectively, and find that it feeds both the upper and lower branches of the SMOC. The difference in the estimates lies in the a priori constraints and mixing representations used in the inverse models.

Karsten and Marshall (2002) and Speer et al. (2000) estimate the rate of upwelling across the ACC by determining the surface Ekman buoyancy and eddy flux components in a residual-mean framework. They infer a surface divergence, and hence a rate of upwelling of water masses into the mixed layer. Karsten and Marshall (2002) project the inferred upwelling down to depth using a simple vertical advective–diffusive balance (assuming a certain vertical diffusivity *D*). This method is applied to the Antarctic Intermediate Water layers only (i.e., the upper branch of the SMOC). Assumptions must be made about the upwelling at a particular mean streamline corresponding to a particular density layer, as contours of sea surface density do not follow streamlines along which the divergence is computed. Olbers and Visbeck (2005) investigate the relationship between Ekman transport, eddy fluxes, and vertical mixing in the Southern Ocean. They apply an a priori estimate of the meridional transport of UCDW and Antarctic Intermediate Water and infer a thickness diffusivity. The thickness diffusivity diagnosed accounts for both eddy variability and large-scale standing eddies and their solution is likely to be sensitive to their description of the Ekman velocity and their zonal averaging.

The transport of UCDW can be related to the along-isopycnal and vertical mixing coefficients through the temperature and salinity fields. Along the ACC there exist strong meridional temperature and salinity gradients on isopycnals. More precisely, density layers are cooler and fresher at the outcropping regions to the south and become warmer and saltier to the north. For these gradients to exist in steady state, there must be a balance between advection, transporting heat, and salt up or down the tracer gradient on isopycnals and the effects of both along-isopycnal and vertical mixing. Along-isopycnal mixing (*K*) acts to mix tracer anomalies on the isopycnal, while vertical mixing destroys or enhances anomalies by transferring temperature and salinity across isopycnals. This advective–diffusive balance is evident from observed tracer distributions.

In this study we determine the transport and spatial structure of UCDW as a function of the vertical and along-isopycnal mixing coefficients using the advective–diffusive balance described earlier (see section 3). As in Zika and McDougall (2008), the advective–diffusive balance is applied by integrating along temperature contours on isopycnal layers.

Using established parameterizations for the bolus flux (i.e., the difference between the mean and thickness-weighted average flow in isopycnal coordinates), we show the dependence of the UCDW transport on the along-isopycnal thickness or potential vorticity mixing coefficient (section 4). The upwelling across isopycnals along the ACC in terms of a vertical advective–diffusive balance is also considered (section 5).

It is shown that below approximately 500 m the ratio of the mean along-isopycnal mixing coefficient *K* to the mean vertical mixing coefficient *D* is *O*(2 × 10^{6}) (section 6). In section 6, the ratio of *K* to *D* is also derived by applying conservation of volume to each layer, reaffirming a value of *O*(2 × 10^{6}).

Section 7 contains a comparison and discussion of these results with previous theoretical and numerical studies of the SMOC. The consistency of a low along-isopycnal to vertical diffusivity ratio in the Southern Ocean is discussed in the context of coarse-resolution numerical models.

Here conservative temperature (Θ) is used and is proportional to potential enthalpy, and represents the “heat content” per unit mass of seawater (McDougall 2003)—that is, where potential temperature (*θ*) would commonly be used as a conservative variable for heat, we use Θ, as it is equivalent to *θ* while being far more conservative. Note that the distinction between conservative temperature and potential temperature and neutral and potential density is not central to this paper. We will frequently refer to Θ as temperature and neutral density layers as isopycnals.

## 2. Water mass equation and cross-contour flow

*S*, conservative temperature Θ, and neutral density (

*γ*) is simply the path where

*S*, Θ, and

*γ*are constant. Currents in such an ocean would closely follow contours of constant temperature and salinity on isopycnals. It is clear that the amount by which seawater chooses not to follow such a path—that is, the flow across temperature contours on isopycnals (

**n**

_{Θ}), and vertically through density surfaces (

*w*)—is determined purely by the magnitude of vertical and along-isopycnal mixing processes. Here,

^{γ}**v**is the absolute 2D velocity,

**∇**

*is the gradient on the isopycnal,*

_{γ}**n**

_{Θ}is the unit vector down the along-isopycnal temperature gradient (

**n**

_{Θ}=

**∇**

*Θ/|*

_{γ}**∇**

*Θ|), and*

_{γ}*h*is the vertical distance between closely spaced neutral density

*γ*surfaces. The equation describing the balance between the cross-contour flow

_{n}**n**

_{Θ}and mixing process in a steady ocean isMcDougall (1984) first derived (1) (in a slightly different form) and described it as the “Water Mass Transformation” equation (see appendix A for a detailed derivation). Cases where the along-isopycnal gradient of

*K*is a significant term in (1) are not considered in this study.

Here, (1) represents a balance between the advection down a temperature gradient on an isopycnal and both along-isopycnal and vertical mixing. This downgradient advection can be thought of as the “nonadiabatic” component of the along-isopycnal flow. It is important to recognize that (1) does not involve the diapycnal velocity component *w ^{γ}*, vertical differences in

*D*, or individual second derivatives of tracers in

*z*. Instead, vertical mixing appears in (1) through the Θ–

*S*curvature (see appendix A), a quantity less sensitive to noise in hydrographic data than

*S*and Θ

_{zz}*individually. In (1),*

_{zz}*λ*are diffusive “scale lengths.” Note that although a singularity exists in (1) when

^{γ}**∇**

*Θ = 0, no contour exists either.*

_{γ}Ignoring, for a moment, the consequences of the nonlinear equation of state, and ignoring the thickness gradient, the first term on the right-hand side of (1) represents the ratio of along-isopycnal curvature of temperature to the along-isopycnal temperature gradient ∇_{γ}^{2}Θ/|**∇*** _{γ}*Θ| (appendix A). As in the 1D vertical balance of “Abyssal recipes” (Munk 1966) where the ratio of the vertical gradient of temperature to the vertical curvature of temperature (and more accurately density) dictates the ratio of diapycnal advection to vertical mixing, similarly here the ratio for tracers Θ and

*S*along-isopycnals dictates the ratio of cross-contour flow

**n**

_{Θ}to along-isopycnal mixing

*K*. One way of understanding this balance is to consider an isopycnal with an along-isopycnal temperature gradient

**∇**

*Θ and curvature ∇*

_{γ}

_{γ}^{2}Θ (Fig. 1). Along-isopycnal mixing acts to smooth out the curvature of temperature and if the curvature is to remain in steady state there must be either up or downgradient advection to maintain it.

The second term on the right-hand side of (1) is proportional to the vertical curvature of temperature and salinity *d*^{2}*S*/*d*Θ^{2}. It is not simply Θ* _{zz}* or

*S*that affects the balance on the isopycnal, but the vertical curvature that involves both Θ

_{zz}*and*

_{zz}*S*. Vertical mixing (

_{zz}*D*) acts to smooth out the Θ–

*S*curvature. In order for it to be maintained, there must be cross-contour advection (

**n**

_{Θ}) or along-isopycnal mixing (

*K*) (Fig. 2). Equation (1) allows each of these effects to be quantified. It also includes the effect of a thickness gradient, as well as nonlinear effects as a result of cabbeling and thermobaricity.

## 3. The Southern Ocean overturning

*K*and

*D*. The total thickness-weighted volume flux across such a contour between a pair of density surfaces provides an estimate of the meridional transport. Integrating (1) yieldswhere

*x*

_{Θ}is oriented along a contour of constant Θ (which is also a contour of constant salinity as it is on an isopycnal). To apply (2) to the Southern Ocean, we define circumpolar tracer contours of constant temperature from the World Ocean Circulation Experiment (WOCE) Hydrographic Atlas (Orsi and Whitworth 2004) compiled as a gridded climatology on neutral density layers (Gouretski and Koltermann 2004; Jackett and McDougall 1997). Each layer represents an interval of

*γ*= 0.1 kg m

_{n}^{3}(i.e., the

*γ*= 27.6 kg m

_{n}^{3}layer is between neutral density surfaces

*γ*= 27.55 kg m

_{n}^{3}and

*γ*= 27.65 kg m

_{n}^{3}). In each layer between neutral densities of

*γ*= 27 kg m

_{n}^{−3}and

*γ*= 28 kg m

_{n}^{−3}three contours are chosen corresponding to a northern, central, and southern contour of the ACC (Fig. 3). Isopycnals above

*γ*= 27 kg m

_{n}^{−3}outcrop and temperature contours on isopycnals below

*γ*= 28 kg m

_{n}^{−3}are interrupted by topography.

UCDW, which is characterized by low oxygen concentration, is sandwiched between overlying fresher and higher oxygen concentrated Antarctic surface water (AASW) and the underlying salinity maximum and higher oxygen concentration of LCDW. Reviewing maps of oxygen and salinity from the WOCE Atlas we define UCDW to be between *γ _{n}* = 27.4 kg m

^{−3}and

*γ*= 28 kg m

_{n}^{−3}. Both Lumpkin and Speer (2007) and Sloyan and Rintoul (2001) also use this range to define UCDW.

Fields of the vertical and along-isopycnal tracer gradients and curvatures are determined from the WOCE climatology. The along-isopycnal mixing and vertical mixing terms in (1) are linearly dependent on *K* and *D*, respectively. Using (1), we estimate the total meridional transport on particular density layers for various values of the along-isopycnal and vertical tracer diffusivities. We consider the case where *K* = 200 m^{2} s^{−1} and *D* = 2 × 10^{−4} m^{2} s^{−1}. Fluxes across the northernmost temperature contours of the ACC are northward for AASW *γ _{n}* < 27.4 kg m

^{−3}and mostly southward for UCDW

*γ*> 27.4 kg m

_{n}^{−3}(Fig. 4). The level of zero cross-contour flow (i.e., the level where the flow is neither to the south nor to the north) is at approximately

*γ*= 27.5 kg m

_{n}^{−3}. Transports closer to the center of the ACC show a similar structure to the northern contour, albeit the level of zero cross-contour flow moves to denser layers. Both the vertical mixing and along-isopycnal mixing terms change sign from southward to northward on

*γ*= 27.6 kg m

_{n}^{−3}across the ACC. This convergence suggests that UCDW feeds the upper and lower limbs of the SMOC between these contours.

The cumulative integral of (2) along a circumpolar path of each temperature contour summed over layers from *γ _{n}* = 27.4 kg m

^{−3}to

*γ*= 28 kg m

_{n}^{−3}gives the spatial variation in the meridional transport of UCDW (Fig. 5). Both the along-isopycnal mixing and diapycnal mixing components of the overturning circulation vary smoothly, giving confidence that the use of second derivatives of the tracer fields is not particularly noisy, however, this may also relate to the smoothing applied to hydrographic data in order to produce the Atlas. For the northernmost contour, the two components are mostly negative (southward) and vary in a similar way along the contour, suggesting that warm anomalies are advected southward and both vertical and along-isopycnal mixing act to mix them across and along isopycnals, respectively. At the southernmost contours, the magnitude of the vertical mixing term is much larger than the along-isopycnal mixing term, suggesting that either vertical mixing dominates the balance or

*K*is large relative to

*D*.

We sum the vertical and along isopycnal mixing terms in (2), again from *γ _{n}* = 27.4 to 28 kg m

^{−3}. This results in the total flux of UCDW in

*K*–

*D*space (Fig. 6). We now review observational estimates of the vertical mixing coefficient

*D*, the along-isopycnal or lateral mixing coefficient

*K*, and the transport of UCDW in the Southern Ocean (presented graphically in Fig. 6).

Munk (1966) estimates *D* to be *O*(10^{−4} m^{2} s^{−1}) in the deep ocean, by considering the mixing necessary to close the global overturning circulation. However, Ledwell et al. (1993) observes a diffusivity of *O*(10^{−5} m^{2} s^{−1}) by releasing a tracer across the pycnocline of the northeast Atlantic. Recent observational estimates in the Southern Ocean suggest that mixing is at the upper end of this range and higher, close to rough topography and in the core of the ACC (Naveira-Garabato et al. 2004; Sloyan 2005; Kunze et al. 2006).

In the Southern Ocean, estimates exist for a surface eddy diffusivity from satellite observations (Marshall et al. 2006) and float measurements have been used to calculate eddy kinetic energy and eddy diffusivity. Reconciling the various estimates that range from less than *O*(100 m^{2} s^{−1}) to *O*(10 000 m^{2} s^{−1}) is difficult, as there are likely to be large differences between buoyancy diffusivities and tracer or potential vorticity diffusivities (see Smith and Marshall 2009). In addition, the grid spacing of inverse models and coarse-resolution ocean models can play a large role in determining the estimated or required diffusivity. Phillips and Rintoul (2000) estimate the lateral diffusivity (*K _{xy}* =

_{z}/

**∇**

*;*

_{z}θ*z*being a constant depth surface) of temperature from a mooring array time series of velocity and temperature placed within the ACC near 50°S, 143°E. Their estimates are in the broad range 100–1000 m

^{2}s

^{−1}for this one geographical location (500–1000 m

^{2}s

^{−1}above 500-m depth and 100–500 m

^{2}s

^{−1}below; Dr. H. Phillips 2008, personal communication). Gille (2003) estimated eddy heat fluxes in the Southern Ocean using Autonomous Lagrangian Circulation Explorer (ALACE) floats and found the lateral mixing coefficient to be between 300 and 600 m

^{2}s

^{−1}(at around 900-m depth).

Estimates from McKeague et al. (2005), based on an inverse model of the ocean circulation on *γ _{n}* = 28 kg m

^{−3}in the South Atlantic, are relevant to our study as they considered the along-isopycnal mixing of tracers, including temperature and salinity, assuming steady state. They find a meridional diffusivity

*K*= 100 ± 50 m

^{x}^{2}s

^{−1}and zonal

*K*= 750 ± 100 m

^{y}^{2}s

^{−1}. As temperature contours are close to lines of constant latitude, the meridional diffusivity is perhaps the most relevant here. However, as the authors suggest, the difference in magnitude may relate to the difference in grid sizing, which again makes interpretation of the eddy diffusivity difficult. Naveira-Garabato et al. (2007) were able to estimate the along-isopycnal mixing coefficient for a passive tracer in the southeast Pacific and southwest Atlantic Oceans along

*γ*= 27.98 kg m

_{n}^{−3}. They measure an along-isopycnal diffusivity of 360 ± 330 m

^{2}s

^{−1}in the frontal regions of the ACC and an area average of 1860 ± 440 m

^{2}s

^{−1}, which is thought to be associated with intensification of eddy-driven mixing in the Scotia Sea relative to ACC-mean conditions. The range 100–1000 m

^{2}s

^{−1}is consistent with that used by coarse-resolution models, higher diffusivities leading to unrealistic ACC transports.

In this study, estimates of vertical and along-isopycnal mixing may be used to infer the southward transport of UCDW. Lumpkin and Speer (2007) and Sloyan and Rintoul (2001), both determine the southward transport of UCDW with an inverse model, inferring 20 and 52 Sv, respectively. Direct comparison of our estimates with those of Lumpkin and Speer (2007) and Sloyan and Rintoul (2001) is not exact, as the transport diagnosed from (2) is across the northern flank of the ACC meandering close to 52.5°S, whereas the inverse estimates are calculated for hydrographic sections between 30° and 40°S.

At the limit where vertical mixing *D* is zero, an overturning circulation of *O*(20–50 Sv) would require an along-isopycnal diffusivity of about 200–500 m^{2} s^{−1}. At this limit, the overturning circulation is driven by Ekman and eddy transport close to the surface—that is, UCDW flows to the south in the presence of along-isopycnal mixing only until it reaches the mixed layer. It is worth noting, however, that the zero vertical mixing case is only possible for the upper branch of the SMOC where UCDW is transformed into Antarctic Intermediate, Subantarctic Mode, and surface waters. The lower branch of the SMOC involves conversion of UCDW and LCDW to Antarctic Bottom Waters (AABW). The lower branch requires abyssal diapycnal mixing in the Southern Ocean or other ocean basins to close the overturning circulation.

For an UCDW transport of *O*(20–50 Sv), there must be either strong vertical or strong along-isopycnal mixing or some combination thereof. An overturning circulation of less than 5 Sv would require small along-isopycnal diffusivities (*K* < 50 m^{2} s^{−1}), and vertical diffusivities between 0 and 10^{−4} m^{2} s^{−1} would make little difference to the size of the overturning circulation (Fig. 6). If a limit of 60 Sv where placed on the transport of UCDW at the northern side of the ACC this would imply an upper bound on *K* of 600 m^{2} s^{−1} and on *D* of 10^{−3} m^{2} s^{−1}, as both have a positive contribution to the southward transport.

Any distribution of *K* and *D* can be applied to (2) to diagnose a transport of UCDW. Various potential distributions of mixing strengths may be considered by reviewing Figs. 4 and 5. Here, if a lateral diffusivity of 400 m^{2} s^{−1} is assumed on a specific layer, the strength of the transport on that layer, as a result of the *K* term, would be double that what was shown in Fig. 4. The longitudinal variation in the transport can be considered in the same way for various distributions of *K* and *D* (Fig. 5). Cases such as stronger along-isopycnal mixing resulting from added kinetic energy provided by the winds close to the surface, or a steering level of baroclinically unstable waves (Smith and Marshall 2009), or stronger vertical mixing on deeper layers, perhaps a result of interaction with topography, could all be considered.

## 4. The residual-mean overturning and bolus velocity

In depth coordinates, and at constant latitude above the depth of the shallowest topographic feature, the circumpolar integral of the geostrophic velocity is zero. Here, we consider the circumpolar integral in isopycnal coordinates, rather than depth coordinates, and integrate along a contour of constant potential vorticity PV = *f*/*h*, rather than latitude. In this case also, the geostrophic component is zero and only an ageostrophic flow remains.

**v**

^{Ek}such thatwhere

**v**is the lateral velocity vector (overbars represent temporal averages),

*f*is the Coriolis frequency, and

**k**is the unit vector in the vertical direction. The actual nature of the streamfunction is not relevant here, merely that one approximately exists. Following from (3), we define the transport within a density layer across a contour of constant potential vorticity PV and decompose the thickness flux (velocity times layer thickness correlated) into a temporal mean and perturbation componentand considering the components down the PV gradient for

**n**

_{PV}=

**∇**

*PV/|*

_{γ}**∇**

*PV| we have*

_{γ}**n**

_{PV}circumpolarly along a PV contour in an isopycnal layer allows the mean geostrophic component to be eliminated:Hence, the meridional transport is purely an ageostrophic one involving eddy and Ekman transports. Here, the PV contours need not remain within the latitude band of Drake Passage. The PV contours only need to be fully circumpolar, as Eq. (6) shows that in isopycnal layers for PV contours, which run along the entirety of the ACC, there is a barrier to the mean geostrophic transport (Fig. 7). The transport in these layers can only come from the Ekman and transient eddy components. In layers where PV contours do not run along the entire ACC there can be a net southward transport because of the mean geostrophic current (i.e.,

*dx*runs along the ACC). The mean geostrophic flow may add significantly to a net along-isopycnal transport across the ACC, even if the net transport is zero on any given pressure level (Fig. 7). The SMOC is defined in density space (Hallberg and Gnanadesikan 2006), therefore mean geostrophic flows can contribute to the overturning in layers where PV contours do not follow the entirety of the ACC—that is, large-scale standing eddies. Olbers and Visbeck (2005) consider the circumpolarly integrated transport of Antarctic Intermediate Water and UCDW in terms of an Ekman and eddy flux. For layers where PV contours do not sufficiently coincide with the streamwise averaging they use, the along-isopycnal mixing coefficient they impose must account for both eddy variability and large-scale correlations between thickness and the geostrophic velocity.

*υ**) such that (following McDougall 1991)where in (8),

*K*

_{PV}is the along-isopycnal potential vorticity mixing coefficient,

**v*** is the bolus velocity,

*β*= ∂

*f*/∂

*y*, and

**j**is the unit vector in the meridional direction. Here, along-isopycnal mixing (

*K*

_{PV}) acts to evenly distribute PV. Cast in terms of a velocity, the first component represents the effect of gradients of

*h*and the second is due to the meridional gradient of

*f*. A different approach is taken by Gent et al. (1995) to parameterize the quasi-Stokes velocity

*υ*

^{+}(the counterpart to the bolus velocity in Eulerian coordinates):This parameterization ignores the

*β*effect and is commonly used in ocean circulation models and is the same as the first term on the right-hand side of (7) for small Δ

*ρ*.

Given (8), we quantify the meridional transport of UCDW in terms of the along-isopycnal potential vorticity diffusivity (*K*_{PV}). This is done in layers where contours of constant PV are continuous around the circumpolar path of the ACC and are sufficiently distant, in the vertical, from the Ekman layer. We define contours of constant PV in *γ _{n}* layers from the WOCE climatology, where

*h*is the thickness of the layer. Contours may only be defined for northern, central, and southern pathways of the ACC in layers

*γ*= 27.5 kg m

_{n}^{−3}to

*γ*= 27.8 kg m

_{n}^{−3}(

*γ*= 27.5 kg m

_{n}^{−3}to

*γ*= 27.7 kg m

_{n}^{−3}in the case of the northern contour). We assume a uniform

*K*

_{PV}of 200 m

^{2}s

^{−1}and quantify both the thickness and

*β*terms in (8).

If *K*_{PV} is assumed to be the same as the along-isopycnal mixing coefficient for tracers *K*, it is expected that the meridional transports derived from (2) and (8) should be similar for contours with similar paths and depths (away from Ekman effects). This is the case for the deepest of layers (Figs. 4 and 8). Northward transports suggested from (2) for *γ _{n}* < 26.7 kg m

^{−3}may be explained by Ekman effects as isopycnals shoal in the southward direction and perhaps a strong gradient of

*K*at the bottom of the mixed layer.

## 5. Diapycnal flow and the 1D balance

Since the work of Munk (1966), it has been common to consider a 1D balance where a vertical mixing “scale length” relates a rate of upwelling to a vertical mixing coefficient.

*w*:where

^{γ}*η*and

^{γ}*D*is a significant term in (10) are not considered in this study. If a linear equation of state is assumed, the along-isopycnal mixing term is zero and (10) reduces to

_{z}*w*=

^{γ}*Dγ*/

_{zz}*γ*(appendix B). This simplification of (10) is commonly used in studies attempting to infer large-scale dynamics in the Southern Ocean from observations (Karsten and Marshall 2002; Naveira-Garabato et al. 2007). Here, we shall retain both the linear and nonlinear components and determine their relative role in cross isopycnal transport.

_{z}The diapycnal velocity is computed for neutral density surfaces from tracer fields of the WOCE climatology and integrated circumpolarly between temperature contours corresponding to the southern and northern sides of the ACC. With a vertical diffusivity of 2 × 10^{−4} m^{2} s^{−1}, the vertical velocity through *γ _{n}* = 27.7 kg m

^{3}, a result of the

*D*term in (10), is determined (Fig. 9). The velocity is of

*O*(10

^{−7}m s

^{−1}) to the north of the ACC and is of

*O*(10

^{−6}m s

^{−1}) to the south (Fig. 9). The accumulated diapycnal transport between the temperature contours is approximately 8 Sv for the

*D*term in (10). The diapycnal transport is spread reasonably even across the entirety of the ACC, with the net effect being an upwelling. When a value for

*K*of 200 m

^{2}s

^{−1}is used, the downwelling resulting from this term is negligible (≤1 Sv). If

*K*were

*O*(1000 m

^{2}s

^{−1}) the downwelling would be first order [see Iudicone et al. (2008), for a comprehensive discussion]. The lateral mixing term is often neglected, even when large along-isopycnal or lateral diffusivities are observed (Karsten and Marshall 2002; Naveira-Garabato et al. 2007).

## 6. The relative role of vertical and along-isopycnal mixing

In this section we present evidence that the ratio of the along-isopycnal and vertical mixing coefficients is *O*(2 × 10^{6}) along the ACC in the density layers of UCDW. This is done by combining the concept of a cross-contour flow **n**_{Θ}, which was discussed in section 2, with the PV parameterizations of section 4. This combination yields a balance of vertical and along-isopycnal mixing, independent of the mean velocity. We next combine the cross-contour transports derived in section 3, with the diapycnal transports in section 5. We then apply continuity to finite volumes around the entire ACC, arriving at a balance of along-isopycnal and vertical mixing.

**n**

_{Θ}). Substituting for (1) we havewhere

**v**

**n**

_{Θ}is the mean velocity down the along-isopycnal temperature gradient. Equation (12) differs from (1) as the left-hand side is only the mean velocity and not the residual one. Away from the Ekman layer the mean velocity is approximately geostrophic, hencewhere the circumpolar integral of

*f*

**v**

**n**

_{Θ}along circumpolar temperature contours is zero. Hence, integrating

*f*times Eq. (12) gives

*S*and Θ are approximately passive tracers, as the act of stirring them locally does not change the stratification (although there may be some effects that result from the nonlinear equation of state). We assume

*K*=

*K*

_{PV}and from (14) the ratio of

*K*to

*D*is

Here, (15) holds for any enclosed temperature contour on an isopycnal that does not interact with the Ekman layer. With above-average depths of 500 m, the ratio of *K* to *D* is zero or negative, suggesting these contours interact with the Ekman layer or lateral gradients of *K*, which becomes important somewhere along their circumpolar path. Below 500 m the ratio is consistently positive and of *O*(2 × 10^{6}) (Fig. 10).

*K*and

*D*. We may apply continuity of volume in order to ascertain the ratio of these two coefficients. Writing the steady continuity equation for a particular volume on a density layer bound by contours of constant temperature to the south and north (Θ

_{1}and Θ

_{2}respectively), we havewhere

*u*and lower

*l*bounding neutral density surfaces between temperature contours (Θ

_{1}and Θ

_{2}respectively). Assuming again that the mixing coefficients are constant in space we may write

Here, Eq. (17) is applied to volumes between contours on the northern side of the ACC below an average depth of 500 m (Fig. 10). We find that the ratio of *K* to *D* is again of *O*(2 × 10^{6}) there.

## 7. Discussion and conclusions

This study has investigated the relationship of along-isopycnal mixing (*K*) and diapycnal mixing (*D*) to the strength of the Southern Ocean meridional overturning circulation (SMOC) as quantified by the southward transport of Upper Circumpolar Deep Water (UCDW). The total transport of UCDW, within isopycnal layers, has been diagnosed from tracer distributions. The transports are inferred directly from observations through a linear relationship with the mixing coefficients. The sensitivity of the overturning transport and spatial characteristics to a range of possible diffusivities is a direct result of the analysis (section 3).

An important aspect of this study is the discussion of cross-contour transport in density-temperature space. The thickness-weighted velocity down the temperature gradient on an isopycnal **n**_{Θ}, discussed in section 2, is dependent on *K* and *D* locally, and thus the spatial structure of the transport of specific water masses can be analyzed as well as their circumpolar integral.

The diffusive scale lengths, *λ ^{γ}*, integrated circumpolarly show that UCDW is transported southward where it feeds the upper and lower cells of the SMOC. Careful comparison of the fluxes inferred from tracer gradients, the bolus transport, and conservation considerations for along-isopycnal and vertical transports suggests a ratio of

*K*to

*D*of

*O*(2 × 10

^{6}). The implications of such a ratio for the overturning, the possible range of mixing coefficients, and in turn for numerical modeling of the Southern Ocean are discussed below.

The southward transport of UCDW inferred from inverse studies (Sloyan and Rintoul 2001; Lumpkin and Speer 2007) suggests a range of 20–52 Sv. We have derived a linear relationship between the southward transport of UCDW and *K* and *D* (2) and we have estimated the ratio of *K* to *D* in the UCDW layers to be 2 × 10^{6} ± 10^{6} (Fig. 10). Thus, we estimate *K* and *D* individually and find *K* = 300 ± 150 m^{2} s^{−1} and *D* = 10^{−4} ± 0.5 × 10^{−4} m^{2} (blue cross-hatching in Fig. 6). Such rates of diapycnal mixing are considered to be large in the midlatitude oceans, but are supported by observations of diapycnal mixing (*D*) in the ACC such as Sloyan (2005), Naveira-Garabato et al. (2004), and Kunze et al. (2006). Our results suggest *D* is *O*(10^{−4} m^{2} s^{−1}) beneath the mixed layer in the Southern Ocean, supporting the hypothesis that vertical mixing, in the ocean interior along the ACC, makes a significant contribution to water mass conversion. The diagnosed along-isopycnal mixing coefficient (*K*) is also within the range estimated by both Phillips and Rintoul (2000) and McKeague et al. (2005).

For along-isopycnal mixing coefficients of *O*(200 m^{2} s^{−1}), the nonlinear terms contributing to the diapycnal transport below the mixed layer in the Southern Ocean are small (section 5). However, for values of *O*(10^{3} m^{2} s^{−1}) as used in Naveira-Garabato et al. (2007), downwelling resulting from cabbeling and thermobaricity is significant. This study has not considered cases where **∇*** _{γ}K* is a significant term in (1). Although this term is not necessarily negligible in the Southern Ocean, sufficient observations of

*K*do not exist to reasonably quantify it.

Although there is mounting evidence that there is intense vertical mixing along the ACC in the Southern Ocean, global climate simulations are yet to include spatially and vertically varying diffusivities. This study presents further evidence that in the Southern Ocean *D* is indeed of *O*(10^{−4} m^{2} s^{−1}) or greater, and that a significant fraction of upwelling UCDW is transported across isopycnals below the mixed layer. This study also suggests that the along-isopycnal mixing coefficient of *O*(10^{2} m^{2} s^{−1}), used in the majority of such simulations, is appropriate. Isopycnal mixing of *O*(10^{3}–10^{4} m^{2} s^{−1}) as measured at the surface (Karsten and Marshall 2002; Sallée et al. 2008) is inappropriate for deep layers as they would give extremely large values for the southward transport of UCDW (Fig. 6).

Along-isopycnal and vertical mixing coefficients inferred from observations are often difficult to compare with those required by coarse-resolution ocean models. Observational studies using moorings, Lagrangian tracers, and satellite altimetry diagnose an effective diffusivity, whereas numerical models require a diffusivity to represent the velocity to property correlations (i.e., *K* is a result of submesoscale eddies, and hence the *K* discussed in this paper is equivalent to that desired in a coarse-resolution ocean model. In continuing work we have developed an inverse technique that diagnoses the downgradient transport and the mixing coefficients *K* and *D* (Zika et al. 2009). This is done by using the integral of (1) along a tracer contour and a form of the thermal wind equation cast in terms of differences in geostrophic streamfunction Ψ* ^{γ}* along such a contour. The technique is validated against the output of a 20-yr average of a 100-yr climate simulation of the Hallberg Isopycnal Model at 1° × 1° resolution. In so doing, we show that the

*K*and

*D*used in (1) is close to that when explicitly applied to a coarse-resolution ocean model.

We thank Drs. Jean-Baptiste Sallée and Steve Rintoul for insightful and helpful comments on a draft of this paper. We also thank two anonymous reviewers for their helpful remarks. This work contributes to the CSIRO Climate Change Research Program and has been partially supported by the CSIRO Wealth from Oceans Flagship and the Australian Research Council Centre of Excellence for Mathematics and Statistics of Complex Systems.

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# APPENDIX A

## Derivation of the Water Mass Equation

*S*and conservative temperature Θ in steady state areThese equations have been written in the advective form, and with respect to neutral density (

*γ*) layers (Jackett and McDougall 1997), so that

_{n}*w*is the vertical velocity through neutral density surfaces (i.e., the diapycnal velocity component of the vertical velocity) and (

^{γ}*h*

*S*and conservative temperature Θ in (A1) and (A2) are the thickness-weighted values obtained by averaging between closely spaced pairs of neutral density surfaces (McDougall and McIntosh 2001). In these equations

*h*

*K*) along the density layer and vertical small-scale turbulent mixing (with coefficient

*D*). We have not included double-diffusive convection or double-diffusive interleaving. We define the cross-contour direction as

**n**

_{Θ}=

**∇**

*/|*

_{γ}S**∇**

*|. Notably,*

_{γ}S**∇**

*/|*

_{γ}S**∇**

*| ≡*

_{γ}S**∇**

*Θ/|*

_{γ}**∇**

*Θ| as we are considering gradients on a neutral density layer. Multiplying (A1) by Θ*

_{γ}*and (A2) by*

_{z}*S*givesSubtracting (A4) from (A3), the

_{z}*D*and

_{z}*w*terms are eliminated and we findwhereThe stability ratio is

^{γ}*R*=

_{ρ}*α*Θ

*/*

_{z}*βS*. On a neutral density surface

_{z}*α*/

*β*= |

**∇**

*|/|*

_{γ}S**∇**

*Θ|, so*

_{γ}*R*≡ |

_{ρ}**∇**

*|Θ*

_{γ}S*/|*

_{z}**∇**

*Θ|*

_{γ}*S*. Hence, neither

_{z}*λ*contains singularities if the water column is stably stratified (i.e., if

^{γ}*R*≠ 1 then Θ

_{ρ}*|*

_{z}**∇**

*| −*

_{γ}S*S*|

_{z}**∇**

*Θ| ≠ 0).*

_{γ}**∇**

*≡*

_{γ}α**∇**

*≡ 0, hence*

_{γ}β_{z}

^{3}(

*d*

^{2}

*S*)/(

*d*Θ

^{2}) = Θ

*−*

_{z}S_{zz}*S*Θ

_{z}_{zz}, 1/

*λ*may be written

^{γ}# APPENDIX B

## Derivation of the Density Equation

**∇**

*Θ| and (A2) by |*

_{γ}**∇**

*| and expanding the along-isopycnal mixing terms in both equations we haveSubtracting (B1) from (B2), the*

_{γ}S**∇**

*(*

_{γ}*hK*) and (

*h*

**n**

_{Θ}terms are eliminated and we findwhereandHere, neither

*η*is singular if the water column is stably stratified. If a linear equation of state is assumed such that

^{γ}*ρ*∝ −

_{z}*α*Θ

*+*

_{z}*βS*, for

_{z}*α*and

*β*constant and again

*α*/

*β*= |

**∇**

*|/|*

_{γ}S**∇**

*Θ|, the vertical mixing term 1/*

_{γ}*η*becomesAlso, as

^{γ}**∇**

*≡*

_{γ}α**∇**

*≡ 0 the along-isopycnal mixing term*

_{γ}β*w*=

^{γ}*Dρ*/

_{zz}*ρ*+

_{z}*D*.

_{z}