## 1. Introduction

At the latitudes of the Antarctic Circumpolar Current (ACC), waters from the Atlantic, Indian, and Pacific Oceans are brought to the surface by the Roaring Forties to be transformed into Subantarctic Mode Waters to the north and Antarctic Bottom Waters to the south (Marshall and Speer 2012). This global transformation of water masses is achieved by intense air–sea exchange of heat, freshwater, carbon, and other chemical tracers in the Southern Ocean and exerts a strong control on Earth’s climate. Above the sill depth of the Drake Passage, the circulation is dominated zonally by the ACC and meridionally by the sum of a wind-driven meridional overturning circulation (MOC) plus a MOC driven by the turbulent eddies generated through instabilities of the ACC (Johnson and Bryden 1989; Speer et al. 2000; Marshall and Radko 2003). The air–sea fluxes and Earth’s climate are therefore very sensitive to oceanic turbulence in the Southern Ocean. The current debate as to whether Southern Ocean carbon uptake will increase or decrease in a warming climate stems from different assumptions about the changes in oceanic turbulence (Russell et al. 2006; Abernathey et al. 2011).

Despite its importance for climate studies, there have not been direct observational estimates of the rate of mixing that drives the eddy-induced circulation across the ACC. Indirect estimates have been made, for example, by Stammer (1998) who used scaling laws and the surface geostrophic velocity from altimetry and by Marshall et al. (2006) who advected numerical tracers with the altimetric velocity field. Phillips and Rintoul (2000) attempted to estimate the fluxes of heat and momentum from mooring data, but not the rate at which tracers are mixed. Here we present the first direct measurements based on the spreading of a tracer deliberately released as part of the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES). The mixing is quantified with an eddy diffusivity, which is defined in terms of the spreading rate of the meridional distribution of the tracer, once it asymptotes to a constant. The eddy diffusivity is a tensor **K** that quantifies the growth of the patch in all three dimensions. Here we will focus on the component of the diffusivity representing the tracer spreading along the neutral density surface (isopycnal mixing) and across the ACC, because this is the component that drives the eddy-induced MOC and plays an important role in setting the strength of both the upper and lower overturning cells in the Southern Ocean.

The anthropogenic tracer was released on an isopycnal surface near 1500-m depth, at the interface between the upper and lower MOC cells, in the Pacific sector of the Southern Ocean 2300 km upstream of the Drake Passage, midway between the Polar Front (PF) and the Subantarctic Front (SAF). Ledwell et al. (2011) estimated that after 1 yr the tracer spread vertically to a Gaussian profile in density with a standard deviation of less than 30 m relative to the target density surface and was thus confined to a very thin layer.

Our analysis focuses on the first year of spreading when most of the tracer remained west of the Drake Passage; numerical simulations suggest that the leading edge of the tracer reached the Drake Passage after somewhat less than 2 yr. We focus on measurements collected in the sector upstream of the Drake Passage, because the ACC jets are mainly zonal there. Past the Drake Passage, the jets strongly meander, and it is difficult to separate along- and across-jet dispersion. Furthermore, the tracer sampling downstream of the Drake Passage may not have been adequate to determine cross-stream isopycnal mixing as it was designed to estimate the diapycnal diffusivity; the tracer was sampled only along the individual transects shown in Fig. 1a with no attempt to map the whole tracer patch.

Because of the temporal and spatial scales involved, measuring the isopycnal diffusivity by sampling a tracer spreading through the ocean is difficult, since only a fraction of the tracer distribution can be directly sampled. Some method must be developed to extrapolate the tracer measurements and infer where the unsampled tracer may have spread. Ledwell et al. (1998) estimated the isopycnal diffusivity at the mesoscale in the North Atlantic pycnocline by fitting a two-dimensional Gaussian to the tracer patch measured 30 months after release. Assuming such a 2D Gaussian is perhaps reasonable in a region with weak mean flows, although even at their site, Ledwell et al. (1998) suspected a role played by gyre-scale strain in the mean flow in enhancing the apparent zonal diffusion. The assumption of 2D Gaussian spreading cannot be used in the Southern Ocean, where the tracer is advected rapidly downstream by the meandering ACC jets, at the same time being dispersed meridionally by the turbulent eddies. Here, therefore, the tracer measurements have been extrapolated by simulating the DIMES tracer release with a numerical model of the region, run at one-twentieth of a degree horizontal resolution. The model output is compared with hydrography and mooring observations (see appendix B) and provides a link between the subsampled tracer distributions and the full tracer distribution.

Using the tracer sampled during the 1-yr tracer survey (called US2), together with the numerical model, we estimate that the tracer experienced a meridional isopycnal diffusivity of 710 ± 260 m^{2} s^{−1} over the first year after release. This value agrees with an independent estimate based on the dispersion of 72 acoustically tracked isopycnal floats, deployed on the same isopycnal surface as the tracer (see LaCasce et al. 2014). The main objective of this paper is to explain how we obtained this estimate.

The isopycnal diffusivity estimated here is an isopycnal tracer diffusivity, not a lateral buoyancy diffusivity. That is, we are discussing the Redi diffusivity, not the Gent–McWilliams diffusivity, using the jargon of noneddy-resolving climate models [see the discussion in the textbook by Griffies (2004)]. The isopycnal diffusivity is also the diffusivity that mixes potential vorticity, thereby driving the overturning ocean circulation (e.g., Plumb 1986). The model suggests that the isopycnal tracer diffusivity increases from about 300 m^{2} s^{−1} in the upper ocean to 900 m^{2} s^{−1} at 2 km and decays rapidly below. The maximum in eddy diffusivity is near the steering level where the phase speed of the eddies equals the mean current speed. This is consistent with the suggestion that the zonal-mean flows suppress mixing in the upper ocean, while the diffusivity is unsuppressed, and thereby enhanced, near the steering level (Smith and Marshall 2009; Abernathey et al. 2010; Klocker et al. 2012b). The values of the diffusivity at the steering level from the present results are on the low side of those reported in the literature that span 1000–3000 m^{2} s^{−1} (Smith and Marshall 2009; Klocker et al. 2012b; Abernathey et al. 2010). DIMES is the first study that relies on direct estimates of tracer spreading, while all previous studies were only indirectly constrained by data. Hence, the DIMES estimates provide ground truth to derive better parameterizations of eddy mixing for climate models.

Our paper is organized as follows: The DIMES tracer release, sampling, measurements, and uncertainty are discussed in section 2. The numerical model and its comparison with observations are discussed in section 3. Section 4 derives our best estimate of the eddy diffusivity based on DIMES data and model output. Section 5 describes the modeled estimates of the vertical dependence of diffusivity using a set of tracers released at different depths. We conclude in section 6.

## 2. The DIMES tracer release

In early February 2009 (cruise US1), 76 kg of a passive chemical tracer [trifluoromethyl sulphur pentaflouride (CF_{3}SF_{5})] were released from the Research Vessel (R/V) *Roger Revelle* on the 27.9 kg m^{−3} neutral density surface (near 1500-m depth) upstream of the Drake Passage (58°S, 107°W) between the SAF and the PF (see Figs. 1a and B1). The tracer was released in a rough “x” pattern in an area about 20 km across. The injection system was maintained within a few meters of the target isopycnal surface via a feedback control system, as described in Ledwell et al. (1998). The tracer distribution was sampled within 2 weeks of the release and found to be confined to within 6 m rms of the target density surface (Ledwell et al. 2011).

The tracer was intentionally released in fluid whose eastward motion was biased low, in order to facilitate initial sampling. The release location was guided by altimetry data indicating a stagnation point at depth, assuming the current to have an “equivalent-barotropic” structure (Killworth and Hughes 2002). Further evidence of a small velocity was obtained from a CTD survey conducted within 2 days of release in a 70-km box centered on the release site. The magnitude of the geostrophic velocity at the center of the tracer patch estimated from this survey, with surface geostrophic velocity from altimetry as reference, was less than 0.03 m s^{−1}. Low velocity of the tracer patch was at least partially confirmed by the observation that all of the stations at which tracer was found during the initial survey, 4 to 14 days after release, were within 10 km of the center of the initial patch.

In kinematic simulations based on the altimetry at the time of the experiment (not shown), with velocity at the tracer depth approximated as 0.38 times the surface geostrophic velocity from the altimeter, the center of mass of the tracer moved slightly to the west at first and did not start moving east until a month after release. Thus, the actual tracer movement might be expected to have been delayed by about a month relative to the mean of an ensemble of numerical releases in other representations of the flow field.

The spread of the tracer was sampled during cruise US2 (see Table 1), a year after the release, using a conventional CTD/rosette system. Water samples were analyzed using a method similar to that described in Ho et al. (2008). The uncertainty (one standard deviation) of individual concentrations was no greater than 0.03 × 10^{−15} mol L^{−1}, or 5% of the concentration, whichever was greater. This uncertainty is small compared to the peak concentration measured during US2 of about 4 × 10^{−15} mol L^{−1}.

Brief information about the DIMES cruises.

Figure 1a shows the location of the initial tracer release on cruise US1 (black dot) and the locations (circles) of the normalized amounts of column-integrated tracer concentration measured (circle area) in the follow-up cruises: US2 (blue), UK2 (purple), UK2.5 (black), and US3 (red). The UK cruise tracks, which sampled multiple transects, have been subdivided into individual transects UK2A, UK2B, UK2C, UK2.5A, and UK2.5B. The areas of the circles in each cruise have been normalized by the maximum amount of tracer measured on that cruise, and the largest circles of each cruise have the same area (except US2 where due to high concentrations the largest circle has 4 times the area).

The column integral at each station was calculated by integrating over a profile obtained by interpolating linearly between the sample levels. Uncertainty of the column integrals is less than 5%, which is very small compared with lateral variations, as assessed from the lateral autocorrelation of tracer integrals (not shown). The closest station spacing was 28 km, along the lines at 93° and 96°W. The autocorrelation of column integrals of all station pairs with separation within 30 km (71 pairs) was only 0.4 ± 0.2. The autocorrelation decreases to 0 ± 0.2 for 121 pairs with separations between 90 and 120 km, which is less than the distance between major survey lines. Hence, accurate interpolation of the data to create a map is not possible even within the bounds of the survey. Furthermore, it is clear from the high levels of tracer found along the northern border of the survey (Fig. 1a) that, although the survey may have delimited the tracer fairly well to the west and south, the patch was not delimited to the north and northeast.

The average of all the vertical profiles obtained during US2 was approximately Gaussian in shape with a standard deviation of 30 m and with virtually all the tracer found within 100 m of the target density surface, as shown in Ledwell et al. (2011). Hence, 1 yr after release, the vertical spread of the tracer was of the same order as the vertical resolution of most ocean circulation models, including the one used in the present study. Incidentally, variations among profiles of the vertical distribution were small enough that the estimate by Ledwell et al. (2011) of the diapycnal diffusivity, and its uncertainty, in the region between the injection location and the US2 survey area were accurate, despite the variability of column integral within the patch and the failure of the survey to delimit the patch.

Figure 2 shows column-integrated tracer concentrations divided by the total amount of tracer released (circles; units m^{−2}) for each of the cruises. Only two transects of cruise US2 are shown: the latitudinal transect at 96°W denoted as “US cruise 2A” and the latitudinal transect at 93°W denoted as “US cruise 2B.” The x’s with error bars shown in Fig. 2 represent simulated concentrations, which will be discussed in section 3b. The largest column integral measured during US2 was 3.46 × 10^{−9} mol m^{−2}, located at 56.66°S, 94°W, which, after normalizing by the 387.6 mols of injected tracer, is 8.92 × 10^{−12} m^{−2}. The maximum relative concentrations during UK2, UK2.5, and US3 were 1.05 × 10^{−12}, 9.55 × 10^{−13}, and 6.30 × 10^{−13} m^{−2}, respectively. The maximum during US2 is an outlier twice as large as the next largest value during US2, which is itself 50% larger than the next 5–10 data points. Notice that the scale of the vertical axis in Fig. 2 decreases in downstream cruises because of dilution by dispersion and also because only the leading edge of the tracer patch is being sampled (UK2B, UK2C, and UK2.5B) or the trailing edge of the tracer is being sampled (US3).

Cruise US2 is the only cruise where the tracer was sampled over a two-dimensional grid; hence, it is the only cruise from which the center of mass of the tracer can be estimated. The blue x in Fig. 1b shows the center of mass of the DIMES tracer during US2, computed as ^{−1} over the first year of dispersal. The trajectory of the center of mass followed very closely a constant streamline from the mean Archiving, Validation, and Interpretation of Satellite Oceanographic data [AVISO; Centre National d’Études Spatiales–Collecte Localisation Satellites (CNES-CLS09), version 1.1; Rio et al. 2011] dynamic topography.

## 3. The Drake Patch model

The simulated tracer data presented here are from a series of virtual tracer releases, which replicate the DIMES release, using a regional setup of the Massachusetts Institute of Technology general circulation model (MITgcm; Marshall et al. 1997a,b), hereinafter referred to as the “Drake Patch.” The model’s horizontal grid resolution is one-twentieth of a degree (a resolution of 3 km × 6 km at the location of the tracer injection), spanning the Drake Passage from 75° to 35°S in latitude and from 160° to 20°W in longitude. The vertical mesh grid is divided into 100 layers of unequal thickness such that the top 70 layers, which span the top 1900 m, are all less than 35 m thick.^{1}

The Interim European Centre for Medium-Range Weather Forecasts (ECMWF) Re-Analysis (ERA-Interim; Simmons et al. 2006) 6-h winds and buoyancy fluxes force the model’s surface, and the Ocean Comprehensive Atlas (OCCA; Forget 2010) provides monthly transports, heat and salt fluxes, as well as sea ice area and thickness at the lateral boundaries. Initial model conditions are an interpolation of the 1° × 1° resolution OCCA state on 1 January 2005, and the model cycles repeatedly over the years for which OCCA is defined (2004–06). The simulations are intended to capture the statistics of the seasonal cycle and mesoscale of the Southern Ocean near the Drake Passage rather than predict the specific ocean state at the time of the DIMES tracer release. The model domain (excluding where restoring is applied to the OCCA state estimate) is shown in Figs. 3 and 4. A more detailed description of the model setup is given in appendix B.

### a. Comparison of the model with observations

We begin by comparing the Drake Passage transports, eddy kinetic energy (EKE), and temperature–salinity hydrography with the Drake Patch simulation. The model vertically integrated zonal transport across the Drake Passage has a mean of 152 Sverdrups (Sv; 1 Sv ≡ 10^{6} m^{3} s^{−1}) and varies between 144 and 162 Sv, with a standard deviation of 3 Sv, consistent with the transport entering from the open western boundary from OCCA (152 Sv; Forget 2010). This transport is somewhat larger than past estimates (137 ± 7 Sv; review by Meredith et al. 2011), but agrees with more recent ones (Firing et al. 2011; 154 ± 38 Sv). The standard deviation is consistent with a recent eddying Southern Ocean state estimate (Mazloff 2008), but much smaller than reported from observations, possibly because models underestimate the current temporal variability or because observational estimates are biased high due to poor temporal sampling especially at depth. We show below that tracers injected in the model move eastward at the same rate as the tracer released in DIMES, further confirming that the model eastward transport is consistent with observations.

The initial and boundary conditions in the Drake Patch are derived from the 1° × 1° OCCA climatology that does not resolve eddies. Upon spinning up, boundary currents, baroclinic and barotropic instabilities, and topographic steering quickly develop, in *O*(50) days, at and downstream of the Drake Passage (east of 75°W), as well as far upstream at the Udintsev and Eltanin fracture zones (between 145° and 135°W). After *O*(100) days, a vigorous mesoscale eddy field is established in these regions. Weaker mesoscale eddies develop locally near the US2 region after *O*(300) days, and a significant amount of eddy kinetic energy is advected into the US2 region from the fracture zones to the west. An earlier model configuration, which had its western boundary at 115°W and so lacked the upstream fracture zones, exhibited only about 60% of the eddy kinetic energy in a region near US2 (60°–55°S and 90°–100°W ) compared to the current configuration. Therefore, a significant amount of the eddy energy between 100° and 80°W is advected into that region from the fracture zones at 140°W, despite the advective time scale for eddies to propagate 50° downstream at 2.3 cm s^{−1} being about 4 yr and the time scale of local baroclinic instability being less than a year (Tulloch et al. 2011). The simulation that includes the Udintsev and Eltanin fracture zones also exhibits relatively more interannual variability of kinetic energy (KE) than the simulation without them and takes about twice as long to equilibrate at the surface (about 800 days vs 400 days to reach 90% of surface KE).

Figures 3 and 4 compare mean and eddy current speeds in the Drake Patch model with AVISO altimetric observations. The model and the altimetric observations agree rather well, although the model’s eddy kinetic energy is about 10% larger than AVISO near the US2 cruise track shown in Fig. 1a. The model’s time-mean flow

The model has a southward-flowing boundary current off the coast of Chile that ejects northwest-propagating anticyclonic eddies into the Pacific Ocean that are absent in the observations. These eddies are generated by the large freshwater fluxes along the Chilean coast^{2} and they propagate away from the DIMES region. On the basis of our examination of water mass exchanges between the Chilean coastal region and the tracer sampling area, we do not expect freshwater fluxes to influence the tracer distribution during the first 2 yr.

Figure 5 compares the vertical structure of simulated root-mean-square current speed with observations from the First Dynamic Response and Kinematic Experiment (FDRAKE) moorings located in the Drake Passage during the late seventies (Pillsbury et al. 1979; Nowlin et al. 1982). The moorings were deployed for an average of about 320 days and are corrected for blowover (Nowlin et al. 1985). They are compared to a 3-yr average in the model. The vertical decay of kinetic energy in the upper 3 km is very similar in both model and observations, although the model is somewhat more energetic than the observations. The good match in the vertical decay of kinetic energy is important to support the analysis of lateral mixing at different depths presented below. The very energetic model vertical profile that lies to the right of all other profiles in Fig. 5 comes from the location of the northernmost mooring, which is close to the model's strong boundary current, visible in Fig. 3b. This outlier profile is probably not very significant, because this current exhibits significant year to year variability in the model. In any case, our analysis focuses on mixing away from this boundary current.

One possible reason for the energy level mismatch is due to missing ocean physics. While the model resolves mesoscale eddies, bottom boundary layer turbulence (Kantha and Clayson 2000) and lee-wave generation (Nikurashin and Ferrari 2011; Nikurashin et al. 2013) are not well resolved, so the modeled eddies experience too little bottom dissipation. It may be possible to reduce the bias by a slight increase in quadratic bottom drag.

Temperature, salinity, and neutral density in the model upstream of the Drake Passage agree well with CTD data from the World Ocean Circulation Experiment (WOCE) and the Climate Variability and Predictability (CLIVAR) programs. In appendix B, sections P18, P19C/S, and A21 are compared with the model solution. The model receives large-scale hydrographic information from OCCA at the western and northern boundaries, so the upstream sections in the model largely resemble OCCA and therefore observations. Within Drake Passage, the Polar Front appears to be shifted north by about 1° and is somewhat more intense. Section A21 appears to slice through a recirculation just north of 58°S in both observations and the model, a feature that is amplified in the model. While these differences may represent model bias, they are within the observed natural variability. For example, the Polar Front has been shown to meander between 57° and 61°S (Dong et al. 2006): it was observed just south of 59°S during the DIMES experiment (St. Laurent et al. 2012), close to 61°S in A21 (at 68°W), and at 60°S in the Drake Patch model. The multiyear sea ice extent shown in Fig. 1a is also in reasonable agreement with observations.

### b. Comparison with DIMES tracer measurements

We repeated 12 tracer injection experiments using the Drake Patch model. In each experiment the tracer was injected at the location of US1 in the DIMES field experiment. They were released 10 days apart from January through March of the sixth year of model integration. The initial tracer distribution was a Gaussian blob in *x*, *y*, and *z* (*σ*_{x} = *σ*_{y} = 20 km, *σ*_{z} = 75 m), with the vertical distribution centered on the 59th model layer (1512-m depth), which is closest to the *ρ*_{n} = 27.9 kg m^{−3} neutral density surface in the model in February.

Figure 1a shows a snapshot of column-integrated tracer concentration (in units of m^{−2}) after 365 days of integration for the tracer blob released on 4 February of the sixth year of model integration. The tracer concentration shown is normalized by the maximum concentration in the domain, as was done for the tracer concentrations measured along each cruise and shown as circles, and all values between 0.5 and 1 have a uniform red tone. This is the same normalization used to display the tracer concentrations measured during the US2 cruise, 1 yr after the DIMES release, and shown as blue circles. Tracer concentrations from later cruises (UK2A, UK2.5, and US3) are also shown for reference.

The model tracer is still streaked into numerous filaments after 1 yr (Fig. 1a). Much of the streakiness is eliminated in Fig. 1b, which shows the distribution of the ensemble average of all 12 releases, 365 days after each of their respective starting times. The blue x in Fig. 1b marks the center of mass of tracer collected during cruise US2 of the DIMES experiment, while the black x (+) marks the center of mass of the model ensemble average tracer sampled along the US2 cruise track (over the whole domain) at *t* = 365 days. The excess zonal distance traveled by the modeled tracer ensemble (1.2°) corresponds to an excess zonal propagation speed of the center of mass of 0.2 cm s^{−1} over the first year, compared to the DIMES tracer propagation speed of 2.3 cm s^{−1}. This difference is consistent with the fact that the DIMES tracer was purposefully released between the fronts in a region where the altimetric velocity was particularly weak—the tracer did not move east until a month after release, as discussed in section 2.

Figure 2 shows transect-by-transect comparisons of tracer concentrations observed in DIMES (gray circles) and the simulated ensemble average (black x’s) for each of the cruises. Note that US2 has been split into its two main transects at 96°W (denoted US2A) and 93°W (US2B). The comparison indicates that, at least until UK2.5, the propagation and dispersion of the observed and simulated tracers are consistent. The ensemble tracer is generally less streaky than the observations because it is an average over 12 tracers. Some differences can be seen for the US3 transect. The model has more tracer north of 59°S than the observations and the observed tracer distribution is multimodal, while the modeled ensemble average concentration appears to be more Gaussian.

The time evolution of the mean and standard deviations of the modeled tracer concentration on the US2 cruise track stations are shown as black lines in Figs. 6a and 6c. The red x’s mark the observed values, normalized by the total amount of tracer released. The mean concentration along a cruise track is defined as *N* is the number of cruise track stations. The concentrations *c*_{i} are obtained by column integrating the raw tracer concentrations, in mol L^{−1}, and then normalizing by the number of mols of CF_{3}SF_{5} injected. The mean concentration reaches a maximum in the first 200 days and then decays as the tracer is advected toward the location of the US2 cruise track stations. The standard deviation, a measure of the tracer streakiness, instead peaks earlier at about 50 days. At the time of US2, the modeled streakiness has decayed to about one-eighth of its initial peak, as a result of lateral homogenization of the streaks. Both the modeled mean and standard deviations agree with observations, that is, the red error bar, defined as a 95% confidence interval using bootstrapping of the observed concentrations (Efron and Tibshirani 1993; Zoubir and Boashash 1998), overlaps the gray shading, which is the range spanned by the modeled ensemble members.

A summary comparison of the modeled and observed mean and standard deviations of tracer concentration along each of the cruise tracks, at the times of each cruise, is in Figs. 6b and 6d. As per Fig. 2, the mean and variance of concentrations on all of the cruises are consistent with observations, although the modeled concentrations are slightly larger for the US3 transect. The excess concentration in the model at the most northwest station of US3 indicates that the DIMES tracer might have taken a slightly more southerly path than the modeled tracer. UK2.5A and UK2.5B in Fig. 2 seem to be in agreement with this hypothesis; however, UK2A and UK2B do not. Figure B1f, seen later in appendix B, shows that the Polar Front in the model is displaced northward compared to observations and probably explains these discrepancies.

## 4. Using passive tracers to estimate dispersion and isopycnal eddy diffusivity

*τ*, within an isopycnal layer of thickness

*z*

_{ρ}= ∂

*z*/∂

*ρ*, evolves according to the equation

**u**

_{i}is the along-isopycnal velocity, and the divergence is taken at constant density. Equation (1) does not include a diapycnal flux because Ledwell et al. (2011) reported very small diapycnal diffusivities of the order of 10

^{−5}m

^{2}s

^{−1}upstream of the Drake Passage at the tracer depth. The Drake Patch model has a similarly low diapycnal diffusivity

*K*< 10

^{zz}^{−5}m

^{2}s

^{−1}(see appendix B). For such small diffusivities, the diapycnal tracer flux is orders of magnitude smaller than the along-isopycnal one and can be ignored at leading order.

*z*

_{ρ}:

**K**is a 2 × 2 along-isopycnal eddy diffusivity tensor and ⊗ is the tensor product of the eddy diffusivity tensor and the tracer gradient vector.

*y*= 0. Under these assumptions, the meridional eddy diffusivity can be estimated multiplying Eq. (4) by

*y*

^{2}and integrating over the density layers and lateral extent of the tracer patch. This gives the equation for the growth rate of the second meridional moment of the vertically integrated tracer concentration, as shown in appendix A:

For a meandering mean flow, one ought to use a coordinate system that tracks the mean streamlines of the ACC in order to separate the eddy diffusivity along and across the mean flow. In appendix A, we show how to extend the expression for the eddy diffusivity to a curvilinear coordinate system (*s*, *ψ*), where *s* is the along-stream coordinate and *ψ* is the cross-stream coordinate. While the cross-streamline eddy diffusivity is mathematically well defined, it depends on curvature terms that are difficult to calculate accurately. Here we chose to restrict the analysis upstream of the Drake Passage, west of 75°W, where the flow is mainly zonal and free of the strong meanders that exist downstream. The analysis in appendix A confirms that the meridional and cross-streamline estimates of the eddy diffusivity are indistinguishable within error bars in the upstream region. In the interest of simplicity, we focus on the estimates of meridional diffusivity *K*^{yy}.

Another important consideration is whether the assumption of small longitudinal and latitudinal variations of *K*^{yy} in the ACC sector is supported by the tracer data. Strong support for this assumption comes from the analysis to follow, which shows that *K*^{yy} does asymptote to a constant value over the first year. The term *K*^{yy} would continue to vary, if the tracer kept sampling regions with different dispersion rates.

### a. Estimates of dispersion from the deliberate tracer release data

First, we estimate the dispersion of the DIMES tracer after 1 yr (US2) using available observations. Since only a fraction of the tracer was sampled during US2, any attempt of inferring the dispersion will be stymied by substantial uncertainty. We attempt to quantify this uncertainty by comparing a number of different approaches to estimating the rate of spreading experienced by the tracer after 1 yr. Furthermore, any estimate of dispersion requires an average over many tracer release experiments as discussed in the previous section. But only one such release was done in the DIMES experiment. We will use the numerical model in the next section to determine how well we can infer dispersion from a single tracer release.

We consider three approaches to estimating the spreading of the tracer given by the centered second *y*-moment *N* is the number of stations occupied in US2, *i* minus the latitude of the tracer center of mass, and *c*_{i} is the vertically integrated tracer concentration measured at that station. In the second method, the binned second moment, we first average all *c*_{i} in latitude bins, that is, we average over longitude to obtain an estimate of the concentration as a function of latitude only. Then the centered second moment is computed from the concentration as a function of latitude. The third method does a least squares Gaussian fit to the tracer concentration binned as a function of latitude, and

Estimates of

Table 2 reports estimates of *K*^{yy} based on Eq. (7) and the three methods outlined above for estimating *K*^{yy} = 407 m^{2} s^{−1}, while for the binned second moment *K*^{yy} = 524 m^{2} s^{−1}, and the least squares fit to a Gaussian gives *K*^{yy} = 708 m^{2} s^{−1}. The second moment *K*^{yy} = 407 m^{2} s^{−1} is shown in Fig. 7 as a red x. The errors bars around the x in Fig. 7 correspond to the bracketed uncertainty ranges in Table 2, which are 95% confidence intervals computed by bootstrapping the sample data 10 000 times (Zoubir and Boashash 1998).

Observed estimates of the average rate of dispersion of the DIMES tracer over the first year on the US2 cruise track (*σ*^{2}/2*t* at *t* = 1 yr; m^{2} s^{−1}). The 95% confidence intervals are determined using bootstrapping. The first three lines report estimates using three different methods to estimate *σ*^{2}(1 yr) in both latitude and streamline coordinates (see section 4a and appendix A). The last two rows report our best estimate of the diffusivity obtained by multiplying the first two rows by a model-derived factor that accounts for the incomplete tracer sampling during the US2 cruise (see section 4c). Bins of ½° width span from 65° to 53°S in latitude coordinates and of 4 × 10^{−3} m^{2} s^{−1} width span from −1.75 × 10^{4} to 8 × 10^{4} m^{2} s^{−1} in streamfunction coordinates.

Values of the eddy diffusivity *K*^{nn} in streamline coordinates are also reported in Table 2. These are obtained applying Eq. (7), but using

The large range in estimates of eddy diffusivity confirms that incomplete sampling of the tracer contributes a large uncertainty. Furthermore, as will become more clear, all estimates ignore initial transients during which the growth of

### b. Estimates of dispersion and diffusivity from numerical tracers

The model is used to address three aspects of tracer dispersion. First, we want to know whether the eddy diffusivity asymptotes to a constant over the first year. Second, we need to know whether we can use Eq. (7) to estimate the diffusivity. Third, we will consider the effect of undersampling the tracer on estimates of the eddy diffusivity.

The blue line in Fig. 8a shows

The spreading of the ensemble-mean tracer, the blue line in Fig. 8a, is not diffusive from day one. There is a small initial transient in the first 100 days when

To assess whether this transient invalidates the use of Eq. (7), we least squares fitted a line to *t* = 100 days and *t* = 500 days (black line in Fig. 8a) and compared it to the red line that simply connects ^{2} s^{−1}, respectively, suggesting that the ansatz of Eq. (7) is accurate to within 10%. Notice, however, that these estimates are based on an ensemble-averaged tracer. In the DIMES experiment we have only one realization. In Fig. 8b, we show, for each tracer release experiment, the half slopes estimated from linear least squares fits between *t* = 100 days and *t* = 500 days, black x, versus the half slopes obtained from Eq. (7), red x. Because of the initial transient, estimates of *K*^{yy} based on Eq. (7) in the individual realizations vary from 718 to 966 m^{2} s^{−1}, whereas the dispersion rate from 100 to 500 days varies 727–861 m^{2} s^{−1}, which is a tighter bound on the diffusivity. Nevertheless, the differences between the two estimates are quite small and on average no larger than in the ensemble mean. We conclude that Eq. (7) can be used to estimate *K*^{yy} from data with perhaps a 20% uncertainty.

A more problematic issue in estimating the diffusivity is the extrapolation of the subsampled tracer on the US2 grid points to the full tracer distribution. Figure 7 shows half the second moment of the US2 subsampled tracer divided by time (red line) and that for the full tracer upstream of the Drake Passage (black line); these are estimates of *K*^{yy} based on Eq. (7) applied at all times instead of only at 1 yr. Second moments for the subsampled tracer are calculated using the first approach described in section 4a, that is, from all the individual column integrals, with no binning. The red line is 60% smaller than the black line implying that the US2 grid samples only a fraction of the tracer distribution. The ratio of the two curves is fairly constant between 250 and 450 days, suggesting that estimates of *K*^{yy} based on sampling the tracers along the US2 grid after 1 yr are biased 60% low.

The analysis presented so far suggests that Eq. (7) is appropriate to estimate *K*^{yy} if the tracer is sampled adequately. Figure A2 (shown later) confirms that the estimate of *K*^{yy} is independent of the specific method used to estimate *K*^{yy} computed using only data on the US2 cruise track. We repeated the same analysis followed for the DIMES observations and used Eq. (7) with the three different approaches to estimate *K*^{yy}. The Gaussian fit method correctly extrapolates the missing tracer when applied to the ensemble-averaged tracer on the US2 grid, but returns widely varying results when applied to a single tracer injection experiment. The inescapable conclusion is that none of the three approaches can be used to infer the spreading rate experienced by the tracer in DIMES, because the uncertainty associated with the missing tracer is too large.

Modeled estimates of average rate of dispersion of the tracer ensemble over the first year using three methods and two coordinate systems (*σ*^{2}/2*t* at *t* = 1 yr in m^{2} s^{−1}). The mean value is based on the ensemble average tracer, while the upper and lower bounds (in brackets) are the maximum and minimum values from the 12 tracer release experiments. Estimates using the full tracer west of 75°W are in the top three rows, and estimates using the subsampled tracer on the US2 grid are in the bottom three rows. Bins of ½° width span from 65° to 53°S in latitude coordinates and of 4 × 10^{−3} m^{2} s^{−1} width span from −1.75 × 10^{4} to 8 × 10^{4} m^{2} s^{−1} in streamfunction space.

Alternatively one can use the model estimate of *K*^{yy}, since the model has been tested against data. However, a comparison of data and model estimates based on tracer data on the US2 cruise track shows that the model estimates are biased high (see Tables 2 and 3 and Fig. 7). Although the error bars are large enough to make all estimates consistent (the model uncertainty is estimated as the range of values obtained from the 12 tracer release experiments, while the DIMES uncertainty is computed using bootstrapping), the high model bias is consistent with the model kinetic energy being somewhat too high as discussed in section 3a. It appears that the best way forward is to extrapolate the *K*^{yy} estimate from the DIMES data on the US2 cruise track using the model to infer the bias introduced because of the subsampling of the tracer. This is done in the next section.

### c. Best estimate of the eddy diffusivity upstream of the Drake Passage at 1500 m

The tracer dispersion estimated from the DIMES data in section 4a is likely an underestimate because only half of the tracer was sampled and large values to the north suggest more dispersion northward. Since the model consistently overestimates the tracer dispersion compared to the DIMES observations, it cannot be used directly to estimate the DIMES diffusivity. We showed that by fitting a Gaussian meridionally to the subsampled tracer, a Gaussian returned a diffusivity of ^{2} s ^{−1}, but the uncertainty in this value is very large spanning the range 358–840 (see Table 2). Alternatively, the model can be used to infer how much of the tracer dispersion was missed by sampling only on the US2 cruise track.

The red x in Fig. 7 marks the eddy diffusivity estimated using data along the US2 stations, while the blue x is the extrapolated value. The last two rows of Table 2 summarize the results. Using this extrapolation we estimate that the meridional eddy diffusivity in the DIMES experiment was 710 ± 260 m^{2} s^{−1} at 1500 m. This value agrees well with that estimated using a least squares Gaussian fit, building confidence in our estimate.

## 5. Estimating the vertical structure of the eddy diffusivity

There is growing evidence that the isopycnal eddy diffusivity of passive tracers varies in the vertical and has subsurface maxima (Treguier 1999; Smith and Marshall 2009; Abernathey et al. 2010; Lu and Speer 2010; Klocker et al. 2012b), unlike the horizontal buoyancy diffusivity that appears to be less variable in the vertical. It is therefore difficult to interpret the significance of the DIMES estimate and compare it to previous work without some information about the vertical variations from the 710 m^{2} s^{−1} value. We use the Drake Patch model to extrapolate the DIMES observations to the rest of the water column.

To assess the vertical variations of eddy diffusivity in the DIMES region, we ran an ensemble of tracers injected on 4 February of the sixth year of model integration at 12 different depths between 500 and 3500 m. The time evolution of *t* = 500 days. Afterward the dispersion accelerates as most of the tracer has reached the Drake Passage (not shown). The red lines are the dispersion experienced by the tracer over the first year and its slope is given by Eq. (7); this is the estimate of the diffusivity used for the DIMES tracer in section 4. The black line shows a linear least squares fit to the dispersion between *t* = 100 days and *t* = 500 days, which attempts to remove the initial transient from the diffusivity estimate. For tracers released in the upper 1000 m, the slopes of the red and black curves are very different because the effect of the initial transient is significant. It is actually difficult to select the time window over which the growth rate of

Figure 10a shows the vertical profile of the diffusivity *K*^{yy} estimated by least squares fitting lines between *t* = 100 days and *t* = 500 days (black line). The figure also shows the range of eddy diffusivity estimates from all 12 ensemble members released at 1500 m (thin horizontal black line) to emphasize that much uncertainty remains when the eddy diffusivity is estimated from a single release experiment. For comparison, the best estimate of the eddy diffusivity from the DIMES tracer release is shown as a blue circle with its uncertainty. The model estimate is biased slightly too high, but well within the observational error bars.

Despite the uncertainty, Fig. 10a shows that the eddy diffusivity has a maximum between 1700 and 2500 m. Naïvely, one may expect the eddy diffusivity to scale with the eddy kinetic energy, which is monotonically decreasing with depth as shown in Fig. 10b. However, Bretherton (1966) and Green (1970) pointed out that mixing is strongly suppressed when eddies propagate at a speed different from the mean flow. Figure 10b shows both the mean flow speed as a function of depth, averaged over the patch extending from 61° to 56°S and 10° to 80°W, and the eddy propagation speed, estimated with a radon transform of the sea surface height in the same region (see Smith and Marshall 2009). The eddy propagation speed is much smaller than the mean flow speed in the upper kilometer, resulting in a suppression of the eddy diffusivity. Close to the steering level, where the mean flow equals the eddy propagation speed, there is no suppression, and the eddy diffusivity is largest. Similar vertical profiles of eddy diffusivity have been reported in recent studies of ACC flows more or less constrained to observations (Smith and Marshall 2009; Abernathey et al. 2010; Lu and Speer 2010; Klocker et al. 2012b).

Based on the model results, we infer that the meridional eddy diffusivity in the DIMES region peaks at around 900 m^{2} s^{−1} between 1700 and 2500 m, while it is smaller than 500 m^{2} s^{−1} in the upper kilometer. While this structure is consistent with recent studies, the absolute values of the diffusivity are less so. In particular, Abernathey et al. (2010) and Klocker et al. (2012a) published larger estimates for the DIMES region. Abernathey et al. (2010) estimated the diffusivity advecting tracers with a state estimate of the Southern Ocean circulation and reported values around 500 m^{2} s^{−1} in the upper kilometer and values in excess of 2000 m^{2} s^{−1} at the steering level. Klocker et al. (2012a) estimated, using an idealized, two-dimensional, zonally reentrant setup driven by surface altimetry, that the eddy diffusivity in the DIMES region peaked at 1000 m^{2} s^{−1} at 1.5-km depth, decreasing to 700 m^{2} s^{−1} at the surface. Most likely these differences stem from the different velocity fields used in the calculation and, in the case of Abernathey et al. (2010), from the use of a different method to compute the eddy diffusivity—they used Nakamura’s definition of the eddy diffusivity. We believe that our estimate is more robust than these previous ones because it is grounded in direct observations.

## 6. Discussion

This paper presents the first direct estimate of the isopycnal eddy diffusivity across the ACC just upstream of Drake Passage. The estimate was computed from the spreading of the DIMES tracer that was released in February 2009. Using tracer sampling at 1 yr after release (cruise US2), we estimated an isopycnal eddy diffusivity of 710 ± 260 m^{2} s^{−1} upstream of Drake Passage at 1500 m. The estimate is based on the tracer spreading measured during US2 supplemented by a numerical model used to infer where the full tracer patch had spread after 1 yr; US2 sampled only half of the tracer that was injected 1 yr earlier.

In a companion paper, LaCasce et al. (2014) find similar values of isopycnal eddy diffusivity from floats released during the DIMES field campaign and floats released in the same numerical model used in our study of tracer dispersion. This builds confidence that our estimate is accurate.

The numerical model further suggests that the isopycnal eddy diffusivity at 1500-m depth is close to its maximum in the water column. Diffusivities above 1000 m and below 3500 m appear to be smaller than 500 m^{2} s^{−1}. The maximum in eddy diffusivity coincides with the steering level where the eddy propagation speed of 2.2 cm s^{−1} matches the zonal-mean flow (Fig. 10).This vertical profile is consistent with the notion that mixing is suppressed in the upper kilometer of the ocean where eddies propagate much slower than the strong ACC flow, while it is large at the steering level where there is no suppression (Bretherton 1966; Green 1970; Ferrari and Nikurashin 2010). The mixing suppression at the surface and enhancement at depth is a robust feature of ocean mixing that has already been reported in idealized studies of channel flows (Treguier 1999; Smith and Marshall 2009), in studies informed by ACC observations (Abernathey et al. 2010; Lu and Speer 2010; Klocker et al. 2012b), and in hydrographic sections (Naveira Garabato et al. 2011).

The present results have important implications for ocean models. The diffusivity estimated here is the Redi isopycnal diffusivity that homogenizes tracers and potential vorticity (Griffies 2004). Our result is that the Redi diffusivity in a sector of the Southern Ocean varies in the vertical with a peak of approximately 900 m^{2} s^{−1} at 2 km. If these variations are not isolated to the region sampled in DIMES, they imply the strongest ventilation at the interface between the upper and lower meridional overturning cells (Marshall and Speer 2012), a region crucial for ocean carbon uptake. The implications for the horizontal buoyancy (Gent–McWilliams) diffusivity are more subtle. Smith and Marshall (2009) and Abernathey et al. (2013) find that the buoyancy diffusivity is more vertically constant than the tracer diffusivity and has a magnitude close to the surface value of the tracer diffusivity. If this holds true in general, our results imply that the buoyancy diffusivity is less than 500 m^{2} s^{−1}, a value smaller than presently used in ocean models used for climate studies. However, we realize that our results apply only to a small sector of the Southern Ocean upstream of the Drake Passage, and one cannot extrapolate the results to the global ocean. Rather our analysis provides a ground truth for developing parameterizations, which can then be used to extrapolate our results to other regions. A new parameterization of eddy mixing based on these results is currently being developed (Bates et al. 2014).

## Acknowledgments

Ferrari and Tulloch wrote the manuscript with input from the other coauthors. Ledwell, Messiah, Speer, and Watson led the tracer field measurements. Ferrari, Jahn, Klocker, LaCasce, Marshall, and Tulloch led the numerical simulation work. We also thank all the scientists and ship crews of DIMES for their many contributions to a very successful experiment. NSF support through Awards OCE-1233832, OCE-1232962, and OCE-1048926 is gratefully acknowledged. Computing resources on Pleiades and Yellowstone proved essential to perform the numerical simulations that are used to interpret the data.

## APPENDIX A

### Computation of Tracer Dispersion

The goal of this paper is to quantify the mixing by geostrophic eddies along isopycnal surfaces and across mean currents. It is thus necessary to use a coordinate system that follows isopycnal surfaces and mean streamlines. We discuss the transformation to isopycnal coordinates first, and then we tackle the transformation into a streamline coordinate system.

#### a. Tracer moments in isopycnal coordinates

*y*

^{2}and integrating over density and in the horizontal beyond where there is any tracer. The final result is given in Eq. (6) in the main text. Here are a few more steps to help follow the full derivation:

#### b. Tracer moments in streamline coordinates

Isopycnal mixing by geostrophic eddies is generally strongly anisotropic, being much larger along mean currents than across. It is therefore necessary to rotate coordinates along and across mean streamlines to properly estimate mixing in the two directions. We could not find a description of how to compute eddy diffusivities in a streamline coordinate system, and so we decided to include in this appendix the details involved in the calculation. The second section of the appendix then compares estimates of the dispersion in streamline and longitude–latitude coordinates for the DIMES region.

*s*,

*ψ*), where

*s*is the along-stream coordinate (with units of length) and

*ψ*is the cross-stream coordinate that increases normal

*ψ*:

*J*is a two-dimensional Jacobian, and

**F**represents the eddy flux of tracer. The flux term in streamline coordinates takes the form

*F*(

*x*,

*y*) over a region encircled by a streamfunction

*ψ*:

*I*(

*ψ*) with respect to

*ψ*, which is the average of

*F*(

*x*,

*y*) along the streamline:

**K**:

**K**=

**K**

^{asym}+

**K**

^{sym}, is

**K**into its tensor components gives

**K**=

**K**(

*ψ*), the

*K*

^{nn}is the only component that evolves the stream-averaged tracer.

*ψ*gives

*ψ*, if either the diffusivity or the mean flow increase with

*ψ*(∂

_{ψ}

*K*

^{nn}> 0 or the streamlines become more packed).

*ψ*gives

*K*

^{nn}is approximately uniform (∂

_{ψ}

*K*

^{nn}→ 0), then the cross-stream diffusivity is approximately

*ψ*∂

_{ψ}(|

**∇**

*ψ*|

^{2}) ≪ |

**∇**

*ψ*|

^{2}, then the expression for

*K*

^{nn}reduces to

**∇**

*ψ*|

^{2}factor in the denominator represents that the conversion between dispersion in

*ψ*coordinates and length coordinates.

Finally, note that if the center of mass of the tracer in streamline coordinates is not at *ψ* = 0, that is, *ψ* = 0 for the streamline along which the tracer was released.

#### c. Estimates of tracer dispersion across streamlines in the Drake Patch

We introduced three different estimators of *ψ* = *gη*/*f*, where *g* is the gravitational constant, *η* is sea surface height, and *f* is the local Coriolis frequency, to define our streamlines. Figure A2 shows estimates of *K*^{yy} (top) and *K*^{nn} (bottom) versus time using the three methods described in section 4a: a second moment that assumes all data points are independent, a binned second moment averaged along the stream (zonally) within cross-stream (meridional) bins, and a least squares fit to a Gaussian distribution using the binned data (left to right). To define the streamlines, the model’s sea surface height was averaged from year 5 to 10, then coarse grain averaged using a Shapiro (1970) filter to remove eddy aliasing. To smooth the diffusivity in time, we plot the time-integrated rate of dispersion *K*^{yy} and the bending would make the curvature term in the denominator of Eq. (A13) for *K*^{nn} significant. Also, narrowing of the stream in and downstream of the Drake Passage likely invalidates the assumption that ∂_{ψ}*K*^{nn} → 0. To alleviate all of these issues, we have restricted the tracer dispersion calculations shown in Fig. A2 to tracer that is west of 75°W, which encompasses nearly all of the tracer shown in Fig. 1 at *t* = 1 yr.

In the left panels of Fig. A2, the dispersion is integrated exactly as defined in the equations above. In the middle panels, the tracer is first averaged zonally along 25 stripes half a degree latitude bins between 65° and 53°S and an equivalent number of stripes in streamline coordinates 4 × 10^{−3} m^{2} s^{−1} wide are defined to bin the tracer before summing across the stream. This calculation is essentially identical to the method on the left, but with less cross-stream resolution. In the right panels, the tracer is first binned as in the middle panels and then fitted to a meridional or cross-stream Gaussian profile via least squares gradient descent, analogously to the method used in Ledwell et al. (1998). Figure A2 shows that the three methods shown agree with each other when the full (upstream) tracer is taken into account and that the latitudinal and cross-stream diffusivities are both approximately *K* = 800 − 1000 m^{2} s^{−1} in the model at *t* = 1 yr. When the full tracer is known, the estimates on the right agree with the estimates on the left in the ensemble mean (thick black line), but there is more uncertainty in the ensemble members (thin gray lines). The middle and left plots also decrease at later times as more of the tracer approaches the Drake Passage where the stream is slightly narrower, while this effect seems to be absent in the least squares fits on the right. The cross-stream diffusivities are a bit larger than the latitudinal diffusivities (Table 3), but the differences are not significant compared with the uncertainties.

#### d. Estimates of tracer dispersion across streamlines in DIMES

Figure A3 shows estimates of eddy diffusivity using the same three methods described in section 4a, but using streamlines coordinates. The second moment of the tracer in streamline coordinates is estimated as

## APPENDIX B

### Model Setup and Comparison with Hydrography

The Drake Patch model is a regional configuration of the MITgcm on a one-twentieth of a degree resolution latitude–longitude grid. Horizontal vorticity is advected with a fourth-order accurate spatial discretization using an enstrophy-conserving (Arakawa and Lamb 1977) and vector-invariant formulation. Horizontal viscosity is biharmonic, with an amplitude that scales according to local grid spacing and stresses (Fox-Kemper and Menemenlis 2008). Vertical viscosity is Laplacian and a quadratic bottom drag is imposed in the lowest layer. Momentum, temperature, and salinity are forced at the surface by reanalysis from the ECMWF ERA-Interim on a 6-hourly time scale and at approximately 0.7° resolution (Dee et al. 2011). The initial hydrography is taken from an average of OCCA’s December 2004 and January 2005 fields (Forget 2010). There is dynamic sea ice, and the freezing temperature is set to *T* = 273.2501 − 0.0575 · *S*. Advection of temperature, salinity, and passive tracers is by a spatially seventh-order accurate, monotonicity-preserving scheme (Daru and Tenaud 2004). The *K*-profile parameterization scheme of Large et al. (1994) is used to parameterize vertical mixing due to boundary layer shear and convective instability. Table B1 summarizes the numerical parameters. The bathymetry was downloaded from

Numerical parameters used in the Drake Patch simulation.

Lateral boundary conditions (*U*, *V*, *S*, *T*, and sea ice) on a monthly time scale and 1° resolution from OCCA are interpolated onto the model’s resolution. A relaxation boundary condition absorbs outgoing flow over a 1° sponge layer [see section 6.3.2 of Adcroft et al. (2014) for details of the MITgcm relaxing boundary condition scheme]. The model cycles repeatedly over the 3 yr for which OCCA is defined (2004–06). Tracers are injected once the model has cycled 1.66 times through the OCCA 3-yr period. The OCCA boundary conditions are interpolated in time to avoid any shocks in the dynamics and tracer evolution.

#### a. Comparison of Drake Patch model against hydrography

Figure B1 compares the model’s hydrography (right plots) with CTD data stored at the CLIVAR and Carbon Hydrographic Data Office (left plots) from sections P18 (top), P19 (middle), and A21 (bottom), which are denoted with gray dashed lines in Fig. 3. The westernmost section, P18 at 103°W, is in a relatively quiescent region of the ACC, near the initial DIMES tracer injection point. The SAF is visible at 55°S, 103°W and PF at 60°S, 103°W) in both the model and in P18. North of 60°S, there appears to be a deeper mixed layer, or mode water, in the model compared to observations. Deeper model mixed layers are expected because the model does not have a submesoscale parameterization for mixed layer restratification (Fox-Kemper and Ferrari 2008). At P19 (88°W), the fronts appear to be sharper south of 60°S in the model than in observations, possibly due to the different sampling resolution of model versus ships or to the model lacking a representation of bottom dissipation processes. There is also more mode water present at P19 in the model than in observations. Within the Drake Passage, at section A21, the SAF appears similar between the model and observations, but the PF is stronger in the model and displaced northward by about half a degree. There also appears to be a bowl of low density water in the model between 60° and 58°S, which does not appear in the observations below 1 km. The bowl of low density water in the model likely results from the path of the ACC in the model along A21, visible in Fig. 3b. The transect appears to run almost parallel to the jet at 58.5°S.

#### b. Vertical diffusivity in the model

Ledwell et al. (2011) showed that diapycnal diffusivity upstream of the Drake Passage is approximately 1.3 × 10^{−5} m^{2} s^{−1} at 1500-m depth. However, many eddying z-coordinate models contain a horizontal bias as isopycnal surfaces become steeply inclined, which can lead to numerically generated diapycnal mixing of the order of 10^{−4} m^{2} s^{−1} (Griffies et al. 2000). Hill et al. (2012) show that this spurious diapycnal mixing can be limited to *K*^{zz} < 10^{−5} m^{2} s^{−1} when the vertical tracer variations are well-resolved and a second-order moment (SOM) advection scheme (Prather 1986) is employed. Specifically, for a tracer with a Gaussian concentration and a vertical standard deviation of 50 m and layer thicknesses of 10 m, they obtain a diapycnal diffusivity of about 0.5 × 10^{−5} m^{2} s^{−1} using the SOM scheme with a flux limiter (their simulation A2). However, when the Gaussian profile is not well resolved, that is, layer thicknesses of 100 m, the flux-limited scheme produces 8 times more diapycnal diffusivity. Without a flux limiter (simulation A1), the diffusivity stays under 10^{−5} m^{2} s^{−1}.

Figure B2 shows the evolution of tracer variance in density space in the Drake Patch model for a single tracer released with a Gaussian initial profile with half-width *σ*_{z} = 75 m, using the SOM advection scheme without flux limiter and a seventh-order, one-step, monotonicity-preserving method (Daru and Tenaud 2004). All layers shallower than 2 km in the Drake Patch are thinner than 35 m, so this tracer, centered at 1500 m, is well resolved in the vertical. Converting from variance in density coordinates to height coordinates using the average neutral density gradient at 1500 m as *dρ*_{n}/*dz* ≈ −3.8 × 10^{−4} kg m^{−4} yields *K*^{zz} < 10^{−5} m^{2} s^{−1} for both advection schemes.

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^{1}

Layer spacing Δ*z* ≤ 35 m allows the vertical grid to resolve Gaussian tracer profiles with a root-mean-square spread as small as 70 m (Hill et al. 2012) and most importantly ensures that spurious numerical diffusion in the vertical is below 10^{−5} m^{2} s^{−1}, consistent with direct estimates of diapycnal diffusivity upstream of the Drake Passage from the DIMES tracer release (Ledwell et al. 2011).

^{2}

An experiment with the atmospheric forcing shifted 20° west resulted in the generation of anticyclones 20° west of the Chilean coast. These anticyclones appeared to be driven by freshwater forcing at the surface, as that region is one of the rainiest in the world; for example, Villa Puerto Edén receives almost 6 m of rain per year. They are likely sensitive to the ECMWF ERA-Interim reanalysis product and its low resolution, which does not limit the heavy rain to the coastline.