#### 3.1. Estimates of the Eddy Mixing Length Scale *L*_{mix} From Hydrographic Sections

[12] Our method for estimating *L*_{mix} rests on the mixing length arguments of *Armi and Stommel* [1983] [see also *Ferrari and Polzin*, 2005], according to which *L*_{mix} may be defined as

where θ_{rms} is the rms potential temperature fluctuation along a neutral surface arising from eddy stirring of the large-scale potential temperature θ_{m}, and ∇_{n} is the gradient operator on the same neutral surface. This definition is formally valid to the extent that tracer fluctuations are generated by local stirring of the large-scale tracer gradient (i.e., advection of tracer variance from regions upstream is assumed to be weak), and insofar as ∣∇_{n}θ_{m}∣ varies slowly over the eddy mixing length *L*_{mix} (i.e., a scale separation between eddy and mean flow scales is assumed). While the first assumption appears to hold in numerical models [*Abernathy et al.*, 2009], it is difficult to extrapolate to the real ocean because models and observations are too coarse to quantify tracer variance budgets over large regions. The second assumption might be violated where jets are particularly narrow and there is no distinction between mean and eddy length scales. We chose to proceed and presume that both assumptions hold. Verification of the assumptions will be done *a posteriori*, to the extent that spatial variations in our estimates of *L*_{mix} can be related to local flow and tracer statistics.

[13] We note that this description of lateral mixing is, by assumption, distinct from that put forward by *Joyce et al.* [1978] in the context of a high-spatial-resolution survey of the PF that was part of the International Southern Ocean Studies (ISOS). Whereas our framework assumes that thermohaline variability on isoneutrals arises passively from the mesoscale eddy-induced filamentation of background θ − *S* gradients (e.g., as found by *Smith and Ferrari* [2009] in the eastern subtropical North Atlantic), *Joyce et al.* [1978] contend that interleaving features actively enhance thermohaline variability through double diffusive processes that cause the features to slope across neutral surfaces. The rationale and justification of our framework and the extent to which it affects our results are discussed in Appendix B.

[14] Along each hydrographic section, we calculate _{m} and θ_{rms} on discrete neutral surfaces separated by an interval of 0.02 kg m^{−3} in the neutral density variable *γ*^{n}of [*Jackett and McDougall*, 1997]. This involves mapping the measured θ profiles, which are provided in a 2 dbar pressure grid, to the selected *γ*^{n} surfaces using linear interpolation. We wish to distinguish between spatial θ anomalies caused by the meandering of streamlines, which in the ACC are aligned with horizontal contours of hydrographic properties to a very good approximation [*Sun and Watts*, 2001], and anomalies associated with the genuine eddy-induced translation of water parcels across streamlines, which ultimately leads to cross-stream mixing when those anomalies are eroded by small-scale mixing processes. This distinction is made by using a baroclinic stream function (the geopotential anomaly at 500 dbar relative to 1500 dbar, defined as ϕ_{500}^{1500} = *δdp*, where *δ* is the specific volume anomaly and *p* is pressure) as the cross-stream coordinate in place of geographical distance. Our results are insensitive to the exact choice of baroclinic stream function, because the ACC streamlines are equivalent barotropic and hence veer little with depth (Appendix A).

[15] The distance between ϕ_{500}^{1500} contours at each hydrographic section is computed as the mean distance between the contours averaged over all repeats of the section. We will refer to this pseudo-distance as *Y*, with the origin chosen as the southernmost station in the section. Since all sections are oriented approximately perpendicular to mean streamlines, *Y* is a reasonably good estimate of cross-stream distance. We opted for a definition of *Y* in terms of ϕ_{500}^{1500} (as opposed to sea level, which would give many more realizations than hydrographic section repeats) because we have greater confidence in the resolution of *in situ* cross-frontal variability in a geopotential anomaly-based reference frame rather than a sea level-based reference frame.

[16] After mapping the θ observations along all sections to a *Y*-*γ*^{n} grid, we calculate the θ_{m} distribution along each transect location by fitting a cubic spline to all the *Y*-θ data pairs sampled on each neutral surface. The choice of a cubic spline in this definition is motivated by the continuity of both the curve and its first derivative, which is implicated in the calculation of *L*_{mix} (see (3)). Any other smooth function with these properties gives very similar results to the ones presented here. An illustration of the calculation is provided by Figure 3. The θ_{rms} at location *Y* is estimated as the one standard deviation of (θ − θ_{m}) for all measurements obtained within *Y* ± Δ*Y*. The thermal anomalies entering the calculation of θ_{rms} have characteristic vertical scales of O(10–100 m), comparable to the dimensions of cross-frontal interleaving features reported elsewhere [e.g., *Joyce et al.*, 1978; *Toole*, 1981]. The width of the interval Δ*Y* is chosen to be in the range 30–150 km, with the exact value depending on the spatial density of sampling and the width of the ACC at each transect location. Our choice is guided by the requirement to have at least 5–10 data points in each calculation interval to approach statistical stability. This calculation of *L*_{mix} exhibits only minimal sensitivity to Δ*Y* values on the order of 10–100 km (Appendix A). In essence, our definition of θ_{m} is identical to that of the gravest empirical mode of *Sun and Watts* [2001], while θ_{rms} measures the extent to which the observed θ departs from that modal structure.

[17] The distributions of θ_{m} arising from the preceding calculations are shown in *Y*-*γ*^{n} space for each of the five transect locations (Figures 4a, 5a, 6a, 7a, and 8a). The main water masses and frontal features of the Southern Ocean can be immediately recognized. A relatively warm (θ_{m} ∼ 1–3°C) and voluminous body of CDW is seen to occupy the bulk of the sections at densities in excess of *γ*^{n} ∼ 27.6 kg m^{−3}, overlying a layer of colder (θ_{m} 3 < 0°C) and denser (*γ*^{n} > 28.27 kg m^{−3}) AABW near the southern end of the WOCE SR1b, I6S, I8S and SR3 transects. In the upper layers, a sub-surface θ_{m} minimum colder than θ_{m} ∼ 2°C is observed in each of the sections to the south of the Polar Front, denoting the core of the wintertime variety of Antarctic Surface Water (referred to as Winter Water). Warmer (θ_{m} > 3°C) upper-ocean waters are found further to the north. These indicate the presence of SAMW and AAIW equatorward of the Subantarctic Front, underlying a thin layer of surface waters.

[18] The cross-stream isoneutral gradient of θ_{m} at each of the transect locations is displayed in Figures 4b, 5b, 6b, 7b, and 8b. These reveal that the thermohaline gradients upon which eddies act are largest in the upper layers of the Polar Front (particularly in Drake Passage) and of the Subantarctic and Subtropical Fronts (in WOCE I6S, I8S and SR3), as well as more generally along the base of the pycnocline. The patterns in the distribution of θ_{rms} (Figures 4c, 5c, 6c, 7c, and 8c) broadly follow those in the ∇_{n}θ_{m} field, indicating that the local rate of thermal variance production by eddy stirring along neutral surfaces is highly dependent on the local isoneutral gradient of θ_{m}. However, the covariation of θ_{rms} and ∇_{n}θ_{m} is not perfect. Rather, it exhibits substantial spatial inhomogeneity, its structure reflecting variations in the mixing length scale *L*_{mix}.

[19] Estimates of *L*_{mix} as defined in (3) are shown in Figures 4d, 5d, 6d, 7d, and 8d. We estimate that *L*_{mix} varies by at least one order of magnitude. Values on the order of 5–10 km are commonly found in the upper layers (uppermost 500–1000 m) of the major ACC frontal jets, identified by their geostrophic velocity expressions in Figures 4e, 5e, 6e, 7e, and 8e, whereas mixing lengths of 50–150 km occur in the jets' deeper layers and in interfrontal regions. The finding that the mixing length scale inferred from isoneutral θ fluctuations is comparable to or smaller than the horizontal eddy length scale of O(100 km) lends further support to our interpretation of those fluctuations as being the product of eddy stirring (Appendix B).

[20] We find a notable suppression of *L*_{mix} in the four frontal jets in western Drake Passage (Figures 4d and 4e); the PF and SBdy in eastern Drake Passage (Figures 5d and 5e); the STF, SAF and SACCF in WOCE I6S (Figures 6d and 6e); the STF, SAF and PF in WOCE I8S (Figures 7d and 7e); and the SAF's southern branch, the two branches of the PF, SACCF and SBdy south of Tasmania (Figures 8d and 8e). Only in three frontal jet sites do we find an obvious absence of eddy mixing length suppression: the SAF in WOCE SR1b (Figures 5d and 5e), the PF south of Africa (Figures 6d and 6e), and the SAF's northern branch in WOCE SR3 (Figures 8d and 8e). The correspondence, or lack thereof, between areas of reduction in *L*_{mix} and fronts is ambiguous in a few of the weaker jets, namely the SACCF in eastern Drake Passage (Figures 5d and 5e), the SBdy in WOCE I6S (Figures 6d and 6e), and the SACCF and SBdy in WOCE I8S (Figures 7d and 7e). To summarize out of 24 frontal jet crossings, 17 exhibit evidence of suppression, such evidence is ambiguous in 4, and only 3 definite exceptions are noted.

[21] We note that the increased magnitude of *L*_{mix} in the deep ACC, below a depth of approximately 1000 m, is of dubious significance as it is associated with the tendency of ∇_{n}θ_{m} toward zero there. This tendency could indeed reflect intense eddy stirring at depth, but it might also be a consequence of the general increase in the ventilation age of water masses with depth. In spite of this caveat, at least one piece of unambiguous evidence can be uncovered that points toward a genuine intensification of eddy stirring with depth at the ACC frontal jets. Such evidence may be found in the quasi-synoptic survey of the PF conducted during ISOS [*Joyce et al.*, 1978], which lies approximately 250 km to the east of the northern end of the WOCE S1 transect (Figure 2a) and was much more densely sampled than any WOCE section, thereby providing a unique view of the current's thermohaline structure. Using this data set, we construct a cross-frontal section with a variable horizontal resolution of 3–15 km. The distributions of θ_{m}, ∇_{n}θ_{m}, θ_{rms}, and *L*_{mix} in *Y*-*γ*^{n} space along this section are displayed in Figures 9a–9d and exhibit many of the properties that we encounter in the WOCE transects. The ISOS section spans across the northern flank of the PF, as denoted for example by the northern terminus of the Winter Water θ_{m} minimum at *γ*^{n} ∼ 27.45 kg m^{−3} near *Y* ∼ 30 km (Figure 9a). The strongest θ_{m} gradients are seen in waters lighter than *γ*^{n} ∼ 27.7 kg m^{−3} in the horizontal proximity of this terminus (Figure 9b), and occur in association with slightly elevated values of θ_{rms} (Figure 9c). It is in this area that eddy stirring is most strongly suppressed and *L*_{mix} values under 10 km are found (Figure 9d). These values are comparable to the cross-frontal length scales of the coherent interleaving features observed in the upper part of the PF during ISOS [*Toole*, 1981]. North of and below the upper layers of the front's northern flank, eddy stirring is no longer inhibited, and *L*_{mix} is typically an order of magnitude larger.

[22] We now wish to direct the reader's attention to a feature in the ISOS section that was not immediately apparent in the WOCE transects: the boundary between the regions of weak and intense stirring in and north of the PF, respectively, is not vertical. Rather, it has a distinct slope of *O*(5 × 10^{−3}) that broadly parallels geostrophic isotachs (Figure 9e). That the area of reduced stirring in the upper layers of the frontal jet is bowl-shaped and has sloping lateral boundaries is further evidenced by the distribution of ∇_{n}^{2}θ_{m} ≈ ∂^{2} θ_{m}/∂*Y*^{2} along neutral surfaces in the ISOS section (Figure 9g). A band of markedly negative values occupies the deeper part of the region of weak stirring (Figure 9d) and mimics the shape of mean flow contours (Figure 9e). Equatorward of this band, ∂^{2}θ_{m}/∂ *Y*^{2} ≈ 0, as would be expected from a region of strong stirring where any curvature in θ_{m} tends to be erased. We conclude that the detailed thermohaline structure of the PF in the ISOS section points to the existence of an inverse relationship between the mean flow speed and the eddy stirring rate, and is thus consistent with a genuine intensification of eddy stirring with depth at the ACC frontal jets.

#### 3.2. Estimates of the Eddy Velocity *U*_{e} From Altimetry

[23] We estimate the cross-stream eddy velocity scale *U*_{e} as the one standard deviation in time of **u** · , where **u** is the horizontal velocity vector, and is the unit vector perpendicular to the time-mean horizontal velocity . The calculation consist of five steps. First, we construct a 15-year time series of weekly maps of sea surface height *η* by interpolating along each section the combined altimetric and absolute dynamic topography products introduced in Section 2. Second, along each transect, we map the *γ*^{n} and surface geopotential anomaly relative to an isobaric surface of pressure *p*, ϕ_{0}^{p}, as a function of *η* at the time of the transect and *p*. Then we fit cubic splines to the *γ*^{n}-*η* and ϕ_{0}^{p}-*η* pairs at each isobaric level. The procedure is analogous to the gravest empirical mode calculation of *Sun and Watts* [2001]. Third, we use these sets of spline fits to obtain weekly, two-dimensional (i.e., as a function of *η* and *p*) estimates of the *γ*^{n} and **u** fields along each hydrographic section location. The surface **u** field is calculated from *η* using thermal wind, and projected to depth using ϕ_{0}^{p}-derived geostrophic shear. Fourth, we map the along-transect distribution of **u**, which is originally estimated on a pressure-based vertical grid, to the *γ*^{n} grid used in the calculation of *L*_{mix}. Fifth, we compute *U*_{e} (*η*, *γ*^{n}) from the expression above and use the mean (of all section repeats) profile of sea level (which is tightly related to ϕ_{500}^{1500}) versus along-transect distance for each section to obtain *U*_{e} (*Y*, *γ*^{n}), i.e., expressed in the same coordinates as *L*_{mix}.

[24] The resulting distributions of *U*_{e} along the sections are shown in Figures 4f, 5f, 6f, 7f, and 8f. These exhibit prominent maxima in a broad region around the SAF and PF, and secondary maxima near some of the other frontal jets. Minima in *U*_{e} are generally found near the current's poleward and equatorward edges. There is also a conspicuous decreasing tendency with depth that stems from the equivalent barotropicity of the ACC. All in all, the *U*_{e} distributions reflect the well-known patterns of EKE in the Southern Ocean (cf. Figure 2b), and imply the existence of a broad anticorrelation between the structure of the *U*_{e} and *L*_{mix} fields. Nonetheless, there are two notable differences between these fields. First, the characteristic lateral (along a neutral surface) width of *U*_{e} maxima is significantly larger (typically by a factor of ∼2) than that of *L*_{mix} minima. Second, the range of variation in *U*_{e} across a front is substantially smaller than the range of variation in *L*_{mix} (a factor of 2–3 in *U*_{e} versus at least an order of magnitude in *L*_{mix}). Consequently, the structure of the *L*_{mix} field prevails over that of *U*_{e} in shaping the anatomy of κ in the Southern Ocean, although vertical variations in *U*_{e} cannot be neglected.

#### 3.3. Estimates of the Isentropic Eddy Diffusivity κ

[25] The dominance of *L*_{mix} in determining κ is explicitly demonstrated by Figures 4g, 5g, 6g, 7g, and 8g, which show the distribution of *U*_{e}*L*_{mix} across each of the meridional sections analyzed here. We remind the reader that, in a mixing length theoretical framework (represented by (2)), *U*_{e}*L*_{mix} is equivalent to *c*_{e}^{−1} κ, where *c*_{e} is the supposedly constant mixing efficiency of eddies. We prefer to discuss *c*_{e}^{−1} κ rather than κ *per se* because of the uncertainty surrounding the magnitude of *c*_{e}. Estimates of *c*_{e} in the literature are based chiefly on the analysis of numerical simulations of a fully developed mesoscale eddy field [e.g., *Holloway and Kristmannsson*, 1984; *Visbeck et al.*, 1997; *Karsten et al.*, 2002], and generally fall in the range 0.01–0.4. This wide range likely reflects differences in the definition of *L*_{mix}. Many theoretical studies test the mixing length argument in terms of a length that scales with, but is not equal to, the mixing length scale proper (e.g., the eddy size). In these cases, *c*_{e} accounts for the relationship between *U*_{e} and the scale of choice. For illustrative purposes, we may use the only observational estimate of *c*_{e} ≈ 0.16 obtained by *Wunsch* [1999] from the analysis of a quasi-global inventory of moored current meter and temperature records. Using this *c*_{e} value yields diffusivities of O(200 m^{2} s^{−1}) in the core of jets, and of O(2000 m^{2} s^{−1}) in inter-frontal regions. Regardless of which *c*_{e} value is adopted, Figures 4g, 5g, 6g, 7g, and 8g indicate that the structure of *c*_{e}^{−1} κ is nearly identical to that of *L*_{mix}, at least in the upper ∼1 km of the water column. At greater depth, the decrease in *U*_{e} becomes significant, and *c*_{e}^{−1} κ is often seen to decay toward the sea floor in a way that *L*_{mix} does not. The *c*_{e}^{−1} κ distributions indicate that eddy stirring is regularly suppressed in the upper layers of the ACC frontal jets, and that it is intense in the deeper part of the jets and in interfrontal regions. We note that the area of mixing suppression associated with each jet often extends beyond the depth range of significant isentropic PV gradients (cf. Figures 4d, 5d, 6d, 7d, and 8d and Figures 4h, 5h, 6h, 7h, and 8h), as may be expected from a kinematic interpretation of the suppression.

[26] As in the analysis of *L*_{mix}, we find three unambiguous exceptions to the generalized reduction of eddy stirring at the core of the ACC frontal jets, with high values of *c*_{e}^{−1} κ found in the upper part of the SAF in WOCE SR1b (Figure 5g), the PF south of Africa (Figure 6g), and the SAF's northern branch in WOCE SR3 (Figure 8g). In those sites, high diffusivities occur in conjunction with sizeable isentropic PV gradients (see Figures 5h, 6h, and 8h).