a. Method: Defining fluxes

ALACE floats provide quasi-Lagrangian measurements of temperature and velocity, averaged over 9–25-day intervals. Details of Southern Ocean ALACE temperature and velocity measurements are reviewed in Part I. The floats are ballasted to a predetermined depth, so their motions are predominantly along constant depth surfaces rather than isopycnals. Southern Ocean ALACE measurements have an average pressure near 900 dbar, and this study will focus on the fields at 900 m.

Analyses of Lagrangian float data often employ dispersion statistics determined from the long-term evolution of extended float trajectories. However, ALACE floats are not suited for this kind of analysis, because they spend 24 hours on the ocean surface at the end of each measurement cycle, where they behave like undrogued drifters that are blown by the wind (Gille and Romero 2003). In addition, ALACEs do not resolve temperature or velocity fluctuations on timescales shorter than a measurement cycle but return only the time-averaged temperature and total position displacement for the duration of the cycle. Interpreting eddy statistics from the data therefore requires an alternate approach. In this paper, each ALACE displacement is treated much like a single Eulerian measurement from a current meter located at the center point of the ALACE trajectory.

At any given point in time, the ACC can be thought of as a series of narrow jets of about 100-km width, characterized by rapid temperature change, as depicted by the dashed lines in Fig. 1. Over time, each jet meanders meridionally with a range of about 100 km so that the instantaneous jet position shifts. In Fig. 1, the dashed lines indicate the positions of the jet at three separate points in time, and the time mean is represented by the solid line. A current meter placed at fixed depth in the middle of the mean jet (the gray line at 0 km in Fig. 1a) will measure the downstream flow of the jet, the cross-stream velocities due to the meandering of the jet, and temperature anomalies resulting from the difference between the time averaged and instantaneous temperatures. For comparison, floats deployed in the jet will be advected downstream and will also move meridionally with the meandering fronts. Figure 1b indicates the temperature that might be recorded by three independent floats deployed at different points in time and at different locations relative to the mean axis of the current. During each measurement cycle, we expect that a typical float will migrate cross stream with the front, maintaining a roughly constant temperature. Thus a single float observation will not provide information about the time-averaged temperature or velocity at a given location, but the combined data from many independent floats will allow us to estimate the mean temperature and velocity over a broad geographic region spanning the full length and width of the current. Note that the approach used here differs substantially from the framework described by Bower (1991) or by Lozier et al. (1996) who examined cross-stream fluxes relative to a coordinate frame that moved with the meandering current. This analysis is strictly concerned with motions relative to the time-mean flow.

This study estimates heat and momentum fluxes by computing differences between individual ALACE measurements and mean fields. Figure 2 illustrates the method schematically. Temperature and velocity measurements from each “instantaneous” ALACE displacement are compared with mapped mean temperature and velocities that have been interpolated onto the locations of the ALACE measurements. The mapped time-mean streamlines are used to project the total ALACE velocity u (solid arrow in Fig. 2) into orthogonal components, u and υ (dashed arrows), in the along-stream and cross-stream components, respectively. At the same location, the time-mean velocity u is also calculated from dynamic height contours to represent the mean circumpolar flow. The along-stream anomaly u′ is u − u, and the cross-stream velocity anomaly υ′ is equivalent to υ since υ is defined to be zero. The potential temperature anomaly θ′ = θ − θ is determined from the difference between ALACE and time-mean estimates. Eddy heat transports (ρC_p〈u′θ′〉) and momentum transports (e.g., 〈u′υ′〉) are then computed, where ρ is density and is taken to be 1035 kg m⁻³, and C_p is the specific heat of water at constant pressure and is taken to be 4000 J kg⁻¹ °C⁻¹.

Here overbars specifically denote time means at a single location, while angle brackets denote averages that can be spatial as well as temporal. Primes are anomalies relative to the local time mean. By definition, the means of u′ and θ′ are zero. Using standard Reynolds averaging, we expect that formally u′θ′ = (u − u)θ = u(θ − θ). Therefore, if perfect data were available, it would be possible to estimate the eddy flux quantities by knowing the time-averaged value of either θ or u. In reality the data coverage is variable, and we work with spatiotemporal averages that do not necessarily have zero mean. Thus this study relies on separate estimates of u and θ, as discussed in the companion paper, Part I.

The presence of a meandering jet, as depicted in Fig. 1, will not automatically result in poleward eddy heat flux. For example, if the instantaneous temperature structure in Fig. 1 is always the same, then northward and southward motions will cancel each other out, and υ′θ′ will be zero. (This is true regardless of whether floats or current meters are used.) In order to have a net heat flux, the average temperature of northward-moving water must differ from that of southward-moving water. These temperature differences can be caused by mixing associated with particle motions within a jet (e.g., Owens 1984; Bower 1991; Lozier et al. 1996) and by the formations of rings and eddies associated with baroclinic instability of the flow. Since float temperatures and velocities are 9–25-day averages, high-frequency components of the eddy flux cannot be determined from floats, and the details of these physical processes cannot readily be explored using ALACE data. Thus the analysis in this paper is focused on bulk eddy flux estimates rather than specific processes.

b. Estimated fluxes

Eddy heat and momentum fluxes estimated from ALACE data are presented in Table 2. Here means are derived either from the objectively mapped fields computed directly from the ALACE data, as presented in Part I, or from gridded hydrographic data produced by Gouretski and Jancke (1998, henceforth GJ). Case A uses a mean based on ALACE observations, mapped using an isotropic Gaussian covariance function with a decorrelation scale of 495 km, a signal-to-noise ratio of 2, and initial-guess fields for the dynamic topography based on atlas data referenced to 3500-m depth. As indicated in Part I, gridded fields are sensitive to the parameters used to map, so a variety of other mean fields are explored. Cases B through F are like case A except that they use differing decorrelation scales. Cases G and H resemble case A except that the signal-to-noise ratios are changed to 1 and 4, respectively. Case I assumes that the temperature and dynamic height fields decorrelate more slowly along streamlines than across them. Cases J and K are like case A but use initial-guess dynamic topography fields referenced to 3000 and 4000 m, respectively. Case M is an optimal combination of case A and atlas data. The final three cases are based solely on GJ's hydrography at 900-m depth, with dynamic topography referenced to 3000, 3500, or 4000 m. The parameters defining these cases are presented in detail in Table 1 of Part I. This wide range of definitions of means is used because eddy flux estimates are sensitive to the choice of mean. The discussion of resulting fluxes will focus on signals that are not strongly dependent on the choice of mean.

For ease in comparing the subsurface heat fluxes with fluxes computed in other studies at a variety of depths, heat fluxes are computed based on potential temperature referenced to the surface. Since salinity measurements are not uniformly available, a constant salinity of 34 psu is assumed in the potential temperature calculations. In practice the difference between fluxes computed from temperature and fluxes computed from potential temperature is negligible.

ALACE measurements are used for this analysis if they come from regions with time-averaged dynamic heights in a range corresponding to the core of the ACC, as described in the caption to Table 2. This dynamic height range was chosen in order to focus this analysis on the portion of the Southern Ocean that is best sampled by float data and to minimize the impact of the subtropical gyres and Agulhas Retroflection, which differ dynamically from the ACC. Within statistical error bars, cross-stream eddy fluxes (ρC_p〈υ′θ′〉 as well as 〈υ′υ′〉) for the full dataset are the same as the ACC core eddy fluxes reported here. Regardless of the choice of mean fields, this analysis implies that the meridional eddy heat transport across the ACC is poleward and between 3.2 and 6.7 kW m⁻².

Error bars represent the standard error about the mean and are equivalent to one standard deviation divided by the square root of the number of observations. Statistical error bars range between 1.3 and 2.8 kW m⁻² for cases using float data as a mean. Estimates based on atlas means have larger errors. This occurs because the atlas means differ substantially from the float means in some locations such as the Agulhas Retroflection region, and as a result the eddy fluxes have larger variances.

Part I reports that float temperatures are systematically warmer than atlas temperatures, implying a long-term warming trend in the Southern Ocean. This temperature bias suggests that fluxes based on the atlas mean will appear larger than fluxes based on the float mean. However, because results based on atlas data have large statistical errors, the differences between atlas and float means are not discernibly different within error bars.

In contrast with the cross-stream fluxes, eddy fluxes that depend on the along-stream anomaly u′ depend strongly on the choice of mean. Along-stream eddy heat fluxes range from −0.8 to 21.2 kW m⁻²; fluxes based on float-derived means do not differ from zero within error bars, while fluxes based on atlas means are somewhat larger. This difference occurs because u is smaller in the atlas fields than in the ALACE-derived fields as discussed in Part I. Long-term temperature trends do not appear to have a strong influence on the difference. Along-stream eddy fluxes outside the core ACC (not shown here) tend to be smaller or more negative than the fluxes in Table 2 because the statistics of the alongstream mean are different inside and outside the jet core.

Mean heat transports are not reported in Table 2 since they depend on the arbitrary scale used to measure temperature; in general they are an order of magnitude or more larger than the eddy heat fluxes. The 〈u′u′〉 and 〈υ′υ′〉 momentum fluxes are comparable to each other in magnitude with values between 40 and 70 cm² s⁻². The covariance terms 〈u′υ′〉 are positive for ALACE means and negative for atlas means, but they are an order of magnitude smaller than the other terms and are not consistently statistically different from zero.

ALACE data coverage is more extensive in the highly energetic Atlantic sector of the Southern Ocean than in the eastern Indian or Pacific Oceans. There is, however, little evidence for a distortion in the eddy heat flux estimates as a result of this geographic bias. Within statistical error bars, eddy heat fluxes based on float means for the region between 60°E and 60°W spanning the Indian and Pacific Oceans are generally equivalent to the fluxes reported here. Eddy heat fluxes based on atlas means in the Indo–Pacific subregion are equatorward on average but have large error bars, so this difference may not be significant. Eddy momentum fluxes in the Indo–Pacific subregion are slightly less energetic than eddy momentum fluxes in Table 2; this occurs in part because the flow is less well sampled, and so the instantaneous velocities may not be as well differentiated from the mean flow.

a. Corrected eddy heat flux

On the basis of the results in section 3, eddy heat and momentum fluxes derived from ALACE in section 2 were readjusted to account for the expected underestimation due to the float sampling. Since the median ratios of filtered to unfiltered current meter fluxes shown in Fig. 4 decrease linearly with increasing averaging times, linear correction functions were defined for eddy heat and momentum fluxes as a function of ALACE sampling length. Then the eddy fluxes computed from each ALACE displacement were individually rescaled by an appropriate factor, depending on the duration of the displacement. Table 4 reports averages of the resulting revised eddy heat and momentum fluxes for the ACC core. Poleward heat fluxes range between 4.7 ± 2.3 and 7.5 ± 2.9 kW m⁻² when ALACE provides the mean and are between 8 and 10(±6) kW m⁻² but with larger error bars when the mean is derived from atlas data. These rescaled heat fluxes will be the focus of the remaining discussion.

Table 4 shows that in the ACC core momentum fluxes, like heat fluxes, have along-stream and cross-stream components that are comparable in magnitude. The along-stream component of eddy kinetic energy 〈u′u′〉 is about 20% larger than the cross-stream component 〈υ′υ′〉, while the term 〈u′υ′〉 is significantly smaller than either 〈u′u′〉 or 〈υ′υ′〉. Variance ellipses are often used to assess the orientation and anisotropy of eddy variability (e.g., Morrow et al. 1994). In this analysis, the major and minor axes of variance ellipses are the same within error bars, and the orientation of the variance ellipses relative to the direction of mean flow is not statistically significant. Even if the data are sorted geographically or by dynamic height contour, no statistically significant trend in 〈u′υ′〉 emerges. The results therefore suggest that, within the limitations of the ALACE measurements, mean eddy variability is isotropic near the core of the ACC, and gradients in 〈u′υ′〉 do not measurably influence the mean flow. This result is not surprising in light of analysis of altimeter data by Hughes and Ash (2001) showing that eddy acceleration and deceleration of the ACC varies on meridional length scales of O(100 km), which are not well resolved by the available ALACE data.

b. Comparing with other methods: Fluxes in the ACC temperature core

How well do the eddy fluxes in Table 4 agree with results obtained using other methods? Typical estimates of poleward middepth heat flux across the ACC, such as those reported in Table 1, range from −3.7 to −40.6 kW m⁻². Reported values vary spatially with lower values occurring on the southern side of the ACC (Nowlin et al. 1985) and with lower estimated fluxes in cases where data are bandpass filtered in time. Altimeter data suggest a similar range of values: Stammer (1998) reported a vertically integrated heat transport in the core of the ACC of O[−5 to −15 (×10⁶ W m⁻¹)]. If this is assumed to be distributed evenly through the upper 1000 m of the water column, then the expected meridional heat flux in the upper ocean is −5 to −15 kW m⁻². These ranges are the same order of magnitude as the cross-stream eddy heat fluxes reported in the second column of Table 4, but the largest estimates from current meters exceed the averages obtained from float data. This is not surprising since the current meter estimates are sensitive to the depths, frequency bands, and coordinate frames used to analyze the data.

Cross-stream eddy heat fluxes from the second column of Table 4 agree within error bars with the mean cross-stream eddy heat flux from current meter data of −7.5 ± 1.3 kW m⁻² reported in Table 3. However, this comparison may not be meaningful since the current meter data come from a wide range of depths and from regions on the periphery of the ACC, well outside the dynamic height band used for the float analysis. In order to remove the impact of current meters located far from the ACC, mean fluxes were also computed for a subset of current meters with mean temperatures between 2° and 4°C and for float data from the same temperature band. In this case, current-meter-derived eddy heat fluxes exceed float-derived fluxes by a factor of 3. This difference in eddy flux may occur because the current meters deployed in this temperature band have preferentially sampled the upper ocean or the northern part of the ACC, where Drake Passage observations would suggest that we might expect larger poleward eddy heat fluxes (Nowlin et al. 1985).

Along-stream eddy heat transports depend strongly on the method used to calculate them. In the float analysis using the ALACE mean, they are zero within error bars, though typical values are about the same order of magnitude as cross-stream eddy fluxes. In the float analysis using an atlas mean, they are positive, implying downstream heat transport. In contrast, current meter data indicate significant upstream transport of about the same size as the cross-stream eddy heat flux.

These results show that in both current meter and float data, cross-stream eddy heat fluxes are uniformly poleward and statistically significant, while along-stream eddy heat fluxes are comparable in size or smaller than the cross-stream fluxes. Together, these two pieces of evidence imply that at 900-m depth cross-stream eddy heat flux is not inhibited, and therefore the ACC is not a strong barrier to meridional eddy heat transfer, at least in the isobaric and time-invariant stream coordinate frame used here.

Eddy momentum fluxes from current meter data in Table 3 are of the same order of magnitude as fluxes from ALACE data in Table 4 but are consistently smaller. This is also true when the current data and float data are screened to represent the 2°–4°C temperature band. This difference most likely occurs because eddy momentum fluxes have strongly nonnormal distributions and because current meters cannot survive in the very high eddy energy regions where ALACE floats tend to provide the best sampling. Given the large geographic variations in fluxes and the sparse sampling provided by current meters, the order-of-magnitude agreement is reassuring and more detailed comparison seems unjustifiable.

c. Estimates of vertically integrated heat transport

Translating eddy heat flux at 900-m depth into full water column heat transport requires some knowledge of the vertical structure of the eddy heat fluxes. In the well-studied Gulf Stream region, floats and hydrography suggest that heat and tracers are readily exchanged below the thermocline but that the Gulf Stream acts as a barrier to strong mixing in shallower waters (Owens 1984; Shaw and Rossby 1984; Bower et al. 1985). Park and Gambéroni (1997) found a similar distinction between thermocline waters and deep waters (below about 1000 m) in the region near the Agulhas Return Current where the subtropical and subantarctic fronts nearly merge. However, in most of the Southern Ocean, the ACC differs from the Gulf Stream because it has coherent flow from top to bottom and does not have a well-defined thermocline. Therefore the two distinct mixing regimes found in the Gulf Stream or the Indian Ocean subtropical front cannot be expected to exist in the Circumpolar Current as a whole.

In this study, current meter observations were used to infer a vertical structure for ACC heat flux. Analyses of Drake Passage current meters have sometimes been interpreted to suggest little vertical variation in eddy fluxes (Johnson and Bryden 1989). Sixty-three of the current meters available for this study were from moorings with three or more current meters. As in the Phillips and Rintoul (2000) detailed analysis of current meters south of Australia, these records show that eddy fluxes at a given location decrease with depth, with roughly exponential structures and e-folding depths of O(1000 m) (not shown here). Since fluxes drop off sharply with depth, ALACE fluxes at 900-m depth are unlikely to be equivalent to fluxes throughout the entire water column.

Assuming that heat flux decreases with depth with an e-folding scale (H_e) of 1000 m and that the length of the ACC (L) is 23 000 km (corresponding to the circumference of the earth at 55°S), then we can approximate the total vertically integrated heat flux as

View Expanded

where H is the ocean depth and z_ref is the depth at which heat fluxes q are observed, 900 m in this case. Based on the third column of Table 4, q(900 m) is between 4.7 and 7.5 kW m⁻² for float means implying a vertically integrated flux between 0.27 ± 0.13 × 10¹⁵ and 0.42 ± 0.16 × 10¹⁵ W. For atlas means, the estimated fluxes between 7.7 and 10.1 kW m⁻² imply vertically integrated fluxes between 0.4 and 0.6 ± 0.3 × 10¹⁵ W. Variations in H_e of ±400 m change the estimated heat fluxes by less than 10%. These numbers are consistent with previously established ranges for meridional heat flux across the ACC as reported in Table 1 (e.g., deSzoeke and Levine 1981; Gordon and Owens 1987; Keffer and Holloway 1988).

Poleward eddy heat fluxes have been linked to the surface wind stress that drives the ACC through the mechanism of interfacial form stress. Johnson and Bryden (1989) suggested that heat flux should be proportional to wind stress multiplied by the vertical derivative of potential temperature; that is,

View Expanded

where τ^x is mean zonal wind stress and is estimated to be about 0.1 N m⁻² with a variance of 0.15 N m⁻² (Gille et al. 2001). In GJ atlas data, the vertical derivative of potential temperature, θ_z, at 900-m depth has a mean value for the Southern Ocean of 4 × 10⁻³ °C m⁻¹ but varies geographically: average values along the core of the ACC are 3 × 10⁻⁴ °C m⁻¹, an order of magnitude smaller than the regional mean. With this low value of stratification, at 55°S, a minimum poleward heat flux of 1 kW m⁻² would be required to balance the mean wind stress. The observed heat flux amply exceeds this minimum value.

d. Spatial variation

How do the fluxes estimated from ALACE vary spatially across the ACC? In Fig. 5 mean heat transport (ρC_p〈υθ〉) and eddy heat fluxes (ρC_p〈u′θ′〉 and ρC_p〈υ′θ′〉) have been averaged along dynamic height contours around the entire ACC. Black lines indicate fluxes computed using the case A ALACE mean, and gray lines used the mean derived from GJ atlas data referenced to 3500 m. (The Southern Ocean–wide means of these fluxes are reported as case A in Table 4.) Other cases show qualitatively similar results. Mean along-stream heat transports are uniformly positive and are largest in the core of the subantarctic front, on the north side of the ACC (but, as mentioned earlier, are dependent on the somewhat arbitrary definition of 0°C). Along-stream eddy heat transports are both positive and negative with no consistent sign and no discernible trend as a function of dynamic height. Cross-stream eddy heat fluxes are about the same order of magnitude as along-stream fluxes but are consistently poleward with values of about 5 kW m⁻² throughout the core of the ACC. Error bars are too large to infer a trend in poleward heat flux as a function of dynamic height contour.

Figure 6 shows bin-averaged cross-stream heat fluxes. In Figs. 6a and 6b, heat fluxes are averaged in 5° latitude by 30° longitude boxes. Figures 6c and 6d show heat fluxes averaged in stream-following boxes delineated by 30° longitude and 0.2 dynamic meter increments. To avoid confusion in places where the mean flow reverses direction, cross-stream fluxes are analyzed only in regions of mean eastward flow.

In Fig. 6, spatial variations in heat fluxes indicate strong poleward eddy heat flux in the western Indian Ocean north of 35°S. The poleward Indian Ocean flux estimates are consistent with basin-scale patterns reported from hydrographic data analyses (Georgi and Toole 1982; Macdonald and Wunsch 1996; Sloyan and Rintoul 2001), suggesting that a major pathway for heat to enter the ACC may exist in the high-eddy region downstream of the Agulhas Retroflection. Moderate poleward heat fluxes occur along the core of the ACC, centered along the 0-m height contour (as shown in Figs. 2c and 2d in Part I). This suggests that heat is transferred poleward along the entire length of the ACC. The color scheme in Fig. 6 suppresses some of the variability in poleward eddy heat fluxes. For example, in Fig. 6d, statistically significant eddy heat flux estimates in the 0 to −0.25 (×10⁵ W m⁻²) band vary between −0.039 × 10⁵ and −0.24 (×10⁵ W m⁻²). Elevated fluxes in the core of the ACC are further evidence that the sloping isopycnals of the ACC are not a strong barrier to meridional eddy heat flux. In some cases equatorward heat fluxes occur near Kerguelen Island (70°E), but these fluxes appear to depend on the choice of mean dynamic height and mean temperature. There is no evidence of large equatorward heat flux anywhere in the calculations that use means computed from float data (cases A–I) and average along streamlines (as in Fig. 6d). In contrast, the poleward heat fluxes associated with the Agulhas Retroflection and along the core of the ACC are robust features in all flux maps.

e. Parameterizing eddy heat flux

Eddy heat fluxes are often parameterized in terms of large-scale temperature gradients so that 〈υ′θ′〉 ∝ 〈T_y〉 (Keffer and Holloway 1988; Stammer 1998). Figure 7a shows uncorrected and corrected ρC_p〈υ′θ′〉 as a function of 〈T_y〉 for 0.2 dynamic meter by 30° longitude bin averages computed from case D. The binning used here is equivalent to what is shown in Fig. 6d. Case D (with decorrelation scales of 440 km in the zonal direction and 220 km in the meridional direction) was selected for this plot, because it has statistically significant correlation coefficients linking eddy heat fluxes with their possible parameterizations. In this case, the correlation coefficient between uncorrected 〈υ′θ′〉 and 〈T_y〉 is −0.35, which is statistically significant at the 95% level. Cases F, J, and K also have statistically significant correlation coefficients between −0.33 and −0.38, while the other cases are not significant. Thus simple parameterizations that assume eddies transfer heat downgradient in proportion to the strength of the large-scale temperature gradient appear likely to account for as much as 40% of the spatial variations in eddy heat flux, though in some situations there may be no statistical relationship between heat flux and large-scale temperature gradient. Values of 〈υ′θ′〉 that have been corrected to account for the duration of ALACE trajectories are not significantly correlated with large-scale temperature gradients, possibly because the statistical correction introduces noise. Along-stream eddy heat fluxes are also not correlated with along-stream temperature gradients at a statistically significant level and will not be discussed.

Least squares fitting a straight line to the uncorrected case-D fluxes (solid circles in Fig. 7) produces 〈υ′θ′〉 = (0.0 ± 0.2) × 10⁻³ m °C s⁻¹ − 422 ± 130 m² s⁻¹ 〈T_y〉 (where heat fluxes have been expressed as temperature fluxes to facilitate comparison with results published elsewhere). The other statistically significant uncorrected fluxes (cases F, J, and K) are the same within error bars, and the cases that are not statistically significant also agree with these values. Because the correlation coefficients for “corrected” fluxes are not statistically significant, those fits are not reported here, but correction factors are expected to increase the regression coefficient from about 400 to 600 m² s⁻¹. These fluxes are comparable to results derived by Stammer (1998) from TOPEX/Poseidon altimeter data; he showed eddy mixing coefficients within the ACC that ranged from about 250 to 1000 m² s⁻¹.

In more detailed eddy flux parameterizations, eddy heat transfer also depends on the strength of the local eddy kinetic energy. For example, Stammer (1998) assumed 〈υ′θ′〉 ∝ 〈E_K〉 〈T_y〉, where E_K = u′u′ + υ′υ′. Figure 7b shows a scatterplot of uncorrected and corrected 〈υ′θ′〉 as a function of 〈E_K〉〈T_y〉. In this study, for case D the correlation coefficient between 〈υ′θ′〉 and 〈E_K〉 〈T_y〉 is −0.54 for uncorrected fluxes and −0.43 for corrected fluxes. Given the scatter in the data, this diffusion parameterization appears unlikely to be easily tuned in order to explain 100% of the observed heat flux. These correlation coefficients are statistically significant at the 95% level for case D, and overall for 8 of 11 uncorrected cases (cases B–F, H, J, and K) and 4 of 11 corrected cases (cases C, D, H, and J). The fact that correlation coefficients are greater when E_K is included in the regression suggests that eddy kinetic energy has a significant impact on the rate of eddy heat transfer in the ocean.

Case D uncorrected fluxes can be described by a linear least squares fit: 〈υ′θ′〉 = 0.8(±1.1) (×10⁻⁴ m °C s⁻¹) − 6.3(±1.4) × 10⁴ s 〈E_K〉 〈T_y〉, and corrected fluxes fit: 〈υ′θ′〉 = −0.8(±1.7) × 10⁻⁴ m °C s⁻¹ − 2.8(±0.6) × 10⁴ s 〈E_K〉 〈T_y〉. Other cases produce the same results within error bars. These results are consistent with estimates obtained by Stammer (1998), who suggested that the coefficient in front of E_KT_y should be 2ατ ≈ 4 × 10⁴ s, where α, a measure of eddy mixing efficiency, is approximately 0.05, and the timescale τ ≈ 5 days in the Southern Ocean.