1. Introduction
The oceans are replete with mesoscale eddies and associated turbulence. These time-dependent motions are an integral part of the general circulation, playing a significant role in the mixing and stirring of tracers. The rate at which mesoscale eddies mix can be characterized in terms of an eddy diffusivity that has a value on the order of 1000 m2 s−1. However, it is clear that this canonical value is only a reference: in reality the ocean’s mesoscale eddy diffusivity is far from constant in space or time but instead exhibits considerable variability (e.g., Davis 1991; Holloway 1986; Ledwell et al. 1998; Marshall et al. 2006; Abernathey et al. 2010; Naveira Garabato et al. 2011).
Eddy transfer is thought to be of leading order importance in dynamical balances in the ocean and the distribution of tracers therein, particularly in the Southern Ocean (see Marshall and Speer 2012). Therefore, coarse-resolution models that do not resolve mesoscale eddies must parameterize their effect. This is typically achieved by mixing tracers along neutral surfaces, as suggested by Redi (1982), and by modifying the advective process by introducing an eddy-induced flow pioneered by Gent and McWilliams (1990). Such schemes often use a spatially and temporally constant diffusivity [see, e.g., the models described in Griffies et al. (2009)]. However, as shown by, for example, Ferreira et al. (2005) and Danabasoglu and Marshall (2007), if one allows eddy diffusivities to vary in space, systemic drifts in climate models can be reduced. Moreover, the response of models to changes in external forcing (such as trends in Southern Ocean winds due to anthropogenic forcing) is found to depend on the form of the eddy closure employed. A further complication arises because the along-isopycnal diffusivity for tracers (Redi 1982) and the diffusivity used to close for the eddy-induced circulation may not be the same, a point emphasized by Smith and Marshall (2009). Here, we focus on an eddy diffusivity that can be used for tracers—including potential vorticity—that depends on the state of the large-scale flow and so can change as the climate changes.
Recently, Abernathey and Marshall (2013) estimated the surface cross-stream eddy diffusivity for passive tracers by diagnosing the part of the downgradient eddy flux associated with irreversible mixing using the Osborn and Cox (1972) relation, as described in section 2. They show overall agreement with the effective diffusivity of Nakamura (1996) that also quantifies the enhancement of mixing due to finescale features created by eddy stirring. Importantly, the diffusivity from the Osborn and Cox (1972) relation and the effective diffusivity both characterize mixing across mean tracer gradients. The spatial map of the effective diffusivity Keff from Abernathey and Marshall (2013) is shown in Fig. 1. It is clear that there is a high degree of heterogeneity with values considerably larger and considerably smaller than the canonical 1000 m2 s−1.
The effective diffusivity (m2 s−1) at the ocean’s surface from Abernathey and Marshall (2013). The gray contours are isolines of 1000 m2 s−1. (left) A map where the color is saturated at Keff = 10 000 m2 s−1. (right) The zonally averaged effective diffusivity.
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
The most direct measurement of the surface diffusivity is from satellite-tracked drifters. The single-particle diffusivity based on drifters is equivalent to that estimated from tracer fluxes if the number of drifters is sufficiently large (Klocker et al. 2012a,b). Zhurbas and Oh (2004) produced a global map of surface diffusivity based on drifter data. Their map does not reveal the suppression of mixing in the core of mean currents reported by Abernathey and Marshall (2013). Klocker et al. (2012b) show that it is extremely difficult to capture the suppression of the diffusivity from Lagrangian data because the suppression is associated with a negative lobe in the velocity autocorrelation function at long lags on the order of a month [see also Swenson and Niiler (1996)]. Hundreds of drifters are required to accurately estimate the velocity autocorrelation function at such long lags, but no more than tens of drifters are ever available in any patch of ocean. Poulain and Niiler (1989) and many authors since have resorted to estimating an upper bound on the diffusivity by ignoring the negative lobe. This is tantamount to ignoring the suppression. We believe that these problems are manifest in the extreme large values of surface diffusivity in the map of Zhurbas and Oh (2004), which often exceed 10 000 m2 s−1. Using drifters, Sallée et al. (2008) proposed to fit an analytical expression to the observed autocorrelation function, which includes a negative lobe. Sallée et al. (2011) also used drifter data to estimate a diffusivity in the Southern Ocean by averaging over days 250–360 to avoid missing the lobes and thereby capturing the effects of suppression. While these approaches are more in line with our work, the statistical uncertainty at long lags remains an issue in the estimation of suppression. While the use of tracer-derived diffusivities have their own associated uncertainty and drawbacks, given that our paper focuses on suppression of mixing over the entire globe, we prefer to compare our results against the surface diffusivity map of Abernathey and Marshall (2013, shown in Fig. 1).
Mixing length theory has been the basis for several attempts to estimate the eddy diffusivity in the ocean both from data and models including, for example, the studies of Visbeck et al. (1997), Eden and Greatbatch (2008), and Ferrari and Nikurashin (2010). Here, we use observationally based estimates of urms and Lmix to yield an eddy diffusivity based on Eq. (1) that is then compared to that diagnosed by Abernathey and Marshall (2013). We find that steering level effects play a very significant role in modulating the mixing length (see Bretherton 1966; Green 1970; Stone 1972; Killworth 1997; Smith and Marshall 2009; Ferrari and Nikurashin 2010; Klocker and Abernathey 2014) and contribute to rich spatial structure in the eddy diffusivity.
Our manuscript is set out as follows. In section 2, the horizontal distribution of K at the ocean’s surface is interpreted in terms of mixing length theory. In particular, we examine the importance of the suppression of mixing due to the steering level effects and discuss the limitations and uncertainty in the associated theory. Section 3 then applies the insights developed for the surface problem to the subsurface, obtaining estimates of the interior distribution of diffusivity. Section 4 uses the resulting three-dimensional distributions of eddy diffusivity to estimate the eddy-induced overturning circulation in the Southern Ocean and compares it to one calculated using a constant diffusivity. Finally, we conclude in section 5.
2. The surface eddy diffusivity
Here, maps of surface eddy diffusivity are computed using Eq. (1) with urms estimated from altimetry and Lmix estimated from the theory and data reviewed below. These are then compared to the map produced by Abernathey and Marshall (2013), which is shown in Fig. 1.
Two tracers were used: one was initially aligned along latitude circles and the other was aligned with the streamfunction for the mean horizontal flow. The minimum of the diffusivity based on each tracer was used, and the map is interpreted as a cross-stream/zonal eddy diffusivity. Because Eq. (2) assumes that the tracer variance is locally generated and dissipated, it is thus meaningful only when averaged over large ocean patches. When using the map of Abernathey and Marshall (2013) to ground our results based on Eq. (1), we will therefore focus on broad-scale patterns rather than small-scale variations.
a. Estimates of eddy velocity, eddy scales, and implied mixing rates
The rms eddy velocity is defined as 
(a) The square of the rms eddy speed 
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
The mixing length is conceptually analogous to the mean free path in thermodynamics; a fluid parcel will conserve its properties for a characteristic length, before mixing with the surrounding fluid. In the oceanographic literature, it is often assumed that the mixing length scales with the size of the energy-containing eddies (e.g., Holloway 1986; Haine and Marshall 1998). However, the mixing length can be much larger than the eddy scale, if eddies are so strongly nonlinear as to trap fluid and travel erratically long distances before mixing it with the surrounding fluid (e.g., Thompson and Young 2006). The mixing length can also be smaller than the eddy size, if eddies are very weakly nonlinear and hardly displace fluid parcels. Altimetric observations show that oceanic eddies are nonlinear (Chelton et al. 2011), but not so strongly as to propagate in an erratic fashion. Recent work by Klocker and Abernathey (2014) provides evidence that, in the absence of mean flows, the mixing length does indeed scale directly with the eddy diameter.
Observations indicate that oceanic eddies typically propagate along and not across potential vorticity contours (Marshall et al. 2006). The oceanic potential vorticity contours are zonal in much of the ocean (particularly the Southern Ocean) and follow the mean currents where there are strong flows. Hence, eddy propagation in the ocean does not act to increase meridional/across mean–current mixing. Rather, Ferrari and Nikurashin (2010) find that the across potential vorticity gradient mixing length is proportional to the eddy scale, but it is suppressed if eddies propagate at different speeds from the mean flow.
To gain insight in what parameters dominate variations in the eddy diffusivity, Fig. 3 compares the meridional structure of the zonally averaged Keff of Abernathey and Marshall (2013), the zonally averaged eddy diameter map of Chelton et al. (2011), and the zonally averaged urms from AVISO. The overall decrease of Keff toward the poles appears to track the decrease in eddy diameter, rather than the less pronounced, irregular variations in rms eddy velocity. Of interest is the sudden decrease of Keff at the equator despite the fact that both the eddy size and rms eddy velocity are largest there. It will be shown below that these variations are consistent the idea that in the deep tropics eddies propagate at speeds that are much larger than mean currents, thus suppressing mixing.
The zonally averaged diffusivity Keff of Abernathey and Marshall (2013, red), the zonally averaged eddy diameter Leddy (solid black) of Chelton et al. (2011), the AVISO rms velocity urms (solid blue), and the Hughes westward zonal eddy phase speed c (i.e., westward is positive; dashed blue). The color of the curves corresponds to the color of the vertical axis (black corresponds to length, red corresponds to diffusivity, and blue corresponds to velocity).
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
As a first step, we calculate the eddy diffusivity using Eq. (1) by setting the mixing length to the eddy diameter multiplied by 0.35 [the factor of 0.35 was empirically determined by Klocker and Abernathey (2014) and is held at this constant value here; see appendix A] and using AVISO to estimate urms. The resulting map of the surface diffusivity is shown in Fig. 4 and should be compared with Fig. 1. It is clear that the eddy diffusivity is overestimated almost everywhere. Moreover, this cannot be “fixed” by simply scaling down the mixing length because the degree of overestimation varies with latitude and is particularly noticeable in the tropics. In the next section, we review the theory that quantifies the reduction in mixing due to steering level effects.
The predicted eddy diffusivity (m2 s−1) from the unmodulated mixing length theory, where the mixing length is set to the eddy diameter multiplied by 0.35 in Eq. (1). Here the eddy velocity urms is derived from AVISO altimetry data; the eddy diameter Leddy is taken from Chelton et al. (2011). (left) A map where the gray line is an isoline of 1000 m2 s−1. (right) The zonal average of the left panel (blue) and the zonally averaged diffusivity of Abernathey and Marshall (2013; see also Fig. 1, red).
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
b. The role of eddy propagation in modulating mixing
Mixing by geostrophic eddies and turbulence can be strongly modulated by interactions with the mean flow. This notion is well established in the atmospheric literature and was pioneered by Bretherton (1966), Green (1970), and Stone (1972). There is a growing, parallel literature in the ocean context (e.g., Bower 1991; Lozier and Bercovici 1992; Bower and Lozier 1994; Smith and Marshall 2009; Abernathey et al. 2010; Ferrari and Nikurashin 2010). Ferrari and Nikurashin (2010) showed that in the Southern Ocean if the mixing length is set equal to the eddy scale, as in Holloway (1986), the resulting diffusivities are inconsistent with diagnostic estimates of the eddy diffusivity found. Most of the oceanographic literature on the suppression of mixing by steering level effects is focused on the ACC, where these effects are significant, and the geometry is simple and somewhat similar to the atmosphere. However, such effects are also likely to be significant in other regions where eddies and jets coexist, such as western boundary currents and equatorial regions. We now briefly review key elements of “mixing theory” in which waves moving zonally with phase speed c along a mean zonal flow 
1) Linear mixing theory
By considering the growth of a linearly unstable baroclinic wave, Bretherton (1966) and Green (1970) derived an eddy diffusivity that depends on eddy kinetic energy, the mean current, the growth rate, and propagation phase speed of the most unstable waves. As a perturbative linear model, it is only able to predict the spatial variation of mixing and not the amplitude. Nevertheless, it emphasizes the tendency of eddy propagation to suppress mixing and the importance of steering levels at which the eddy propagation speed matches the mean flow speed.
2) Stochastic mixing theory
Klocker and Abernathey (2014) provide an estimate of the parameters Γ and b1 (see also appendix A) in a region of the eastern Pacific and confirm that the unsuppressed diffusivity is proportional to the eddy size. They arrive at this conclusion by driving a model of the eastern Pacific with geostrophic eddy velocities inferred from AVISO altimetry and by varying the mean zonal velocity. The mean zonal velocity where the diffusivity is a maximum corresponds to an unsuppressed diffusivity (i.e., 
c. Refined predictions of the surface eddy diffusivity
The suppression factor is evaluated at the surface and shown in Fig. 5. The eddy phase speed was provided by Hughes who estimated it from altimetry (see Hughes et al. 1998; Hughes 2005; Tulloch et al. 2009), the zonal-mean velocity 
The suppression factor 
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
We wish to gain some insight into the skill of Eq. (6). Accordingly Fig. 6 compares estimated probability density functions of the effective diffusivity of Abernathey and Marshall (2013), Keff, with that of our predicted diffusivity with no suppression (b1 = 0) and an increasing amount of suppression (b1 = 1 and 4). We plot the probability density functions for four regions: the southern and northern high latitudes, midlatitudes, and the tropics, as indicated in the legend. Including the suppression factor eliminates the significant bias toward large diffusivities in all regions. However, the optimal value of b1 appears to vary somewhat regionally with a suggestion that a smaller value (b1 = 1) is more appropriate in the tropics, and a medium value (b1 = 4) is more appropriate in the southern high latitudes. Given that we do not currently have a robust theory to estimate b1 regionally, we choose to keep it globally constant and adopt b1 = 4 as our optimum choice. This does a reasonably good job in all regions and particularly in the southern high latitudes where we know eddy processes, and thus parameterization of those processes, is the most critical.
The probability density function of diffusivity for (a) high northern latitudes (north of 40°N), (b) midlatitudes (40°–20°S and 20°–40°N), (c) the tropics (20°S–20°N), and (d) southern high latitudes (south of 40°S). The diffusivity of Abernathey and Marshall (2013, black) and the diffusivity using Eq. (6) with b1 = 0 (corresponds to no suppression; blue), b1 = 1 (green), and b1 = 4 (red) are shown.
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
In summary we draw the following broad conclusions:
The inclusion of steering level effects reduces the systematic overestimation implied by the unsuppressed diffusivity. When compared to the map of Abernathey and Marshall (2013), including suppression eliminates the significant bias toward large values in all regions. Quantitative comparison suggests that b1 = 4 gives an optimum level of suppression. This value is broadly consistent with prior estimates in Klocker and Abernathey (2014) and Ferrari and Nikurashin (2010).
Broad patterns in the diffusivity emerge that are a consequence of suppression and not merely associated with the spatial distributions of urms. Compare Figs. 1, 4, and 7.
Eddy diffusivities are strongly suppressed in the tropics (Fig. 5) because
The eddy diffusivity (m2 s−1) resulting from the modulated mixing length theory, as described by Eq. (6). (left) A map, where the color is saturated at 10 000 m2 s−1, and the gray line is an isoline of 1000 m2 s−1. (right) A zonal average of (left) is given in blue and the zonally averaged diffusivity of Abernathey and Marshall (2013; see also Fig. 1) is given in red.
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
3. The subsurface eddy diffusivity
A full theory of mixing by geostrophic eddies requires that we extend the analysis of the surface eddy diffusivity to the whole ocean column. However, the scaling law in Eq. (6), which is the cornerstone of this paper, has not yet been validated against numerical simulations or observations. Here, we take the bold move of assuming that Eq. (6) holds at all depths in the ocean and we will compare its predictions against the observations of the vertical structure of the eddy diffusivity in the real ocean. Consistent with the interpretation of Eq. (6) given in the previous sections, we assume that the eddy size Leddy and the phase speed c are depth independent, because they cannot vary with depth if the eddy is to propagate as a coherent structure over the water column. The mean and urms velocities are instead taken to be depth dependent and will be computed from the ECCO (Wunsch et al. 2009) and ECCO phase II (ECCO2; Menemenlis et al. 2008) state estimates respectively, averaged onto our 1° grid.
a. Estimating the subsurface eddy diffusivity
As shown in section 2, suppression at the surface plays an important role in modulating the eddy diffusivity. But what might its effect be in the ocean interior? Figure 8 shows the zonally averaged suppression factor [Eq. (8)], which modulates the mixing length. At mid- and high latitudes there is significant suppression at the surface implying a mixing length that is typically only half the eddy scale. Suppression at depth is less strong, except in the tropics. Indeed, in the tropics there is strong suppression at all depths. The broad geography of suppression in the extratropics is consistent with that of steering levels in the global ocean, which tend to be shallow in equatorial latitudes and deepen moving poleward. The overall geography of steering levels is in accordance with expectations from theory explored in previous studies (e.g., Smith and Marshall 2009; Tulloch et al. 2011). Strong suppression in the tropics is expected because it is a region of strong wave activity with 
The zonally averaged suppression factor 
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
The implied zonal average of diffusivity is shown in Fig. 9. The influence of the Kuroshio and Gulf Stream is clear at 40°N or so where relatively weak suppression and a large surface rms velocity conspire to produce a relatively strong eddy diffusivity over the upper 1000 m of the ocean. In the Southern Ocean, steering levels are also at work with suppression offsetting strong surface urms but weaker suppression at depth. This leads to eddy diffusivities that have significant magnitudes even at depth.
The zonally averaged predicted eddy diffusivity (m2 s−1), calculated using the ECCO2 urms, ECCO mean currents, an observed eddy length scale, and an observed eddy phase speed. The color is saturated at 2500 m2 s−1 and white contours are shown at 0.5 × 103, 1.0 × 103, 1.5 × 103, and 2.0 × 103 m2 s−1.
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
The robust feature of Fig. 9 is that the eddy diffusivities are surface intensified over the majority of the ocean and decay rather rapidly with depth, to the extent that below 1 km they have values less than 500 m2 s−1. It is difficult to make detailed comparisons with observations as there have only been two attempts to infer the vertical profile of the eddy diffusivity, namely, the North Atlantic Tracer Release Experiment (NATRE) and the Diapycnal and Isopycnal Mixing Experiment in the Southern Ocean (DIMES). Figure 10 presents our estimates of eddy diffusivity in the NATRE and DIMES regions. We present calculations with b1 = 0 (no suppression), b1 = 1, and b1 = 4 and compare them with independent estimates reported in Ferrari and Polzin (2005) and Tulloch et al. (2013, manuscript submitted to J. Phys. Oceanogr.), respectively.
Vertical profiles of diffusivity averaged over the (a) NATRE and (b) DIMES regions. Estimates from Ferrari and Polzin (2005) for NATRE and Tulloch et al. (2013, manuscript submitted to J. Phys. Oceanogr.) for DIMES are the solid dots, while the prediction using Eq. (6) is shown as the lines—solid for b1 = 0 (no suppression), dashed for b1 = 1, and dashed–dotted for b1 = 4. Note the different horizontal and vertical axis scales between (a) and (b).
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
In the NATRE region, the diffusivity calculated using Eq. (6) matches the general shape found by Ferrari and Polzin (2005) reasonably well for all values of b1. The magnitude is underestimated with both b1 = 1 and b1 = 4, while it is mostly overestimated without any suppression (b1 = 0). In this instance, a value of b1 = 0.5 produces the best fit. This indicates that in the NATRE region the decay of diffusivity is largely related to the decay in urms with suppression diminishing the magnitude of the diffusivity throughout the water column.
In the DIMES region the interplay between the suppression factor and the rms eddy velocity is more complicated. Unlike in NATRE, the overall shape of the diffusivity is sensitive to the choice of b1. Indeed Tulloch et al. (2013, manuscript submitted to J. Phys. Oceanogr.) argue that the eddy diffusivity in DIMES may have a subsurface maximum, as plotted in Fig. 10b, which is associated with the steering level of the baroclinic waves in the ACC being at a considerable depth [see Smith and Marshall (2009)]. This offsets the decay of urms from the surface. Our application of Eq. (6) shows that increasing the influence of suppression by increasing b1 does indeed lead to vertical structures that are more similar to the diffusivity inferred by Tulloch et al. (2013, manuscript submitted to J. Phys. Oceanogr.). However, we can only achieve a subsurface maximum in diffusivity for b1 ≫ 4 that, given the results in section 2c, would appear to be unrealistically large. However, it is clear that the application of Eq. (6) to the subsurface ocean captures the suppression of K in the upper kilometer of the DIMES region, which is the most glaring failure of the application of mixing length theory sans steering level effects.
For the development of an eddy diffusivity parameterization, it is unclear how important it is to capture the subsurface maxima apparent in the observational estimates of K because the results of Abernathey et al. (2013) indicate that the eddy stress is very sensitive to the bulk value of K in the thermocline but not as much to its vertical variations at depth. Thus, it is reasonable to anticipate that the most important feature for a parameterization to capture is suppression in the upper ocean, which Fig. 10 indicates the parameterization Eq. (6) captures. Having said that, there is clearly a knowledge gap in mixing length theory for the subsurface ocean, and the discrepancies in Fig. 10 are clear motivation for further studies, observational, modeling, and theoretical, on the vertical structure of the diffusivity in the ocean.
b. Implied heat transport due to isoneutral diffusion
The diffusivity in Eq. (6) and discussed above is the cross-stream eddy diffusivity for a passive tracer [and not, e.g., buoyancy, as discussed in Smith and Marshall (2009) and Ferrari and Nikurashin (2010)]. This makes it suitable for use in eddy mixing parameterizations for passive tracers along neutral surfaces, such as that of Redi (1982).
Figure 11 compares the meridional eddy heat flux due to isoneutral mixing when a constant and a variable diffusivity is used. Large differences are evident. It should be noted, however, that the data used to obtain Fig. 11 are derived from ECCO that itself employs an isoneutral diffusion parameterization. Absent from the variable diffusivity estimate is the feedback between the parameterization and the density structure. However, our purpose here is to demonstrate that the use of a variable diffusivity might lead to rather substantial changes in parameterized fluxes. As is perhaps to be expected, the region where the greatest effect occurs is in the Southern Ocean.
The poleward heat transport (PW) from eddy mixing, using a constant diffusivity of K = 1000 m2 s−1 and the variable diffusivity in Eq. (6).
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
4. The eddy-induced advective velocity and overturning streamfunction
The wind-driven overturning in the Southern Ocean, often referred to as the Deacon cell (e.g., Döös and Webb 1994), is largely balanced by an eddy-induced circulation; the resulting net circulation is referred to as the residual circulation (e.g., Marshall and Radko 2003; Marshall and Speer 2012). In ocean models that do not resolve eddies, it is critically important to parameterize the eddy-induced circulation to capture the residual upwelling flux (Danabasoglu et al. 1994). Here, we therefore focus on the eddy-induced component of the circulation only.
The expansion of the parameterized eddy force Ξ (which we truncate at M = 6) is shown in Fig. 12 using the variable K calculated from Eq. (6). It is compared to that obtained when a constant K = 1000 m2 s−1 is assumed. The large-scale pattern of the quasigeostrophic potential vorticity gradient is consistent with previous studies (e.g., Tulloch et al. 2011). While the zonally averaged expansion in Figs. 12c and 12d does not exactly match the corresponding fields in Figs. 12a and 12b, this is to be expected. Discarding the barotropic mode enforces the integral constraint Eq. (12) by essentially removing the column mean. This manifests itself in Fig. 12 as a constant shift in value (and hence color) at each latitude when comparing Figs. 12a and 12b to Figs. 12c and 12d. Also, note how the expansion usefully acts as a low-pass filter to smooth the field and remove much of the noise.
Estimates of the quasigeostrophic potential vorticity eddy flux 
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
The implied eddy-induced streamfunction for a constant and variable K are shown in Fig. 13. Both expansions are truncated at M = 6 (as for Fig. 12). The most striking difference between the two estimates is the position of the overturning cells. Using a variable K broadens and moves the center of the cell north compared to that obtained with a constant K.
The Eulerian mean overturning streamfunction [Sverdrup (Sv); 1 Sv ≡ 106 m3 s−1] for the eddy-induced flow. Zonally averaged constant buoyancy surfaces are represented by the white contours. (a) A constant K = 1000 m2 s−1 is used, while (b) a variable K, calculated using Eq. (6), is used. Both are truncated at M = 6 and correspond to the quasi-potential vorticity eddy fluxes shown in Figs. 12c and 12d. Note that the color is saturated at ±30 Sv.
Citation: Journal of Physical Oceanography 44, 6; 10.1175/JPO-D-13-0130.1
Note how the use of a variable diffusivity has a nontrivial effect. That is, the maxima of the overturning cell in Fig. 13b now depend on local isopycnal slope and urms. The northernmost maximum in Fig. 13b is at ~40°S and corresponds to the latitude of the highly energetic region downstream of the Agulhas retroflection and the Argentine Basin (see Fig. 2a). The remainder of the cell, from ~45° to ~60°S, covers the latitudinal extent of the ACC, including the regions with the steepest isopycnals. In contrast, the constant K streamfunction is centered only on the latitudes of the steepest isopycnals. The depth of the maximum value of the overturning streamfunction is also shallower than when a constant K is used. Again, this is likely the influence of the eddy kinetic energy distribution, with larger values of urms closer to the surface.
5. Discussion and conclusions
In this study, we have used mixing length theory to test an expression for the mesoscale eddy diffusivity that attempts to represent the suppression of mixing due to steering level effects. It represents a synthesis of ideas that have and are being explored in the literature and is a first step toward applying them to eddy parameterizations in global ocean climate models.
We explored a form for the eddy diffusivity given in Eq. (6), taken from Ferrari and Nikurashin (2010, see section 2). The quantities that enter in the equation are estimated from global observations of mean and eddy currents. The resulting distributions are compared to the maps of eddy diffusivity diagnosed by Abernathey and Marshall (2013) using tracer advection methods driven by satellite altimetry. We found that the suppression of the surface eddy diffusivity by steering level effects is very significant and plays a role that is at least as important as the spatial distribution of urms. Suppression is most active where the ratio 
Attempts to quantify the quality of the agreement between “predicted” [using Eq. (6)] and “estimated” diffusivity [from Abernathey and Marshall (2013)] enable us to explore the sensitivity of our expression to uncertain parameters that control the degree of suppression, that is, b1 in Eq. (6). This suggests that a global value of around 4 is optimum, consistent with prior estimates of Ferrari and Nikurashin (2010) and Klocker and Abernathey (2014).
The resulting maps reveal the following:
Inclusion of suppression due to steering level effects significantly improves our ability to reconstruct the spatial distributions of eddy diffusivity found by Abernathey and Marshall (2013).
Broad patterns in the diffusivity emerge that are a consequence of suppression and not merely associated with spatial distributions of urms.
Eddy diffusivities are strongly suppressed in the tropics and the ACC because
We applied the theory developed for the ocean surface to the subsurface ocean. Evaluating the efficacy of applying the theory to the entire water column is made difficult by the paucity of subsurface measurements of mixing. Our results display some features that would be expected from previous works on steering level effects on mixing such as strong suppression in mixing in the upper kilometer along the path of the ACC (Smith and Marshall 2009; Abernathey et al. 2010; Ferrari and Nikurashin 2010); however, other aspects, such as anticipated subsurface maxima, indicate that the theory for subsurface mixing is incomplete and requires additional development. To help bridge this knowledge gap in our understanding of how the vertical structure of K is set, numerical studies and observational campaigns like DIMES will be invaluable in making further progress.
Finally, we developed a methodology, inspired by the work of Ferrari et al. (2010), to obtain the eddy-induced advective velocity implied by potential vorticity mixing (as opposed to buoyancy mixing), in which a vertical projection onto orthogonal baroclinic modes is carried out. The approach guarantees that eddies only redistribute momentum in the vertical. The resulting eddy-induced overturning streamfunction shows that steering level effects, as encapsulated in our spatially varying diffusivity, play an important role in setting the amplitude and pattern of eddy-induced overturning in the Southern Ocean.
We note in conclusion that the approach outlined here lends itself to a parameterization of mesoscales that can be used in primitive equation ocean models as they integrate forward in time. In a future study, we will bring the ideas explored here in to a full parametric representation of mesoscales and evaluate it as a global ocean model.
Acknowledgments
We thank Joe LaCasce, Ryan Abernathey, and Andreas Klocker for very helpful discussions. We are also grateful to the ECCO and ECCO2 projects for making their data available, as we are to Chris Hughes for the use of his eddy phase speed data. We also wish to thank two anonymous reviewers for their constructive comments that improved the manuscript. This study was supported by the Polar Programs division of NSF, by the MOBY project of NSF and by NSF Grant 1233832.
APPENDIX A
A Discussion on the Cross-Stream Diffusivity
The constants b0 and b1 need to be determined empirically. Ferrari and Nikurashin (2010) estimated them by fitting Eq. (A5) to surface estimates of 
As stated above, if the eddy field were purely monochromatic, the two constants should be related as 
APPENDIX B
Expansion of the Eddy Flux of Quasigeostrophic Potential Vorticity
REFERENCES
Abernathey, R., and J. C. Marshall, 2013: Global surface eddy diffusivities derived from satellite altimetry. J. Geophys. Res. Oceans, 118, 901–916, doi:10.1002/jgrc.20066.
Abernathey, R., J. C. Marshall, M. Mazloff, and E. Shuckburgh, 2010: Enhancement of mesoscale eddy stirring at steering levels in the Southern Ocean. J. Phys. Oceanogr., 40, 170–184, doi:10.1175/2009JPO4201.1.
Abernathey, R., D. Ferreira, and A. Klocker, 2013: Diagnostics of isopycnal mixing in a circumpolar channel. Ocean Modell., 72, 1–16, doi:10.1016/j.ocemod.2013.07.004.
Bower, A. S., 1991: A simple kinematic mechanism for mixing fluid parcels across a meandering jet. J. Phys. Oceanogr., 21, 173–180, doi:10.1175/1520-0485(1991)021<0173:ASKMFM>2.0.CO;2.
Bower, A. S., and M. S. Lozier, 1994: A closer look at particle exchange in the Gulf Stream. J. Phys. Oceanogr., 24, 1399–1418, doi:10.1175/1520-0485(1994)024<1399:ACLAPE>2.0.CO;2.
Bretherton, F. P., 1966: Critical layer instability in baroclinic flows. Quart. J. Roy. Meteor. Soc., 92, 325–334, doi:10.1002/qj.49709239302.
Cerovečki, I., R. A. Plumb, and W. Heres, 2009: Eddy transport and mixing in a wind- and buoyancy-driven jet on the sphere. J. Phys. Oceanogr., 39, 1133–1149, doi:10.1175/2008JPO3596.1.
Charney, J. G., and M. E. Stern, 1962: On the stability of internal baroclinic jets in a rotating atmosphere. J. Atmos. Sci., 19, 159–172, doi:10.1175/1520-0469(1962)019<0159:OTSOIB>2.0.CO;2.
Chelton, D. B., M. G. Schlax, and R. M. Samelson, 2011: Global observations of nonlinear mesoscale eddies. Prog. Oceanogr., 91, 167–216, doi:10.1016/j.pocean.2011.01.002.
Danabasoglu, G., and J. C. Marshall, 2007: Effects of vertical variations of thickness diffusivity in an ocean general circulation model. Ocean Modell., 18, 122–141, doi:10.1016/j.ocemod.2007.03.006.
Danabasoglu, G., J. C. McWilliams, and P. R. Gent, 1994: The role of mesoscale tracer transports in the global ocean circulation. Science, 264, 1123–1126, doi:10.1126/science.264.5162.1123.
Davis, R. E., 1991: Observing the general circulation with floats. Deep-Sea Res., 38A (Suppl. 1), S531–S571, doi:10.1016/S0198-0149(12)80023-9.
Döös, K., and D. J. Webb, 1994: The Deacon cell and the other meridional cells of the Southern Ocean. J. Phys. Oceanogr., 24, 429–442, doi:10.1175/1520-0485(1994)024<0429:TDCATO>2.0.CO;2.
Eden, C., and R. J. Greatbatch, 2008: Diapycnal mixing by meso-scale eddies. Ocean Modell., 23, 113–120, doi:10.1016/j.ocemod.2008.04.006.
Ferrari, R., and K. L. Polzin, 2005: Finescale structure of the T–S relation in the eastern North Atlantic. J. Phys. Oceanogr., 35, 1437–1454, doi:10.1175/JPO2763.1.
Ferrari, R., and M. Nikurashin, 2010: Suppression of eddy diffusivity across jets in the Southern Ocean. J. Phys. Oceanogr., 40, 1501–1519, doi:10.1175/2010JPO4278.1.
Ferrari, R., S. M. Griffies, A. J. G. Nurser, and G. K. Vallis, 2010: A boundary-value problem for the parameterized mesoscale eddy transport. Ocean Modell., 32, 143–156, doi:10.1016/j.ocemod.2010.01.004.
Ferreira, D., J. C. Marshall, and P. Heimbach, 2005: Estimating eddy stresses by fitting dynamics to observations using a residual-mean ocean circulation model and its adjoint. J. Phys. Oceanogr., 35, 1891–1910, doi:10.1175/JPO2785.1.
Gent, P. R., and J. C. McWilliams, 1990: Isopycnal mixing in ocean circulation models. J. Phys. Oceanogr., 20, 150–155, doi:10.1175/1520-0485(1990)020<0150:IMIOCM>2.0.CO;2.
Greatbatch, R. J., 1998: Exploring the relationship between eddy-induced transport velocity, vertical momentum transfer, and the isopycnal flux of potential vorticity. J. Phys. Oceanogr., 28, 422–432, doi:10.1175/1520-0485(1998)028<0422:ETRBEI>2.0.CO;2.
Green, J. S. A., 1970: Transfer properties of the large-scale eddies and the general circulation of the atmosphere. Quart. J. Roy. Meteor. Soc., 96, 157–185, doi:10.1002/qj.49709640802.
Griffies, S. M., and Coauthors, 2009: Coordinated Ocean–Ice Reference Experiments (COREs). Ocean Modell., 26, 1–46, doi:10.1016/j.ocemod.2008.08.007.
Haine, T. W. N., and J. C. Marshall, 1998: Gravitational, symmetric, and baroclinic instability of the ocean mixed layer. J. Phys. Oceanogr., 28, 634–658, doi:10.1175/1520-0485(1998)028<0634:GSABIO>2.0.CO;2.
Holloway, G., 1986: Estimation of oceanic eddy transports from satellite altimetry. Nature, 323, 243–244, doi:10.1038/323243a0.
Holloway, G., and S. S. Kristmannsson, 1984: Stirring and transport of tracer fields by geostrophic turbulence. J. Fluid Mech., 141, 27–50, doi:10.1017/S0022112084000720.
Hughes, C. W., 2005: Nonlinear vorticity balance of the Antarctic Circumpolar Current. J. Geophys. Res.,110, C11008, doi:10.1029/2004JC002753.
Hughes, C. W., M. S. Jones, and S. Carnochan, 1998: Use of transient features to identify eastward currents in the Southern Ocean. J. Geophys. Res., 103, 2929, doi:10.1029/97JC02442.
Jackson, J. D., 1998: Classical Electrodynamics. 3rd ed. Wiley, 808 pp.
Killworth, P. D., 1997: On the parameterization of eddy transfer. Part I. Theory. J. Mar. Res., 55, 1171–1197, doi:10.1357/0022240973224102.
Klocker, A., and R. P. Abernathey, 2014: Global patterns of mesoscale eddy properties and diffusivities. J. Phys. Oceanogr., 44, 1030–1046, doi:10.1175/JPO-D-13-0159.1.
Klocker, A., R. Ferrari, and J. H. LaCasce, 2012a: Estimating suppression of eddy mixing by mean flows. J. Phys. Oceanogr., 42, 1566–1576, doi:10.1175/JPO-D-11-0205.1.
Klocker, A., R. Ferrari, J. H. LaCasce, and S. Merrifield, 2012b: Reconciling float-based and tracer-based estimates of eddy diffusivities. J. Mar. Res., 70, 569–602, doi:10.1357/002224012805262743.
Lagerloef, G. S. E., G. T. Mitchum, R. B. Lukas, and P. P. Niiler, 1999: Tropical Pacific near-surface currents estimated from altimeter, wind, and drifter data. J. Geophys. Res., 104, 23 313–23 326, doi:10.1029/1999JC900197.
Large, W. G., G. Danabasoglu, S. C. Doney, and J. C. McWilliams, 1997: Sensitivity to surface forcing and boundary layer mixing in a global ocean model: Annual-mean climatology. J. Phys. Oceanogr., 27, 2418–2447, doi:10.1175/1520-0485(1997)027<2418:STSFAB>2.0.CO;2.
Larichev, V. D., and I. M. Held, 1995: Eddy amplitudes and fluxes in a homogeneous model of fully developed baroclinic instability. J. Phys. Oceanogr., 25, 2285–2297, doi:10.1175/1520-0485(1995)025<2285:EAAFIA>2.0.CO;2.
Ledwell, J. R., A. J. Watson, and C. S. Law, 1998: Mixing of a tracer in the pycnocline. J. Geophys. Res., 103, 21 499–21 529, doi:10.1029/98JC01738.
Lozier, M. S., and D. Bercovici, 1992: Particle exchange in an unstable jet. J. Phys. Oceanogr., 22, 1506–1516, doi:10.1175/1520-0485(1992)022<1506:PEIAUJ>2.0.CO;2.
Marshall, D. P., J. R. Maddison, and P. S. Berloff, 2012: A framework for parameterizing eddy potential vorticity fluxes. J. Phys. Oceanogr., 42, 539–557, doi:10.1175/JPO-D-11-048.1.
Marshall, J. C., 1981: On the parameterization of geostrophic eddies in the ocean. J. Phys. Oceanogr., 11, 257–271, doi:10.1175/1520-0485(1981)011<0257:OTPOGE>2.0.CO;2.
Marshall, J. C., and T. Radko, 2003: Residual-mean solutions for the Antarctic Circumpolar Current and its associated overturning circulation. J. Phys. Oceanogr., 33, 2341–2354, doi:10.1175/1520-0485(2003)033<2341:RSFTAC>2.0.CO;2.
Marshall, J. C., and K. Speer, 2012: Closure of the meridional overturning circulation through Southern Ocean upwelling. Nat. Geosci., 5, 171–180, doi:10.1038/ngeo1391.
Marshall, J. C., E. Shuckburgh, H. Jones, and C. Hill, 2006: Estimates and implications of surface eddy diffusivity in the Southern Ocean derived from tracer transport. J. Phys. Oceanogr., 36, 1806–1821, doi:10.1175/JPO2949.1.
Menemenlis, D., J.-M. Campin, P. Heimbach, C. Hill, T. Lee, A. Nguyen, M. Schodlok, and H. Zhang, 2008: ECCO2: High resolution global ocean and sea ice data synthesis. Mercator Ocean Quarterly Newsletter, No. 31, Mercator Ocean, Ramonville Saint-Agne, France, 13–21.
Nakamura, N., 1996: Two-dimensional mixing, edge formation, and permeability diagnosed in an area coordinate. J. Atmos. Sci., 53, 1524–1537, doi:10.1175/1520-0469(1996)053<1524:TDMEFA>2.0.CO;2.
Naveira Garabato, A. C., R. Ferrari, and K. L. Polzin, 2011: Eddy stirring in the Southern Ocean. J. Geophys. Res.,116, C09019, doi:10.1029/2010JC006818.
Osborn, T. R., and C. S. Cox, 1972: Oceanic fine structure. Geophys. Fluid Dyn., 3, 321–345, doi:10.1080/03091927208236085.
Plumb, R. A., and J. D. Mahlman, 1987: The zonally averaged transport characteristics of the GFDL general circulation/transport model. J. Atmos. Sci., 44, 298–327, doi:10.1175/1520-0469(1987)044<0298:TZATCO>2.0.CO;2.
Poulain, P.-M., and P. P. Niiler, 1989: Statistical analysis of the surface circulation in the California Current system using satellite-tracked drifters. J. Phys. Oceanogr., 19, 1588–1603, doi:10.1175/1520-0485(1989)019<1588:SAOTSC>2.0.CO;2.
Prandtl, L., 1925: Bericht über Untersuchungen zur ausgebildeten Turbulenz. Z. Angew. Math. Mech., 5, 136–139.
Redi, M. H., 1982: Oceanic isopycnal mixing by coordinate rotation. J. Phys. Oceanogr., 12, 1154–1158, doi:10.1175/1520-0485(1982)012<1154:OIMBCR>2.0.CO;2.
Rhines, P. B., and W. R. Holland, 1979: A theoretical discussion of eddy-driven mean flows. Dyn. Atmos. Oceans, 3, 289–325, doi:10.1016/0377-0265(79)90015-0.
Rhines, P. B., and W. R. Young, 1982: Homogenization of potential vorticity in planetary gyres. J. Fluid Mech., 122, 347–367, doi:10.1017/S0022112082002250.
Sallée, J. B., K. Speer, R. Morrow, and R. Lumpkin, 2008: An estimate of Lagrangian eddy statistics and diffusion in the mixed layer of the Southern Ocean. J. Mar. Res., 66, 441–463, doi:10.1357/002224008787157458.
Sallée, J. B., K. Speer, and S. R. Rintoul, 2011: Mean-flow and topographic control on surface eddy-mixing in the Southern Ocean. J. Mar. Res., 69, 753–777, doi:10.1357/002224011799849408.
Salmon, R., 1998: Lectures on Geophysical Fluid Dynamics. Oxford University Press, 378 pp.
Smith, K. S., 2007: The geography of linear baroclinic instability in Earth’s oceans. J. Mar. Res., 65,655–683, doi:10.1357/002224007783649484.
Smith, K. S., and J. C. Marshall, 2009: Evidence for enhanced eddy mixing at middepth in the Southern Ocean. J. Phys. Oceanogr., 39, 50–69, doi:10.1175/2008JPO3880.1.
Stewart, R., and R. E. Thomson, 1977: Re-examination of vorticity transfer theory. Proc. Roy. Soc., 354, 1–8, doi:10.1098/rspa.1977.0053.
Stone, P. H., 1972: A simplified radiative-dynamical model for the static stability of rotating atmospheres. J. Atmos. Sci., 29, 405–418, doi:10.1175/1520-0469(1972)029<0405:ASRDMF>2.0.CO;2.
Swenson, M. S., and P. P. Niiler, 1996: Statistical analysis of the surface circulation of the California Current. J. Geophys. Res., 101, 22 631–22 645, doi:10.1029/96JC02008.
Taylor, G. I., 1922: Diffusion by continuous movements. Proc. London Math. Soc.,s2-20, 196–212, doi:10.1112/plms/s2-20.1.196.
Thompson, A. F., and W. R. Young, 2006: Scaling baroclinic eddy fluxes: Vortices and energy balance. J. Phys. Oceanogr., 36, 720–738, doi:10.1175/JPO2874.1.
Thomson, R. E., and R. Stewart, 1977: The balance and redistribution of potential vorticity within the ocean. Dyn. Atmos. Oceans, 1, 299–321, doi:10.1016/0377-0265(77)90019-7.
Treguier, A.-M., I. M. Held, and V. D. Larichev, 1997: Parameterization of quasigeostrophic eddies in primitive equation ocean models. J. Phys. Oceanogr., 27, 567–580, doi:10.1175/1520-0485(1997)027<0567:POQEIP>2.0.CO;2.
Tulloch, R., J. C. Marshall, and K. S. Smith, 2009: Interpretation of the propagation of surface altimetric observations in terms of planetary waves and geostrophic turbulence. J. Geophys. Res.,114, C02005, doi:10.1029/2008JC005055.
Tulloch, R., J. C. Marshall, C. Hill, and K. S. Smith, 2011: Scales, growth rates, and spectral fluxes of baroclinic instability in the ocean. J. Phys. Oceanogr., 41, 1057–1076, doi:10.1175/2011JPO4404.1.
Vallis, G. K., 2006: Atmospheric and Oceanic Fluid Dynamics: Fundamentals and Large-Scale Circulation. Cambridge University Press, 745 pp.
Visbeck, M., J. C. Marshall, T. W. N. Haine, and M. Spall, 1997: Specification of eddy transfer coefficients in coarse-resolution ocean circulation models. J. Phys. Oceanogr., 27, 381–402, doi:10.1175/1520-0485(1997)027<0381:SOETCI>2.0.CO;2.
Welander, P., 1973: Lateral friction in the oceans as an effect of potential vorticity mixing. Geophys. Fluid Dyn., 5, 173–189, doi:10.1080/03091927308236114.
White, A. A., 1977: The surface flow in a statistical climate model—A test of a parameterization of large-scale momentum fluxes. Quart. J. Roy. Meteor. Soc., 103, 93–119, doi:10.1002/qj.49710343506.
Wunsch, C., 2011: The decadal mean ocean circulation and Sverdrup balance. J. Mar. Res., 69, 417–434, doi:10.1357/002224011798765303.
Wunsch, C., P. Heimbach, R. M. Ponte, I. Fukumori, and ECCO-GODAE Consortium Members, 2009: The global general circulation of the ocean estimated by the ECCO-Consortium. Oceanography, 22, 88–103, doi:10.5670/oceanog.2009.41.
Young, W. R., 2012: An exact thickness-weighted average formulation of the Boussinesq equations. J. Phys. Oceanogr., 42, 692–707, doi:10.1175/JPO-D-11-0102.1.
Zhurbas, V., and I. S. Oh, 2004: Drifter-derived maps of lateral diffusivity in the Pacific and Atlantic Oceans in relation to surface circulation patterns. J. Geophys. Res.,109, C05015, doi:10.1029/2003JC002241.
The form that Klocker and Abernathey (2014) uses is 
