1. Introduction
Interactions at the ocean surface form an integral part of the variability of the earth system and in particular its climate. These interactions include thermodynamically mediated changes to the ocean heat budget; changes to the ocean salinity budget via evaporation and precipitation; exchanges of gases such as oxygen, carbon dioxide, and nitrous oxide; and processes influencing biological productivity. Ocean mesoscale eddies may modulate such interactions, particularly in eddy-rich regions such as the Gulf Stream, Kuroshio. and Antarctic Circumpolar Current (ACC; Tandon and Garrett 1996; Greatbatch et al. 2007). In this paper, we will focus on the role that eddies play in determining the distribution of sea surface temperature (SST). Our results also have implications for the distribution of other surface tracer fields such as salinity and chlorophyll. Eddies contribute to the budgets of such fields through their role in lateral transport. This transport, however, is intimately connected to irreversible processes such as lateral small-scale mixing and damping processes associated with air–sea fluxes (Zhai and Greatbatch 2006a,b; Greatbatch et al. 2007). It is a quantification of the latter process that is the focus of attention here.
Figure 1 shows a wintertime instantaneous (Fig. 1a) and monthly-mean (Fig. 1b) net air–sea heat flux obtained from a global ⅛° eddy-resolving model driven by observed atmospheric fields through bulk formulas that allow the evolving SST to modulate air–sea fluxes (see appendix for details). Only the eddy-rich Southern Ocean is depicted. The instantaneous field reveals two scales: one associated with the prevailing atmospheric synoptic-scale systems (∼1000 km) and the other controlled by the ocean’s mesoscale variability (∼20 km). The monthly-mean air–sea flux averages out the rapid synoptic-scale variability imposed by the atmosphere to reveal the smaller spatial-scale and longer time-scale modulation of air–sea fluxes by the ocean mesoscale. This modulation is very clear in the local zoom of monthly-mean patterns shown in Fig. 1c. The imprint of the ocean eddies is large, resulting in anomalous fluxes that often exceed ±100 W m−2.
The modulation of air–sea fluxes on the eddy scale acts to damp eddies, as can be seen in Fig. 1d, which plots the damping rate α = −
Although the model results presented in Fig. 1 are used here only to illustrate the physics at play, it is worth briefly examining their relevance. The model might exaggerate the heat flux damping because it does not employ an atmospheric boundary layer scheme. Indeed, in the real world, air temperature would adjust to the SST anomalies, hence reducing the air–sea temperature contrast and anomalous fluxes (for a simple model to rationalize the “reduced heat flux” damping due to the air–sea adjustment, see Barsugli and Battisti 1998). Using the Comprehensive Ocean–Atmosphere Data Set (COADS), Frankignoul et al. (1998) estimate that the surface air temperature adjustment reduces the heat damping by about a factor of 2, from 50 to 20 W m−2 K−1. The value α = 20 W m2 K−1 is probably a lower bound for the real damping rate because the calculation was performed “locally” on each 5° × 5° (latitude × longitude) grid box of the dataset, not at the mesoscale, and the heat flux damping is likely to increase with decreasing spatial scales (Bretherton 1982; Zhai and Greatbatch 2006a). The Frankignoul et al. (1998) maximum estimate of α = 50 W m−2 K−1 (no air temperature adjustment) provides an upper bound on this. The model damping rates we find here are consistent with those broad ranges. More importantly, although the exact rate is somewhat uncertain, it is the very fact that mesoscale SST anomalies are damped by air–sea heat fluxes, which is key here. This is a robust feature that does not depend on the details of the heat flux scheme in the model and is supported by observations (Bourras et al. 2004).
These results corroborate a standard assumption made in models (see Haney 1971) and also adopted here, in which an advected tracer representing SST is damped by a simple restoring boundary condition with a relaxation time scale λ−1 (where λ = α/ρOCpH: ρO is a reference ocean density, Cp is the specific heat of seawater, and H is the mixed layer depth). Moreover, patterns that form in this type of modeled SST-like tracer from the combined influence of stirring by mesoscale eddies and damping–dissipative effects are consistent with those found in SST from satellite observations (Abraham and Bowen 2002). Comparison of the spatial patterns in model and observational data from the southwest Tasman Sea has indicated a relaxation time scale of 20 days (Abraham and Bowen 2002). Simple bulk estimates suggest a time scale on the order of a few months (Bracco et al. 2009), and studies based on direct analysis of limited ship- and satellite-derived heat flux data for the Southern Ocean indicate time scales of 1–10 months depending on season and location (e.g., Park et al. 2005). We return in section 2a to a discussion of damping time scales implied by Fig. 1d in the Southern Ocean.
Figure 2 describes the process by which SSTs may be influenced by mesoscale ocean eddies at the sea surface. As the eddies sweep water meridionally (Fig. 2a), anomalously warm (cold) water is moved poleward (equatorward). Mixing and anomalous air–sea fluxes (Fig. 2b) result in the warm water cooling and the cold water warming. Thus a lateral eddy flux of heat through the mixed layer is achieved that is intimately tied to mixing and anomalous air–sea fluxes induced by the eddies themselves (Fig. 2c). Considering the streamwise average, eddies act to reduce meridional gradients of temperature through both stirring and the modulation of air–sea fluxes, and the gradients are then restored by air–sea interaction acting on the large scales (Fig. 2d). It is this “passive” coupling mechanism that will be investigated in this paper through a kinematic study of an idealized SST-like tracer. Other potential mechanisms for an “active” coupled feedback response involve the dynamical influence of the eddies on the wind stress curl that results from the SST gradients associated with the eddies, as discussed by, for example, Bourras et al. (2004), Spall (2007), Jin et al. (2009), and Hogg et al. (2009).
The modulation of air–sea heat and freshwater fluxes by ocean eddies is likely to be important for the large-scale circulation. For example, theoretical work (Marshall et al. 2002; Radko and Marshall 2004) has indicated a possible role of near-surface diabatic eddy fluxes in the maintenance of the main thermocline, and recent work by Iudicone et al. (2008) has highlighted the role of surface forcings and mixing in water mass formation and transformation in the Southern Ocean. More generally, air–sea interaction with the mesoscale eddy field will likely play an important role in biogeochemical cycles and ecosystem evolution through the influence on the upwelling of dissolved gases and nutrients into the surface ocean.
In this paper, we introduce new diagnostics to characterize the lateral eddy heat flux associated with (i) stirring by eddies and (ii) eddy modulation of air–sea interaction, and we discuss the large-scale implications. The study considers the evolution of an idealized SST-like tracer advected by surface geostrophic velocities derived from altimetric data. The domain considered is the Southern Ocean, and particular attention is given to the influence of eddy processes in the distinct dynamical regimes of the core of the ACC, its flanks, and the region farther equatorward.
The paper is organized as follows: In section 2, we outline the difficulties associated with traditional approaches to quantifying eddy fluxes. In particular, eddy fluxes typically include a (hard to remove) large rotational component that plays no role in the tracer budget (see Marshall and Shutts 1981). Then we set out a new theoretical framework for application to SSTs, based on an extension to the “effective diffusivity” formalism of Nakamura (1996). The effective diffusivity approach focuses on determining the irreversible mixing effect of eddies on tracers, which results from the divergent eddy fluxes. New diagnostics are presented to quantify the effective diffusivity associated with eddy stirring and eddy modulation of air–sea interaction. In section 3, we apply the effective diffusivity formalism to the surface Southern Ocean and quantify/discuss the augmenting effects of eddy stirring and eddy damping by air–sea interaction in determining the lateral eddy diffusivity. In section 4, we discuss the application of our effective diffusivity approach to other fields such as salinity and chlorophyll. Finally, we conclude and discuss the implications of our results in section 5.
2. Theoretical framework
a. Traditional approach
We now go on to discuss how we propose to use a Nakamura tracer-based framework (Nakamura 1996) to quantify the impact of damping of eddies by air–sea interaction on surface eddy diffusivities.
b. Using a tracer-based framework
Equation (7) can be transformed to a coordinate system based on the area Ai contained within contours T = Ti of the tracer (the area Ai = A(Ti) = ∫T≤Ti dA is represented by the blue shading in Fig. 2a). In this framework, the diffusive effects of the eddies can be clearly identified because only diffusion, not advection, can change the area that a particular tracer contour encloses. We refer the reader to earlier papers (Nakamura 1996; Marshall et al. 2006; Shuckburgh et al. 2009a) and to the appendix for a full explanation and derivation.
3. Surface effective diffusivity from altimetric observations
a. Model results
We used the same numerical framework as Marshall et al. (2006) and Shuckburgh et al. (2009a,b) to calculate the surface effective diffusivity. The infrastructure of the Massachusetts Institute of Technology general circulation model (MITgcm; Marshall et al. 1997a) was employed to evolve a tracer according to Eq. (7) with the velocity field v being the lateral near-surface geostrophic velocity field derived from altimetry data (for more details, see Marshall et al. 2006). A horizontal resolution of 
We first verified that the calculation of the effective diffusivity is not strongly sensitive to the chosen value of numerical diffusion k. This has already been shown to be true for the case of no relaxation (Marshall et al. 2006; Shuckburgh et al. 2009a). In the appendix we present the results at equilibrium for a strong relaxation time scale of λ−1 = 12 days with two values of numerical diffusivity, k = 50 m2 s−1 and k = 100 m2 s−1. Here, Kλ and KNak are found to be nearly identical for the two values of k. A similar result was found for other values of the relaxation time scale.
For the case of no relaxation, it was found (Shuckburgh et al. 2009a) that the calculation of KNak reached an equilibrium value after an initial spinup time of about 3 months for a value of k = 50 m2 s−1. Those authors noted that this adjustment time was inversely related to the value of the numerical diffusivity. Here, we find that Kλ reaches equilibrium after a time scale of about λ−1. Consequently, we choose to present the results for Keff after an integration of at least 1 yr, with longer integrations for the longer relaxation time scales.1
The results of calculations at equilibrium for relaxation times of 12 days, 6 months, and 5 yr are shown in Fig. 3 with KNak (dotted line), Kλ (dashed line), and Keff = KNak + Kλ (solid line). The results are plotted against equivalent latitude.2
For short relaxation time scales (λ−1 = 12 days; Fig. 3a), the value of Keff (solid line) is dominated by the contribution from Kλ (dashed line), whereas, for long relaxation time scales (λ−1 = 5 yr; Fig. 3c), the value of Keff is dominated by the contribution from KNak (dotted line). For λ−1 = 6 months (Fig. 3b), KNak and Kλ provide approximately equal contributions to Keff.
Figure 4a presents the results of effective diffusivity for various values of the relaxation time. The results are averaged over the equivalent latitude bands used by Shuckburgh et al. (2009a), which are representative of the core of the ACC (49°–56°S, black line), the flanks of the ACC (41°–49°S, dark gray line), and equatorward of the ACC (33°–41°S, light gray line). In each band, the values of Keff = KNak + Kλ show a maximum at approximately λ−1 = 10 days and the values of KNak and Kλ are found to be equal in each band at a relaxation time of approximately 200 days.
b. Scaling of effective diffusivity with damping time-scale and flow-field parameters
We now explore how Keff may be expected to vary with the damping rate λ and flow-field parameters. Previous studies (Shuckburgh et al. 2001; Marshall et al. 2006) have argued that in mixing regions the Nakamura effective diffusivity is expected to scale as 
Equation (18) predicts the following (see Figs. 4b,c for illustration): (i) Keff will converge to KNak (0) for large λ; (ii) Keff becomes very small for small λ (in this limit the damping is so strong that the eddy field cannot deform the mean SST contours and thus create filaments); and (iii) between these two limits, Keff peaks at a damping time scale of λp−1 = τ/(1 − 2Sτ), with the peak value being dependent on KNak (0), S, and τ. This can be interpreted as follows: For somewhat weak damping (λ−1 ≥ λp−1), the SST variance is mainly generated by the chaotic advection of the eddy field, and the air–sea heat flux provides, alongside the small-scale mixing, an additional mechanism to destroy variance; hence, the effective diffusivity increases above KNak (0). As the strength of the damping increases, the SST variance is reduced, hampering the ability of the eddy fields to generate filaments. Ultimately, for very large damping, the SST field is “pinned down” to T*, the eddy field cannot create SST anomalies, T ′ → 0, and the eddy diffusivity converges to k.
The stretching rate S can be estimated from a calculation of finite-time Lyapunov exponents (Marshall et al. 2006). The results for the three equivalent latitude bands are S = 2.13 month−1 for the ACC, 2.01 month−1 for the flanks of the ACC, and 1.88 month−1 equatorward of the ACC. We take the value of τ as the damping time scale at which Kλ peaks and this gives values of τ = 0.29 (ACC), 0.34 (flanks), and 0.27 month (equatorward). These values, which are in the range 8–10 days, are broadly consistent with the Lagrangian decorrelation times found by Veneziani et al. (2004) for the northwest Atlantic. The presence of coherent structures in the flow (meandering jets and vortices) alters the decorrelation time, making it longer where trajectories exhibit looping (Richardson 1993). This likely explains the slightly larger value of τ found on the flanks of the ACC.
These S and τ values are used to estimate the values of KNak, Kλ, and Keff according to Eq. (18) with a = ⅙. The results are presented in Fig. 4b as blue curves. It can be seen that the estimate provides a remarkably good fit to the diffusivities.
As a final test of the scaling, we consider the case where the tracer is advected only by eddies with the mean flow set to zero. The stretching rate, which scales with the eddy kinetic energy (EKE; Waugh et al. 2006), is expected to remain similar. On the other hand, the typical Lagrangian decorrelation time τ may be expected to be 1) longer, because of the presence of more looping trajectories,3 and 2) more uniform across the latitude bands, because of the absence of the influence of jets in some regions. The results for the effective diffusivities in the case of no mean flow are presented in Fig. 4c. Marshall et al. (2006) found that the Nakamura effective diffusivity [KNak (0)] in this case varied little in latitude. Consistent with this, KNak [which we suggest scales only with KNak (0) and τ] is seen to be similar for each of the latitude bands. The maximum values of Kλ and Keff are shifted to longer damping times. Again, taking the value of τ as the damping time scale at which Kλ peaks, we find values of τ = 0.32 (ACC), 0.5 (flanks), and 0.59 month (equatorward). This is consistent with the anticipated longer Lagrangian decorrelation time without the mean flow. When we use these values of τ to reestimate the values of KNak, Kλ, and Keff according to Eq. (18), we again find a good fit (blue curves in Fig. 4c). This further confirms the utility of our scaling.
We now use Eq. (18) to estimate the values of Keff of relevance to the Southern Ocean. We take representative values for the stretching rate and Lagrangian decorrelation time scales of S = 2 month−1 and τ = 0.3 month. For the damping time scale λ−1, we take values in the range 2–8 months. Because we expect a shorter damping time scale in regions of strong eddy activity, we use the streamwise average of (EKE)−1 to set the spatial variability within this range (using 10 times the value of EKE in m2 s−2 gives a value of λ in months−1 of about the right magnitude). The EKE is largest on the flanks of the ACC (with an average value of 0.029 m2 s−2) and this gives an average value of our estimated λ−1 of 3.83 months. Equatorward the average value of EKE is smaller (0.017 m2 s−2), and this gives an average value of λ−1 of 5.97 months.
The results for 16 October 19984 are presented in Fig. 5a. The effective diffusivity calculated for a “conserved tracer” (by which we mean a tracer for which the reaction rate λ is zero) as in Shuckburgh et al. (2009a) is plotted for comparison in Fig. 5b (black curve). As previously, a smoothing has been applied to remove unrealistic high-frequency noise. The latitudinal distribution for the total effective diffusivity for SST and the conserved tracer are very similar with low values in the core of the ACC and higher values equatorward. The values for SST are typically about 500 m2 s−1 larger than those for a conserved tracer, ranging from about 1500 to over 3000 m2 s−1. In the core of the ACC, KNak and Kλ contribute about equally to the total effective diffusivity, whereas equatorward KNak contributes about two-thirds and Kλ contributes about one-third. The values of Keff for SST are broadly in line with those found by Zhai and Greatbatch (2006a). They found values in the range 1000–2000 m2 s−1 within the Gulf Stream, with some “hot spots” of 104 m2 s−1 to the south. Although we do not find such large hot spots, it should be remembered that our values are a streamwise average. It should also be noted that our values represent a minimum effective diffusivity, because they do not account for, for example, interactions at the base of the mixed layer.
Finally, Fig. 6 presents the values of Keff calculated for Fig. 5a, plotted on the relevant equivalent latitude contours. This figure is to be compared with the results of the Nakamura effective diffusivity for a conserved tracer presented in Fig. 1 of Shuckburgh et al. (2009a). The same basic pattern of low values in the ACC and higher values on its flanks can be clearly observed in both cases.
4. Effective diffusivity for other tracers
We now consider the relevance of our results for other tracer fields: namely, sea surface salinity (SSS), phytoplankton, zooplankton, and various dissolved gases.
a. Salinity
The case of sea surface salinity (SSS) is particularly interesting because, as described below, our results suggest that, depending on the relative directions of the temperature and salinity gradients, air–sea interaction could enhance or diminish the effective eddy diffusivity of salinity. Returning to Fig. 2, consider the case where temperature and salinity gradients are in the same direction (as in the ACC, where both point equatorward). As a warm and salty water parcel moves southward, it experiences a cooling, partly achieved through latent heat flux–evaporation. Hence, although temperature anomalies are damped, salinity anomalies are reinforced, generating an up-gradient flux as they return northward. Thus, we expect that, in such a case, KλS for salt would be negative. If, however, temperature and salinity gradients are opposed to one another (as in the subtropics), KλS is expected to be positive.
Making some simple approximations, the term in the salinity variance equation associated with freshwater exchanges at the air–sea interface can be expressed in a form analogous to that seen in the temperature case. This in turn allows us to relate the air–sea eddy diffusivity of salt KλS to that of temperature KλT.
The value of αL/αT is not known for the Southern Ocean. However, Frankignoul and Kestenare (2002) estimated for the Northern Hemisphere that the radiative contribution to the total heat flux damping is small, typically less than 10%, and can be neglected. Observations for the Northern Hemisphere also suggest that the ratio of the sensible to the latent heat fluxes, the Bowen ratio, ranges from ¼ in midlatitudes to 1 at high latitudes. Overall, this suggests that αL/αT ≃ 0.5–0.8. Let us assume αL/αT ≃ 0.65 and that SO = 34 psu and Cp = 4000 J kg−1 K−1 (and, of course, ρO/ρF ≃ 1). Typically, ∂y
It should be emphasized that, because of the contribution of 
b. Biogeochemistry
Our results also have relevance for simple descriptions of biogeochemical processes in the ocean. A number of studies have emphasized the importance of horizontal eddy stirring in determining the surface distribution of, for example, phytoplankton (Lévy 2003) and the partial pressure of carbon dioxide at the sea surface (pCO2; Resplandy et al. 2009). Equation (7) has been used to model the carrying capacity field in a simple system describing the evolution of phytoplankton and zooplankton (Abraham 1998), where the carrying capacity is the maximum phytoplankton concentration attainable within a fluid parcel in the absence of grazing. This carrying capacity is assumed to represent the effect of a limiting nutrient or to represent variations in mixed layer depth. As a parcel moves through the domain, the carrying capacity continually relaxes toward a spatially varying background nutrient value, which may be determined by, for example, mixed layer entrainment or wind-driven upwelling. Abraham (1998) took the relaxation profile to be a smooth function of latitude, similar to the relaxation profile used in this study. In both cases, spatial variability is injected into the model at the large scale. Further, Bracco et al. (2009) have used equations of the form of (7) with different values of the relaxation time scale λ−1 as a simple description of the evolution of the phytoplankton and zooplankton to understand the structure of their spatial distributions. Mahadevan and Archer (2000) have also used similar expressions to consider tracers such as dissolved organic carbon (DOC) and hydrogen peroxide (including the effect of vertical transport).
Bracco et al. (2009) assumed a value of λ−1 = 4 days for phytoplankton, 12 days for zooplankton, and 40 days for SST (a value broadly in line with the values we have used above), Mahadevan and Archer (2000) used a long relaxation time scale (60 days) for DOC and a short time scale (3 days) for hydrogen peroxide. Considering the results presented in Fig. 4, it can be seen that the values of eddy diffusivity for λ−1 = 4 days (phytoplankton) are close to those for λ−1 = 12 days (zooplankton), with both being strongly dominated by the values of Kλ. This is consistent with the finding of Bracco et al. (2009) that the addition of turbulent diffusion does not significantly modify the spectral slope of tracers with reaction times shorter than the Lagrangian decorrelation time scale. On the other hand, for the longer reaction time scales of relevance to SST, turbulent diffusion was observed by Bracco et al. (2009) to influence the spectral slope, consistent with our finding of a significant contribution by KNak to the total effective diffusivity. From Fig. 3a, it can be seen that the values of Keff of relevance to phytoplankton or zooplankton range from about 2000 m2 s−1 in the ACC to about 5000 m2 s−1 equatorward.
5. Conclusions and discussion
In this paper, we have presented a new technique that is able to robustly quantify the effective eddy diffusivity for tracers subject to advection, diffusion, and a simple reaction consisting of a relaxation to a large-scale background profile. The effective diffusivity is expressed as a streamwise average. We have chosen to relax the tracer back to a profile that is aligned with the time-mean streamlines. This assumption will be valid when the time scale for along-stream advection is shorter than the relaxation time scale. This is evidently true for the Southern Ocean, where the mean SST contours are observed to be broadly aligned with the mean streamlines.
We find, for example, that air–sea damping can augment the lateral diffusivity within the mixed layer by, depending on the assumed SST damping time scale, a value on the order of 500 m2 s−1 (see Fig. 5). In frontal regions such as the Gulf Stream, Kuroshio, or ACC, where SST can change on the order of 5°C in 100 km, this is equivalent to an air–sea flux of 100 W m−2 acting over a mixed layer depth of 100 m. This is a very significant flux, which would be absent in coarse-resolution models unless explicitly accounted for. Our results may therefore help inform model parameterizations thus improving the fidelity of coarse-grained models.
Our results indicate that, near the surface, the total eddy diffusivities associated with different tracers (e.g., temperature, salinity, dissolved gases, and chlorophyll) may differ significantly in magnitude. We find values in the ACC ranging from about 200 m2 s−1 or less for salinity, through 1500 m2 s−1 for temperature, up to about 2000 m2 s−1 for chlorophyll. The values equatorward of the ACC are larger, but strong differences between tracer species remain with the total eddy diffusivity being ∼800 m2 s−1 for salinity, ∼3000 m2 s−1 for temperature, and ∼5000 m2 s−1 for chlorophyll. This has implications for model parameterizations as it suggests that, near the surface, different values of eddy diffusivity may be required for different tracers.
The approach we have presented in this paper evidently neglects many physical, chemical, and biological processes that may influence surface fields. Nevertheless, it constitutes a powerful new technique that quantifies the mixed layer lateral eddy fluxes mediated by air–sea interaction. In this way, it can be used to provide valuable information concerning the evolution of any surface field (from observations or models) that exhibits variability correlated with the mesoscale eddy field and that is influenced by air–sea interactions.
Acknowledgments
EFS has been supported by an NERC postdoctoral fellowship. The MIT group would like to acknowledge support of both NSF (Physical Oceanography) and NASA (ECCO2).
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APPENDIX
Model and Method Description
The eddy-resolving model used was the MITgcm (Marshall et al. 1997a,b). The simulation was conducted as a part of the Estimating the Circulation and Climate of the Ocean, Phase II (ECCO2) project and is freely available on the Internet (available online at http://ecco2.org). The ocean is forced from April 2002 to March 2005 by the National Centers for Environmental Prediction (NCEP) Reanalysis-1surface atmospheric state (Kalnay et al. 1996). Sea surface heat fluxes are computed using a classic set of bulk formulas (Large and Yeager 2004). A surface relaxation to monthly Levitus sea surface salinity is applied with a relaxation time constant of 44.5 days (Levitus and Boyer 1994). The simulation also includes a full dynamic–thermodynamic sea ice model (for more details, see online at http://mitgcm.org). The resolution of the model is 50 vertical levels and ⅛° both in latitude and longitude: that is, about 14 km at the equator decreasing to about 7 km at high latitudes. The model is run globally but the domain of analysis for this study was limited to the Southern Ocean from 20° to 80°S. The model eddy temperature variance field follows the distribution of the eddy kinetic energy because of the mesoscale activity of the Southern Ocean. It is realistically maximum (values from 6° to 10°C2) south of the Cape of Good Hope (on the poleward flank of the Agulhas current), downstream of the Drake Passage (where the ACC merges with the Brazilian Current in the South Atlantic subtropical gyre southwest corner), along the Brazilian Current off the South American coast, eastward of the New Zealand north coasts, and finally all along the ACC path.
Effective diffusivity derivation
Previous studies (Marshall et al. 2006; Shuckburgh et al. 2009a) have investigated dependence of the Nakamura effective diffusivity on the value of the diffusivity k. The results indicated that, when the Péclet number 
Here, we investigate the dependence of Keff(λ, k) = KNak(λ, k) + Kλ(λ, k) on the value of k. In the limit of small λ, then Kλ,k → 0 and the above result concerning the independence of the effective diffusivity on the value of k holds. We therefore investigate the case of large λ.
In Fig. A1, we present the results at equilibrium for a strong relaxation time scale of λ−1 = 12 days with two values of numerical diffusivity k = 50 m2 s−1 (black line) and k = 100 m2 s−1 (gray line). It can be seen that the results for Kλ (dashed line) and KNak (dotted line) are nearly identical for the two values of k. A similar result is found for other values of the relaxation time scale. We conclude that Keff is not strongly dependent on the value of the numerical diffusivity for small enough values of k and thus that we can consider Keff = Keff (λ), KNak = KNak (λ), and Kλ = Kλ (λ).
In a flow with time-varying eddy diffusivity, the geometric structure of a tracer at any instant will depend on the history of the flow (weighted toward the recent past, defined by some “memory time”). The memory time will be shorter when k is larger. Temporary changes in the eddy diffusivity of the flow will be fully represented by Keff only if they persist for longer than the memory time. We believe that the memory time implied by a numerical diffusivity of k = 50 m2 s−1 is sufficiently small to allow Keff to resolve variations in the mixing ability over time scales of about a month or so (Shuckburgh et al. 2009a).
(a) Daily mean of the sea surface heat flux Q for 5 May 2003 of the ⅛° ECCO2 simulation. (b) Monthly mean of Q for May 2003. (c) A local zoom of (b), in the eddy-rich region around 60°E along the ACC and indicated by the red box in (b). Superimposed are SST contours for the same period (black thin: every 1°C; thick: every 5°C). (d) May 2002–April 2005 mean of α = −
Citation: Journal of Physical Oceanography 41, 1; 10.1175/2010JPO4429.1
Schematic diagram of SST fluctuations associated with meandering ocean currents. (a) A temperature contour Ti = 〈
Citation: Journal of Physical Oceanography 41, 1; 10.1175/2010JPO4429.1
Latitude dependency of surface eddy diffusivity with Kλ (dashed line), KNak (dotted line), and Keff = Kλ + KNak (solid line). Results are plotted, at the equilibrium state, for a relaxation time scale of (a) λ−1 = 12 days, (b) 6 months, and (c) 5 yr. (Note the different vertical scales)
Citation: Journal of Physical Oceanography 41, 1; 10.1175/2010JPO4429.1
Dependency of surface eddy diffusivity on relaxation time scale inferred from the tracer analysis averaged over the equivalent latitude bands: KNak (dotted lines), Kλ (dashed lines), and Keff = KNak + Kλ (solid lines). (a) Results for three equivalent latitude bands: 33°–41°S (light gray lines), 41°–49°S (dark gray lines), and 49°–56°S (black lines). (b) Overplotted in blue are the results of the analytical estimate of Keff given by Eq. (18). (c) The results for a calculation where the mean flow is set to 0 (i.e., the flow field consists only of the eddies).
Citation: Journal of Physical Oceanography 41, 1; 10.1175/2010JPO4429.1
Effective diffusivity for 16 Oct 1998 (a) for SST, with KNak (dotted line), Kλ (dashed line), and Keff = KNak + Kλ (solid line), and (b) for a conserved tracer (black line) and SSS (gray line), both with Keff.
Citation: Journal of Physical Oceanography 41, 1; 10.1175/2010JPO4429.1
(a) Sea surface height (SSH) anomalies and (b) effective diffusivity Keff (ϕe) for SST for 16 Oct 1998 with overplotted streamlines with values (from equator to pole) of −9, −5, 0 (bold), and 6 × 104 m2 s−1 [time-mean streamlines in (a), instantaneous streamlines in (b); these mark the equivalent latitude bands used to denote the ACC, its flanks, and equatorward]. Note that Keff is a function of equivalent latitude ϕe only; therefore this two-dimensional plot contains only one-dimensional information (see text for further explanation). Latitudes from 30°S to the pole are plotted.
Citation: Journal of Physical Oceanography 41, 1; 10.1175/2010JPO4429.1
Fig. A1. Latitudinal dependency of surface eddy diffusivity inferred from the tracer analysis with Kλ (dashed line) and KNak (dotted line), for a relaxation time scale of λ−1 = 12 days. Results are for k = 50 m2 s−1 (black line) and k = 100 m2 s−1 (gray line).
Citation: Journal of Physical Oceanography 41, 1; 10.1175/2010JPO4429.1
For these calculations, we used altimetry data from 1997, annually repeating where required.
The equivalent latitude, ϕe(T, t), is related to the area A within a tracer concentration contour by the identity A = 2πr 2(1 − sinϕe), with r being the radius of the earth. For each tracer contour, the equivalent latitude is therefore the latitude the contour would have if it were remapped to be zonally symmetric while retaining its internal area.
Veneziani et al. (2004) found significantly more looping trajectories and longer decorrelation times north of the Azores current, where there is only a weak eastward mean flow.
Note that the diffusivity calculated for this day will reflect the influence of the eddies on the tracer field over the recent past defined by some memory time. See appendix for further details.
Because part of the anomalous evaporation is rained out locally, 〈
