How Does the Antarctic Circumpolar Current Affect the Southern Ocean Meridional Overturning Circulation?

Christopher C. Chapman∗ LOCEAN-IPSL Université de Pierre et Marie Curie mailto:[email protected] Jean-Baptiste Sallée† LOCEAN-IPSL CNRS/Université de Pierre et Marie Curie

December 6, 2016

Abstract

The Meridional Overturning Circulation (MOC) in the Southern Ocean is investigated using hydrographic observations combined with satellite observations of sea-surface height. A three-dimensional (spatial and vertical) estimate of the isopycnal eddy-diffusivity in the Southern Ocean is obtained using the theory of Ferrari & Nikurashin (2010), that includes the influence of suppression of the diffusivity by the strong, time-mean flows. It is found that the eddy diffusivity is enhanced at depth, reaching a maximum at the “critical layer” near 1000m. The estimate of diffusivity is used with a simple diffusive parameterization to estimate the meridional eddy volume flux. Together with an estimate of the meridional Ekman transport and the time-mean meridional geostrophic transport, the eddy volume flux is used to reconstruct the time-mean overturning circulation. By comparing the reconstruction with, and without, suppression of the eddy diffusivity by the mean flow, the influence of the suppression on the overturning is illuminated. It is shown that the suppression of the eddy diffusivity results in a large reduction of interior eddy transports, and a more realistic eddy induced overturning circulation. Finally, a simple conceptual model is used to show that the MOC is influenced not only by the existence of enhanced diffusivity at depth, but also by the details of the vertical structure of the eddy diffusivity, such as the depth of the critical layer.

I. Introduction subduction of atmospheric CO2, has a large inﬂuence on the climate system [Talley et al., he Meridional Overturning Circulation 2003, Marshall and Speer, 2012]. In the South- arXiv:1612.01115v1 [physics.ao-ph] 4 Dec 2016 (MOC), is a global scale circulation that, ern Ocean, the overturning is related to the rate Tdue to its important role in the redistri- that deep carbon-rich waters are ventilated at bution of heat, salt and biogeochemical tracers the surface where they can communicate with from warmer to colder latitudes, and in the the atmosphere, and the rate at which surface waters are in turn subducted into the ocean in- ∗ Corresponding author address: C. C. Chapman, terior [Sallée et al., 2013]. Thus, changes in the LOCEAN-IPSL, Université de Pierre et Marie Curie, Paris CEDEX ,France. rate of the overturning have been hypothesized E-mail: [email protected] to lead to a reduction in the Southern Ocean’s †Corresponding author

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ability to absorb and sequester CO2 [Le Quéré ence on the stirring ability of meso-scale eddies et al., 2007]. Understanding the dynamic con- and hence their ability of move water poleward trols of the MOC in the Southern Ocean, as [Bates et al., 2014]. The capacity of eddies to well as how it will respond to external changes induce a downgradient flux is often measured in the climate system, is therefore a pressing by the eddy diffusivity, K, which relates the question in physical oceanography. eddy-flux of some tracer with concentration C, Motivated by the widely acknowledged im- to the large scale gradient of that tracer: portance of the Southern Ocean for the global 0 0 MOC, intense focus on this region has led to u C = K∇C (2) significant advances in our understanding of Although baroclinic eddies are ubiquitous in the structure of the MOC and dominant dy- the Southern Ocean, certain regions, referred namical mechanisms that lead to its formation. to as “hot-spots" or “storm tracks", which In particular, a description of the Southern arise from the interaction of the ACC with Ocean based on the Transformed Eulerian Mean bathymetry [Williams et al., 2007, Chapman (TEM) formulation has shown that, on the large et al., 2015], cause localised increases in K [Sal- scale, the Southern Ocean overturning results lée et al., 2008]. The ACC also modulates the from a competition between a northward wind vertical structure of eddy diffusivity, which is driven Eulerian mean overturning cell, Ψ, and known to be enhanced at depth, reaching a a southward eddy induced overturning, Ψ? maxima at the “steering-level" or “critical layer [Johnson and Bryden, 1989, Döös and Webb, depth" [Ferrari and Nikurashin, 2010, Klocker 1994]. The mean and eddy-induced overturn- and Abernathey, 2014], associated with the ing are thought to be of similar magnitude, yet fastest growing linear waves [Smith and Mar- opposite sign, such that only a small residual shall, 2009]. Linear stability analysis places transport remains. The resulting overturning, this level at around 1000 m depth [Smith and commonly expressed as an overturning stream- Marshall, 2009]. Although it has been shown function, is written: that including a spatially varying K can reduce Ψres = Ψ + Ψ?. (1) bias in coarse resolution climate models [Fer- reira et al., 2005, Danabasoglu and Marshall, Because of the delicate balance between the 2007], and while the implications of a three- eddy and mean overturning, the residual over- dimensional K on the broad scale flow have turning is sensitive to even small changes in been briefly discussed in several studies [Mar- one or the other component resulting from shall et al., 2006, Shuckburgh et al., 2009, Smith changes in surface forcing [Viebahn and Eden, and Marshall, 2009, Naveira Garabato et al., 2010, Abernathey et al., 2011, Meredith et al., 2011, Bates et al., 2014], a detailed understand- 2012, Downes and Hogg, 2013]. ing of the physical implications of spatially The Southern Ocean also hosts the Antarc- varying K for the large-scale overturning circu- tic Circumpolar Current (ACC), a system of lation is still lacking. currents that are among strongest on Earth. Al- In this study, we seek to characterize and though the ACC is primarily zonally oriented, quantify the influence of “storm-tracks" and it has direct and indirect roles in shaping the the suppression of the eddy diffusivity by the Southern Ocean MOC. For instance, the inter- mean-flow on the Southern Ocean overturn- action of the ACC with bathymetry results in a ing circulation. We will explore the impact significant, but frequently ignored, geostrophic of geostrophic mean-flow on the MOC, and interior overturning circulation that is distinct in particular the impact of the strong currents from the mean ageostrophic overturning asso- of the ACC in modulating the eddy overturn- ciated with Ekman currents [MacCready and ing. To achieve our goals, we reconstruct Rhines, 2001, Mazloff, 2008, Mazloff et al., the overturning circulation from a large obser- 2013]. In addition, the ACC has a strong influ- vational dataset that combines hydrographic

2 Submitted for publication in the Journal of Physical Oceanography data from Argo floats, oceanographic cruises II. The Southern Ocean and instrumented elephant seals, with sea Meridional Overturning surface height altimetry. The observational Circulation datasets are used to develop a direct estimate of the Eulerian mean overturing that includes Here we briefly revise the basic theory of the both ageostrophic Ekman currents and the im- MOC in the Southern Ocean, the theory of portant deep geostrophic currents that arise mean-flow suppression of eddy diffusivity, and from the interaction of ACC with the bottom the formulation of the TEM model equations. bathymetry. In addition, we produce a three- On an isopycnal layer, γ, with thickness dimensional estimate of K based on the theory h = −∂z/∂γ, the time-mean meridional vol- of Ferrari and Nikurashin [2010] that allows a ume flux is given by: reconstruction of an eddy overturning streamfunction from the downgradient diffusion of hv = hv + h0v0, (3) potential vorticity [Treguier et al., 1997]. We will investigate the influence of spatial varia- where v is the meridional velocity, (·) is the tion of K and the suppressing influence of the time-averaging operator, and the flow has been background flow on the reconstructed eddy decomposed into time-mean and eddy com- and residual streamfunctions. By reconstruct- ponents. The primed quantities are perturba- 0 ing Ψ and Ψ? from observations, we show that tions from the time-mean, such that v = v + v 0 the ACC can impact significantly both of these and v = 0. We can further decompose v into terms, and has therefore a key role in shaping geostrophic, vg, and ageostrophic, vag compo- the residual overturning circulation. In tandem nents: with this reconstruction, we employ a simple = + + 0 0 + 0 0 conceptual model, based on the TEM approach hv hvg hvag h vg h vag. (4) of Marshall and Radko [2003] and Marshall The meridional transport can then be vertically and Radko [2006] to guide our interpretation integrated on across isopycnal layers to de- of the observational-based reconstruction. termine the time-mean isopycnal overturning streamfunction [Döös and Webb, 1994]:

Z γ res 0 ? ? Ψ (x, y, γ) = hv dγ = Ψag + Ψg + Ψg + Ψag . The remainder of this paper is organized 0 | {z } | {z } as follows: the theoretical framework used mean eddy for building the reconstruction of the MOC (5) from observations, including the procedure for estimating the horizontal (isopycnal) dif- i. The ageostrophic transport fusivity K, will be presented in Section II. The observational data set we employ will be de- In the Southern Ocean, the overwhelming ma- scribed in Section III. Our estimate of the three- jority of the ageostrophic transport, hvag oc- dimensional eddy diffusivity will be presented curs due to surface Ekman currents. The 0 0 in Section IV and our estimated MOC recon- ageostrophic eddy transport, h vag, although struction, along with a comparison of the the not completely negligible, is much smaller than results obtained with and without the influence the time-mean Ekman transport [Mazloff et al., of the background flow in Section V. The influ- 2013]. Since we are unable to estimate this term ence of a vertically varying K on the overturn- from the data used in this study we will not ing will then be elucidated using a conceptual discuss it further, although, for reference, Ma- model in Section VI. Finally, we will bring to- zloff et al. [2013] finds a southward transport of gether the observational and theoretical results approximately 5 Sv contained almost entirely of this study in Section VII. in the surface layers. The time-mean Ekman

3 Submitted for publication in the Journal of Physical Oceanography velocity can be determined from the surface et al., 2013, Dufour et al., 2015]. However, we wind stress using the equations for an Ekman are unable to estimate this term directly from spiral [Dutton, 1986, pg. 449] with a constant observations, and hence it must be parame- Ekman layer depth, here taken to be 100 m, terized. Following Marshall et al. [1999], we consistent with observations in the Southern start by noting that the geostrophic eddy flux Ocean [Lenn and Chereskin, 2009]. f +ζ of Ertel potential vorticity (PV), q = h , can be written as: ii. Time-mean geostrophic transport f 0 0 = − 0 0 vgq 2 h vg, (7) The time-mean geostrophic transport, hvg h arises due to the outcropping of the isopy- assuming planetary geostrophic scaling such cnal surfaces either with the ocean surface that q ≈ f /h and h0/h << 1. Thus, the or the ocean floor [Ward and Hogg, 2011, geostrophic eddy volume flux can be written Mazloff et al., 2013]. On pressure surfaces, as: the geostrophic velocity can be determined h from hydrography by computing the dy- h0v0 = − v0 q0. (8) g q g namic height anomaly, which gives an exact geostrophic streamfunction. On an isopycnal We now employ the simple down-gradient dif- layer, there is no exact geostrophic streamfunc- fusive closure for v0q0 ≈ K∂q/∂y, described in tion [McDougall, 1989]. However, McDougall detail by Treguier et al. [1997] and discussed in and Klocker [2010] have formulated an approx- numerous papers thereafter [Killworth, 1997, imate streamfunction on a neutral density sur- Marshall et al., 1999, Wardle and Marshall, face [Jackett and McDougall, 1997], that can 2000, Roberts and Marshall, 2000, Wilson and be computed from hydrography. The formula- Williams, 2004, Plumb and Ferrari, 2005] to tion of this streamfunction takes into account give: the non-linearity in the equation of state and h ∂q h0v0 = −K . (9) allows for temperature to vary quadratically g q ∂y with pressure along the neutral surface. The geostrophic velocities are then related to the Eqn. 9 is a crude parameterization of the true streamfunction, M, by: eddy fluxes with numerous failings [Roberts and Marshall, 2000, Wilson and Williams, 2004]. f ug = k × ∇M, (6) However, it has been shown to work well when one is interested only in the large-scale flow where f = 2ΩE sin φ is the Coriolis parameter. [Marshall et al., 1999, Plumb and Ferrari, 2005, The meridional geostrophic transport, hvg, Kuo et al., 2005]. does not necessarily integrate to zero around a circumpolar circuit due to outcropping of the iv. Eddy diffusivity K isopycnals. In fact, the geostrophic transport is of first order importance to the zonally av- With the closure for the eddy volume flux in eraged interior mass balance [MacCready and terms of the large scale PV gradient, we are Rhines, 2001, Koh and Plumb, 2004, Mazloff able to reconstruct the geostrophic eddy vol- et al., 2013]; a point we will return to in Section ume flux with knowledge of the eddy diffu- V. sivity K, which is known to be dependent on the eddy kinetic energy, the spatial and tem- iii. Geostrophic eddy transport poral scales of the meso-scale eddies, and on the state of the background mean flow [Smith 0 0 The geostrophic eddy transport h vg is also and Marshall, 2009, Abernathey et al., 2010, found to be of first order importance in a num- Ferrari and Nikurashin, 2010, Klocker and ber of studies [Abernathey et al., 2011, Mazloff Abernathey, 2014, Bates et al., 2014]. Ferrari

4 Submitted for publication in the Journal of Physical Oceanography and Nikurashin [2010], using the assumptions the zonally averaged TEM equations in the of simple isotropic turbulence modeled by a ocean interior: white-noise process, derived the following ex- res pression for K, that takes into the account the Jyz Ψ , b = 0, (11) effects of the mean-flow: Ψres(y, b) = Ψ(y) + Ψ?(y, b), (12) K K˜ = 0 , (10) , 2 2 2 ? ∂b ∂b 1 + k τ cp − u(x, y, z) Ψ = −K , (13) eddy eddy ∂y ∂z where the term K0 represents the eddy diffusivity which is unmodified by the mean-flow, where b = b(y, z) is the buoyancy, Jyz is the ˜ K is the modified eddy diffusivity (sometimes Jacobian operator, and ∂b / ∂b is the isopycnal called the effective diffusivity or the suppressed ∂y ∂z slope. Here, the mean overturning is taken to diffusivity for reasons that will soon become ap- be simply that associated with the Ekman flow parent), k is the zonal eddy wavenumber, eddy Ψ = −τwind(y)/ f . Rearranging Eqns 12 and τ c eddy is eddy decorrelation time scale, and p 13 and substituting for Ψ gives: is the eddy phase speed. As u → c , the denominator of Eqn. 10 " # p τwind(y) ∂b ∂b approaches unity, which means that K˜ → K . + Ψres + K = 0. (14) 0 f ∂z ∂y In the case where u 6= cp, the denominator of Eqn. 10 is greater than unity and K˜ < K0. For this reason, the denominator of Eqn. 10 Eqn. 14 can be solved numerically using the is called the “suppression factor" [Klocker and method of characteristics, as detailed in Ap- Abernathey, 2014]. This reaches its minima at pendix B. The boundary conditions are identi- the critical level, i.e. where u = cp, which is cal to those of Marshall and Radko [2006]: the thought to lie at about 1000 m of depth [Smith buoyancy field is set at the base of the homo- and Marshall, 2009]. The suppression factor geneous mixed layer (here taken as z = 0) and also varies throughout the Southern Ocean: on the northern boundary: tracer diffusivity is suppressed more in regions y where mean currents are strong and less where b(y, 0) = ∆bsurf , (15) they are weak [Klocker and Abernathey, 2014]. Ly z With Eqns. 9 and 10 we can reconstruct the b(Ly, z) = b(Ly, 0)e he , (16) eddy component of the MOC. Investigating the role that the spatial variability and suppression where Ly = 2000 km is the meridional scale of of K play in the MOC is the primary goal of the ACC, he = 1000 m is the e-folding depth, this paper. and ∆bsurf = 0.007 m.s−2 is the buoyancy gain across the ACC. Additionally, the surface wind v. Simple conceptual model of the stress, τwind(y) is taken to be a simple sinu- Southern Ocean overturning soidal profile: In order to guide our analysis based on the observation-based reconstructed overturning, wind π τ (y) = τ0 0.3 + sin y , (17) we will use a simple conceptual model, with Ly the aim to further understand the influence −4 that vertically-variable diffusivity has on the with τ0 =1.5×10 . The residual overturning stratification and the overturning (Section V). streamfunction Ψres, is not specified as a sur- Our model has essentially the same form as face boundary condition, but is instead calcu- that of Marshall and Radko [2003] and Mar- lated as part of the solution using the iterative shall and Radko [2006]. Specifically, we solve technique of Marshall and Radko [2006].

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III. Observational Data i. Hydrographic Data The primary data used for this study are approximately 250,000 profiles of temperature and salinity from the surface to 2000 db, collected from 1,956 autonomous Argo floats [Roemmich et al., 2009, Riser et al., 2016], between 80◦S and 30◦S of latitude, from the 1st of January 2006 to the 31 of December 2014. Argo floats provide broad scale coverage of the Southern Ocean, shown in Fig. 1a, with suffi- cient spatial and temporal resolution to resolve the large-scale circulation and seasonal variability. Unfortunately, the relatively large distances between observations (approximately 200 km in the Southern Ocean) and the insufficient number of temporally simultaneous measure- Figure 1: Spatial distribution of hydrographic profiles in ments means that the Argo array is not capable the Southern Ocean used in this study. The of directly resolving the instantaneous meso- number of (a) Argo profiles ; (b) ship based scale. However, McCaffrey et al. [2015] have WOD profiles; and (c) instrumented elephant shown that is is possible to use Argo floats to seals profiles, in each 0.5◦× 0.5◦ grid box. measure the statistics of the meso-scale turbulence. −3 The Argo array has a limited number of ob- and to about γ=28.0 kg.m . We note that −3 servations along the southern border of the although waters denser than γ=28.0 kg.m ACC, and along the seasonal sea-ice edge. To (Antarctic Bottom Water) are sampled in this supplement the Argo data, we additionally em- dataset, the coverage is patchy, being sampled ploy hydrographic profiles obtained from vari- primarily in the Atlantic Sector. It can be seen ous research cruises, assembled in the Wold in Fig. 2c that the addition of instrumented Ocean Database (WOD) [Boyer et al., 2009] seals to the database significantly improves and 223,426 profiles collected from 513 instru- winter data coverage. mented southern elephant seals [Roquet et al., For all datasets, only profiles that have 2013, 2014]. These animals forage throughout passed quality control checks are used. Ad- the Southern Ocean, but prioritize regions that ditional quality control was carried out using are generally further south of those sampled automated outlier detection algorithm based by the Argo floats. Data coverage of Argo, on an interquartile range filter and density in- WOD and instrumented seals are shown in version filter, as in Schmidtko et al. [2013]. Data 1. The combined dataset samples all of the from 2006 to 2014 are used in this study, as major water mass classes within the South- there is insufficient data in preceding years to ern Ocean, as shown in Fig. 2. This figure provide coverage of the entire Southern Ocean, shows histograms of the deepest depth (Fig. as shown in Fig. 2c. 2a) and densest neutral density (Fig. 2b) sampled by each of the three data sources. The ma- ii. Satellite Data jority of Argo profiles sample to 2000 m and to about γ=27.8 kg.m−3; the majority of the instru- In addition to the hydrographic data, we mented seals profiles sample to around 500 m also employ satellite derived estimates of depth, with some profiles deeper than 1000 m, sea-surface dynamic topography in order

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pling times by 3-dimensional linear interpolation (that is, spatially and temporally). Thus, for every hydrographic profile we have an associated estimate of the ADT. Profiles obtained in regions or at times where ADT data are not available (which occurs frequently in winter in the far south of the domain) are flagged and excluded from the analysis.

iii. Surface Wind Forcing To calculate the meridional circulation due to Ekman currents we use the daily mean output of surface momentum flux from the National Center for Environ- mental Prediction (NCEP) reanalysis product (http://www.esrl.noaa.gov/psd/data/reanalysis/reanalysis.shtml), described in Kalnay et al. [1996], to determine Figure 2: Depth, density and temporal sampling of in the wind stress τ. the hydrographic data in the Southern Ocean. (a) histogram showing the deepest depth sam- iv. Climatology of the Southern pled by hydrographic profiles sourced from the Argo floats (blue), ship based WOD (red) and Ocean instrumented elephant seals (black); (b) as in Using the hydrographic and satellite data prod- Fig. 2a but for the densest neutral density ucts described above, we develop a climatology γ sampled for each data source; and (c) The of the Southern Ocean. In particular, from the number of profiles from each data source, each month, between 2005 and 2014. temperature, salinity and pressure profiles we compute the neutral density, γ, the isopycnal potential vorticity (IPV), q, and the absolute geostrophic streamfunction, M. to provide a “reference" velocity. Here Profiles of γ are computed from our hydro- we use the Archiving, Validation, and graphic database using the software described Interpretation of Satellite Oceanographic by Jackett and McDougall [1997]. In order data (AVISO) daily gridded absolute dy- to compute the IPV, we make the planetary namic topography (ADT) from Ssalto/Duacs, geostrophic approximation, which is a good downloaded from Copernicus Marine Ser- approximation of the Ertel PV in the Southern vices (http://marine.copernicus.eu/web/69- Ocean interior [Thompson and Naveira Gara- interactive-catalogue.php). We use delayed- bato, 2014]: mode dynamic topography provided on f ∂γ a 1/4◦ Mercator grid, obtained by opti- q ≈ . (18) ρ0 ∂z mally interpolating the alongtrack data series To compute profiles of isopycnal streamfunc- based on the REF dataset, which uses two tion, defined in McDougall and Klocker [2010] satellite missions [Ocean Topography Exper- (see Eqn. 6), we use version 3 of the TEOS-10 iment(TOPEX)/Poseidon/European Remote software [McDougall and Barker, 2011]. From Sensing Satellite(ERS) or Jason-1/Envisat or hydrographic data we can only obtain the rela- Jason-2/Envisat] with consistent sampling over tive streamfunction, that is, the streamfunction the 21-yr period. relative to some reference level γ = γref: The AVISO ADT is then calculated at each u (x, y, γ) = u (x, y, γ) − u (x, y, γ ) ... of the hydrographic profile locations and sam- grel g g ref

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1 weaker meridional gradients, which according = k × ∇Mrel. (19) f to Eqn. 7, can indicate regions of enhanced To determine the absolute streamfunction we eddy volume transport. Finally, Fig. 3c shows follow Kosempa and Chambers [2014] and ref- the geostrophic current speed computed from erence our streamfunction to the surface. Since the gradient of the absolute geostrophic stream- the ADT can be interpreted as the surface function gradient. The currents appear realistic: streamfunction: they form jets, and show steering by topography. The strength of these mean currents is g ug(x, y, γ ) = k × ∇ADT, (20) important for the suppression of eddy volume surf f fluxes, as will become apparent in sections IV the absolute streamfunction is computed by and V. adding the estimated ADT at each hydrographic profile location to the relative stream- IV. The Three-Dimensional Eddy function referenced to the surface: Diffusivity M = ADT + M . (21) abs rel In this section, we use the hydrographic pro- Finally, profiles of neutral density, IPV and files to determine a three-dimensional esti- absolute geostrophic streamfunction are inter- mate of both the suppressed K˜ and unsup- polated to a regular longitude/latitude grid pressed K0 eddy diffusivity, following the the- using the CARS–LOWESS (CSIRO Atlas of Re- oretical framework described by Ferrari and gional Seas robust LOcally Weighted regrES- Nikurashin [2010], described in Sec. II. Recall Sion) software [Ridgway et al., 2002]. The neu- Eqn. 10, in which the total eddy diffusion is tral density is mapped on depth surfaces from written as an unsuppressed diffusivity K0 mul- the surface to 2000 m, with a vertical spac- tiplied by a suppression factor that describes ing of ∆z=50 m. The IPV and streamfunction the influence of the mean flow on the eddy are mapped on a set of isopycnal layers from stirring. γ=26.0 kg.m−3 to γ=28.5 kg.m−3, with a verti- In order to compute the unsuppressed diffu- cal spacing of ∆γ=0.05 kg.m−3. For consistency sivity, we use the expression introduced by Hol- with the altimetric observations, we use a hor- loway [1986] and Keffer and Holloway [1988] izontal grid spacing of 0.25◦×0.25◦, although that relates the root-mean-square of the stream- the effective resolution of the hydrographic function fluctuations to K0: data is coarser (the average distance between Γ 1/2 K = M0 M0 (22) Argo floats profile locations in the Southern 0 f Ocean is approximately 200 km). An example of our climatology is shown in where Γ is a constant mixing efficiency, usu- Fig. 3, here for the isopycnal γ =27.9 kg.m−3. ally taken to be 0.35 [Klocker and Abernathey, The depth of this isopycnal is shown in Fig. 3a, 2014]. We compute the RMS of the geostrophic which reveals, as expected, isopycnals shoaling streamfunction by first computing the stream- towards higher latitudes and eventually out- function fluctuations by subtracting the mean cropping with the surface near the Antarctic geostrophic streamfunction, M, from each of continent. We note that although this isopyc- the instantaneous profiles of M. The square nal is well represented in our dataset (see Fig. of M0 is then computed for each profiles, and 2b), it is deeper than 2000 m over much of mapped using the CARS–LOWESS software region north of the ACC. Fig. 3b shows IPV on a regular longitude/latitude grid (see Fig. on the same isopycnal, which increases pole- 4a). A highly zonally assymetric field is pro- ward, as expected. However, it is worth noting duced, with elevated streamfunction variance that the IPV structure is not zonally homoge- found in regions downstream of large bathy- neous, and there are regions of stronger and metric features, in western boundary currents,

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Figure 3: The climatology of the Southern Ocean, determined from combined hydrographic/satellite data set, on isopycnal γ=27.9 kg.m−3. (a) The depth of the isopycnal; (b) the isopycnal potential vorticity (IPV); and (c) the zonal current speed. Blanked out areas indicate regions with fewer than 200 data points in the interpolation process, or, where the isopycnal is deeper than 2000m. Note the logarithmic colorscale used for the IPV in Fig. 3b in the Agulhas region (∼ 20–60◦E), and at the of the linearized quasi-geostrophic equation us- central Pacific Fracture Zone (∼ 140◦W), con- ing the finite difference scheme of Smith [2007] sistent with previous studies [Sallée et al., 2008, and our gridded interpolated neutral density. Klocker and Abernathey, 2014, Roach et al., The maps of LD (not shown) produced by this 2016]. calculation are very similar to those of Chel- To compute the suppression factor, we re- ton et al. [1998], although due to the more quire an estimate of the time-mean current ve- complete data coverage provided by the Argo locity, u, the eddy phase speed, cp, the eddy floats, there are fewer regions with missing decorrelation time-scale, τeddy, and the eddy data and we find a larger deviation of contours of constant L near large bathymetric features. wavenumber, keddy. u is obtained from the D The eddy decorrelation time-scale, τ is absolute geostrophic streamfunction M, as de- eddy taken to be a constant 4 days, as found by scribed in Sec. IIIiv. cp is calculated using Rossby wave dispersion relationship, Doppler Klocker and Abernathey [2014]. Finally, the shifted by the depth mean flow, as suggested eddy length scale, used in the calculation of by Klocker and Marshall [2014]: eddy wavenumber keddy = 2π/Leddy, is es- zt 2 timated by assuming a constant ratio between cp = u − βL , (23) D the eddy size and LD, which is approximately where β is the meridional gradient of the Cori- valid for strongly non-linear eddies, such as olis parameter, uzt is the depth averaged zonal those found in the Southern Ocean [Klocker velocity and LD is the first baroclinic defor- and Abernathey, 2014]. We set this ratio to 2.5, = mation radius. To compute LD, we solve the so that Leddy 2.5LD. Sturm-Liouville problem for the neutral-modes

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With all the ingredients assembled, we com- 500 m2s−1 in the ACC latitudes. However pute the suppression factor, which is plotted on the suppressed diffusivity still peaks down- isopycnal γ=27.9 kg.m−3 in Fig. 4b. Several re- stream of large bathymetric features, reaching gions of heavily suppressed diffusivities (with its maximum values downstream of the Pacific a suppression factor between 0 and 0.25) are Antarctic Rise (∼140◦W), and downstream of found in regions of strong zonal jets (compare Drake Passage (∼60◦W; Fig. 5). The suppress- with Fig. 3c), as expected. The suppression fac- ing effect of the mean-flow is perhaps most tor computed here shows a strong qualitative clearly seen at Southeastern Indian Ridge and ◦ resemblance to those computed by Ferrari and the Campbell Plateau (∼150–170 E), where K0 Nikurashin [2010] and Klocker and Abernathey peaks at about 2500 m2s−1, but the suppressed [2014] at the surface using altimetry alone. diffusivity does not rise above 1500 m2s−1; a lo- 2 −1 The geographical distribution of the sup- cal effective suppression of about 1000 m s . pressed eddy diffusivity, once again for the The spatial structure of our estimate is simi- isopycnal γ=27.9 kg.m−3, is plotted in Fig. 4c. lar to that of Roach et al. [2016] shown in Fig. Our map appears realistic and shows similar 5b, who used the dispersion of Argo floats at features to the estimate of Ke by Cole et al. 1000 m to directly estimate cross-stream dif- [2015] using estimate of the mixing length ob- fusivity. Roach et al. [2016]’s estimate shows tained by considering the decorrelation length- peaks in similar locations to ours, with sim- scale of salinity fluctuations measured by Argo ilar magnitudes, although our estimates of floats. In particular, we note enhanced regions Ke are substantially lower than theirs at the of Ke downstream of large topographic features Campbell Plateau once the suppression factor where both streamfunction fluctuations are is applied. The differences between our esti- strong and time-mean flows are weak. When mates may arise due to the different formula- zonally integrated and mapped back to depth tion used in the estimate, but more likely due coordinates, as shown in Fig. ??, we see that to the fact that the Roach et al. [2016] estimate was made at 1000 m, whereas the isopycnal the unsuppressed diffusivity K0 is strong at the −3 surface and decreases with depth (Fig. ??a). In γ=27.9 kg.m is closer to 1500 m in the ACC contrast, Ke is enhanced at depth, reaching a (see Fig. 3a). peak at about 1000m. This peak in Ke is found very close to the steering level (where cp ≈ u) predicted by Smith and Marshall [2009] using linear theory, and that observed by Cole et al. [2015], although it is shallower than steering We compare our estimated diffusivities with level found in Abernathey et al. [2010]’s eddy several estimates made near Drake Passage by permitting simulation (found at about 1750m). direct measurement [Naveira-Garabato et al., To underscore the important role that bottom 2007, Faure and Speer, 2012, LaCasce et al., bathymetry plays in controlling the diffusivity, 2014, Tulloch et al., 2014]. We find that our esti- we plot K0 (red) and Ke (black) on the isopycnal mate of the effective diffusivity agrees broadly γ=27.9 kg.m−3 in Fig. 5a, but now meridion- with these other estimates, although we note ally averaged from the southern boundary of that our estimates are significantly higher than the ACC to the northern boundary of the ACC those of Faure and Speer [2012] and somewhat (determined by finding contours of MDT that lower than those of Naveira-Garabato et al. correspond to the Southern ACC Front and [2007]. However, given the difficulty in esti- the Subantarctic Front, as in Sokolov and Rin- mating certain parameters in the suppression toul [2007], plotted as solid lines in Fig. 4). The factor, the reasonably close agreement between zonal mean of the K0 and Ke (dashed lines in Fig. our estimate of Ke and previous local or re- 5) show that the suppressing effect of the mean- gional estimates gives us some confidence in flow acts to reduce the diffusivity by about our maps of eddy diffusivity.

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Figure 4: Meridional isopycnal diffusivity in the Southern Ocean on isopycnal γ=27.9 kg.m−3: (a) the variance of the isopycnal geostrophic streamfunction M0 M0; (b) the diffusivity suppression factor 2 1/ 1 + k2 τ2 c − u ; and (c) the suppressed eddy diffusivity K˜. Solid lines indicate the po- eddy eddy p sitions of the northern and southern ACC boundaries.

V. Reconstruction of the MOC time-mean isopycnal layer thickness h by simply taking the difference in the depths of the As described in Section II the MOC can be isopycnal layer interfaces: decomposed into an time-mean Ekman compo- h = z − z (24) nent hvEkman, a time-mean geostrophic com- j j+1/2 j−1/2 ponent hv , and the transient eddy component g where j is the index of the jth isopycnal layer. 0 0 h vg. In this section, we compute each of these Finally, the results are integrated zonally and components from observations in order to re- vertically to give the mean overturning stream- construct the residual overturning streamfunc- functions, ΨEkman, Ψg and the total mean tion and described how it is inﬂuenced by the overturning Ψ = ΨEkman + Ψg. These stream- spatial variation and suppression of the diffu- functions are plotted in Fig. 6. sivity. The zonally integrated Ekman driven overturning ΨEkman, shown in Fig. 6a, consists of i. Eulerian Mean Overturning a single clockwise overturning cell that transports around 20 Sv of water northwards at The components of the time-mean overturn- the surface, and drives a strong upwelling being are determined by computing the Ekman tween 65◦S and 55◦S, which corresponds to ageostrophic velocity from the equations for the unblocked latitudes of Drake Passage. The an Ekman spiral (Eqns. ?? and ??), and by Ekman circulation is largely opposed by the computing the time-mean geostrophic velocity mean geostrophic overturning, Ψg (Fig. 6b), ug from the absolute geostrophic streamfunc- that consists of a counter-clockwise overturn- tion, M and Eqns. 6. We then determine a ing cell with a peak transport of more than

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streamfunction computed from the combined altimetry/hydrography is giving realistic results. The total mean overturning Ψ, (Fig. 6c) shows the effect of compensation of the Ek- man driven overturning by the geostrophic overturning. As noted by Mazloff [2008] and Mazloff et al. [2013], in much of the region north of Drake Passage (i.e. north of 55◦S), the geostrophic component of the overturning dominates the Ekman transport, leading to a net southward transport of Circumpo- lar Deep Waters (those waters denser than about 27.5 kg.m−3), although it must be emphasized that within Drake Passage latitudes, the Ekman driven upwelling still dominates the geostrophic downwelling, and that the northward transport due to the Ekman currents remains dominant near the surface in the lighter water classes. Our results strongly echo those Figure 5: Variation of diffusivity with longitude: (a) The of Mazloff [2008], and stress the importance unsupressed diffusivity K (red) and the su- 0 of the interior geostrophic component for the pressed diffusivity K˜ (black) averaged over the ACC envelope (solid lines) and their zonal overturning. We note that this component is average (dashed lines) on isopycnal γ = often ignored in analyses of the overturning, 27.9 kg.m−3. For comparison, the estimates of and is not well incorporated into TEM theories the diffusivity from previous studies are shown of the overturning. as solid markers; and (b) the one and two par- ticle estimates of the diffusivity from Roach et al. [2016] at 1000 m depth, included for ii. Eddy Overturning comparison. We now discuss the contributions of transient geostrophic eddies to the MOC. Here we employ the simple downgradient diffusive closure 40 Sv near γ =28.0 kg.m−3. The geostrophic given by Eqn. 9. To understand the influence overturning cell additionally drives a weak of the suppression of the eddy diffusivity by downwelling in the Drake Passage latitudes. the mean flow on the overturning, we recon- Both the geostrophic and Ekman overturning struct the eddy volume flux using both the un- cells show strong similarity to those obtained suppressed diffusivity K0, and the suppressed from the data-assimilating Southern Ocean diffusivity Ke. State Estimate (SOSE) model [Mazloff, 2008, The longitudinal/vertical structure of merid- Mazloff et al., 2013] (in particular, see Fig. ional IPV gradient, and its relationship with 4-5 of Mazloff [2008]). The concordance be- the parameterized eddy fluxes is plotted in Fig. tween SOSE and our estimates is perhaps not 7, which shows the meridional IPV gradient surprising, given that SOSE assimilates both (Fig. 7a), and estimates of the eddy volume flux Argo hydrographic data and satellite altime- using both suppressed and unsuppressed dif- try. However, comparing our results to the fusivities (Fig. 7b,c), averaged over the ACC en- SOSE output does give some confidence that velope. Despite the argument that IPV should the analysis of the hydrographic profiles has be relatively homogenized in the ocean interior been conducted correctly and that the absolute [Marshall et al., 1993], we find substantial IPV

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ports computed with the unsuppressed (Fig. 7b) and suppressed (Fig. 7c) diffusivities. As expected, the eddy volume transports are much larger when the unsuppressed diffusivity is used in the reconstruction. This is particularly evident in the region near the base of the thermocline where the IPV gradient changes sign. When mean-flow suppression is taken into account, the majority of the near surface transport disappears. Additionally, the vertical structure of the transport varies between the suppressed and unsuppressed cases. The unsuppressed transport shows a vaguely equiva- lent barotropic structure, while the suppressed eddy transport shows minimal interior transports away from the Pacific-Antarctic Rise (between 150◦W and 130◦W) and Drake Passage (between 40◦W and 30◦W). While the unsuppressed transport is typically strongest near the surface, the interior suppressed transport Figure 6: The time-mean overturning streamfunction: is intensified near the critical layer (at approx- (a) the Ekman overturning ΨEkman; (b) the imately 1000 m depth, indicated by the solid geostrophic overturning Ψ ; and (c) the total g black line in Fig. 6). Deep transports are, in time-mean overturning Ψ = Ψ + Ψ . Ekman g general, southward. Positive (negative) values indicate clockwise (counter-clockwise) transport, as indicated by In Fig. 8, we plot the parameterized zon- the arrows. ally integrated eddy overturning streamfunction computed using the unsuppressed (Fig. 8a) and the suppressed (Fig. 8b) diffusivi- gradients in certain regions, particularly down- ties, as well as the difference between them stream of large bathymetric features, a fact that (Fig. 8c). We note that although the param- has been remarked upon by previous authors eterization used here is extremely crude, we [Thompson and Naveira Garabato, 2014]. As a are able to capture a surprisingly large degree result, both the unsuppressed and suppressed of the eddy-overturning streamfunction com- eddy fluxes (Fig. 7b,c) are concentrated in re- puted from the eddy-permitting SOSE model gions donwstream of topography, which is also [Mazloff, 2008, Mazloff et al., 2013]. In particu- consistent with previous work [Thompson and lar, the overturning streamfunction is generally Sallée, 2012, Dufour et al., 2015, Chapman and clockwise in a latitude-density plane, for both Sallée, 2016]. Additionally, we note that there is suppressed and unsuppressed diffusivities. We a change in the sign of the IPV gradient in the note a weak northward flow in the light, near- lighter, surface waters, leading to a northward surface waters that generally reinforce the Ek- volume transport near the surface, in contrast man currents, with upwelling in the Drake to the southward eddy transport in the inte- Passage latitudes. In contrast to the SOSE out- rior. The northward eddy flux of light waters, put, our calculations show a general increase consistent with the negative near-surface IPV in the strength of the eddy overturning stream- gradients, was discussed in depth by Mazloff function with depth, although this feature is [2008]. not as strong (i.e. more consistent with SOSE) The suppression of the eddy-flux by the when the diffusivity is suppressed. We find mean-flow can be seen by comparing the trans- southward overturning transports of around

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Figure 7: The relationship between the meridional IPV gradient and the parameterized eddy ﬂux averaged over the ACC latitude envelope: (a) The meridional IPV gradient; (b) the derived geostrophic eddy-volume ﬂux h0v0, computed from Eqn. 9 using the unsuppressed eddy diffusivity K0; and (c) as in 7b, using the effective eddy diffusivity K˜. Solid black lines denote the average 1000m depth contour, the approximate critical layer depth. Note the differing colorscales between the unsuppressed (panel b) and suppressed (panel c) transport.

10 Sv at γ = 27.0 kg.m−3 at 55◦S when com- clear from Fig. 9a that the vertical structure puted with suppressed diffusivities, increasing of the meridional transport obtained using K0 to around 45 Sv at γ = 28.0 kg.m−3 when us- resembles those obtained using constant diffu- ing the unsuppressed diffusivity. These trans- sivities, with relatively strong southward transport are different by about a factor two for the ports in the ocean interior that peak at γ =26.7, suppressed diffusivity case (K = Ke): around 26.9 and 27.5 kg.m−3. 5 Sv at γ = 27.0 kg.m−3 at 55◦S, increasing to In contrast, the interior southward trans- around 20 Sv at γ = 28.0 kg.m−3. For com- port determined using the suppressed eddy- parison, Mazloff [2008] finds maximum eddy diffusivity is much more modest and has a dif- overturning transport of between 10 and 25 Sv, ferent vertical structure, reaching a peak near depending on the season. the critical layer, which occurs at approximately To further investigate the influence of the 27.8 kg.m−3. Although the suppressed trans- three-dimensional diffusivity on the MOC, we port shows a peak transport near 26.9 kg.m−3, compare the zonally integrated eddy volume similar to the unsuppressed case, the sup- transport v0h0 (i.e. the transport itself, not the pressed transport on this isopycnal is factor streamfunction) computed using our diffusiv- of 4 smaller in magnitude than that of the unity estimates, to the transport obtained assum- suppressed transport. In short, the mean flow ing constant diffusivities of between 500 and of the ACC strongly suppress the intensity of 3500 m2s−1, meridionally averaged over the eddy-diffusion, which dramatically reduces the ACC envelope (Fig. 9a). A similar zonal and southward interior geostrophic eddy-induced meridional averaging is applied to the spatially transport, and concentrates it in the denser wa- variable diffusivities K0 and Ke (Fig. 9b). It is ter masses near the critical layer. Assuming

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Figure 9: The meridional geostrophic eddy transport 0 0 h v h = K q ∂q/∂y, using variable and con- Figure 8: The geostrophic eddy overturning streamfunc- stant diffusivities. (a) the meridional trans- tion computed from Eqn. 7. (a) Ψ? computed port, zonally integrated and averaged over the ? using the unsuppressed diffusivity K0; (b) Ψ ACC latitudes, obtained from Eqn. 9 using the computed using the suppressed diffusivity K˜ suppressed and unsuppressed spatial variable (c) The difference between the eddy overturn- diffusivity (thick black lines, K = K0 dashed, ing streamfunctions computed using K0 and K = Ke solid) and using a constant diffusiv- K˜. Positive (negative) streamfunction values ity (thin colored lines) between 500 m2s−1 denote counter-clockwise (clockwise) flow - as and 2500 m2s−1. Southward transports are indicated by the black arrows. Note the differ- negative, northward positive, zero transport ent colorscale used for panel (c). is indicated by the thin grey line; and (b) the suppressed (solid) and unsuppressed (dashed) diffusivities, zonally averaged over the entire that the simple parameterization used here is Southern Ocean and meridionally avereaged valid, it is clear that the modification of the ver- in the ACC lattitudes. tical structure of the diffusivity has important implication for the Southern Ocean overturning. overturning streamfunctions show numerous features in common with those computed from iii. The Residual Overturning sophisticated numerical models [Dufour et al., 2012, Mazloff et al., 2013, Zika et al., 2013]. For With the time-mean and eddy components of both the suppressed and unsupressed diffusiv- the overturning in hand, we are now able to ities, the residual overturning streamfunction reconstruct the total residual meridional circu- is generally clockwise, with upwelling in the lation (Eqn. 4) and its overturning streamfunc- Drake Passage latitudes and northward flow tion (Eqn. 5). The zonally integrated residual in the lighter water masses near the surface. overturning streamfunction Ψres is shown us- Although our dataset does not sufficiently sam- ing the unsuppressed diffusivities in Fig. 10a, ple waters denser than about 28 kg.m−3 and using the suppressed diffusivities in Fig. 10b, thus cannot resolve the northward abyssal cell, and the difference between them in Fig. 10c. there is some suggestion of northward flow Firstly, we note that our estimated residual closing the clockwise cell at about 60◦S, and

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tion [Mazloff, 2008, Mazloff et al., 2013], we find that when using the suppressed diffusivities, the clockwise deep overturning cell is weaker in the region to the north of Drake Passage, and that the zero Sverdrup transport line that separates the deep overturning cell from the Antarctic Bottom Water (AABW) cell is shallower in our observation-based estimate, contrary to the overturning obtained using the unsuppressed diffusivities where the southward cell is stronger and deeper than the SOSE- based estimate. It is likely that the eddy volume flux of the present study estimated using the unsuppressed diffusivities is too strong, while that estimated using the suppressed diffusivities is too weak. However, we note that the strong decrease of the eddy flux that occur when using a suppressed diffusivity leads to a residual overturning in closer agreement with the results of numerical models, particularly in Figure 10: The residual overturning streamfunction the Drake Passage latitudes. Ψres reconstructed from the observations. (a) The residual overturning streamfunction clac- We can explore the distribution of the merid- ulated using the unsuppressed diffusivity K0; ional volume transport throughout the South- (b) as in 10a, this time calculated using the ern Ocean by calculating the vertically inte- suppressed diffusivity K˜; and (c) the differ- grated cumulative transport (determined by in- ence between the Ψres and Ψres. Arrows K0 K˜ tegrating the transport along lines of constant indicate the sense of the overturning. latitude) for each of the contributing components, averaged over the ACC envelope, shown in Fig. 11. Here, we can gauge the influence ◦ in the blocked latitudes north of about 50 S. of the Southern Ocean’s bathymetry on the We find a peak overturning transport of about meridional transport, as well as see how the 60 Sv when using the unsuppressed diffusivi- diffusivity suppression influences the transport ties, and a more realistic 30 Sv when using the around the Southern Ocean. We see that, sim- suppressed diffusivities. Unsurprisingly, the ilarly to the cross-frontal transport computed clockwise overturning cell is much stronger from a high-resolution numerical model by Du- with unsuppressed diffusivities, and the north- four et al. [2015], the time-mean geostrophic ward flow is found in denser (deeper) levels, transport (solid blue line) is concentrated in re- as can be seen in Fig. 10c. gions close to large bathymetric features, with Although our reconstructions show numer- step-changes in the transport near the Kergue- ous realistic features, it is clear that a perfect len Plateau, the Campbell Plateau, and the reconstruction eludes us. One illustration of Pacific Antarctic Rise (indicated in Fig. 11). this imperfection is that our residual stream- As discussed previously, the mean geostrophic function is quite noisy, although it should be transport is primarily southward, balancing noted that Mazloff [2008] also produced a noisy the northward Ekman transport (solid ma- residual streamfunction when attempting a re- genta line). However, there is a large north- construction from each of the individual com- ward mean transport that occurs as the ACC ponents from SOSE model output. When com- passes through Drake Passage, largely associ- paring our results with the SOSE reconstruc- ated with the strong western boundary current

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that forms along the coast of South America. This northward transport balances the majority of the accumulated southward transport, resulting in effectively zero total time-mean geostrophic transport. In contrast, both the suppressed and unsuppressed eddy transport show a relatively uniform southward transport throughout the Southern Ocean, with a series of step changes of enhanced southward transport near certain bathymetric features, most clearly seen in the unsuppressed transport1. The eddy volume ﬂux concentrated in the step-changes downstream of large bathymetric features corresponds to the locations “storm tracks” or “mixing hot spots” identiﬁed by previous studies [Thompson and Sallée, 2012, Dufour et al., 2015, Chapman et al., 2015]. However, these concentrated regions of southward transport are generally limited in magnitude, being between 2-5 Sv for the unsuppressed diffusivities and 5-10 Sv for the suppressed diffusivities.

VI. Influence of a Vertically Varying K in a Simple Conceptual Model

In lieu of a constant diffusivity K, we employ in Eqn. 14 a diffusivity with a simple vertically varying structure:

2 − (z−zc) ? 2h Ke(z) = K0 + K e K . (25)

In this form, the diffusivity is enhanced at depth, with a peak amplitude of K? at the critical level zc. The vertically varying part of K is superposed over a constant background diffusivity K0, reminiscent of the diffusivity calculated from the observations (see Fig. ??b). The model is run over a broad parameter space, with critical layers ranging from 750m to 2000m depth, and peak K? from 500 to 3500 m2.s−1, a similar range to those suggested by Abernathey et al. [2010]. The background diffusivity K0 is

1We note that if the eddy transport is computed across the mean streamlines as opposed to meridionally, the constant southward transport disappears and the eddy transport instead manifests as a series of step changes.

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Figure 11: The longitudinal variation of the vertically integrated cumulative meridional transport and each of its components, averaged over the ACC latitudes. The ageostrophic Ekman transport vEKh (magenta); the 0 0 h mean geostrophic transport hvg, the geostrophic eddy transport vgh ≈ K q ∇q (red) for the suppressed 0 0 (solid) and unsuppressed (dashed) diffusivities; and the total transport vh = vEKh + hvg + vgh (black) for the suppressed (solid) and unsuppressed (dashed) diffusivities.

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2 −1 set to 250 m .s and the vertical scale, hK, is set to 500 m. Additionally, we run the model with a constant, vertically invariant K, ranging from 500 to 3500m2.s−1. Fig. 12 contrasts the results of the TEM model using a constant eddy diffusivity K=1500 m2.s−1 and using the vertically varying K with peak amplitude K? of 1500 m2.s−1. Both the buoyancy field (Fig. 12a) and the residual overturning streamfunction (Fig. 12b) show the same basic structure for the constant diffusivity (Fig. 12i) and the vertically varying diffusivity (Fig. 12ii), but with some important differences. Principally, the isopycnal inclination is greater in the case with vertically varying Ke than with constant K. Secondly, the maximum Ψres is about 10 Sv larger in the vertically varying case. Since the time-mean overturning is iden- Figure 12: The effect of a vertically varying diffusivity tical in both cases, we must conclude that the on the stratification and overturning in a opposing eddy-overturning is weaker for verti- conceptual model. The simulated zonally cally varying Ke, despite the increased isopycnal averaged buoyancy field (a); and residual tilt that should, by Eqn. 13, lead to a higher overturning streamfunction Ψres (b) from eddy volume fluxes. As such, it seems that the TEM model for a case with (i) a con- the principle result of the suppression of the stant eddy diffusivity K = 1500 m.s−2; eddy-diffusivity in the ocean interior is to re- and (ii) vertically varying eddy diffusiv- duce the eddy induced overturning, which in ity, given by Eqn. 25 with a maximum ? −2 turn results in a steeper isopycnal slope. diffusivity of K = 1500 m.s . Solid black lines indicate contours of (a) buoyancy To underline this point further, Fig. 13 shows (CI: 1.0×10−3 m.s−2); and (b) overturning the influence of the varying the critical layer streamfunction (CI: 10Sv). The black arrows ? zc and the peak eddy diffusivity K on the indicate the sense of the overturning. stratification and the residual overturning, and how the results using a vertically varying diffusivity differ from those with a constant dif- “felt" on this isopycnal (who’s depth is con- fusivity. Fig. 13a shows the depth of a repre- strained to be 1200 m on the northern bound- sentative isopycnal (in this case b=0.2 m.s−2, ary), at this latitude, increases, resulting in a which is found at 1200m depth on the north- flattening of the isopycnal. The exact response ern boundary) at approximately 45◦S. It can be of the isopycnal depth depends on the choice seen clearly that this isopycnal shallows with of isopycnal, and where it lies in relation to the increasing diffusivity and that the shallowing critical layer. As such, changes in the zc can of isopycnals appears to approach a limit with modify vertical density gradient in the interior increasing K. It can also been seen in Fig. 13a and, hence, the thickness of isopycnal layers. that as the critical layer depth increases (solid The residual overturning streamfunction is colored curves), the depth of the isopycnal also also sensitive to changes in zc and K. Fig. increases. When K is constant (dashed curve) 13b shows the maximum value of the resid- the isopycnal depth tracks closely the curve as- ual streamfunction at the northern boundary sociated with the shallowest critical layer con- for each model run. Here, we note that in gen- sidered here (750 m), except at low values of eral, the overturning streamfunction decreases K. As the critical layer deepens, the diffusivity approximately linearly with increasing K for

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both the vertically varying K cases and the constant K case. Indeed, weaker eddy diffusivity reduces the efficiency of the eddy transport to counterbalance the mean transport, which results in higher residual overturning. How- ever, unlike the isopycnal depth shown in Fig. 13a, the sensitivity of the overturning transport to changes in K varies depending on the critical layer depth zc. For example, with a deep critical layer, of zc=2000 m (black solid line in Fig. 13b), the slope of the line is approximately -5×10−4 Sv/(m2.s−1), indicating almost no sensitivity to changes in K, while with a shallow critical layer of zc=750 m, the streamfunction is highly sensitive to changes in K: the slope is approximately -1×10−1 Sv/(m2.s−1), almost 3 orders of magnitude higher than than when zc=2000 m. As with the isopycnal depth, the constant K case shows the strongest similarity with the vertically varying K at the shallowest critical level, although the constant K shows a steeper curve that, at high diffusivities, results in the eddy overturning dominating the time-mean overturning and a reversal of the overturning sense. As in the observational part of our study, the principle effect of the introducing a vertical structure to the eddy diffusivity is to suppress the southward interior transports by eddy-fluxes. However, the details of this effect of this suppression on the resulting overturning depend critically on the depth of the critical layer, where the diffusivity is still large enough to enable substantial eddy fluxes. For example, when the critical layer is very deep (zc =2000 m), the enhanced diffusivity is deeper than the depths with substantial isopycnal slopes and hence the eddy induced transport is weak. When the critical layer is shallower, say at 750m or 1000m, as found in our observations, the enhanced diffusivity is found at levels with large isopycnal slopes or IPV gradients and a substantial eddy overturning can be supported, and hence the resultant residual overturning is sensitive to changes in the value of K. As the critical layer depth varies throughout the southern ocean [Smith and Marshall, 2009, Abernathey et al., 2010, Cole et al., 2015],

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Figure 13: The response of the stratiﬁcation (a) and the overturning streamfunction (b) to changes in the critical level depth and the eddy diffusivity K?. (a) The depth of the 0.2 m.s−1 buoyancy surface at 50◦S; and (b) the maximum residual overturning transport. Each solid curve corresponds to simulations run with with a different critical layer depth zc in Eqn. 25 (see legend in Fig. 13b). The dashed black curve (with ‘?’) corresponds to simulations run with a constant eddy diffusivity.

21 Submitted for publication in the Journal of Physical Oceanography this result could have important implications SOSE model, described in Mazloff et al. [2013]). for the local eddy-flux and its parameterization We find that the parameterized eddy fluxes, as in climate models. well as the time-mean geostrophic flows, are zonally asymmetric, being concentrated near, or downstream, of bathymetric features, in re- VII. Discussion and Conclusions gions corresponding to the mixing “hot spots” or “storm tracks” identified in previous stud- In this study, we have investigated how the ies. three-dimensional structure of the eddy diffusivity and its suppression by the time-mean One inherent limitation of our observation- flow, can influence the MOC in the Southern based approach is that diffusivity and stratifi- Ocean. Combining hydrographic observations cation are intrinsically related, while here we obtained with satellite altimetry, we have es- apply differing diffusivity on a fixed stratifica- timated the isopycnal eddy diffusivity, K, us- tion. To go beyond this limitation and further ing the framework of Ferrari and Nikurashin explore how the overturning responds to the [2010], in three-dimensions and including the depth varying structure of the eddy diffusiv- effect of suppression by the time-mean flow. ity, we use a simple conceptual model of the We obtain a K field that is highly spatially vari- Southern Ocean, based on that of Marshall and able. Large values of diffusivity are found in Radko [2003] and Marshall and Radko [2006]. regions downstream of large topographic fea- We find that, as in the observational part of tures, and K is suppressed in regions of strong this study, the addition of a vertical structure time-mean flow. When suppression is taken to the eddy diffusivity acts to suppress the into account, the diffusivity K reaches a peak interior southward eddy transport when com- at the critical layer, which we find to be at pared to model runs performed using a ver- about 1000 m. Using the estimate of the eddy tically constant K. The resulting stratification diffusivity, we are able to estimate the eddy and overturning circulation is also sensitive to volume flux on an isopycnal as a downgradi- the depth of the critical layer. As the critical ent diffusion of isopycnal potential vorticity. layer becomes shallower, the overturning trans- Additionally, using the approximate isopyc- port becomes more sensitive to changes in the nal streamfunction of McDougall and Klocker magnitude of the peak diffusivity, as the criti- [2010], we are able to estimate the time-mean cal layer and its associated region of high eddy geostrophic meridional circulation. Together diffusivities, is more likely to coincide with re- with the ageotrophic Ekman transport, we are gions with large isopycnal slopes or potential able to reconstruct the full upper ocean merid- vorticity gradients. ional circulation. The principle result of this study is that the We have focused on the effect of the suppres- mean flow of the Antarctic Circumpolar Cur- sion of K by the mean-flow on the resulting rent is critical in shaping the interior Southern overturning. By comparing our reconstruc- Ocean overturning circulation, not only driving tions of the overturning circulation with, and a significant time-mean geostrophic overturn- without, the effects of the time-mean flow sup- ing, a point emphasized by Mazloff [2008] and pression, we are able to show that the primary Mazloff et al. [2013], but also modulating the effect of the suppressed diffusivity is to dramat- efficiency the resulting eddy overturning circu- ically reduce the interior eddy flux, particularly lation. We find that the details of the overturn- in the intermediate and upper-circumpolar ing and interior stratification are sensitive to deep waters. Reconstructing the eddy overturn- both the magnitude of K, and also the depth of ing using either the unsuppressed diffusivity, the critical layer, which both depend on a sub- or a constant diffusivity, strongly overestimates tle balance between eddy characteristics and these interior volume fluxes (at least when com- mean-flow. The corollary of this result is that in pared to the output from the eddy-permitting order to reconstruct an overturning circulation

22 Submitted for publication in the Journal of Physical Oceanography using a downgradient parameterisation, cor- The output is provided annually. Addition- rectly representing the interior suppression of ally, the first two seasonal harmonics are esti- eddy diffusivity by the mean-flow is crucial. mated and are included in the output. In contrast, the zonal asymmetry, although important for the localisation of the volume Appendix B: Numerical Method for transport, is of second order importance when the Conceptual Model considering the zonally averaged circulation, as revealed by the fact that the structure of The primary equation that needs to be solved the zonally averaged meridional volume flux for the implementation of the conceptual TEM computed using the (spatially varying) unsup- model (Eqn. 14) has the form: pressed diffusivity is similar to that obtained using constant values of the K (see Fig. 9). The ∂b ∂b A(y, b, Ψres) + B(y, b, Ψres) = 0 (26) fact that the vertical structure of the diffusivity ∂y ∂z plays such an important role in parameterized eddy flux may have important implications for where A and B are coefficients that are func- coarse-resolution ocean models used for long- tions of the surface wind stress and the resid- period climate studies, as these models still ual overturning streamfunction, together with rely on downgradient turbulence closures such Dirichlet boundary conditions for b at z = and as Gent-McWilliams. Further research will fur- y = Ly. Using the method of characteristics, ther explore the role of the vertical diffusivity this linear partial differential equation can be structure in the response to climate change, written as the set of coupled ordinary differen- as well refining our estimate of the overturn- tial equations: ing circulation through the use of new data, dy = A(y, b, Ψres) (27) parameterizations and analysis techniques. ds dz = B(y, b, Ψres) (28) Acknowledgments ds db = 0 (29) The authors thank Dhruv Balwada, Jessica ds Masich, Andreas Klocker and Christopher dΨres = 0 (30) Roach for useful discussions and to Christo- ds pher Roach for helpfully providing the diffusivity data from Roach et al. [2016] for compar- where s is the distance along the characteristic ison with our calculations. CC was supported curve which, in this case, is simply the isopyc- by an NSF Division of Ocean Sciences post- nal b, together with the boundary conditions: doctoral fellowship Grant No. 1521508. J.B.S. b(y, z = 0) = g(y) (31) received support from Agence Nationale de la b(y = L , z) = f (y) (32) Recherche (ANR), ANR-12-PDOC-0001. y Eqns. 27–29 are solved using a 4th order Appendix A: Data Availability Runge-Kutta method. Boundary conditions are imposed using the shooting method: using All interpolated fields used in this study, in- large initial guesses of Ψres = ±100Sv, and cluding the estimates of the suppressed and starting at z = 0, Eqns. 27–29 are integrated unsuppressed eddy diffusivity; the neutral den- until y = Ly. We then compare the depth of sity, the approximate isopycnal geostrophic the isopycnal to the depth of that isopycnal ex- streamfunction and its variance; and the isopy- pected from the boundary conditions Eqn. 32 cnal potential vorticity, are available for down- and apply the bisection method to systemati- load in NetCDF format from the following cally adjust the guess of Ψres until convergence URL: to a predefined error tolerance (here, 5m).

23 Submitted for publication in the Journal of Physical Oceanography

Computer code to implement this model, Cole, S. T., C. Wortham, E. Kunze, and W. B. written in the open-source, Python program- Owens, 2015: Eddy stirring and horizon- ming language, is available under an open tal diffusivity from argo ﬂoat observations: source MIT license from CC’s Github account: Geographic and depth variability. Geophys- https://github.com/ChrisC28 ical Research Letters, 42 (10), 3989–3997, doi: 10.1002/2015GL063827.

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