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  • View in gallery
    Fig. 1.

    Eddy kinetic energy at 1000 dbar. The color shading shows (u2 + υ2)/2 in log scale. The thick contours show the depth contour of 3000 m from ETOPO5. Thin contours show sea surface height as calculated by the method described in section 2b. From south to north, the contours represent the Southern ACC Front, southern branch (SACCF-S); Southern ACC Front, northern branch (SACCF-N); Polar Front, southern branch (PF-S); Polar Front, middle branch (PF-M); Polar Front, northern branch (PF-N); Subantarctic Front, southern branch (SAF-S); and Subantarctic Front, middle branch (SAF-M) as defined in Sokolov and Rintoul (2007).

  • View in gallery
    Fig. 2.

    Spatial distribution of zonal form stress [(9)] at 1000 dbar. The dashed lines show latitudinal circles at 56.5° and 62.5°S. Positive stress indicates a downward flux of eastward momentum. Form stresses with magnitude less than ρ0fΔhΔu were not plotted, where Δh and Δu are the uncertainties in heave and in drift velocity (section 2a), respectively.

  • View in gallery
    Fig. 3.

    Zonal distribution of the form stress integrated between 56.5° and 62.5°S and in the upper 1000 dbar. The form stress was first integrated at each longitude and accumulated eastward from 0°, so that the value at 360° longitude was the form stress integrated in the box (8.1 × 1011 N). When integrated, the uncertainty due to measurement error was small (0.5 × 1011 N) compared to the uncertainty due to sampling (1.9 × 1011 N). The dotted line shows the similarly accumulated zonal wind stress (17.1 × 1011 N when integrated). The dashed–dotted line shows the planetary vorticity advection by the geostrophic component of the meridional velocity in (13), multiplied by 0.2 for scaling purposes (0.7 × 1011 N, unscaled, when integrated). The bottom topography meridionally averaged at each longitude is also shown (Kerguelen = Kerguelen Plateau, S.E.Ind. = South East Indian Ridge, Pac.A. = Pacific Antarctic Ridge, and Drake = Drake Passage).

  • View in gallery
    Fig. 4.

    Interface height of a two-layer, quasigeostrophic analytical model of a wind-driven current over a meridional ridge [after Abernathey and Cessi (2014, their Fig. 10) © American Meteorological Society. Used with permission.].

  • View in gallery
    Fig. 5.

    Form stress accumulated after sorting. Estimated form stress in each grid was sorted in ascending order and integrated: 6.4 × 1012 m2 of grids had negative values and 8.9 × 1012 m2 of grids had positive values. These comprised 41% and 59% of the total area, respectively, between 56.5° and 62.5°S. Negative grids integrated to a form stress of −7.8 × 1011 N with sampling error estimated by the bootstrap method of (1.3/−1.1) × 1011N. From this negative contribution, −2.2 × 1011 N came from those grids with values less than the observation error of ΔhΔυ. Positive grids integrated to a form stress of 15.9 × 1011 N with sampling error estimated by the bootstrap method of (1.4/−1.6) × 1011 N. From this positive contribution, 2.7 × 1011 N came from those grids with values less than the measurement error of ΔhΔυ. Overall form stress was 8.1 × 1011 N with sampling error of ±1.9 × 1011 N and measurement error of 0.5 × 1011 N.

  • View in gallery
    Fig. 6.

    Streamlines (thick lines) are given as F(x, y) = const. A local circle (dashed line) sharing a point and a tangent with the streamline defines the radius of curvature R. Length Δl of an infinitesimal segment of the streamline is given by Δl = RΔθ.

  • View in gallery
    Fig. 7.

    Spatial distribution of curvature [(18)] of streamfunctions of geostrophic flow field in the upper 1000 dbar. Thick contours show depth contour of 3000 m. Thin contours show sea surface height representing fronts, as in Fig. 1. Color shading indicates curvature calculated by (18) from the geostrophic transport in the top 1000 dbar. Positive curvature is cyclonic or clockwise. Green boxes designate seven topographic barriers that ACC surmounts.

  • View in gallery
    Fig. 8.

    Spatial distribution of perturbation Δκ to curvature [(19)] by eddies indicated by color shading. Gray indicates areas where data are missing. Thin contours show sea surface height representing the fronts as in Fig. 7. Thick contours show depth contour at 3000 m. The region for each panel corresponds to the green boxes in Fig. 7.

  • View in gallery
    Fig. 9.

    Comparison of eddy kinetic energy (abscissa) of approximately transient eddies (transient with spatial scales less than 22 km zonally and 11 km meridionally) and that (ordinate) of eddies used in the main analysis (transient with spatial scales less than 300 km). Colors indicate the box the eddy belongs to, defined in Fig. 7.

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Eddies Observed by Argo Floats. Part II: Form Stress and Streamline Length in the Southern Ocean

Katsuro KatsumataResearch and Development Center for Global Change, Japan Agency for Marine-Earth Science and Technology, Yokosuka, Japan

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Abstract

Argo floats measure horizontal current velocities at the parking depth and vertical profiles of temperature and salinity. These data were used to study the roles that eddies play in the dynamics of the Antarctic Circumpolar Current (ACC) in the Southern Ocean. A zonal momentum budget was quantified in a box spanning the latitudes of the Drake Passage and bounded by the sea surface and the 1000-dbar depth. The input of eastward zonal momentum from the wind (17.1 × 1011 N) was approximately twice the downward transfer of eastward momentum across the isopycnal whose mean depth was 1000 dbar, which was mediated via form stress carried by eddies [(8.1 ± 1.9) × 1011 N]. The zonal momentum budget was closed to within uncertainty, meaning that the momentum not accounted for by eddies was explained by the Coriolis term associated with meridional transport. The form stress was spatially concentrated near meridional ridges, particularly on their eastern flanks. The localization was extreme: 7% of the total area contributed about 90% of the form stress. Lengths of streamlines were stretched around steady standing meanders. Seven major meanders were found at large topographic barriers along the ACC, with cyclonic meander collocated with the peaks of the topographic barriers. Eddies were found to lengthen the streamlines mostly on the eastern flanks of the meridional ridges, where the eddy transport was southward. Poleward eddy transport on the eastern flanks of meridional ridges is thus highlighted in the ACC dynamics in transferring eastward zonal momentum downward and in adjusting to wind changes by stretching streamlines.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Katsuro Katsumata, k.katsumata@jamstec.go.jp

Abstract

Argo floats measure horizontal current velocities at the parking depth and vertical profiles of temperature and salinity. These data were used to study the roles that eddies play in the dynamics of the Antarctic Circumpolar Current (ACC) in the Southern Ocean. A zonal momentum budget was quantified in a box spanning the latitudes of the Drake Passage and bounded by the sea surface and the 1000-dbar depth. The input of eastward zonal momentum from the wind (17.1 × 1011 N) was approximately twice the downward transfer of eastward momentum across the isopycnal whose mean depth was 1000 dbar, which was mediated via form stress carried by eddies [(8.1 ± 1.9) × 1011 N]. The zonal momentum budget was closed to within uncertainty, meaning that the momentum not accounted for by eddies was explained by the Coriolis term associated with meridional transport. The form stress was spatially concentrated near meridional ridges, particularly on their eastern flanks. The localization was extreme: 7% of the total area contributed about 90% of the form stress. Lengths of streamlines were stretched around steady standing meanders. Seven major meanders were found at large topographic barriers along the ACC, with cyclonic meander collocated with the peaks of the topographic barriers. Eddies were found to lengthen the streamlines mostly on the eastern flanks of the meridional ridges, where the eddy transport was southward. Poleward eddy transport on the eastern flanks of meridional ridges is thus highlighted in the ACC dynamics in transferring eastward zonal momentum downward and in adjusting to wind changes by stretching streamlines.

© 2017 American Meteorological Society. For information regarding reuse of this content and general copyright information, consult the AMS Copyright Policy (www.ametsoc.org/PUBSReuseLicenses).

Corresponding author: Katsuro Katsumata, k.katsumata@jamstec.go.jp

1. Introduction

In Katsumata (2016; hereinafter Part I), it was shown that temperature and salinity profiles and trajectory data from an Argo float can provide estimates of water velocity and density at the parking depth (typically 1000 dbar). With an appropriate definition of a mean field, eddies can be defined as the deviations from the mean, and their contribution to upper-ocean general circulation can be calculated. Part I examined eddy mass transport in the upper ocean in detail and found strong eddy transport along the Northern Hemisphere western boundary currents, in the southern Indian Ocean, and in the Southern Ocean.

Eddies play important roles in shaping large-scale circulation, both zonal and meridional, particularly in the Southern Ocean. The roles include diffusion of momentum and tracers (e.g., Klocker et al. 2012), meridional overturning circulation without a large-scale zonal pressure gradient [see Marshall and Speer (2012) for a review], and winter subduction (e.g., Sallée et al. 2010). In this Part II, we focus on two metrics of eddy activity in the Southern Ocean: form stress and length of streamlines.

Form stress transfers zonal momentum downward. The momentum is injected at the surface from the eastward wind stress and is eventually transferred to the solid earth through bottom topography (Munk and Palmén 1951). It is necessary to resolve eddies to study form stress. For this reason, previous analyses of form stress have been limited to regional observations (e.g., Johnson and Bryden 1989; Phillips and Rintoul 2000) or studies of numerical simulation studies (e.g., Stevens and Ivchenko 1997) and state estimates (e.g., Masich et al. 2015). Because the integral of the momentum along a contour ∂Ω enclosing an area Ω is equal to the areal integral of vorticity through Stokes’ theorem
eq1
one could classify vorticity analysis as a way of studying momentum budgets. The latter have, again, been mostly regional (e.g., Firing et al. 2016) or numerical (e.g., Marshall et al. 1993). In section 3, we study the distribution of form stress in the circumpolar Southern Ocean at middepth (1000 dbar) by using data from Argo floats. Although determined at one fixed depth, the momentum budget in the upper 1000 dbar is discussed quantitatively. A subtle difference between the momentum balance in the whole water column and in the upper 1000 dbar is discussed based on the results.

Another quantity of interest is the length of streamlines around standing meanders. Using the equivalent barotropic nature of the Antarctic Circumpolar Current (ACC), Hughes (2005), in one of the rare observational works that are not regional, studied the vorticity balance in the Southern Ocean by using surface velocity observations from drifters. He demonstrated that these standing meanders can be explained as nondivergent barotropic Rossby waves trapped by the eastward flow of the ACC. Thompson and Naveira Garabato (2014) studied standing meanders in an eddy-resolving numerical simulation and found a link between the length of buoyancy contours and barotropic adjustment to increased wind forcing through vertical velocity. The lengthening can be interpreted as an increase in amplitude and/or wavelength of the trapped Rossby wave to accommodate changes in advection in response to the increase of wind forcing. Abernathey and Cessi (2014) also found lengthening of buoyancy contours with increased wind forcing in an idealized model of the ACC over a meridional ridge. In section 4, we investigated whether this perturbation to streamline length can be found in the eddy field estimated from the Argo data.

We note that because the ACC is generally equivalent barotropic (Killworth 1992; Kim and Orsi 2014) and in geostrophic balance, the buoyancy contours at constant depth are generally parallel to the streamlines, although recently some local departures from equivalent barotropic structure have been reported (Graham et al. 2012; Phillips and Bindoff 2014). Our discussion is necessarily qualitative because of the limitation (i.e., density) of the data, but we argue that eddies around seven major standing meanders tend to stretch the streamlines, particularly on the eastern flanks of meridional barriers. We summarize the discussion in section 5.

2. Data

Online locations of the data are listed in the acknowledgments section of this paper.

a. Argo data

From two consecutive Argo float hydrographic profiles, an estimate of the velocity at the parking depth (mostly 1000 dbar) at the midpoint of the trajectory between the two profiles can be calculated. The estimated velocity is the displacement between the two surface positions divided by the surfacing time interval (mostly 10 days; Davis 1998, 2005). The estimated temperature and salinity profile at the midpoint between the two profiles is equated to the average of the two profiles. In the region south of 30°S, 253 990 such pairs of velocity and density estimates were calculated using the December 2015 version of the trajectory data, YoMaHa (Lebedev et al. 2007), and the Argo hydrography dataset. The uncertainty in eddy statistics has two components: the measurement uncertainty in each datum owing to uncertainties in the measurement technique and the sampling (or statistical) uncertainty because the number of data is not infinity. The measurement uncertainties for the velocity and density were 0.5 cm s−1 and 0.020 kg m−3, respectively. Sampling uncertainty was estimated using the bootstrap method (Efron and Gong 1983) at a confidence interval of 68%. The reader is referred to Part I (section 2a) for more details of the Argo data processing and uncertainty estimates.

We estimated eddy statistics on a grid. A circle with a radius of 150 km was used to define a mean field centered on a grid point. This choice was made to separate two distinct length scales in the ACC dynamics, namely, standing meander scales of 300–500 km and transient eddy scales of 10–20 km (Hughes 2005). At the same time, we needed to have sufficient data density to carry out the quantitative analysis in section 3 and to resolve the spatial structure of eddy fluxes in section 4. In accordance with Davis (2005), the radius of the averaging circle was adjusted by a factor of , where H1 and H2 are the water depths on the circle and at the center satisfying H1H2 > 0. These two depths were used to account for the longer correlation scale in the long-isobath direction than in the cross-isobath direction. The number of data was not sufficient for temporal analysis, and we averaged all data from July 1997 (first trajectory data) to December 2015 within the circle. Based on this averaging, we defined an eddy as a perturbation with a length scale less than approximately 300 km and a time scale shorter than decadal. This definition includes both transient and standing components, which could have been clearly separated in the output of a numerical model. The grid interval, which was 1° in latitude and longitude, was shorter than the averaging radius, and data from neighboring grids were statistically correlated. When the sampling error was calculated (e.g., in Fig. 5), this correlation was accounted for by reducing the degrees of freedom by the ratio of grid area to averaging area.

Suppose that n = 1, 2, …, N of velocity data un were found within an averaging circle centered at a grid point (i, j). The mean velocity at (i, j) is
e1
The eddy kinetic energy is
e2
which is plotted in Fig. 1. Eddies were active where the ACC interacted with subtropical western boundary currents (Falkland/Malvinas Current at 60° to 30°W and Agulhas Current at 30°E) and where the ACC was steered by bottom topographic barriers (Kerguelen Plateau at 50° to 70°E and Southeast Indian Ridge to Campbell Plateau from 140°E to 170°W).
Fig. 1.
Fig. 1.

Eddy kinetic energy at 1000 dbar. The color shading shows (u2 + υ2)/2 in log scale. The thick contours show the depth contour of 3000 m from ETOPO5. Thin contours show sea surface height as calculated by the method described in section 2b. From south to north, the contours represent the Southern ACC Front, southern branch (SACCF-S); Southern ACC Front, northern branch (SACCF-N); Polar Front, southern branch (PF-S); Polar Front, middle branch (PF-M); Polar Front, northern branch (PF-N); Subantarctic Front, southern branch (SAF-S); and Subantarctic Front, middle branch (SAF-M) as defined in Sokolov and Rintoul (2007).

Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0072.1

b. Satellite altimeter data

As discussed in the introduction, the sea surface height contour is a good proxy for the streamlines of the ACC (Killworth 1992; Kim and Orsi 2014). Following Sokolov and Rintoul (2007), sea level height anomalies from the AVISO (delayed time, merged gridded) products were added to the climatological dynamic heights referenced to 2500 dbar. The daily altimeter data were averaged from 1 January 2000 to 31 December 2015. The dynamic height was calculated from the Commonwealth Scientific and Industrial Research Organisation (CSIRO) Atlas of Regional Seas 2009 (CARS2009) climatology (Ridgway et al. 2002) by using the International Thermodynamic Equation of Seawater—2010 (TEOS-10; IOC et al. 2010).

c. Climatological transport in the upper 1000 dbar

Climatological transport in the upper 1000 dbar was calculated from geostrophic velocities estimated from the climatological absolute pressure field pA. The pressure pA is the sum of the dynamic height referenced to 1000 dbar and the geostrophic pressure at 1000 dbar. The former was calculated from the temperature and salinity in the CARS2009 climatological data (Ridgway et al. 2002). The latter was a grid product G-YoMaHa estimated with an optimal interpolation method constrained by the Argo drift data (Katsumata and Yoshinari 2010).

d. Atmospheric reanalysis

Sea surface wind stress was obtained from the ERA-Interim synoptic monthly means (Dee et al. 2011) and further averaged from January 2000 to December 2015.

3. Form stress at 1000-dbar depth

a. Formulation

We employ the temporal residual mean (TRM) formulation (McDougall and McIntosh 2001) to separate the mean and eddy. In the TRM averaging, the basic nature of oceanic flows that adiabatic flow follows neutral density surfaces is preserved such that eddy flux across a mean neutral surface appears only when there are genuine diabatic processes. As a result, the total (mean plus eddy induced) advecting velocity is the same for all tracers and momentum. Under simple Eulerian averaging, in contrast, the advecting velocity is generally different for different tracers.

The TRM form of the momentum equation has been derived by McDougall and McIntosh [2001, their (66)]. The zonal component is
e3
where the momentum source is , f is the Coriolis parameter, px is the zonal pressure gradient, ρ0 is a typical density (1025 kg m−3), and we have neglected the third- and higher-order terms in the perturbation amplitude [O(α3) in McDougall and McIntosh 2001]. Here, the TRM velocity is defined as
e4
We used two methods to average velocity: the temporal Eulerian mean at a fixed position and the thickness-weighted averaging between a pair of closely spaced density surfaces. The difference between these two is the quasi-Stokes velocity and is expressed by a streamfunction :
e5
where V = (u, υ) is the horizontal component of the three-dimensional velocity U = (u, υ, w), and is the instantaneous height of the isopycnal surface located at a height of za when averaged over time. We used an approximate form of (5) correct to O(α3):
e6
The diffusivity tensor has been defined by Redi (1982):
e7
where ρ is locally referenced potential density with subscripts denoting derivatives, and the isopycnal and diapycnal diffusivities are A and εA, respectively.
Our data density restricted us to study phenomena with typical spatial scales of 100 km or larger. At these scales, it has been demonstrated using both simulations (e.g., Stevens and Ivchenko 1997) and observations (e.g., Phillips and Rintoul 2000) that the advection term [second term on the left-hand side of (3)] is negligible. We also ignored the diffusive effects of horizontal flows with scales smaller than eddies by letting → 0 for such flows. Wind stress drives the currents . With these simplification, (3) reads, after integrating between a depth of H(=1000 dbar) and the sea surface,
e8
where the surface boundary condition Ψy(z = 0) = 0 was used (McDougall and McIntosh 2001, their section 8). The second term on the right-hand side of (8) is the eddy pressure perturbation multiplying the interfacial slope because
e9
eq2
eq3
where geostrophic balance was assumed. As illustrated in Johnson and Bryden (1989, their Fig. 4), a positive pressure anomaly p′ > 0 pushing an isopycnal interface shoaling eastward transfers eastward momentum from the upper to the lower layer.
Using the same notation as (1) and (2), the form stress term [(9)] on the right-hand side of (8) was estimated from N data of meridional velocity Vn and isopycnal depth zan around a grid point as
eq4
Although the second term on the right-hand side in the approximation [(6)] was included in the actual calculations, the contribution from this term was negligibly small. as shown in Part I. The wind stress term of (8) was estimated from the atmospheric reanalysis data.

b. Form stress at 1000 dbar

The distribution of the form stress [(9)] is shown in Fig. 2. Several in situ observations are available for comparison with the interfacial form stress estimated with the Argo float data. Phillips and Rintoul (2000) measured velocity and temperature with four moorings near 50.5°S, 143.0°E from March 1993 to January 1995. The interfacial form stress at 1150 dbar was 0.51 N m−2. This point measurement does not agree with the nearest grid value (−0.08 N m−2 at 50.5°S, 143.5°E) but is similar in magnitude to the grid values slightly upstream (0.17 N m−2 at 51.5°S, 142.5°E and 0.97 N m−2 at 51.5°S, 141.5°E). The difference might be due to the different averaging methods used, as suggested by the fact that using a 300-km averaging radius resulted in much better agreement (Part I).

Fig. 2.
Fig. 2.

Spatial distribution of zonal form stress [(9)] at 1000 dbar. The dashed lines show latitudinal circles at 56.5° and 62.5°S. Positive stress indicates a downward flux of eastward momentum. Form stresses with magnitude less than ρ0fΔhΔu were not plotted, where Δh and Δu are the uncertainties in heave and in drift velocity (section 2a), respectively.

Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0072.1

Strong form stress was localized to steep topographies such as the Drake Passage (60°W), Kerguelen Plateau (80°E), Macquarie Ridge (165°E), and Pacific Antarctic Ridge (150°W). This localization is similar to the zonal distribution of particle crossing (Thompson and Sallée 2012, their Fig. 3d) and the “shallow” topographic form stress in the state estimate model (Masich et al. 2015, their Fig. 6), although particle crossing and form stress are physically different quantities. That the form stress discussed in Masich et al. (2015) is a topographic form stress, not interfacial form stress, explains the difference between the significant contribution found at 1000 dbar here and the values that are nearly zero in their Fig. 5.

c. Zonal momentum balance in a zonal band across the Drake Passage

Next, we examined the zonal momentum budget in a box bounded by the sea surface and the 1000-dbar depth. The simplest case is a box spanning a zonal band across the Drake Passage (dashed lines in Fig. 2). We integrated (8) between latitudes of 62.5°S (=yS) and 56.5°S (=yN):
e10
where is the spatial integration . The equation states that a change in the zonal momentum in the box (first term on the left-hand side) is balanced by the meridional momentum via the Coriolis force (second term on the left-hand side), momentum input from the upper surface (second term on the right-hand side), and momentum output from the lower (1000-dbar depth) surface by the form stress (first term on the right-hand side). Using the Argo float data, climatological hydrographic data, and reanalysis wind, the terms in (10) were evaluated (Fig. 3). The input of zonal momentum from the wind was 17.1 × 1011 N. Roughly half, (8.1 ± 1.9) × 1011 N, of this momentum was transferred downward across the 1000-dbar surface.
Fig. 3.
Fig. 3.

Zonal distribution of the form stress integrated between 56.5° and 62.5°S and in the upper 1000 dbar. The form stress was first integrated at each longitude and accumulated eastward from 0°, so that the value at 360° longitude was the form stress integrated in the box (8.1 × 1011 N). When integrated, the uncertainty due to measurement error was small (0.5 × 1011 N) compared to the uncertainty due to sampling (1.9 × 1011 N). The dotted line shows the similarly accumulated zonal wind stress (17.1 × 1011 N when integrated). The dashed–dotted line shows the planetary vorticity advection by the geostrophic component of the meridional velocity in (13), multiplied by 0.2 for scaling purposes (0.7 × 1011 N, unscaled, when integrated). The bottom topography meridionally averaged at each longitude is also shown (Kerguelen = Kerguelen Plateau, S.E.Ind. = South East Indian Ridge, Pac.A. = Pacific Antarctic Ridge, and Drake = Drake Passage).

Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0072.1

The meridional transport on the left-hand side of (10) can be further decomposed into Ekman , eddy , and geostrophic components, where the Ekman transport is given by
e11
where HE is the depth of the bottom of the Ekman layer (0 < HE < 1000 dbar). The eddy transport is defined by
e12
Substituting (11) and (12) into (10), we obtain a simple balance of
e13
which expresses the well-known fact that steady geostrophic transport cannot exist without meridional boundaries that support a zonal pressure gradient. The second term of (13) was also plotted in Fig. 3.

That the sampling error was not negligible underscored the limitation imposed by the data density. However, the measurement error was an order of magnitude smaller than the integrated form stress.

A comparison between the wind and form stresses revealed the difference of their spatial scales: the wind stress increased smoothly, whereas the form stress increased at much smaller spatial scales concentrated near the topographic barriers. A closer look showed that the form stress “step” was located on the eastern flanks of the meridional ridges.

The ACC tends to surmount a topographic ridge by first deflecting north on the western flank and then back to the south on the eastern flank (Fig. 4), a pattern reproduced in simple models (Ward and Hogg 2011; Abernathey and Cessi 2014) and consistent with planetary potential vorticity f/H conservation in the Southern Hemisphere. This behavior explains the dashed–dotted line ρf in Fig. 3. Decreases of the planetary vorticity were found on the western flanks of topographic ridges and increases were found on the eastern sides (note f < 0 here). As expected from (13), the integrated geostrophic Coriolis term was much smaller (0.7 × 1011 N) than the other two terms, downward flux of eastward momentum [(8.1 ± 1.9) × 1011 N] and wind input of eastward momentum (17.1 × 1011 N), in agreement with the balance expressed by (13).

Fig. 4.
Fig. 4.

Interface height of a two-layer, quasigeostrophic analytical model of a wind-driven current over a meridional ridge [after Abernathey and Cessi (2014, their Fig. 10) © American Meteorological Society. Used with permission.].

Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0072.1

The extent of the localization was examined by sorting grid values of the form stress (Fig. 5). Approximately the top 7% (1 × 1012 m2) of the total area contributed 7.2 × 1011 N or 90% of the overall form stress (8.1 × 1011 N). Half of the total form stress (4.0 × 1011 N) was accounted for by only 2% (0.2 × 1012 m2) of the total area.

Fig. 5.
Fig. 5.

Form stress accumulated after sorting. Estimated form stress in each grid was sorted in ascending order and integrated: 6.4 × 1012 m2 of grids had negative values and 8.9 × 1012 m2 of grids had positive values. These comprised 41% and 59% of the total area, respectively, between 56.5° and 62.5°S. Negative grids integrated to a form stress of −7.8 × 1011 N with sampling error estimated by the bootstrap method of (1.3/−1.1) × 1011N. From this negative contribution, −2.2 × 1011 N came from those grids with values less than the observation error of ΔhΔυ. Positive grids integrated to a form stress of 15.9 × 1011 N with sampling error estimated by the bootstrap method of (1.4/−1.6) × 1011 N. From this positive contribution, 2.7 × 1011 N came from those grids with values less than the measurement error of ΔhΔυ. Overall form stress was 8.1 × 1011 N with sampling error of ±1.9 × 1011 N and measurement error of 0.5 × 1011 N.

Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0072.1

d. Discussion

Form stress vertically transfers the wind stress applied at the surface to the bottom to maintain a steady momentum balance between the wind stress and topographic form stress. The balance can be immediately seen by integrating the zonal momentum balance (8) from the surface to the bottom z = −B(x, y):
e14
Under steady state (∂t → 0), the two terms on the right-hand side balance. This balance in the entire water column is different from the balance in the upper 1000 dbar [(10)] in three ways: (i) the Coriolis term does not exist because there is no net meridional transport
e15
(ii) the pressure gradient term does not vanish when integrated and leaves the bottom form stress term because pressure is generally different across meridional ridges, and (iii) the eddy form stress disappears because of the boundary condition Ψy = 0 at z = 0 and −B (McDougall and McIntosh 2001, their section 8).

Johnson and Bryden (1989) first estimated the magnitude of zonal momentum transferred vertically through a water column from temperature and velocity measurements from moorings in the Drake Passage. They found that the momentum flux was similar in magnitude to the wind stress and of the right sign to transfer eastward momentum downward. Quantitative discussion of the balance was not possible until the distribution of each term in the zonal momentum equation was estimated by numerical simulation. Marshall et al. (1993) used output from an eddy-resolving, quasigeostrophic, four-layer numerical simulation of the Southern Ocean to find a balance among the wind input, downward flux of momentum through each of the interfaces, and bottom topographic form stress. In that model, the northern boundary at 40°S was closed by a solid wall such that (15) held.

This constraint was removed by the Fine Resolution Antarctic Model (FRAM), which had an open northern boundary (Stevens and Ivchenko 1997). In the surface layer, the Coriolis and wind stress terms balanced. The balance was among the advection, Coriolis, and dissipation terms in the intermediate layers, where these three terms were at least an order of magnitude smaller than the Coriolis term in the surface and bottom layers that balanced the wind stress term and the bottom form stress term, respectively. This result corresponds to our result (Fig. 3) where (11) is the near-surface balance and (13) represents the balance of the intermediate layers. Keep in mind that we neglected the advection and dissipation terms because they were small. Stevens and Ivchenko (1997) identified the difference between the wind input and downward transfer of momentum as the planetary vorticity carried by the residual circulation [their (33)]. In their model, the contribution of the residual circulation was small around 1000 m (their Fig. 10), but our results showed that the Coriolis term that corresponds to their residual circulation term was as much as roughly half of the total wind input. This difference is not unexpected considering the different definitions of eddies and the heavy smoothing in our analysis.

Warren et al. (1996) criticized the form drag concept as “obscurantist physics” because the mass balance (15) is sufficient to assure the top-to-bottom momentum balance (14). Their formulation did not involve averaging and eddy mean decomposition, the result being that the eddy form stress was implicit. Use of the eddy explicit formulation (8) makes it apparent that the eddy form stress is not negligible in the upper-layer momentum balance, but it does not account for all momentum transfer from the surface to the bottom. The rest of the transfer is explained by the Coriolis term . It is worth remarking that because the Coriolis term is not in a flux form (i.e., ∂/∂y) nor a storage form (i.e., ∂/∂t), it does not import/export zonal momentum across the integrating boundaries nor does it store the momentum. Consequently, there is some arbitrariness in its interpretation. The situation is confounded by the definition of the eddy form stress equations [(9) and (12)]: form stress is equal to vorticity carried by meridional eddy transport. Thus, it is possible to account for the eddy form stress as a term in the upper momentum balance equation [(10)] or in the mass balance equation [(15)]. If we consider (10) as the key equation, our result was that the zonal momentum was transferred downward by eddy form stress, and the magnitude of the zonal momentum transferred was approximately half the wind input. In contrast, the result can be paraphrased in terms of (15) that the meridional transport in the box consisted of southward eddy transport equal to approximately half of the northward surface Ekman transport.

Our eddy explicit analysis gives a different answer to another question raised by Warren et al. (1996) regarding whether the ACC is relevant to the integrated zonal momentum balance or not. Figure 3 shows that the eddy form stress was enhanced on the lee side of the meridional ridges, which can be considered as an indirect effect of the ACC in the zonal momentum balance through generation of the eddies. We also note that confirmation of a steady balance is a different question from establishing a steady state. To fully understand the role of the ACC in the zonal momentum balance, it seems necessary to study a time-dependent problem, as was done by Ward and Hogg (2011). The result that the Coriolis term associated with the meridional transport is an important term in the momentum budget suggests that a closed northern boundary might lead to behavior of the modeled ACC quite different from that of the real ACC.

4. Streamline length at stationary meanders

a. Formulation

In this section, we examine the role that eddies play in the Southern Ocean dynamics in terms of the length of streamlines at standing meanders using observed data. Length Δl of a segment of a streamline around a point spanning a small azimuth of Δθ (Fig. 6) is related to the radius of curvature R by
e16
Using this relationship, we discuss the behavior of curvature R as a substitute for l.
Fig. 6.
Fig. 6.

Streamlines (thick lines) are given as F(x, y) = const. A local circle (dashed line) sharing a point and a tangent with the streamline defines the radius of curvature R. Length Δl of an infinitesimal segment of the streamline is given by Δl = RΔθ.

Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0072.1

Based on geometric considerations, local curvature, defined as the reciprocal of the radius of the circle that locally approximates a curve F(x, y) = const is given by
e17
where the subscripts denote partial differentiation. We assume that F is the streamfunction for the geostrophic transport in the top 1000 dbar such that
eq5
eq6
where u and υ are the transport per unit width in the top 1000 dbar in the zonal and meridional directions, respectively. Equation (17) becomes
e18
where the meridional scale of the Coriolis parameter variability f/β = f/(∂f/∂y) is assumed to be much longer than the typical meridional scales of standing meanders. As explained in section 2c, (18) can be evaluated using a climatology for the absolute geostrophic velocity field in the upper 1000 dbar. A perturbation to the curvature is
e19
The key to the present approach is to identify the perturbation transports (i.e., Δu, Δυ, Δυx, Δuy, and Δυy) as the eddy transport observed by Argo floats. The derivatives Δυx, Δuy, and Δυy were estimated by finite differencing. This means, as emphasized previously in section 2a, that the perturbation [(19)] measures the effect of both transient and standing eddies with a typical length scale of less than 300 km.

b. Standing meanders and curvature of streamlines

An analytical solution (Abernathey and Cessi 2014) demonstrates that an eastward-flowing ACC surmounts a meridional ridge first by deflecting northward on the western flank and then gradually relaxing on the eastern flank (Fig. 4). In terms of curvature, this behavior is manifested as a zonal dipole with a negative (counterclockwise, anticyclonic) curvature on the western side and a positive (clockwise, cyclonic) one on the east. Seven such structures (Table 1) were identified in the Southern Ocean (Fig. 7). The magnitude and spatial distribution of curvature varied, but all positive peaks corresponded well to the topographic barriers shallower than 3000 m. From the definition (16), a measure of the magnitude of curvature in the box would be , which is difficult to calculate from gridded data. Instead, a simple average of the curvature κ was calculated within the boxes (Table 1). The average was somewhat sensitive to the choice of the upper bound of uncertainty but was generally positive; a result consistent with the fact that overall the ACC is a cyclonic circulation.

Table 1.

Topographic barriers. Extent of boxes shown in Figs. 7 and 8, and averages of curvature (Fig. 7) and perturbation (Fig. 8) are shown. In the averaging, grid values of eddy transport in the upper 1000 dbar smaller than the measurement uncertainty ΔhΔu (section 2a) were not used. Moreover, grid values smaller than 1 and 2 m2 s−1 were not used in calculating the values in the columns under >1 and >2 m2 s−1, respectively.

Table 1.
Fig. 7.
Fig. 7.

Spatial distribution of curvature [(18)] of streamfunctions of geostrophic flow field in the upper 1000 dbar. Thick contours show depth contour of 3000 m. Thin contours show sea surface height representing fronts, as in Fig. 1. Color shading indicates curvature calculated by (18) from the geostrophic transport in the top 1000 dbar. Positive curvature is cyclonic or clockwise. Green boxes designate seven topographic barriers that ACC surmounts.

Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0072.1

c. Role of eddies

The role of eddies was calculated by identifying the perturbations in (19) as eddy transport. The result is shown in Fig. 8 after applying a 9-point smoother (3 × 3 grid points with a weight of 0.2 in the middle and of 0.1 at the other points) to suppress noise caused by spatial differentiation. Because of the spatial derivatives, the perturbation had smaller spatial scales than the curvature over the topographic barriers. Equation (19) does not allow for intuitive interpretations, and Fig. 7 shows that the eddies have effects that both increase and decrease the curvature κ. A decreased curvature κ means an increased R, the result being a lengthening of the streamline via (16). The average perturbation within the box was generally negative (Table 1) except for the South Scotia Ridge where missing data around topographies may have compromised the averaging (Fig. 8a). This negative average suggests that in most cases eddies increased the length of streamlines. A closer look shows negative peaks corresponding to the poleward eddy transport on the eastern flanks of the ridges, except for the Conrad Rise, which is more of a seamount than a ridge, the localized structure of which might be the reason for the lack of poleward eddy transport. As discussed in section 3, poleward eddy transport is important to downward transport of eastward momentum. The results in Fig. 8 suggest that poleward eddy transport also tends to lengthen streamlines.

Fig. 8.
Fig. 8.

Spatial distribution of perturbation Δκ to curvature [(19)] by eddies indicated by color shading. Gray indicates areas where data are missing. Thin contours show sea surface height representing the fronts as in Fig. 7. Thick contours show depth contour at 3000 m. The region for each panel corresponds to the green boxes in Fig. 7.

Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0072.1

d. Discussion

As stressed in section 2a, our definition of eddy inevitably included both transient and standing (with length scales less than about 300 km) components. A division between the two is possible in numerical simulations, and the contribution from each component depends on the setting. For example, in a four-layer idealized model of the ACC (Treguier and McWilliams 1990, their Fig. 9), the ratio of transient to standing eddy kinetic energy varied from 3 to 20 depending on the topography. In a three-layer quasigeostrophic model of the ACC (Hogg and Blundell 2006, their Fig. 11), the ratio in the top layer was closer to 2. In an attempt to extract the transient component, we redefined an eddy in a much smaller area than the area used in the main analysis: 0.4° in longitude (about 22 km at 60°S) and 0.1° in latitude (about 11 km). Admittedly, 11–22 km is still comparable to the Rossby deformation radius of the first mode, or typical eddy size, but 11 × 22 km2 was the smallest area to obtain a meaningful sample size; 30 grids had at least 5 data points within the standing meander regions defined in Fig. 7. We then calculated eddy kinetic energy as a measure of transient eddy activity. Figure 9 shows this “transient” eddy kinetic energy compared with the eddy kinetic energy (Fig. 1) from the nearest grid of the main analysis. The contribution of the transient component varied, but most of the kinetic energy was transient for strong eddies with kinetic energies >0.01 m2 s−1, particularly from the Southeast Indian Ridge box. The sample size was very limited, but if this observation can be extrapolated to other boxes, the stretching of streamlines by eddies apparent in Fig. 7 would seem to imply that transient eddies tend to stretch the streamlines, a conclusion that is consistent with the eddy-resolving simulation of Thompson and Naveira Garabato (2014, their Fig. 11).

Fig. 9.
Fig. 9.

Comparison of eddy kinetic energy (abscissa) of approximately transient eddies (transient with spatial scales less than 22 km zonally and 11 km meridionally) and that (ordinate) of eddies used in the main analysis (transient with spatial scales less than 300 km). Colors indicate the box the eddy belongs to, defined in Fig. 7.

Citation: Journal of Physical Oceanography 47, 9; 10.1175/JPO-D-17-0072.1

We also note that from 1997 to 2015, the period during which we averaged the Argo data, the westerly wind over the Southern Ocean as measured by the southern annular mode index was increasing (Swart and Fyfe 2012). Therefore, we hypothesize that the results summarized in Fig. 9 represent responses to an increase rather than a decrease of the wind.

5. Conclusions

The roles that eddies play in the dynamics of the ACC were studied by using hydrographic and trajectory data from Argo floats. We used two metrics: form stress across the 1000-dbar isopycnal and changes in the curvature of standing meanders. Eddy activities were very localized near major topographic barriers. The Coriolis forces associated with meridional transport and form stress contributed about equally to balancing the wind input in the circumpolar zonal momentum budget at the latitudes of the Drake Passage. The eddies, and particularly the transient components thereof, tended to lengthen the streamlines by increasing the radii of curvature of the standing meanders.

This study was limited by the number of data. If more float data were available southward of the ACC (e.g., Klatt et al. 2007), it would have been possible to discuss the momentum balance south of the ACC, including the subpolar wind-driven gyres. Recent studies have pointed out the importance of the westward wind near Antarctica in controlling ACC dynamics (e.g., Stewart and Thompson 2012; Zika et al. 2013). The data density also limited our study to analysis of mixed transient and standing eddies. The discussion of streamlines was necessarily qualitative because of the lack of spatial resolution.

Despite its importance in both zonal and meridional global ocean circulation, the Southern Ocean is a challenging region for numerical simulations because of the occurrence of subgrid-scale physical phenomena such as eddy mixing, deep-water production, and ice–ocean interactions. Though limited by data density, the present study offers a benchmark against which eddy parameterizations in numerical simulations can be compared for improvements. This remark also applies to the quasi-global analysis presented in Part I.

Acknowledgments

Argo data were collected and made freely available by the International Argo Program and the national programs that contribute to it (http://www.argo.ucsd.edu; http://argo.jcommops.org). The Argo Program is part of the Global Ocean Observing System. The list of data and their online resource locations are as follows: Argo hydrographic data (snapshot as of December 2015, http://doi.org/10.17882/42182#42348; ftp://ftp2.jamstec.go.jp/pub/argo/ifremer/); quality-checked Argo trajectory data (YoMaHa’07; http://apdrc.soest.hawaii.edu/projects/yomaha/); a gridded product of YoMaHa’07 (G-YoMaHa; http://www.jamstec.go.jp/ARGO/argo_web/argo/?page_id=86&lang=en); CARS2009 (http://www.marine.csiro.au/~dunn/cars2009/); ERA-Interim (http://www.ecmwf.int/en/research/climate-reanalysis/era-interim); and bottom topography data (ETOPO5; https://www.ngdc.noaa.gov/mgg/global/etopo5.HTML).

The Ssalto/Duacs altimeter products were produced and distributed by the Copernicus Marine and Environment Monitoring Service (CMEMS; http://marine.copernicus.eu/copernicus.eu). Computing software for the International Thermodynamic Equation of Seawater—2010 (TEOS-10) is available online (at http://www.teos-10.org/software.htm). I very much appreciate the community efforts to maintain the availability of these data. This work was supported by the Japan Society for the Promotion of Science (KAKENHI) Grant Number 24540475.

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