1. Introduction
The Southern Ocean is a unique and dynamic component of Earth’s climate system. As an important site of mode-water, intermediate-water, and deep-water formation, the Southern Ocean hosts a dominant transport pathway between the atmosphere and the interior ocean (Russell et al. 2006). This pathway is set in part by steeply tilted isopycnal surfaces that outcrop at high southern latitudes. Through this window, atmospheric carbon is exchanged with the interior ocean, potentially slowing the buildup of anthropogenic carbon dioxide in the atmosphere while altering the biogeochemistry of the global ocean (Takahashi et al. 2009). Since mean stratification (and the closely related slope of Southern Ocean isopycnals) is likely to impact abyssal circulation and deep-water transport, the large-scale density structure of the Southern Ocean could be considered an important “state variable” of both global ocean circulation and the larger climate system.
The steep tilt of isopycnals in the Southern Ocean is established and maintained at least in part by (i) convergences and divergences of wind-driven Ekman flow in the surface ocean and (ii) the planetary-scale meridional buoyancy gradient. The zonal mean forcing pattern varies with latitude in such a way that isopycnals are tilted up toward the Antarctic continent (i.e., they outcrop in the south and plunge into the interior toward the subtropics). They join smoothly with the subtropical density structure, which is characterized by relatively flat isopycnals. Climatological mean potential density is shown in Fig. 1.

Zonal mean potential density referenced to 2000 dbar (σ2; kg m−3) for the indicated longitude ranges and from 40° to 60°S. Contours are spaced every 0.1 kg m−3. Data are from World Ocean Atlas 2013.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1

Zonal mean potential density referenced to 2000 dbar (σ2; kg m−3) for the indicated longitude ranges and from 40° to 60°S. Contours are spaced every 0.1 kg m−3. Data are from World Ocean Atlas 2013.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
Zonal mean potential density referenced to 2000 dbar (σ2; kg m−3) for the indicated longitude ranges and from 40° to 60°S. Contours are spaced every 0.1 kg m−3. Data are from World Ocean Atlas 2013.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
Tilted isopycnals like those found in the Southern Ocean are susceptible to baroclinic instability. Mesoscale eddies generated by baroclinic instability have a tendency to homogenize potential vorticity (PV) along isopycnals via downgradient thickness fluxes, flattening constant density surfaces in the process (Marshall and Speer 2012). The mean slope of interior Southern Ocean isopycnals is thought to be set by a balance between the steepening effect of wind stress and buoyancy fluxes and the flattening effect of energetic, mesoscale eddies (Karsten and Marshall 2002). This balance affects many facets of global ocean circulation across different time scales, including the sensitivity of the overturning circulation to changes in Southern Hemispheric wind stress and the strength of the Southern Ocean carbon sink (Lovenduski and Ito 2009; Abernathey et al. 2011; Munday et al. 2014).
In this work, we investigate the influence of a planetary–geometric constraint on the equilibrium slope of tilted buoyancy surfaces in the Southern Ocean. Using zonal mean theory with quasigeostrophic dynamics, we derive a relationship between the large-scale, depth-averaged vertical gradient of isopycnal slope and the ratio β/f0 [where β = df/dy, f = 2Ω sin(ϕ) is the Coriolis parameter, Ω is the planetary rotation rate in radians per second, ϕ is latitude, and f0 is f evaluated about a reference latitude ϕ0]. This relationship suggests that if the depth-averaged quasigeostrophic potential vorticity (QGPV) gradient is small relative to β, then the vertical gradient of isopycnal slope (i.e., the rate of change of slope with depth) is set by β/f0. We find that in climatological Southern Ocean reanalysis fields, the vertical gradient of isopycnal slope is of order β/f0, and we use this condition to derive a scale for the vertical gradient of isopycnal slope. The Pacific Ocean features the weakest vertical gradients of isopycnal slope relative to the Indian and Atlantic sectors of the Southern Ocean. Finally, we use an idealized sector model to examine the sensitivity of stratification to changes in surface wind forcing and eddy activity. The vertical gradient of isopycnal slope is relatively insensitive to surface wind stress in the presence of permitted mesoscale eddies.
2. Theoretical justification
a. Basic assumptions and definitions
We consider an idealized ocean with no longitudinally oriented barriers such that zonal mean theory is generally applicable (Marshall and Radko 2003). Zonal mean buoyancy [simply written as b = b(y, z, t) in this section in order to keep the notation from getting cluttered] can be decomposed into the average buoyancy,









b. The large-scale vertical gradient of buoyancy surfaces







Values of r < 1 indicate that the buoyancy surfaces are “undertilted” with respect to β/f0. In this regime, depth-averaged potential vorticity increases with latitude (i.e.,


Furthermore, the planetary–geometric constraint only influences the rate of change of the slope of buoyancy/density surfaces with depth; it does not fix the actual value of slope, which is strongly influenced by air–sea interactions and mixing. The term s(Hd) in Eq. (7) is not necessarily zero, since we have chosen our domain to lie above the bathymetry and sea floor. Similarly, s(0) is only the slope of buoyancy surfaces at the top of the domain, which is taken to be well below the mixed layer; it is not intended to represent the slope at the surface of the ocean.
In the following sections, we estimate r and estimate a depth scale over which slope changes significantly using both observationally derived datasets and an idealized interhemispheric sector model.
3. Data and methods
a. Climatology
We employ objectively analyzed climatological mean temperature and salinity fields derived from nearly six decades of in situ profile data (Locarnini et al. 2010; Zweng et al. 2013). The fields are on a 1° × 1° global grid with 102 vertical levels from the surface down to 5500 m. Each climatological field is a nearly six-decadal mean taken over the period 1955–2012. Therefore, the fields only represent the long-term, large-scale average structure of temperature and salinity, and some interpolation has been used to fill in missing values. These fields are part of the World Ocean Atlas 2013 (WOA13) suite, which for this work have been retrieved from the National Oceanographic Data Center (NODC; http://www.nodc.noaa.gov/OC5/woa13/). Density is calculated using the modified United Nations Educational, Scientific and Cultural Organization (UNESCO) polynomial of Jackett and Mcdougall (1995, hereinafter JMD95). Since WOA13 has relatively low spatial resolution and does not include velocity fields, we also use 6-yr-averaged (2005–10) zonal and meridional velocity fields from the Southern Ocean State Estimate (SOSE), which are on a ⅙° horizontal grid with 42 vertical levels (Mazloff et al. 2010).
b. Sector model setup
The sector model is an idealized configuration of the MIT general circulation model (MITgcm) designed to allow a large number of numerical experiments to be run to equilibrium at a range of resolutions and wind forcings (Marshall et al. 1997a,b). Full details of the configuration are given in Marshall et al. (1997a,b), Munday et al. (2013), Hogg and Munday (2014), and Munday et al. (2014), although a brief exposition follows.
The sector model domain stretches from 60°S to 60°N and is 20° in longitude wide. A “circumpolar” channel extends over the southernmost 20° of latitude and an extra 1 grid point, or 1° in longitude, whichever is greater, forms the model’s “Southern Ocean.” Within this extra section, the depth is 2500 m, but is otherwise 5000 m throughout the rest of the domain. The step is sufficiently high so as to block all f/H contours. When the model grid spacing is fine enough to permit or resolve the mesoscale eddy field, this allows surface wind stress to be balanced by bottom form stress in the expected momentum balance for the Southern Ocean (e.g., Munk and Palmén 1951). In this balance, interfacial eddy form stresses transmit momentum vertically through the water column.
The sector model is driven at the surface by an idealized profile of wind stress that places the peak wind stress within the circumpolar channel [see Fig. 2 of Munday et al. (2013)]. Surface forcing of temperature and salinity is carried out using restoring toward idealized profiles based on observations of the Atlantic. The restoring time scale for temperature is 10 days, and the restoring time scale for salinity is 30 days. The structure of the surface forcing does not vary with model grid spacing and is designed so that the surface density at the northern boundary is intermediate between that at the southern boundary and the northern edge of the circumpolar channel. This ensures that the model analog of North Atlantic Deep Water sinks to middepth and upwells within the confines of the circumpolar channel.
At a grid spacing of 2°, the sector model uses the Gent and McWilliams (1990) parameterization of the mesoscale eddy field with a constant diffusivity coefficient of 1000 m2 s−1. At finer grid spacings, the coefficient is greatly reduced, such that the permitted/resolved mesoscale eddy field is undamped by the parameterization [see Munday et al. (2013) for details].
At each model grid spacing, a total of 12 different wind stress values were used, ranging from a peak value of 0.0 N m−2 to a peak value of 1.0 N m−2 with a control value of 0.2 N m−2. The 2° model was run for 30 000 years in total and is at equilibrium in under half of this (roughly 10 000 years). The ⅙° model was run for 400 years, which is long enough for the isopycnal slopes and “Drake Passage” transport to be reasonably equilibrated. The control experiment, and both extreme wind perturbations, were run for another 400 years and the (small) change in 10-yr mean circumpolar transport was well within the variability.
The ⅙° model has a sufficiently fine grid so as to permit a vigorous mesoscale eddy field. At the northern edge of the circumpolar channel, the first baroclinic Rossby radius is around 40–60 km, roughly 4–5 grid boxes. The deformation radius is much smaller near the southern boundary and the flow commensurately less well resolved. The eddy kinetic energy (EKE) is comparable to that found in state-of-the-art ocean simulations, such as the coupled climate models in Delworth et al. (2012) or the latest iteration of the SOSE (Mazloff et al. 2010). Details of the spatial distribution and sensitivity to wind stress changes of the EKE can be found in Munday et al. (2013).
4. Observational and numerical tests of constraint
a. Vertical gradient of isopycnal slope
Along several large sections of the ACC, far from large bathymetric features (e.g., in the eastern Pacific sector), the depth-averaged slope gradient (i.e.,

(top) Southern Ocean bathymetry with three fronts of the Antarctic Circumpolar Current, namely the Polar Front (PF, southernmost), Subantarctic Front (SAF, middle), and the northern extension of the Subantarctic Front (SAF-N, northernmost). Depth scale is shown in meters. (bottom) Value of
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1

(top) Southern Ocean bathymetry with three fronts of the Antarctic Circumpolar Current, namely the Polar Front (PF, southernmost), Subantarctic Front (SAF, middle), and the northern extension of the Subantarctic Front (SAF-N, northernmost). Depth scale is shown in meters. (bottom) Value of
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
(top) Southern Ocean bathymetry with three fronts of the Antarctic Circumpolar Current, namely the Polar Front (PF, southernmost), Subantarctic Front (SAF, middle), and the northern extension of the Subantarctic Front (SAF-N, northernmost). Depth scale is shown in meters. (bottom) Value of
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
The mean value of r1 across all longitudes is 2.4, the median value is 2.0, and the standard deviation is 1.2. The values of r1 and r2 are broadly similar across all longitudes, with some exceptions where the ACC passes over the east Pacific Rise. The mean value of r2 across all longitudes is 2.8, the median value is 2.4, and the standard deviation is 1.4. The value of r is somewhat sensitive (within roughly a factor of 1.5–2) to the choice of upper boundary (500–800 m), but it remains order 1 as predicted by quasigeostrophic scaling (which we do not expect to hold exactly). The ratio r tends to be larger near bathymetric features due to the stabilizing topographic β effect. That is, the effective topographic β is larger near a topographic slope, which stabilizes the flow and allows for steeper isopycnals. The inclusion of the topographic β effect might lower the value of r, but detailed analysis of this effect is beyond the scope of this paper.
We also calculated r1 and r2 using the Gibbs Seawater (GSW) Oceanographic Toolbox. The GSW mean values remained within 10% of the values obtained using JMD95, and the standard deviations changed by less than 1%. GSW and JMD95 produced very similar profiles of r with longitude (not shown).
b. Vertical and horizontal stratification


In Fig. 3, we plot M2 versus N2 for three different sections of the Southern Ocean between 50°–60°S latitude and 750–3000-m depth. Although there is some scatter, there is a linear component in the relationship between M2 and N2 in each basin (i.e., p < 0.01 for a linear model). The coefficient of determination R2 is 0.8 for the Atlantic basin, 0.8 for the Pacific basin, and 0.6 for the Indian basin. Using Eq. (8) and the known value f0, we can estimate the vertical scale H for each of the three basins by linear regression. In the Atlantic sector (20°W–0°) H ≈ 3000 m, in the Indian sector (90°–110°E) H ≈ 2600 m, and in the Pacific sector (130°–110°W) H ≈ 4900 m. Since H is the scale over which the slope of density surfaces changes significantly, H can be larger than the actual depth of the ocean. Larger values of H imply a slope that changes little with depth, relative to regions with smaller values of H. In the limit where the slope of density surfaces is uniform with depth, the scale H approaches infinity.

Horizontal stratification M2 vs vertical stratification N2 in the Atlantic (20°W–0°), Indian (90°–110°E), and Pacific (130°–110°W) sectors of the Southern Ocean. Each point represents a sample from a chosen depth and latitude from a zonal mean buoyancy field.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1

Horizontal stratification M2 vs vertical stratification N2 in the Atlantic (20°W–0°), Indian (90°–110°E), and Pacific (130°–110°W) sectors of the Southern Ocean. Each point represents a sample from a chosen depth and latitude from a zonal mean buoyancy field.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
Horizontal stratification M2 vs vertical stratification N2 in the Atlantic (20°W–0°), Indian (90°–110°E), and Pacific (130°–110°W) sectors of the Southern Ocean. Each point represents a sample from a chosen depth and latitude from a zonal mean buoyancy field.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
We can estimate the degree to which M2 and N2 are linearly related by estimating the coefficient of determination (i.e., R2) at each point on a latitude–longitude grid (Fig. 4). The variable R2 can serve as a measure of the linear component in the relationship between N2 and M2 over broad patches of the Southern Ocean. The linearity between N2 and M2 is fairly strong (i.e., R2 > 0.7) over most of the ACC, and is especially high in the Pacific (e.g., in the Bellingshausen basin, just upstream of Drake Passage). The coefficient R2 becomes noticeably smaller (i.e., R2 < 0.4) near large bathymetric features (e.g., Kerguelen Plateau, Campbell Plateau, and Falkland Plateau), and R2 is especially small in the Ross and Weddell Seas, where sea ice melt and formation impose strong controls on the stratification. Near the Antarctic continent, the assumptions of quasigeostrophic theory are violated; the isopycnals are especially steep, and gyres dominate the dynamics. Since our scaling assumes a zonal mean structure, it should be most applicable across the ACC.

(left) Values of R2 for linear regressions between M2 and N2 from WOA13 six-decadal climatological temperature and salinity fields. Density is calculated using JMD95. Three ACC fronts are shown in solid black lines (SAF-N, SAF, and PF). Values are only plotted where the linear relationship is statistically significant at the 95% level or above (i.e., f test p < 0.05). (right) Vertical scale depth H (m) from WOA13 climatology. Values are only displayed where R2 > 0.5 and p < 0.05 for linear regressions between M2 and N2 at each latitude and longitude. The calculation is carried out in a moving window that is 20° wide in longitude and 10° wide in latitude.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1

(left) Values of R2 for linear regressions between M2 and N2 from WOA13 six-decadal climatological temperature and salinity fields. Density is calculated using JMD95. Three ACC fronts are shown in solid black lines (SAF-N, SAF, and PF). Values are only plotted where the linear relationship is statistically significant at the 95% level or above (i.e., f test p < 0.05). (right) Vertical scale depth H (m) from WOA13 climatology. Values are only displayed where R2 > 0.5 and p < 0.05 for linear regressions between M2 and N2 at each latitude and longitude. The calculation is carried out in a moving window that is 20° wide in longitude and 10° wide in latitude.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
(left) Values of R2 for linear regressions between M2 and N2 from WOA13 six-decadal climatological temperature and salinity fields. Density is calculated using JMD95. Three ACC fronts are shown in solid black lines (SAF-N, SAF, and PF). Values are only plotted where the linear relationship is statistically significant at the 95% level or above (i.e., f test p < 0.05). (right) Vertical scale depth H (m) from WOA13 climatology. Values are only displayed where R2 > 0.5 and p < 0.05 for linear regressions between M2 and N2 at each latitude and longitude. The calculation is carried out in a moving window that is 20° wide in longitude and 10° wide in latitude.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
c. Vertical scale of the slope gradient
The Pacific sector of the Southern Ocean features especially weak vertical slope gradients. The Pacific basin is somewhat less topographically constrained than the other two basins, with relatively flat bottoms and no large plateaus. Subtropical stratification, which tends to be flatter and more uniform with depth, extends relatively far southward compared with the other two basins. It is interesting to note that the largest values of H are found just south of (or slightly within) the Polar Front of the ACC.
d. Sensitivity to averaging depth
To test the sensitivity of the relationship between M2 and N2 to various parameters, we employ an idealized sector model as described in section 3. Zonal mean density (i.e., potential density referenced to roughly 2000 m) is depicted in Fig. 5a, wherein the averaging window is indicated with a dashed white line. The relationship between M2 and N2 depends on depth (Fig. 5b). If the averaging window is moved to the upper 300 m of the domain, then the vertical scale H approaches zero (i.e., the slope changes extremely rapidly with depth thanks to the steep tilt). Below 1000 m, the model ocean becomes nearly unstratified. As a result, both M2 and N2 tend toward zero. In this abyssal region, there is a strong linear relationship between the horizontal and vertical gradient in buoyancy. The quasigeostrophic scaling used in this paper is most appropriate between roughly 300 and 1000 m in the sector model, as indicated by the white box in Fig. 5a. The box is chosen to intersect the steeply tilted density surfaces of the circumpolar current while avoiding both the mixed layer and the weakly stratified deep ocean. Note the rapid slope change at approximately 1000 m, which divides the vertical domain into an upper, rotation-dominated region and a lower, stratification-dominated region.

(a) Zonal mean density (kg m−3) for the sector model. (b) Horizontal stratification vs vertical stratification for three different choices of vertical averaging scale. Values of M2 vs N2 for various values of maximum surface wind stress for a sector models with (c) 1° horizontal resolution and (d) ⅙° horizontal resolution. The white dashed line in (a) indicates the vertical averaging domain for plots (c) and (d). Since we are interested in large-scale features in this analysis, the ⅙° model results were interpolated onto a 2° grid.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1

(a) Zonal mean density (kg m−3) for the sector model. (b) Horizontal stratification vs vertical stratification for three different choices of vertical averaging scale. Values of M2 vs N2 for various values of maximum surface wind stress for a sector models with (c) 1° horizontal resolution and (d) ⅙° horizontal resolution. The white dashed line in (a) indicates the vertical averaging domain for plots (c) and (d). Since we are interested in large-scale features in this analysis, the ⅙° model results were interpolated onto a 2° grid.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
(a) Zonal mean density (kg m−3) for the sector model. (b) Horizontal stratification vs vertical stratification for three different choices of vertical averaging scale. Values of M2 vs N2 for various values of maximum surface wind stress for a sector models with (c) 1° horizontal resolution and (d) ⅙° horizontal resolution. The white dashed line in (a) indicates the vertical averaging domain for plots (c) and (d). Since we are interested in large-scale features in this analysis, the ⅙° model results were interpolated onto a 2° grid.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
e. Sensitivity to wind stress and eddy activity
Finally, we examine the sensitivity of the relationship between the horizontal and vertical stratification to resolved/permitted eddy activity (i.e., horizontal resolution) and wind stress. In Fig. 5c, we plot M2 versus N2 for three different maximum values of the surface wind stress (0.0, 0.2, and 0.4 N m−2). In comparing the coarse-resolution case and the eddy-permitting case (i.e., Figs. 5c and 5d, respectively), we find that the scale H is less sensitive to surface wind stress in the model with higher horizontal resolution (see Table 1 for estimates of H for the six different cases). The eddy-permitting model explicitly resolves large-scale eddies and is thereby better able to compensate for any wind-induced changes in isopycnal tilt. In the coarse-resolution model, eddies are parameterized following Gent and McWilliams (1990). In the presence of resolved/permitted mesoscale eddies, isopycnal slope is less sensitive to wind stress than when eddies are parameterized (Munday et al. 2013).
Values of the depth scale H (m) obtained by the scaling H = s/|β/f0|, where s is the slope of the M2/N2 lines from sector model sensitivity experiments. Plots of M2 vs N2 are shown in Figs. 5c and 5d.


f. Revisiting the assumptions
In deriving Eq. (6), we made two assumptions in addition to those inherent to quasigeostrophic theory. First, we assumed that the ratio

Depth-averaged relative vorticity gradient in the Southern Ocean scaled by β. Zonal and meridional velocity fields from the Southern Ocean State Estimate (SOSE).
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1

Depth-averaged relative vorticity gradient in the Southern Ocean scaled by β. Zonal and meridional velocity fields from the Southern Ocean State Estimate (SOSE).
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
Depth-averaged relative vorticity gradient in the Southern Ocean scaled by β. Zonal and meridional velocity fields from the Southern Ocean State Estimate (SOSE).
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
It should be noted that although the above scaling suggests that relative vorticity can be neglected in this analysis, relative vorticity can in principle have an impact on local potential vorticity gradients, leading to homogenization or even sharpening (Hughes 2005). The nonlinear component of vorticity advection can be important on smaller scales than those considered here (Hughes and Cuevas 2001).
The second assumption that we made while deriving Eq. (6) was

Idealized schematic of zonal mean potential vorticity gradients (shading) and isopycnals (solid gray lines) in the Southern Ocean, adapted from Tulloch et al. (2011). The light shaded area indicates the region where the potential vorticity gradient is positive (∂yq > 0), and the dark shaded area indicates a region where the potential vorticity gradient is negative (∂yq < 0). The dashed boxes illustrate three different choices for vertical scales over which to average. The potential vorticity gradient is close to zero if the vertical averaging scale is chosen appropriately (i.e., for box C). Averages taken over boxes A and B would have nonzero mean gradients.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1

Idealized schematic of zonal mean potential vorticity gradients (shading) and isopycnals (solid gray lines) in the Southern Ocean, adapted from Tulloch et al. (2011). The light shaded area indicates the region where the potential vorticity gradient is positive (∂yq > 0), and the dark shaded area indicates a region where the potential vorticity gradient is negative (∂yq < 0). The dashed boxes illustrate three different choices for vertical scales over which to average. The potential vorticity gradient is close to zero if the vertical averaging scale is chosen appropriately (i.e., for box C). Averages taken over boxes A and B would have nonzero mean gradients.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
Idealized schematic of zonal mean potential vorticity gradients (shading) and isopycnals (solid gray lines) in the Southern Ocean, adapted from Tulloch et al. (2011). The light shaded area indicates the region where the potential vorticity gradient is positive (∂yq > 0), and the dark shaded area indicates a region where the potential vorticity gradient is negative (∂yq < 0). The dashed boxes illustrate three different choices for vertical scales over which to average. The potential vorticity gradient is close to zero if the vertical averaging scale is chosen appropriately (i.e., for box C). Averages taken over boxes A and B would have nonzero mean gradients.
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1

Zonal mean quasigeostrophic potential vorticity from WOA13 six-decadal mean temperature and salinity. The values have been scaled by β. Zonal mean potential density contours are shown in black for σ2 = 36.0, 36.5, 37.0, and 37.1 kg m−3. The reference buoyancy profile
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1

Zonal mean quasigeostrophic potential vorticity from WOA13 six-decadal mean temperature and salinity. The values have been scaled by β. Zonal mean potential density contours are shown in black for σ2 = 36.0, 36.5, 37.0, and 37.1 kg m−3. The reference buoyancy profile
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
Zonal mean quasigeostrophic potential vorticity from WOA13 six-decadal mean temperature and salinity. The values have been scaled by β. Zonal mean potential density contours are shown in black for σ2 = 36.0, 36.5, 37.0, and 37.1 kg m−3. The reference buoyancy profile
Citation: Journal of Physical Oceanography 45, 12; 10.1175/JPO-D-15-0034.1
In potential vorticity homogenization theory, eddies act to reduce potential vorticity gradients along isentropes, leading to meridional potential vorticity gradients that are much smaller than β. The r = 1 condition is equivalent to the “strong constraint” that potential vorticity is completely homogenized. We have seen that the “strong” r = 1 constraint does not hold in the Southern Ocean. However, a “weak” form of the constraint r = O(1) does hold. In the Southern Ocean, β itself is small, hence a potential vorticity gradient of order β is still weakly consistent with the idea of potential vorticity homogenization.
5. Discussion
In this work, we have presented a planetary–geometric constraint that relates the depth-averaged vertical gradient of isopycnal slope to the ratio β/f0. We have shown that this constraint holds, to first order, in the climatological Southern Ocean across the ACC, at least far from bathymetric obstructions. This result is broadly consistent with Jansen and Ferrari (2012), in which the authors discuss the sensitivity of stratification to planetary parameters (e.g., rotation rate). Our results are also consistent with detailed studies of the potential vorticity structure of the Southern Ocean [e.g., Marshall et al. 1993; Tulloch et al. 2011]. We used the relationship between horizontal and vertical stratification to derive a depth scale for the slope gradient. In this section, we discuss possible implications of the planetary–geometric constraint for large-scale ocean circulation and sensitivity.
a. Thermal wind scaling




















b. Eddy flux parameterization







The GM90 scheme is built using an f-plane approximation (i.e., GM90 neglects the variation of the Coriolis parameter f with latitude). On an f plane, the isopycnal potential vorticity is clearly linked to the thickness of isopycnal surfaces. To be more specific, the QGPV gradient (neglecting relative vorticity as we have done throughout this paper) simplifies to




Once the isopycnal eddy diffusivity is known [which can be calculated using mean flow and eddy properties; e.g., Klocker and Abernathey (2014)] and appropriate boundary conditions are chosen, it should be possible to derive a physically meaningful GM diffusion coefficient that flattens isopycnals only to the r = 1 limit (i.e., where the slope gradient is constrained by the planetary–geometric parameter β/f0) instead of to the r = 0 limit (i.e., where the slope changes uniformly with depth). This approach has not yet been implemented in any ocean model, but this scheme would very likely lead to a much better representation of the ACC in coarse-resolution global climate models by correcting the equilibrium isopycnal slope across the ACC.
6. Conclusions
If the meridional gradients of relative vorticity (i.e., ∂yζg) and potential vorticity (i.e., ∂yq) are small relative to β, then quasigeostrophic theory predicts that the isopycnal slope s is related to latitude ϕ0 and planetary radius a by ds/dz = β/f0 = cot(ϕ0)/a, or equivalently r ≡ |∂zs/(β/f0)| = 1, where r is the depth-averaged criticality parameter. For large-scale climatological observations, we find that the strict r = 1 condition holds over specific averaging volumes that include regions of both positive and negative meridional gradients of potential vorticity (i.e., ∂yq) in roughly equal measures. A weaker r = O(1) condition for depth-averaged values is generally satisfied along much of the Antarctic Circumpolar Current and throughout the wider Southern Ocean, particularly away from large bathymetric features. In these regions of the Southern Ocean, the large-scale average rate of change of slope with depth is constrained by β/f0, a ratio of purely geometric parameters.
It is important to note that this scaling does not set isopycnal slope, but only its average vertical derivative. A change in forcing (e.g., an addition of buoyancy at high latitudes paired with a loss of buoyancy at low latitudes) may change the slope of isopycnal surfaces across the domain (e.g., box C in Fig. 7), but eddy activity will tend to restore the potential vorticity structure such that r = O(1). This adjustment can in principle involve isopycnal steepening is some parts of the domain and flattening in others, which can change the baroclinic structure of the current; as long as the large-scale average slope gradient is of order β/f0, the r = O(1) constraint is satisfied.
The concept of the depth-averaged balance in Eq. (6) (i.e., the r = 1 regime) provides a useful limiting case for understanding what sets isopycnal slope in rapidly rotating fluids with meridional potential vorticity gradient reversals. Although the Southern Ocean is an interesting test case for this concept, the r = 1 balance may be useful for understanding changes in isopycnal slope in other systems with meridional potential vorticity gradient reversals. Possible applications may be found in paleoceanography and exoplanetary oceanography, which feature a wide range of surface buoyancy and wind forcing profiles. A more thorough exploration of the consequences of the r = O(1) regime would make for a welcome addition to this discussion.
Acknowledgments
The authors thank Antoine Venaille, Andrew Meijers, Laure Zanna, Ryan Abernathey, Emma Boland, Trevor McDougall, and David Marshall for discussions and comments that greatly improved the quality of this paper. We also thank Jean-Baptiste Sallée for providing mean positions of the fronts of the Antarctic Circumpolar Current. DJ was supported by the Natural Environment Research Council (NERC Grant NE/J007757/1). This study is part of the British Antarctic Survey Polar Science for Planet Earth Programme. TI was supported by the National Science Foundation (1142009). AK was supported by an Australian Research Council Discovery Early Career Researcher Award (DE140100076). We gratefully acknowledge the thoughtful comments of two anonymous reviewers.
APPENDIX A
Alternate Form of the Constraint












APPENDIX B
Vertical Velocity Scales









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