1. Introduction
Potential vorticity has been extensively considered as a tracer for ocean circulation. Easily derived from the vorticity and tracer conservation equations, potential vorticity (PV) behaves differently than a passive tracer. It has two features that make it particularly attractive for analyzing ocean circulation. First, if relative vorticity is small, then PV may be computed directly from the vertical density structure, as measured in hydrographic surveys. Second, in an unforced and nondissipative system, PV is conserved along geostrophic streamlines (see Pedlosky 1987). Thus, if flow is along geostrophic streamlines (as the quasigeostrophic approximation would dictate) and either the isopycnal surface is deep enough so that no direct forcing acts or else forcing is dissipated in the same geographic locations where it acts, then PV will be constant along streamlines. Under different circumstances, if streamlines are closed and a small amount of dissipation is present, then PV may be diffused along isopycnal surfaces in which case it should be constant, not only on streamlines but over the entire isopycnal surface (Rhines and Young 1982). Alternatively, if forcing is substantial or if dissipation is more than just a uniform effect due to the background eddy field, then PV will undergo dramatic changes in value along streamlines; the geographic locations of these changes may be useful indicators of the leading mechanisms controlling the dynamics.
Several recent studies have taken advantage of PV to probe Southern Ocean dynamics. Marshall et al. (1993) used the concept of PV homogenization outlined by Rhines and Young (1982) as a starting point. They noted that, if nothing forces a layer of the ocean, as they suggested might be expected below the wind-forced surface layer of the Antarctic Circumpolar Current (ACC), then as flow loops around closed streamlines, background eddy processes should homogenize PV on isopycnal surfaces. Based on hydrographic data, they suggested that PV varies linearly with potential density. Employing this linear functional relationship and hydrographic measurements at one depth only, they predicted dynamic topography in the Atlantic sector of the ACC. Discrepancies between their results and more detailed hydrography, they suggested, indicated locations where the uniform PV assumption was less satisfactory.
Marshall (1995) further explored the implications of determining PV as a function of potential density. By also requiring that PV be conserved along streamlines, that density be constant along streamlines on the sea floor, and that the linear vorticity balance apply, he derived an analytic expression for streamlines. Using realistic bathymetry, but applying no surface wind stress curl, he predicted the path of the ACC to be substantially steered by topography.
This investigation will take an alternative tack. Rather than assuming PV is conserved on isopycnal streamlines or homogenized over isopycnal surfaces, this study will examine the variation of PV along streamlines in order to probe the forcing and dissipation of the ACC.
Potential vorticity is considered along potential density surfaces in numerical model output in section 2. Since PV turns out to be neither homogenized nor conserved along streamlines, section 2c examines the specific mechanisms responsible for its fluctuations in the numerical model. In section 3 hydrographic data will be examined to show how observations resemble and differ from the numerical model output. Finally, the results are summarized in section 4.
2. Potential vorticity on isopycnals in numerical model output
a. Defining potential vorticity
The Semtner–Chervin model is a global general circulation model based on the Cox–Bryan primitive equation code (Semtner 1986). Although the specifics of the resolution and forcing differ, the basic model is the same as the Fine Resolution Antarctic Model (FRAM) (FRAM Group 1991), and the resulting momentum balances are qualitatively similar (Ivchenko et al. 1996; Stevens and Ivchenko 1997; Gille 1997). Wells and de Cuevas (1995) looked at the vorticity balance along vertically integrated streamlines in the FRAM output, concluding that surface wind stress is balanced by bottom pressure torque localized at Drake Passage; Killworth and Nanneh (1994) considered the momentum balance in isopycnal coordinates to examine how surface wind stress is transferred through the water column as interfacial form drag; however, neither study merged the two to consider potential vorticity on isopycnal surfaces.
For this study, results were analyzed from the “quarter-degree” version of the Semtner–Chervin Parallel Ocean Climate Model, which implemented several refinements from the earlier half-degree version (Semtner and Chervin 1992). The horizontal resolution is 0.4° in the zonal direction and 0.4° cos(ϕ) in the meridional direction, which is marginally eddy-resolving. In addition, a free sea surface was included based on the formulation by Killworth et al. (1991). The model was initialized with the output from the half-degree model following 22.5 years of spinup and 10 years of forcing with Hellerman and Rosenstein (1983) winds. During the last 5 years of the half-degree run, all deep relaxation to the Levitus (1982) climatology was eliminated (Semtner and Chervin 1992; A. Semtner 1994, personal communication). At one-quarter degree resolution, the model was then run for one year using climatological winds. Finally, it was forced with ECMWF operational forecast model winds (Trenberth et al. 1989) for the period from January 1986 to December 1989. Surface buoyancy forcing was based on the Levitus monthly climatologies with a one-month relaxation timescale. In addition, temperature and salinity were relaxed to “annual mean Levitus” climatology in the upper 2000 m, within 7° of the northern and southern boundaries of the model domain and within 7° of the Strait of Gibraltar using a three-month timescale. Unlike older implementations of the model, this version included no relaxation to the Levitus climatology below 2000 m. The final 3.75 years of the model run, starting with April 1986, were continuously time-averaged to produce statistics, including the means, variances, and covariances of all model variables.
Each of the terms in (9) accounts for different physical processes in the PV balance: J(
In Fig. 3 Q for the mean Semtner–Chervin model output on the σ1 = 32.3 surface is shown. It varies by about a factor of 2 across the Southern Ocean, with a substantial meridional gradient across the core of the ACC at the southern limit of the contoured values. The inhomogeneities in Q in turn suggest that inhomogeneous forcing is stronger than background eddy diffusion so that PV is not homogenized on this isopycnal.
b. Is potential vorticity conserved on streamlines?
Although diffusion is insufficient to homogenize Q, we might nonetheless predict that PV could be nearly conserved along streamlines. Consider the steady-state case, so that the tendency terms are zero. If PV values on streamlines are reset at one location along their circumpolar path, perhaps due to a localized topographic forcing, then we predict that background diffusion and eddy processes will not be able to homogenize PV on the isopycnal surface. Under these conditions, assuming no diapycnic effects and no ageostrophic velocity, Q should be constant along streamlines. In this case, (9) would reduce to J(
In order to examine to what extent PV is conserved on streamlines, PV is computed along contours of constant Montgomery streamfunction on the potential density surface σ1 = 32.3 in the Semtner–Chervin model output. Figure 4a shows that PV varies by about 25% along two representative streamlines, increasing steadily across the Pacific between 180° and 60°W and dropping sharply just downstream of Drake Passage around 40°W. PV variations across the Atlantic and Indian Oceans are more gradual, but indicate noteworthy rapid fluctuations near Kerguelen Island at 60°E and in the eddy-active region near the Macquarie Ridge and Campbell Plateau from 150°E to 180°.
In contrast with the gradual changes in PV along streamlines in Fig. 4a, PV undergoes dramatic fluctuations along lines of constant latitude, particularly where the isopycnal rises into the thermocline at the southern limit of the region and at Drake Passage (60°W), as shown in Fig. 5. Thus, in comparison with its large zonal variations, PV is nearly conserved along streamlines. The variations that occur are indications that even on this subsurface streamline, the curl of the net forces driving the mean ocean is nonzero. The remainder of this subsection will examine how PV changes qualitatively in response to this forcing.
Figures 4b–d show the relative importance of each of the components of PV. The major variations are attributed to changes in the layer thickness (∂z/∂σ), while variations in latitude (f) have less significant effects and relative vorticity is negligible. That PV variations along streamlines are dominated by stratification changes rather than Coriolis parameter changes has clear implications for simple PV models of the ACC. Although Killworth (1992) noted that the vertical structure of the Southern Ocean may easily be represented using a self-similar profile, the dominant changes in PV, even at middepth, appear as changes in the vertical density profile. Thus, simple models that take advantage of this self-similar structure (e.g., Gille 1995; Marshall 1995; Krupitsky et al. 1996) are unlikely to represent the leading processes governing changes in PV.
In the vertically integrated vorticity balance analyzed by Wells and de Cuevas (1995), basic Sverdrup dynamics apply: wind stress curl is balanced by the barotropic meridional velocity multiplied by β (that is, U·∇f ≈ curl τ/ρo), and in the Drake Passage region, bottom pressure torque removes vorticity input by the wind. In contrast, on this isopycnal surface, stratification changes along streamlines provide the major compensation for the forcing.
If we integrate (9) vertically, we predict that a Sverdrup-like balance should hold:
The middepth surface σ1 = 32.3 is between 1500 and 500 m deep along the 0.2-m and 0.3-m streamlines, as shown in Fig. 6. Thus, the isopycnal layer is far below the ocean’s surface and is not directly wind forced; surface wind forcing is conveyed indirectly to the middepth layers of the ocean model. Figure 6b indicates that the layer thickness decreases as the layer shoals so that PV is larger where the isopycnal surface is shallower. This is consistent with a simple concept of the wind-driven flow in the Southern Ocean. Since the stratification and alongstream velocity are of constant sign, the alongstream derivative of PV, ∂Q/∂s, should be proportional to the curl of the wind stress. In addition, wind stress curl should induce an Ekman pumping vertical velocity, so that curl(τ) ∝ − fwEk, where wEk is the vertical velocity at the base of the Ekman layer. The vertical velocity w throughout the water column is assumed to be roughly wEk and may be written as w = us∂d/∂s, where d is the depth of an isopycnal. The terms us and f are of constant sign and slowly varying. This leads to curl(τ) ∝ ∂Q/∂s ∝ ∂d/∂s. This relation therefore suggests that the depth of the isopycnal should vary with Q and, since Q variations are dominated by changes in stratification (or the reciprocal of layer thickness), stratification should increase (and correspondingly layer thickness decrease) roughly in locations where the wind stress curl brings isopycnals toward the surface, as the model results in Fig. 6 indicate.
c. Forcing terms on streamlines
Figure 7a shows a comparison of
Figure 7b indicates that the PV balance is significantly affected by the transient eddy processes due to tracer advection, transient eddy processes due to momentum advection, the tracer tendency, horizontal momentum diffusion, and horizontal tracer diffusion. The transient tracer flux and transient momentum flux terms (identified as tracer advection and momentum advection in the figure caption) are calculated as residuals from the tracer and momentum balances, respectively, since the model archiving scheme does not include sufficient detail to recover the transient component of the PV balance. The importance of the transient tracer and momentum advection terms suggests that a balance that neglected these terms would lack essential portions of the physics [as Killworth and Nanneh (1994) also noted for the isopycnic momentum and volume budgets]. The results found for this 0.2-m streamline closely resemble those on adjacent streamlines: No single process appears to explain the entire PV balance in Fig. 7b. However, note that many of the small-scale variations (<10° longitude) in PV in Fig. 7a are explained by the tracer advection term, while the large-scale trends (>50°) are partially captured by the sum of the tracer evolution, transient tracer advection, and transient momentum advection terms. In addition the large-scale momentum and tracer diffusion variations often balance each other, so that the net influence of diffusion may be less than the apparent influence in either the momentum or tracer equations. The term designating the PV changes induced because isopycnic and in situ density surfaces are not coincident also influences the PV balance; in essence, this term is a measure of the error inherent in investigating PV along isopycnals. Finally, the discretization error is uniformly small. In addition to the terms plotted in Fig. 7b, recall that the vertical viscosity term, relative vorticity tendency, ageostrophic advection, and diapycnal flux terms were judged to have no significant impact on the PV balance and are not shown.
Because the tracer tendency term plays a significant role in the overall alongstream PV balance, these results suggest that the Semtner–Chervin model may not have spun up sufficiently, so the middepth isopycnals did not reach steady state. If the model reached a quasi-stationary state it might have a different PV balance. However, to the extent that the numerical ocean does resemble the real ocean, this analysis suggests that transient eddy effects play a substantial role in the PV balance along mean streamlines. In particular, transient tracer fluxes appear directly responsible for small-scale PV variations along streamlines. Since transient tracer processes are often associated with the vertical transfer of momentum input by the wind via interfacial form drag (Johnson and Bryden 1989; Marshall et al. 1993), this might be an indication that indirect wind forcing is one of the factors responsible for changes in PV along mean streamlines at middepth in the ACC.
3. Potential vorticity from Southern Ocean hydrography
For comparison with the numerical results, PV may also be calculated using hydrographic data compiled by Olbers et al. (1992, hereafter OGSS). OGSS produced objectively mapped data fields with 1° resolution at 38 depth levels; however, for this study their compiled hydrographic data were used so that all mapping could be done along isopycnal surfaces. This was done specifically to avoid the types of averaging problems discussed by Lozier et al. (1994) who pointed out that smoothing quantities along level surfaces and then mapping them onto isopycnals would produce substantially different results than would smoothing the data along isopycnal surfaces. [Nonetheless, the results in this section would not be changed qualitatively if PV and streamlines were instead computed directly from the gridded quantities produced by OGSS (Gille 1995).] At each hydrographic station, PV and the Montgomery streamfunction were computed on the isopycnal surface σ1 = 32.3. Since PV varies in the upper ocean due to mixed layer processes, data points were retained only if they were deeper than 500 m; as a result, data points at the southern limit of the domain were eliminated. Following the general methodology outlined by OGSS, for both PV and Montgomery streamfunction, outliers were removed if they differed from the mean by more than three standard deviations. To reduce errors associated with the erroneous values initially in the dataset, the outlier test was repeated. The resulting dataset consisted of 2571 points scattered throughout the Southern Ocean, but concentrated primarily in the South Atlantic. Unlike the numerical model output, which includes a known sea surface topography, for the hydrographic data a reference surface must be assigned. The Montgomery streamfunction was referenced to the slightly deeper isopycnal surface, σ1 = 32.4; this shallow reference density was selected because using a denser reference layer would result in extensive data dropout wherever the deeper isopycnal surface intersects the ocean bottom. Therefore, instead of imagining the velocity to be zero on the reference layer, consider it to attenuate smoothly below this depth.
The resulting data were objectively mapped using a Gaussian correlation function, with a 450-km decorrelation length in the zonal direction and 350 km in the meridional direction, as done by OGSS. Gille (1994) noted that these length scales are almost double what the mean fields reconstructed from altimeter data indicate, and they may therefore result in fields smoother than the oceanic mean fields. Extensive smoothing should help to minimize problems associated with nonsynopticity of the data and sparse sampling in the Southern Ocean.
Figure 8 shows the mapped Montgomery streamfunction on the isopycnal surface σ1 = 32.3. PV on the σ1 = 32.3 surface from the hydrographic data is shown in Fig. 9. Despite the expectation that the objectively mapped fields should be smooth renditions of the hydrographic measurements, the objectively mapped PV shows substantial variability over relatively short length scales, in contrast with the numerical model PV (Fig. 3). This may indicate that the historic hydrographic record is not extensive enough to represent accurately the climatological mean state of the Southern Ocean, but it could also indicate that the dominant length scales in the numerical model output are longer than in the real ocean. Because PV is computed on isopycnals directly from hydrographic data and then objectively mapped, rather than being determined from the objectively mapped fields, the results shown in Fig. 9 appear different than PV maps produced by Marshall et al. (1993). However, like the numerical model PV (Fig. 3), the PV shown in Fig. 9 is consistent with that plotted by Marshall et al. and by You and McDougall (1990) in showing that PV may change by a factor of 2 or more in the circumpolar portion of the Southern Ocean.
Potential vorticity variations along Montgomery streamlines (shown in Fig. 10) are comparable in magnitude to the variations seen in numerical model output, but the variations occur over much shorter distances and could represent rather different physical processes. As in the numerical model results in Fig. 4, alongstream changes in PV are predominantly due to stratification changes rather than Coriolis parameter changes as indicated by the large fluctuations in Fig. 10b compared with Fig. 10c.
If the atlas data are representative of Southern Ocean stratification, then the results indicate that the ACC undergoes substantial changes in PV over relatively short distances. The high wavenumber variations in PV on streamlines could be an indication that PV is input by the wind at higher wavenumbers than ECMWF winds would suggest or that it is dissipated due to bottom pressure torques associated with midsized topographic features that the model does not seem to feel, as Gille’s (1997) momentum balance analysis suggests. Thus, the differences between Figs. 4 and 10 might be explained by identifiable differences between the model and the real ocean.
Alternatively, rather than thinking about model−ocean differences, note also that nonsynopticity and smoothing make the atlas data a suboptimal proxy for PV and streamfunctions, so the differences between Figs. 4 and 10 may simply be a result of the sparseness of the hydrographic record. The transient eddy processes shown in the model results are not distinguished by the data sampling, and in the limited available hydrographic record, temporal variability may be aliased to appear as a spatially varying mean field. Thus, readers who are uncomfortable interpreting the hydrographic data as indicating high-wavenumber variations in PV may prefer to conclude that the limited hydrographic measurements south of 30°S resemble instantaneous snapshots of an eddy rich field, and the sampling has not been sufficiently frequent to resolve the mean structure of the Southern Ocean.
4. Discussion and summary
The results of this investigation have several implications for Southern Ocean studies.
First, within statistical error bars PV does not appear to be uniform, neither in the Semtner–Chervin model output nor in hydrographic data. PV differs by 50% over 1000-km length scales and has a substantial mean gradient across the width of the ACC. These results suggest that forcing and dissipation of PV at middepth in the ACC are too irregular to result in uniformly homogenized PV.
Second, in addition to being inhomogeneous on isopycnal surfaces, middepth PV is also not conserved along mean streamlines on isopycnal surfaces, suggesting that some apparent forcing drives PV and that the influence of this forcing is not dissipated precisely in the locations where it enters. In both numerical model output and hydrographic data, changes in PV are largely due to stratification changes rather than changes in planetary vorticity along streamlines. Stratification changes are not precisely related to the locations of bathymetry, so are not easily represented as a simple topographic torque due to sea floor topography. Thus, simple self-similar models of the ACC that assume uniform stratification and predictable responses to bathymetry may not adequately capture dominant PV variations along the path of the ACC.
Third, variations in PV are qualitatively consistent with a simple notion that the middepth system responds in part to changes in wind stress curl. In a detailed analysis of the numerical model PV balance, no single factor emerges as the dominant mechanism responsible for PV changes along streamlines. Thus, to answer the question implicit in this paper’s title, potential vorticity is not conserved along mean streamlines in the Semtner–Chervin Southern Ocean, because a half dozen terms in the PV equation all conspire to keep it from being conserved. Significant contributions to the PV balance come from transient tracer fluxes, the tracer tendency term, transient momentum fluxes, tracer diffusion, momentum diffusion, and a term associated with the isopycnal surface not being identical to a constant in situ density surface. The transient tracer flux contribution captures many of the high-wavenumber PV variations; since this term is sometimes associated with the vertical flux of wind stress via interfacial form drag, its apparent importance may be an indicator that the middepth ACC does feel surface wind forcing, at least indirectly, but that this process only accounts for a portion of the alongstream PV balance.
Acknowledgments
Many thanks to Bert Semtner and Bob Chervin who made their model results readily available and to Tony Craig and Robin Tokmakian who helped in extracting and interpreting the data. This study benefited from discussions with Kathie Kelly, Mike McCartney, John Marshall, John Toole, Terry Joyce, and Russ Davis. Trevor McDougall and several anonymous reviewers made invaluable suggestions, which have helped to shape the analysis and presentation. Joe LaCasce, Stefan Llewellyn Smith, and Lynne Talley also provided useful comments on an earlier version of the manuscript. This work was supported by National Aeronautics and Space Administration Contract NAGW-1666 and by National Oceanic and Atmospheric Administration Award NA47GP0188 to the Lamont/Scripps Consortium for Climate Research.
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Contours of constant Montgomery streamfunction divided by gρo (in m) on the potential density surface σ1 = 32.3, determined from the Semtner–Chervin quarter-degree numerical model. The contour interval is 0.1 m, and the 0-m height contour is fixed arbitrarily. Data are not shown where σ1 = 32.3 rises above 500-m depth.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1286:WPVINC>2.0.CO;2
Contours of sea surface height (in m) from the Semtner–Chervin quarter-degree numerical model. The contour interval is 0.2 m, and the 0-m contour is fixed arbitarily.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1286:WPVINC>2.0.CO;2
PV, (f + ζ)∂σ/∂z, (in kg s−1 m−4 × 108) on the σ1 surface 32.3 determined from mean quarter-degree Semtner–Chervin model output. The σ1 surface outcrops at the southern limit of the domain and is several kilometers deep at the northern limit. Points where σ1 = 32.3 is shallower than 500 m have been eliminated.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1286:WPVINC>2.0.CO;2
(a) PV (in kg s−1 m−4 × 108) along two Montgomery streamfunction contours, M, from the Semtner–Chervin quarter-degree mean model output. Contours used correspond to 0.2 and 0.3 m (which have mean depths of 970 and 1170 m, thus bracketing the 1000-m reference depth.) Sea surface height increases toward the equator, so that the 0.3-m contour is north of the 0.2-m contour. (b) PV along Montgomery streamfunction, when f + ζ is fixed so that only changes in the stratification ∂σ/∂z are assessed. (c) PV along Montgomery streamfunction, where stratification and relative vorticity are held constant, so that only changes in latitude are assessed. (d) PV along Montgomery streamfunction, when f and ∂σ/∂z are held fixed so that only the effect of relative vorticity is considered.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1286:WPVINC>2.0.CO;2
PV (in kg s−1 m−4 × 108) along latitude lines, computed from Semtner–Chervin model output. Latitude lines shown correspond to 45°S, 50°S, and 55°S.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1286:WPVINC>2.0.CO;2
(a) Depth of the σ1 = 32.3 isopycnal surface in kilometers along streamlines. (b) Thickness of the isopycnal layer corresponding to 32.25 < σ1 < 32.35, in km.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1286:WPVINC>2.0.CO;2
(a) PV (in kg s−1 m−4 × 108) along the 0.2-m Montgomery streamfunction (solid line) compared with PV estimated by integrating the quantity [J(
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1286:WPVINC>2.0.CO;2
Montgomery streamfunction M [as defined in (2) in m × 0.1] on the isopycnal surface σ1 = 32.3 referenced to the isopycnal surface σ1 = 32.4. The contour interval is 0.02 m, and the total range is small because of the relatively shallow reference layer but should nonetheless be representative of physical processes. Gray areas indicate regions with no mapped data and regions where the formal error from the objective mapping procedure exceeds 80% of the overall data variance, which is about 0.006 m.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1286:WPVINC>2.0.CO;2
PV, f∂σ/∂z (in kg s−1 m−4 × 108), on the σ1 surface 32.3 determined from hydrographic data. Gray areas indicate regions with no mapped data and regions where the formal error from the objective mapping procedure exceeds 80% of the data variance, which is 1 × 108 for the entire domain.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1286:WPVINC>2.0.CO;2
(a) PV (in kg s−1 m−4 × 108) along Montgomery streamlines determined from the OGSS atlas data. Height contours correspond to the contours labeled 0.04 and 0.02 in Fig. 8. Gray regions indicate the one standard deviation formal error bars from objective mapping. (b) PV along Montgomery streamlines when f is held fixed so that only changes in stratification are assessed. (c) PV along Montgomery streamlines when ∂z/∂σ is held fixed so that only the effects of changing latitude are assessed.
Citation: Journal of Physical Oceanography 27, 7; 10.1175/1520-0485(1997)027<1286:WPVINC>2.0.CO;2
