Journal of Marine Research, 63, 59–93, 2005
The all-Atlantic temperature-salinity-pressure relation and patched potential density
by Roland A. de Szoeke1 and Scott R. Springer2
ABSTRACT The relation between temperature, salinity, and pressure in the Atlantic Ocean is examined. Most of the Atlantic resolves itself into three two-dimensional manifolds of three-dimensional thermody- namic space: a northern, more saline, branch, and a southern, fresher, branch, each quite independent of pressure, and between them a bridge, on which density is uniform at constant pressure. The properties of the branches are crucial to the construction of joint potential density surfaces, patched together at 1000 db intervals. By resolving more finely in pressure (illustrated with 200 db spacing), a finer system of patched potential density surfaces can be obtained, and indeed the continuous limit can be taken. This limit gives a form of orthobaric density, regionally differentiated because it is based on the duplicate regional branches. A mapping can be devised, using the properties of the bridge waters, that links the southern and northern forms of orthobaric density across the boundary between their respective regions of validity. The parallel of patched potential density surfaces to orthobaric density surfaces permits the use of measures developed for the latter to estimate quantitative measures of the material nature (or otherwise) of the former. Simply put, within the waters of the respective branches the patched isopycnals, or orthobaric isopycnals, are very nearly material, limited only by inherent irreversible mixing processes. However, where these isopycnals cross the bridge waters, significant, reversible, material exchange across them may occur. A difficulty may be encountered with coarsely resolved, regionally differentiated, patched potential density. This is that there exist ranges of density which cannot be consistently linked across the regional boundary. A solution for the difficulty, suggested by the continuous form (regionally differentiated orthobaric density), is proposed.
1. Introduction The regularity of the relation between potential temperature and salinity S in the ocean has long been remarked on (Helland-Hansen and Nansen, 1926; Wu¨st, 1933; Iselin, 1939; Sverdrup et al., 1942; Montgomery, 1958; Reid, 1981). Volumetric censuses of water types (finely resolved temperature-salinity combinations) have been published by Worth- ington (1981). Understanding the ϪS relation is key to the mechanisms of the thermohaline circulation and mixing in the ocean (Stommel and Csanady, 1980; McDougall,
1. College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon, 97331, U.S.A. email: [email protected] 2. College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon, 97331, U.S.A. email: [email protected] 59 60 Journal of Marine Research [63, 1
1987). Water properties are frequently charted on core layers, defined by extrema of water characteristics such as salinity or oxygen, that are clearly delineated on ϪS and other property-property relations (Wu¨st, 1933). These core layers are held to reveal the directions of spreading of water masses from their source regions. Alternatively, potential density surfaces, with which core layers often coincide, have been used instead to chart water mass spreading and mixing, particularly as density surfaces indicate dynamically neutral internal horizons in the ocean, and their slopes are related to geostrophic currents (Montgomery, 1938; Reid, 1965, 1981). However, this practice has raised the difficulty of potential density’s undue reliance on the thermal expansion coefficient at the reference pressure (Ekman, 1934). Reid and Lynn (1971) tried to overcome this difficulty by using patched potential density surfaces. These are composed of leaves of potential density surfaces in restricted ranges of pressure (typically 1000 db), referenced to the center of that range, and patched to neighboring leaves so as to minimize dislocation at the overlaps of the pressure ranges. De Szoeke et al. (2000) showed that the patching procedure is equivalent to selecting standard salinity functions of specific volume at the overlap pressures. Alternatively, these can be expressed as salinity vs. temperature functions at fixed pressure. Jackett and McDougall (1997) devised ways of calculating neutral density surfaces which they held to be the continuous analogue of patched potential density surfaces (see also Eden and Willebrand, 1999). These considerations motivate an examina- tion of ϪS relations at fixed pressure to understand how the variable-reference potential density idea might work. Accordingly, in this paper we undertake a refinement of the traditional view of the ϪS relation by differentiating the relation further by pressure (or depth). This reveals insights into the structure of the ocean perhaps more interesting than the original motivation, as we will show. The data we use is the WOCE Atlantic hydrographic set. The three-dimensional temperature-salinity-pressure relation is shown, from a number of points of view. We show how the Atlantic Ocean data resolve into three two-dimensional manifolds of thermody- namic space, and confirm that Reid’s (1994) potential density patching method relies on the empirical relations of vs. S on two of these manifolds. We also confirm that there is a correspondence between patched potential density and regionally differentiated orthobaric density predicated on the same ϪS relations (de Szoeke et al., 2000), and show how quantitative estimators of the degree of materiality of orthobaric density can be used to evaluate either kind of density.
The Atlantic ؊S ؊ p relation .2 We used the 10-decibar hydrographic data file made available by the international World Ocean Circulation Experiment (WOCE) project to plot diagrams of potential temperature vs. salinity for pressures between 300 decibars (db) and 4000 db in the North and South Atlantic oceans (Fig. 1; maps of station locations are shown in Fig. 5). McDougall and Jackett (2005) show similar diagrams derived from atlas data (Gouretski and Koltermann, 2004). 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 61
These diagrams show a number of interesting features.
(i) A lower salinity branch extends from warm temperatures associated with the subtropical gyre to the salinity minimum associated with the Antarctic intermediate water, turns sharply toward higher salinity between 2.0–2.5°C, and then turns sharply again to join the cold, salty Antarctic bottom water. It is convenient to call this set of waters, usually called the south central waters (Sverdrup et al., 1942), the southern branch. By overlaying the diagrams for varying pressure, one sees that the form of this branch is virtually independent of pressure. (ii) A parallel branch, though without the twists at low temperature, of waters about 1–1.5 psu saltier than the southern branch, extends almost to the warmer end of the Antarctic bottom water. These waters—the north central waters—constitute the northern branch. Like the southern branch, the form of this branch is independent of pressure. (iii) Straddling the southern and northern branches, at each pressure from 300 db to about 2000 db, there is a bridge. At the low salinity end of the bridge, before reaching the southern branch, the bridge joins to a spur extending from and nearly parallel with that branch. The bridge shifts with pressure. At pressures deeper than 900 db, the bridge, the southern branch (of which only the line joining the Antarctic intermediate water to the Antarctic bottom water remains), and the northern branch form a triangle which gradually contracts to a small volume at 2000 db, although the line of the bridge is still discernible. (iv) At pressures deeper than 500 db, the bridge extends through and beyond the northern branch, with density increasing with salinity for pressures shallower than 1000 db. This bridge extension is a direct link to the Mediterranean outflow.
Specific volume anomaly (SVA), defined by ␦ϭ␣(S, , p) Ϫ␣(35, 0, p), has been re-plotted vs. salinity (Fig. 2). The diagrams show that the bridge at any pressure deeper than 500 db has an almost constant specific volume. At pressures shallower than 500 db, SVA increases slightly with salinity on the bridge. The extension of the bridge beyond the northern branch appears to have lower SVA. What seems most remarkable about the diagrams of Figure 1 is how little of ϪS Ϫ p space the Atlantic Ocean actually occupies, as McDougall and Jackett (2005) also point out. The ocean is pretty flat in thermodynamic space, being composed, in idealized approximation, of the union of three two-dimensional manifolds, the northern and southern branches which are independent of pressure, and the bridge, joining the two and forming a ramp of SVA as a function of pressure. Deeper than a few hundred decibars, the pressure dependence of SVA of the bridge is described well by the profile shown in Figure 3. The value given by this profile is indicated on each panel of Figure 2. Variation from the idealized forms clearly occurs. In the bridge, this could be attributed to adiabatic heaving of isopycnals about their mean positions by internal motions that are aliased in the synoptic WOCE data. Such an explanation does not suffice in the southern 62 Journal of Marine Research [63, 1
Figure 1. Potential temperature vs. salinity S for numerous values of pressure, from the WOCE data. The southern and northern water masses describe a vertical shaft (independent of pressure) in ϪS Ϫ p space, while a bridge joining them, and extending to higher salinity beyond the northern branch, forms a sloping ramp. Note the change of scale for pressures greater than 2000 db. Colors, for pressures Յ 1000 db: red denotes the bridge between the northern and southern branches; green (magenta), the parts of the northern and southern branches warmer (colder) than the bridge; dark blue, the high-salinity extension of the bridge; light blue, the intersection of the bridge with the northern branch. 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 63
Figure 1. (Continued) 64 Journal of Marine Research [63, 1
Figure 2. Specific volume anomaly (SVA; units, m3 kgϪ1) ␦ vs. salinity S, like Figure 1. SVA is nearly constant at each pressure on the bridge. The constant-SV lines through the bridge give the values shown in Figure 3. 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 65
Figure 2. (Continued) 66 Journal of Marine Research [63, 1
Figure 3. The SVA profile of the bridge (m3 kgϪ1). and northern branches, whose forms are independent of pressure, so that adiabatic motions would preserve the forms of the branches. Rather, the scatter around these branches, or “spiciness,” must be the result of irreversible mixing among diverse water parcels (Munk, 1981). The random appearance of the scatter about the idealized branches could be reduced by averaging (e.g., Gouretski and Koltermann, 2004; McDougall and Jackett, 2005). Figure 4 shows ϪS diagrams from individual WOCE stations on section A16, which runs nominally along 20W in the North Atlantic and 25W in the South Atlantic. The progression of stations shows a fairly smooth transition from northern water masses to southern water masses. At temperatures between 3° and 6°C, south of about 20N, there is a fairly sharp transition from relatively fresher, lighter waters to saltier, denser waters. Between 20S and 9N on this section, it is notable how isothermal this transition is. For example, at 2.6S the transition layer has a temperature of 4.6°C between 600 and 900 db. These transition waters are all sampled from the bridge waters of Figure 1. At a single station, the waters are sampled from the northern and southern branches and from a range of pressures from the bridge waters between them. The totality of stations making up the section may give the impression, as in Figure 4, of a continuum in ϪS space between the northern and southern branches, and of a ϪS relation evolving more or less continuously with latitude. This view, too, is defensible and has been employed in a theoretical study of how thermobaricity and ϪS relation variation affect planetary wave propagation (de Szoeke, 2004). But for the purposes of this paper we shall emphasize the oceans’ contracted shape in ϪS Ϫ p space. 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 67
Figure 4. vs. S profiles from several WOCE stations along section A16. The color bar gives the station latitudes.
The correspondence between the distribution of water types in ϪS Ϫ p space, and in physical space, is seen by comparing Figures 1 (or 2) and 5. In these figures, color coding has been used to indicate the various branches and subparts of the branches. (Red denotes the bridge between the northern and southern branches; green denotes branch waters less dense than the bridge at the given pressure; magenta denotes branch waters denser than the bridge; dark blue denotes the salty extension of the bridge beyond the northern branch; light blue denotes a rather indistinct region where the bridge crosses the northern branch. Little confusion arises from using the same color for waters of the northern or southern branch, since the respective waters occupy distinct geographic domains.) Figure 5a, at 500 db, shows saline Mediterranean water spreading from Gibraltar, merging into colder, fresher, denser branch water to the north, and along North Africa into bridge water to the south. To the west it connects to lighter, subtropical, northern branch waters (green), occupying the latitude band 20–40N. The bridge waters at this pressure occupy 20S–20N. South of this band the southern branch waters are found. At greater depths and pressures (Fig. 5b, at 1000 db), a similar pattern is found, though Mediterranean waters extend farther west and north, while light northern branch waters are largely absent (though one must bear in mind that the choice of boundaries between colors in thermodynamic space in Figs. 1 and 5 is somewhat arbitrary). The bridge region is somewhat wider, 25S–25N. At 1500 db (Fig. 5c), the intrusion of salty Mediterranean water is very evident, merging into waters of the northern branch-bridge intersection and the bridge, though the distinction 68 Journal of Marine Research [63, 1
Figure 5. Maps showing horizontal distribution of water types at (a) 500 db, (b) 1000 db, (c) 1500 db on the WOCE station network in the Atlantic. Colors correspond to the water types in Figures 1, 2.
between the latter masses is very hazy (see Fig. 1). The densest northern and southern branch waters (magenta), very close yet distinct on the ϪS diagram, are far apart in opposite hemispheres in Figure 5c.
3. Patched potential density The northern and southern branches of the Atlantic ϪS Ϫ p relation are encapsulated in the potential density patching method of constructing isopycnals (Reid and Lynn, 1971). Reid (1994) published a table, reproduced here in modified form as Table 1, of potential density values, 0, 1, 2, 3, 4 (with respect to the subscripted reference pressure, in hectobars [hb]), which, taken together over their respective pressure ranges (within Ϯ0.5 hb of the reference pressure) define a “patched” potential density surface—one surface for each row in the table—for each of which Reid (1994) shows maps and sections in his paper. The surfaces are joined at the pressures half way between the reference ϭ ϭ pressures of neighbors in the row, e.g., 0 27.30 joins onto 1 31.938 in the South ϭ Atlantic at p 0.5 hb. The choice of which values of 0 and 1 to pair is determined by plotting the values against one another (e.g. Fig. 6a) and fitting a line by least squares. Note that the data fall roughly along two lines, corresponding to the northern and southern branches. Also evident is the bridge, which appears as a narrow cloud of data points between the two branches. Note, however, that the entries of Table 1 contain no intimation of the bridge. A similar procedure is followed to define pairs at the other transition pressures, p ϭ 1.5, 2.5, 3.5 hb. Another way to see this is by examining the intersections of the pϪ0.5 and pϩ0.5 contours on a ϪS diagram (Jackett and McDougall, 1997). These intersections identify a 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 69
Table 1. Patched potential density surfaces for the Atlantic [after Reid (1994)]. 0 1 2 3 4 North 26.550* 30.850* 26.750 31.090 27.440 31.938 36.300* 27.630 32.200 36.640* 27.757! 32.376 36.880* 27.824 32.456 36.980 41.395 27.846 32.485 37.017 41.440 45.760* 27.874 32.523 37.067 41.500 45.838 27.892 32.548 37.099 41.539 45.880 27.901 32.560 37.115 41.562 45.907 27.908 32.569 37.126 41.572 45.920 27.930* 32.600* 37.179* 41.639* 46.002*
South 26.365* 30.700* 26.750 31.188* 35.450* 27.100* 31.680* 36.130* 27.300 31.938 36.460* 27.563& 32.200 36.731† 27.695!! 32.355 36.890* 41.330* 27.755†† 32.425 36.980 41.420$ 27.770 32.445 37.004% 41.440 45.770 27.787 32.476 37.041 41.500 45.840 27.800 32.487 37.062# 41.528@ 45.880 27.804 32.498 37.074 41.547 45.907 27.815 32.502 37.080 41.553 45.920 27.876* 32.585* 37.196† 41.710† 46.120† *Additional entries. †From Reid (1989). !Changed from Reid’s (1994) entry of 27.777. !!Changed from Reid’s (1994) entry of 27.675. ††Corrected Reid’s (1994) entry of 27.775. &Adopted Reid’s (1994) alternate value for 60–90S. %Adopted Reid’s (1994) alternate value for 30–90S. #Changed from Reid’s (1994) entry of 37.057. @Adopted Reid’s (1994) alternate value for 30–40S. $Changed from Reid’s (1994) entry of 41.400. (Changes and alternate values from Reid’s (1994) were motivated by closer Fits to data on Fig. 6; additional entries by extension of ϪS branches [Fig. 8].) Boldface entries indicate Reid’s (1994) labels for these surfaces. 70 Journal of Marine Research [63, 1
ϭ Figure 6. pϪ0.5, pϩ0.5 pairs from Table 1 for p 0.5, 1.5, 2.5, 3.5 hb (crosses, north branch; circles, south branch; solid line, Reid’s (1994) pairs; dashed line, supplementary pairs from this paper) and WOCE Atlantic data in grayscale. corresponding , S pair (Fig. 7, an enlargement, for clarity, of the 0.5 hb panel of Fig. 1). Taken together, these , S pairs define a standard ϪS relation for each transition pressure. These water types are marked with circles and crosses, corresponding to southern and northern branches respectively, on Figure 8a, connected by straight lines, and plotted on top of the WOCE data points, to demonstrate Reid’s (1994) standard ϪS relations for the 0.5 hb transition. The other panels of Figure 8 show Reid’s (1994) standard water types, determined in a similar way, for the other transition pressures, p ϭ 1.5, 2.5, 3.5 hb. Viewed either way, Figure 6 or Figure 8, Reid’s (1994) northern and southern ϪS relations are reflected in the WOCE data very well.
Virtual compressibility In an attempt to gain finer resolution of patched potential density than the 1.0 hb that Reid (1994) used, we calculated potential densities referenced to 0.2, 0.4, 0.6, 0.8, 1.0 hb ϭ Ϫ from the WOCE data, and at p 0.3, 0.5, 0.7, 0.9 hb plotted them as ( pϩ⌬p/2 ⌬ ϩ ⌬ ϭ pϪ⌬p/2)/ p vs. ( pϩ⌬p/2 pϪ⌬p/2)/ 2 for p 0.2 hb (Fig. 9a–d). (If one plotted ⌬ ϭ simply pϩ⌬p/2 vs. pϪ⌬p/2, as in Figure 6 for p 1.0 hb, one would scarcely detect any variation from a unit-slope line for this small pressure interval.) The diagrams of Figure 9 resolve themselves into the three branches identified above. The upper branch is 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 71
Figure 7. vs. S diagrams at 500 db, showing WOCE Atlantic data in grayscale, and intersections of constant 0 (solid) and 1 (dashed) pairs from Table 1: 2nd through 4th, north branch; 3rd through 5th, south branch.
the southern, the lower the northern, and the link joining them in the middle is the bridge. ϭ The entries in the 0, 1 columns of Table 1 permit a similar calculation for p 0.5 hb, ⌬p ϭ 1.0 hb, which are shown as crosses and circles in panels a–d of Figure 9. This coarser calculation matches the finer-scale calculation quite well at p ϭ 0.5 hb (Fig. 9b). When shifted by simple functions of pressure on both ordinate and abscissa, it matches other nearby pressures also ( p ϭ 0.3, 0.7, 0.9 hb—Figs. 9a, c, d). The higher-pressure panels of Figure 9 show extensions, with no corresponding data, of the coarse-resolution branches to lower densities. This is merely a carryover of the forms of the lines from low pressures. A similar calculation has been done at p ϭ 1.3, 1.5, 1.7, 1.9 hb for ⌬p ϭ 0.2 hb (Fig. 9e–h, shown on expanded scales), and compared to a calculation from the 1, 2 columns of Table 1 for p ϭ 1.5 hb, ⌬p ϭ 1.0 hb. The good agreement between the coarse-scale calculation and the fine-scale is again evident. The coarse-scale lines of Figure 9a–d have been re-drawn on Figure 9f as a dotted line. Evidently, the forms of these lines are remarkably independent of pressure from the surface to about 2 hb. These diagrams give estimates of the virtual compressibility function d/dp ϭ ⌫ 0 ( p, ) (de Szoeke et al., 2000), at the discrete pressures listed above. Actually, it gives 72 Journal of Marine Research [63, 1
Figure 8. vs. S diagrams at p ϭ 0.5, 1.5, 2.5, 3.5 hb, showing WOCE Atlantic data in grayscale, and intersections of pϪ0.5, pϩ0.5 pairs from Table 1 connected by line segments to form reference ϪS relations; meanings of symbols and lines as in Figure 6.
two such functions, corresponding to the southern and northern branches. (The third branch, corresponding to the bridge, will be dealt with below.) Starting from a value of, ϭ ϭ ϭ ϭ say, pϪ⌬p/2 0 at p 0.1 hb, [or pϩ⌬p/2 1 at p 0.9 hb] we could algebraically solve, on either branch, ͑ Ϫ ͒ ⌬ ϭ ⌫ ͑ ͑ ϩ ͒ ͒ pϩ⌬p/2 pϪ⌬p/2 / p 0 p, pϩ⌬p/2 pϪ⌬p/2 /2 (1)
™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™™3 ⌬ ⌬ Ϫ ϩ Ϫ2 Ϫ Figure 9. Virtual compressibility function anomaly, / p (c0 c1p) vs. p,in ϭ ϭ Ϫ1 ϭ ϫ grayscale, for p 0.3, 0.5, 0.7, 0.9, 1.3, 1.5, 1.7, 1.9 hb [c0 1500 m s , c1 2 10Ϫ2 msϪ1 dbϪ1, ϭ4.566 ϫ 10Ϫ3 kg mϪ3 dbϪ1]. Panels b, f are replottings of Figure 6a, b. Note the superposition of Table 1 data (meanings of symbols and lines as in Fig. 6) on panels b, f. The Table 1 data from panel b [f] have been transferred to panels a, c, d, [e, g, h] for comparison. Table 1 data from panel b have been plotted on panel f for comparison (hatched). Note scale change in panels e–h. 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 73
Figure 9 74 Journal of Marine Research [63, 1
for pϩ⌬p/2 [ pϪ⌬p/2], and so on, generating a chain of finely resolved potential density leaves (like the entries on each line of Reid’s table) whose concatenation in physical space over pressure intervals of width ⌬p will furnish a patched potential density surface (Reid and Lynn, 1971; Reid, 1994), which it is convenient to identify by the starting value of 0 [ 1] at its reference pressure 0 hb [1 hb]. Thus one can generate such patched potential density surfaces, not only on a finer pressure grid, but also as a continuous function of the starting value 0 [ 1]. The patched surfaces so generated are independent of which r value, at whatever pressure r, one takes as starting point: one may say they are invariant to reference pressure. (But see the comments below for regionally variable patched potential density.) The potential density leaves need not match perfectly at the boundaries between the pressure intervals. These discordances are sites for leakage of material across the joint, patched surfaces. This leakage is the discrete analogue of the nonmateriality of the continuous limit of patched potential density surfaces, namely, orthobaric isopycnals (de Szoeke et al., 2000; see below).
4. Orthobaric density The limiting form of the algebraic relation (1) is a differential equation,
d ϭ ⌫ ͑p, ͒, subject to ϭ at p ϭ r, (1)Ј dp 0 r
ϭ whose solution is ˆ ( p; r, r), which generates the limit, of in situ density as a function of pressure p, on a continuously patched potential density surface, which we call Ј an orthobaric density surface. Figure 10 shows solution trajectories ˆ ( p;0, 0) of (1) for both northern and southern virtual compressibility functions. One may label the surface by Ј its value of in situ density r at reference pressure r.If rЈ, r occur on the same orthobaric ϭ Ј density surface, i.e., rЈ ˆ (r , r; r), then one might as readily label the surface by its Ј value rЈ at r . Whatever the label, the form of orthobaric density surfaces in physical ϭ space does not depend on reference pressure. The inverse function, r ( p, r; ), maps a value of in situ density observed at pressure p into a value of orthobaric density r with respect to reference pressure r. (If it is agreed to use always the same reference pressure (zero, say), the dependence on r may be suppressed in the notation.) The orthobaric density function is easily understood graphically from Figure 10. One simply locates an in situ density value at its ambient pressure p on Figure 10; then one traces the trajectory through that point to the reference pressure to read off the orthobaric density. Orthobaric ⌫ density surfaces predicated on the virtual compressibility 0 are the continuous limit of patched potential density surfaces. In effect, the data in Table 1, originally presented in that form by Reid (1994), specify such a virtual compressibility function, even if on a coarse pressure and density grid (de Szoeke et al., 2000). Because the data of Table 1 provide such a good fit to the more finely resolved data over the upper 2 hb (Fig. 9), we used them as piecewise linear functions to define the virtual 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 75
Figure 10. Orthobaric density defined by the piecewise linear data from Table 1, shown in Figure 9; southern type (solid), and northern (dashed). Orthobaric density of a water parcel is found by looking up its in situ density and pressure and following the trajectory it lies on to the reference pressure (zero).
compressibility. We then solved (1)Ј using this virtual compressibility function to obtain contours of constant orthobaric density as a function of p and . The orthobaric density surfaces defined in this manner are the continuous limit of the patched potential density surfaces defined by Reid (1994). a. Regional differentiation ⌫ As there are not merely one but two 0’s, generating two distinct sets of orthobaric density surfaces, a southern type and a northern (Fig. 10), a difficulty arises. How does one link the two sets of surfaces? In physical space, as in thermodynamic (ϪS Ϫ p) space, the northern and southern water masses are linked through the bridge transition regions, shown in Figure 5, on a given pressure surface. The first step in resolving the difficulty of two separate orthobaric densities in two disparate regions, is simply to “cut” the bridge at a point intermediate between the northern and southern branches, which we may as well 76 Journal of Marine Research [63, 1
Figure 11. North-south orthobaric isopycnal matching function [Eq. (2)], solid line. Abscissa is S, southern orthobaric density referenced to the surface; N is northern orthobaric density referenced Ϫ N ϪS S to the surface; ordinate is the difference N S. Dashed line: 0 0 vs. 0 for modified patched potential density (see Appendix); open circles correspond to Reid’s (1994) standard isopycnal surfaces.
choose to be at the geographic equator,3 and assign to the respective branches the adjacent sections of the bridge. Because isopycnals are on average level in the bridge region, i.e., ϭ B( p) (Fig. 3), we can associate unique but different values of northern and southern 7 orthobaric density with each other: N( p, B( p)) S( p, B( p)). By eliminating the parameter pressure, this gives the following mapping,
N ϭ fB͑S͒ (2) (Fig. 11), and so a way of constructing surfaces that are continuous on average from south to north. Observe how the density distribution in the bridge mediates this mapping. It should be stressed that the continuity of the mapping is possible only because of the level-isopycnal bridge region. Even so, as waters heave up and down in eddies and waves, and in situ isopycnals deviate from their mean levels in the bridge, discontinuities in the joint orthobaric isopycnals will intermittently appear at the equator. By associating northern and southern orthobaric density by means of (2), contours of regionally variable orthobaric density are on average continuous across the equator, e.g., along WOCE section A16, shown in Figure 12, though with jumps between northern and southern orthobaric density of order 0.1 kg/m3.
3. Alternatively, the cut may be defined thermodynamically as a boundary in ϪS Ϫ p space, i.e., on the panels of Figure 1, rather than at a geographic boundary such as the equator. 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 77
Figure 12. Regionally differentiated orthobaric density along WOCE section A16. The southern (northern) compressibility function is used in the southern (northern) hemisphere. They are matched at the equator using the function shown in Figure 11. b. Nonmateriality of orthobaric density Orthobaric density surfaces are not perfectly material. The substantial rate of change of orthobaric density, and hence the mass flow across orthobaric density surfaces, may be 4 ϭ ϩ obtained as follows (de Szoeke et al., 2000). From the equation of state, 0 ( p, , S), written in terms of potential temperature , one obtains the differential expression
d Ϫ ⌫dp ϭ Ϫ␣d ϩ dS ϵ dq, (3) , ϭץ/ץp is adiabatic compressibility, and ␣ϭϪϪ1ץ/ץwhere ⌫ϭcϪ2 ϭ S are thermal expansion and haline contraction coefficients. From the orthobaricץ/ץϪ1 density function ( p, ) one may obtain ץ ץ ϭ ͩ ͪ ϩ ͩ ͪ ϭ Ϫ1͑ Ϫ ⌫ ͒ ( d d 0dp (4ץ dp ץ d p p ץ ץ ⌫ ϭץ ץ Ϫ1 ϭ where ( / )p, and 0 ( / p) is the same as the virtual compressibility which appears in (1)Ј. Combining (3) and (4),
ϭ Ϫ1͑ ϩ ͑⌫ Ϫ ⌫ ͒ ͒ d dq 0 dp . (5) Per unit time, the rate of change of following a water parcel is
4. The derivation shown in de Szoeke et al. (2000) is for orthobaric specific volume rather than density, resulting in slightly different expressions for some quantities. 78 Journal of Marine Research [63, 1
˙ ϭ Ϫ1 ϩ Ϫ1͑⌫ Ϫ ⌫ ͒ q˙ 0 p˙ , (6) ˙ϭ ϭ ϩ ⅐ ٌ ϩ where p˙ Dp/Dt pt͉ u p p, etc., expressed with respect to isopycnal coordinates, and q˙ ϭ Ϫ␣˙ ϩS˙ is the sum of irreversible, diabatic, buoyancy sources due to heat and salt transfer and diffusion. The last equation may be re-written
Ϫ1 Ϫ1 Ϫ1 Ϫ1 (g p ˙ ϭ g p q˙ ϩ ͑ Ϫ 1͒g ͑pt͉ ϩ u · ٌ p͒, (7a where
ϭ ͕ Ϫ Ϫ1͑⌫ Ϫ ⌫ ͒ ͖Ϫ1 1 0 p . (7b) Eq. (7a) is the orthobaric diapycnal mass flux, in units of kg mϪ2 sϪ1. Using the EOS to eliminate S, we may write the compressibility as a function ⌫(p, , ), and the virtual ⌫ ϭ⌫ Ϫ compressibility in terms of this as 0 (p, , 0(p, )). (Here the S relation, rendered by 0(p, ), may be thought of, in terms of each panel of Figure 1 (at constant p), as temperature as a function of density.) The first term in a Taylor expansion of compressibility is ⌫ץ ⌫ Ϫ ⌫ Х ͩ ͪ ͑ Ϫ ͑ ͒͒ ( 0 p, , (8aץ 0 p, where, by a change of variables, the coefficient can be shown to be
2 2 Ϫ ץ ץ ץ ⌫ץ ⌫ץ ⌫ץ Ϫͩ ͪ ϭ Ϫ ͩ ͪ ϩ ͩ ͪ ϭ Ϫ ϩ ϭ ͩ ͪ. (8b) p ץ p SץSץ p ץץ Sץ ץ ץ p, p,S S p, S S This is Tb, where Tb is McDougall’s (1987) thermobaric parameter, or / gH␣, where H␣ is Akitomo’s (1999) depth-scale. (Since the pressure dependence of S is less than 5% 2 / ץthat of , to this degree of approximation this coefficient may be replaced by Ϫ p.) Thus one may writeץץ
Х ͕ Ϫ Ϫ1 ͑ Ϫ ͑ ͒͒͑ ͒Ϫ1 ͖Ϫ1 Ј 1 0 p, gH␣ p . (7b) The second term of (7a) is the reversible diapycnal flow associated with the deviation of ץ ץ ϭ⌫ a water parcel’s true adiabatic compressibility ( / p),S from the virtual compress- ⌫ Ј ibility 0( p, ) used in the calculation (1) of orthobaric density; its importance is measured by the degree to which the buoyancy gain factor differs from 1. This difference multiplies the apparent vertical motion of the orthobaric isopycnal. Comparison of the definition (1) of patched potential density with that of orthobaric density, (1)Ј, where each is predicated on the same virtual compressibility function, or ϪS relation, shows that the former is a discrete analogue of the latter. Each leaf of potential density is quasi-material—only irreversible, diabatic processes can cause flow across it—analogous to the second term in (3a). But adjacent leaves do not match perfectly at their bounding pressure levels. The mismatch is proportional to the local departure from the ideal ϪS relation. The discrete analogue of the reversible nonmateriality of orthobaric density—the first term of (3a)—is to be found in the leaking of material through these discordances (de Szoeke et al., 2000). 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 79
⌫Ϫ⌫ The relation of compressibility anomaly 0 of a water parcel specified by , S, p to its displacement from the ϪS relation may be seen graphically on Figure 1. For the relevant pressure, one simply measures the temperature displacement of the water parcel from the idealized ϪS relation (having chosen the appropriate branch) along an SVA contour, and multiplies it by the thermobaric parameter p. Water parcels taken from near the respective southern or northern branches of the ϪS Ϫ p relation have a small ⌫Ϫ ⌫ Ϫ 0, and the factor 1 in the second term in (7a) is consequently small. On the other ⌫Ϫ⌫ hand, for water parcels taken from near the bridge, 0 may be large, and larger the further along the bridge from the respective branch. Hence the buoyancy gain factor may differ very significantly from 1, and the second term of (7a) makes a significant contribution to the diapycnal flow. Thus regionally differentiated orthobaric density surfaces may be significantly leaky as they cross the bridge waters. As the “equator” is ⌫ crossed, the virtual compressibility 0( p, ) used in the calculation of (7) must be changed from northern to southern, or vice versa. This results in a discontinuity and a sign change in ⌫Ϫ⌫ Ϫ 0, and so 1. These remarks are illustrated in Figure 13, which shows the geographical distribution of along section A16. Both northern and southern versions are shown. When the water sample and the virtual compressibility function are concordant (i.e. from the same branch/hemisphere), the values of ͉Ϫ1͉ are usually less than 0.1. Note, however, a region of ͉Ϫ1͉ Ͼ 0.2 near 38N and between 1000 db and 2000 db in both figures, corresponding to an intrusion of salty Mediterranean water. When they are discordant, the value of can differ quite substantially from 1. The regionally differentiated orthobaric density, patched across the equator, corresponds to using from Figure 13a in the North Atlantic, and from Figure 13b in the South Atlantic, the better parts of each, though with a discontinuity at the equator (even if regionally differentiated orthobaric density surfaces are continuous). In addition to measuring materiality, the quantity Ϫ1 provides a measure of the neutrality of an orthobaric density surface. The angle between the gradient of orthobaric density and the local dianeutral vector is adequately approximated by
␣ Ϸ͉ Ϫ 1͉ ٌ͉ p͉  g I
de Szoeke et al. (2000). The dianeutral direction is held to be significant because it is often supposed that the diffusivity tensor is diagonal with respect to principal axes oriented to the dianeutral vector. If this assertion is correct, then the diffusion tensor rotated with axes oriented to a coordinate system defined by the orthobaric density gradient results in a false dianeutral 2 diffusion given by KH, where KH represents the epineutral diffusivity (Redi 1982), which Ϸ must be compared with the true dianeutral diffusion, KD. For typical oceanic values, KD/KH O(10Ϫ8), so values of 2 in excess of this figure indicate significant non-neutrality. Using the values of appropriate for each hemisphere, a composite section (Fig. 14) shows that in most of the upper 2000 db, 2 is less than 10Ϫ8. A notable exception is the intrusion of Mediterra- nean water which has a large ͉Ϫ1͉ and sloping isopycnals. On the other hand, 2 is not 80 Journal of Marine Research [63, 1
Figure 13. Buoyancy gain factors N, S along section A16, using northern and southern orthobaric density, respectively.
particularly large near the equator, despite the rather large values of ͉Ϫ1͉ there because the isopycnal surfaces are nearly flat (i.e. have little horizontal gradient).
5. Regional differentiation of patched potential density: some anomalies The patched isopycnal surfaces specified by Table 1 generally cross the equator in the domains of the potential density leaves indicated by the boldface values, with which Reid (1994) labels them. These leaves cross more or less close to their reference pressures. This 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 81
Figure 14. Logarithm (base 10) of the index of neutrality, 2, defined in the text, along section A16. A value ϾϪ8 signifies significant non-neutrality of orthobaric density. The concordant value of is used in each hemisphere.
is a fortuitous property of these surfaces, obviously not pertinent to all conceivable patched isopycnal surfaces intermediate between those of Table 1. We will show now that patched surfaces crossing the equator near one of their transition pressures, where the reference pressure of a leaf is about to change, evince particularly anomalous behavior. In Table 2 we combine some of the information from Table 1 about the association of 0, ϭ 1 leaves from the northern and southern branches of patched surfaces. For example, 0 ϭ ϭ 26.75 joins to 1 31.09, 31.188 in the north and south, respectively, while 1 31.938
Table 2. Some interpolations (in italics) of Table 1. North South 0 1 1 26.75 31.09 31.188 27.099 31.519 31.678 27.229 31.678 31.846 27.30 — 31.938 27.44 31.938 — 82 Journal of Marine Research [63, 1
Ϸ Ϸ Figure 15. Potential density surfaces, 0 ( 27), 1 ( 31), near the equator and near 500 db on section A16. Note the idiosyncratic transitions required by Table 2 for unmodified regionally ϭ ϭ differentiated surfaces. Modified isopycnal ( 1, 0) (31.678, 27.229) (north) links to ( 1, 0) (31.742, 27.149) (south, Table 3).
ϭ joins to 0 27.44, 27.30 in the north and south (shown on two rows of Table 2). Between ϭ ϭ these we have interpolated the associations of 1 31.678 with 0 27.229 (north), 27.099 (south). We also show the alternate 1 associations, again interpolated, of these 0 ϭ leaves, should they be encountered in the opposite hemisphere. The isopycnal 1 31.678 was chosen from WOCE section A16 (Fig. 15), because it rises from the north to intersect 500 db as nearly as can be determined at the equator. Figure 15 demonstrates a particular difficulty with regional patching. The surfaces Յ Յ 31.519 1 31.678 intersect 500 db from higher pressure between the equator and 20N. Յ Յ They should join onto 27.099 0 27.229 for pressures less than 500 db. But this density range is barely found shallower than 500 db. (Perhaps the lighter end 27.099/ 31.519 can be patched in the north. But soon 27.099, as it fluctuates, might go below 500 db again. What should it patch to, as 31.519 is not to be found below 500 db south of 20N?) This significant, broad range of isopycnals can find no home shallower than 500 db because its members are crossing this level in the bridge region at ϪS values far Ϫ different from those of the northern branch of the S relation at which the 0, 1 pairs of Table 2 and Figure 6a are determined. 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 83
a. Bifurcation of isopycnals ϭ Further, the 1 31.678 surface must bifurcate in the following sense as it crosses the ϭ equator. The isopycnal infinitesimally lighter than 1 31.678 crosses 500 db first in the ϭ north and so must join onto 0 27.229, while the isopycnal infinitesimally denser than ϭ 1 31.678 first crosses the equator, and then crosses 500 db in the south, and so must join ϭ onto 0 27.099. If both these isopycnals were to be found above 500 db in the south, one might wonder, first that a 0.13 density difference had suddenly opened up between the infinitesimally close isopycnals crossing from the north, and second that the dense ϭ isopycnal had joined onto the light, and vice versa. As it is, the 0 27.229 isopycnal is ϭ found much deeper than 500 db in the south. Perhaps one could anticipate that the 1 ϭ 31.678 isopycnal, crossing 500 db slightly to the north, and associated with 0 27.229, ϭ which is in turn associated with 1 31.846 in the south (Table 2), will transition virtually to the latter isopycnal in the south. This merely accentuates the bifurcation/reversal difficulty noted above. One could imagine two floats, programmed to follow patched isopycnals, one slightly ϭ lighter than 1 31.678, the other slightly denser, being swept up through 500 db by the horizontal currents from the north as they follow their respective isopycnals. The first float, ϭ crossing 500 db slightly north of the equator, must attempt to find the 1 31.846 surface eventually in the south.5 On the other hand, the second float, crossing 500 db slightly south ϭ of the equator, must attempt to find the 0 27.099 surface. So the two floats cross and then diverge to reach their target isopycnals (Fig. 15).6 Each float must non-materially cross a finite range of density as it maneuvers. Thus, on this account, one cannot regard the patched potential density surfaces, which the floats are following, to be material. Situations can become even more complicated. Suppose the floats are carried by a fluctuating ϭ meridional motion back and forth through the equator. Take the float on the 0 27.099 ϭ surface. If it re-crosses 500 db south of the equator, it goes back to 1 31.678; if north, it ϭ must pass to the lighter isopycnal— 1 31.5185. The float has spiraled upward through a significant range of density, even though carried in a perfectly adiabatic, reversible motion field. This behavior is purely an artifact of implementing the rule for following regionally variable potential density surfaces—specifically of using the 0, 1 pairings predicated on the northern and southern branches of the ϪS relation in the bridge region far from the region of their applicability. ϭ By contrast, the 1 31.938 isopycnal, which Reid (1994) shows, crosses the equator and the whole bridge region near the center of its range, 500–1500 db, rising across 500 db in the north and south within the region of validity of the northern and southern branches of
5. The regional boundary may be defined thermodynamically, as suggested in Footno1te 3, rather than geographically. Then it would not be necessary for the floats to know their geographic position, as the discussion here seems to presume. 6. Some provision must be made for a float seeking a surface denser than the ambient 0 at 500 db to wait at that pressure until it encounters its target density. A float seeking a lighter density than ambient may be programmed to rise to its level. 84 Journal of Marine Research [63, 1
Ϫ ϭ the S relation so that it can join quite smoothly onto 0 27.44, 27.30 respectively. This innocuous behavior is misleading. The range of 1’s that cross 500 db in the bridge ϭ region will inevitably suffer the kind of difficulties described above. The fact that 1 31.938 seems to avoid these difficulties is an artifact of the wide spacing, 1000 db, of the ϭ levels between the isopycnal leaves, and that 1 31.938 falls fortuitously near the middle of such a wide range. This immunity disappears if smaller spacings are employed.
b. A resolution of the difficulty We have dwelt on this anomalous example because it seems essential that regionally differentiated, coarsely patched, potential density surfaces should be well defined, no matter at what pressure level they cross the equator. Their good behavior should not depend on crossing near the reference pressure. Accordingly, we describe, in the Appen- dix, a way to modify the regional matching of potential density at the equator to remove these anomalies. It amounts to introducing small discontinuities in the surfaces at the interregional boundary (the equator)—small in the sense that they vanish as the pressure spacing of the potential density leaves vanishes. Thus it is the discrete analogue of the regional matching method of continuous orthobaric density, described above.
6. Global orthobaric density De Szoeke et al. (2000) defined orthobaric density using a virtual compressibility function which was globally averaged over all oceans, and which is shown on the panels of Figure 16. The calculation took no account as such of northern or southern branches, or bridge. The standard ϪS relations associated with this virtual compressibility function are shown in Figure 17. At 500 db, these are similar to the ϪS relation associated with the southern branch of the Atlantic ϪS relation shown in Figure 8, though displaced to slightly higher salinities owing to the influence of the saltier northern Atlantic branch below 10°C. Above 10°C, the fresher, warmer waters of the other ocean basins dominate the salty north Atlantic in the global average. One can then predict that the degree of immateriality of the globally defined orthobaric density surfaces, as measured by the global buoyancy factor G, is similar to that found when applying the southern virtual compress- ibility everywhere. A plot of G along A16 confirms this (see de Szoeke et al., 2000). At first sight this seems to suggest that global orthobaric density is less material than regional orthobaric density, but it must be recalled that there are no interregional discontinuities associated with global orthobaric density, so all nonmateriality is related solely to the G factor. Whether this amounts to more or less diapycnal flow than regionally defined orthobaric density remains an unanswered question.
7. Discussion The Atlantic Ocean occupies an essentially two-dimensional sub-manifold of ϪS Ϫ p space, consisting of northern and southern branches of waters, with a bridge between 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 85
Figure 16. The virtual compressibility of global orthobaric density, solid dots and line (de Szoeke et al. [2000]), superimposed on the WOCE-Atlantic data points (grayscale). Cf. Figure 9. At pressures lower than 2000 db, the global compressibility lies between the southern and northern branches of Figure 9, though closer to the former.
them. The main lines of the oceans’ ϪS Ϫ p relations, the branches, are a staple of oceanographic lore, and their explanation in terms of end members determined by air-sea-ice interaction at formation sites, and mixing through the ϪS plane, is standard. The unexpected simplicity of the structure of the bridge waters of the Atlantic in ϪS Ϫ p space, lying on isopycnals at constant pressure, is intriguing. The bridge occurs because of the high salinity sources of the Mediterranean. The Atlantic’s northern and southern branches reflect strikingly the choices Reid (1994) made for linking potential isopycnals across adjoining 1-hb spans. De Szoeke et al. (2000) argued that Reid’s (1994) choices amounted to selecting, albeit on a coarse grid, virtual compressibility functions (a northern and a southern) which could be used to construct a regionally differentiated orthobaric density. In the upper 2 hb of the Atlantic we refined Reid’s method by calculating and plotting ⌬/⌬p vs. over 0.2-hb intervals, giving the required virtual compressibility (Fig. 9). Each point of the sets of data that make up the panels of Figure 9—calculated solely from an observation of , S on the relevant pressure surfaces, via the seawater equation of state—reflects its thermodynamic state, not its location. The uniformity of the virtual compressibility functions with respect to pressure, at 86 Journal of Marine Research [63, 1
Figure 17. vs. S relation at several pressures, equivalent to de Szoeke et al.’s (2000) globally averaged virtual compressibility (Figure 16), solid dots and line, superimposed on the WOCE- Atlantic data points (grayscale). Cf. Figure 8.
least in the upper 2 hb, and their concurrence with Reid’s (1994) estimates [Fig. 9b, f] is remarkable. The use of regionally differentiated orthobaric density introduces the complication of matching surfaces calculated from the respective compressibility functions across the boundaries between the regions. This difficulty is resolved by taking advantage of the observation that density is nearly constant on pressure surfaces in the bridge region of the ϪS Ϫ p relation. Then one may uniquely associate northern orthobaric density with southern orthobaric density, so overcoming an objection that de Szoeke et al. (2000) raised to such regional differentiation. This results in quasi-continuous surfaces across the regional boundary. We say quasi-continuous, because adiabatic vertical oscillations of density surfaces from their mean levels at the regional boundary may open up temporary discontinuities in the joint surfaces. Also, the sections of the orthobaric isopycnals where they cross the bridge region are prone to being relatively highly nonmaterial—indicated by significantly nonunit values of the buoyancy gain factor. An advantage of using orthobaric density—a function of density and pressure only—is that, relative to orthobaric density () surfaces, the pressure gradient acceleration and 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 87 gravitational acceleration in the momentum balance may be replaced by an “acceleration potential” gradient, i.e., the Montgomery function (M) gradient:
Ϫ1 Ϫ1 ϩ ⌸͑p͒ץ/Mץz ϩ g ϭ Ϫץ/pץ Ϫ͑, S, p͒ ٌHp ϭ ϪٌM, Ϫ͑, S, p͒
(de Szoeke, 2000). Relative to surfaces of any other variable, say Ј, not a function of density and pressure only, these replacements must include additional terms, expressible as
Ј (or equivalent), so that the separation of horizontal balances from theץ/SץٌЈS, buoyancy balance is not so clear. While this complication can be dealt with (Sun et al., 1999), it means, for example, that the simple thermal wind balance, geostrophic shear vs. slope of Ј surfaces, does not hold. A remnant of this complication does survive with regionally variable orthobaric density when the surfaces are discontinuous at the regional boundary. Then the additional terms have the form of an impulsive force applied at that boundary. The matching of regionally differentiated orthobaric density across the bridge between northern and southern branches suggests a modification of Reid’s (1994) method of calculating regionally differentiated patched potential density to eliminate an anomaly in that concept which we noted in Section 5, associated with the coarseness of its vertical resolution, and the lack of recognition of the properties of the bridge region. While it seems preferable on grounds of vertical continuity and ease of use to employ regionally differentiated orthobaric density rather than (modified) patched potential density, we have, because of the wide use of patched potential density in descriptive physical oceanography, shown how to calculate the potential density re-mappings required by the modification (see the appendix). These re-mappings imply discontinuities in the surfaces at the interregional boundaries, which may be sites for mass exchange (“diapycnal flow”) analogous to the non-materiality of the continuous orthobaric density analogue. De Szoeke et al. (2000), eschewing “regionally differentiated” orthobaric density as described above, chose a unique, global-average virtual compressibility function, which is shown on Figure 16, to define a global orthobaric density function. This lies between the northern and southern branches of the Atlantic virtual compressibility, somewhat nearer the southern because of the influence of the other oceans in making up the global average. ͉ Ϫ ͉ ͉ Ϫ ͉ As a result, G 1 tends to be slightly larger in the southern hemisphere than S 1 ͉ Ϫ ͉ while in the northern hemisphere it is significantly larger than N 1 . However, it is continuous across the equator and has no leakiness associated with any matching there, unlike regionally differentiated orthobaric density. The potential leakiness, or nonmateriality, of regionally differentiated orthobaric density and its coarsely resolved analogue, patched potential density, in bridge waters, which are quite extensive, must be kept well in mind. Reid (1994) warned that “the [patched] isopycnal surfaces mapped are not truly isentropic or ideal neutral surfaces. However, . . . these surfaces can be useful approximations to such idealized surfaces.” We concur in this warning and recommendation, and offer the discussion given in this paper as a 88 Journal of Marine Research [63, 1 systematic examination of the departure of these surfaces from the “ideal,” which we take to mean their proneness to nonmateriality. Jackett and McDougall (1997), in a study of an empirical method for calculating neutral density, used Reid’s (1994) patched potential density surfaces as a standard of comparison. Their method relies on using a “neutrally labeled” reference global hydrographic data set from which the displacement of a subsequent observation of temperature and salinity at an ambient pressure and location is calculated. This displacement is used to relate the reference neutral density to the observation. Since the reference data hews closely to the respective branches when in the northern and southern waters, one may expect Jackett and McDougall’s neutral density surfaces to coincide closely in these water masses with Reid’s patched potential density surfaces and their continuous limit, regionally differentiated orthobaric density surfaces based on the ϪS relations of these waters. In the transition between these waters—the bridge—in the subtropical-tropical Atlantic, where we have had to modify patched potential density to make it self-consistent, the correspondence is not so clear, despite Jackett and McDougall’s (1997) unqualified assertion of the equiva- lence of neutral density and patched potential density. We recommend the use of regionally differentiated orthobaric density over modified patched potential density, first because of the elimination of the coarse grid of multiple reference pressures, but also because of the far greater ease of use. The calculation of the northern or southern orthobaric densities from in situ density and pressure, in effect by looking up the isopleths of Figure 10, is straightforward and easily automated. The buoyancy gain factor of regional orthobaric density should always be calculated, and examined as an indicator of its degree of reversible materiality. (The difference of buoyancy gain factor from unity is only one factor of the contribution to non-materiality; ϩ ⅐ ٌ ϭ ץ the other is the apparent motion, ( tp u p), of the orthobaric density surface, constant.) We have demonstrated how to devise a regionally variable orthobaric density for the North and South Atlantic Oceans. Inspection of ϪS Ϫ p relations for the global ocean reveals interesting features for the other basins as well (Sverdrup et al., 1942). Like the Atlantic, the Pacific has two branches, a southern branch almost identical with the Atlantic’s southern branch and a northern branch with even fresher waters; though the Pacific has no bridge. The Indian Ocean has the Atlantic’s southern branch solely, with a “bridge to nowhere” owing to the salinity influence of the Red Sea. The Indian bridge waters, like the Atlantic’s, are uniform in density on constant pressure surfaces, though lighter than in the Atlantic. By extending the methods described herein, these features could be incorporated into a more comprehensively defined, regionally differentiated orthobaric density that distinguished different provinces of the world’s oceans.
Acknowledgments. This work was supported by the National Science Foundation under grant 0220471. We are indebted to Drs. T. McDougall and D. Jackett for bringing the thinness of the ocean to our notice. Nick Fofonoff’s careful work on seawater thermodynamics has been an inspiration. 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 89
APPENDIX Regional differentiation of coarsely resolved patched potential density The difficulty in matching the regionally differentiated forms of patched potential density will be resolved in the following. Though continuous, regionally differentiated, orthobaric density makes patched potential density obsolete and is easier to calculate as a practical matter; still, it is instructive to see how the discrete form can be made consistent and universal (well defined for any starting density), and to see how the inevitable inter-regional gaps in the surfaces arise. A patched potential density surface in the northern hemisphere is specified by a multiplet of numbers such as
⌺N ϭ ͕N N N N ͖ 0 , ⌬p, 2⌬p, 3⌬p,... (A1) N where each r is a value of potential density referenced to r, and refers to a surface, called r Ϫ 1 ⌬p Յ p Յ r ϩ 1 ⌬p a “leaf,” of that density for pressures in the range 2 2 . The patched surface ⌺N is the concatenation of all these leaves. (A standard choice for ⌬p is 1 hb.) Any N ϭ ⌬ one of the r values, say for r n p, generates all the others from repeated application of
N N ͑ Ϯ ͒⌬ Ϫ ⌬ 1 1 n 1 p n p ϭ ⌫Nͩͩ Ϯ ͪ⌬ ͑N ϩ N ͒ͪ n p, ͑ Ϯ ͒⌬ ⌬ (A2) Ϯ⌬p 2 2 n 1 p n p
where ⌫N is the virtual compressibility function for the northern hemisphere (cf. (1)). Patched potential density surfaces ⌺S in the southern hemisphere are specified in a similar way. The in situ density of the bridge, as a continuous function of pressure, is B(p) [see Fig. 3]. This means that at pressure p the isopycnal with value B( p) intersects the northern branch of the ϪS relation (Fig. 1) at a point whose virtual compressibility is well approximated by
N N 1 1 N N ⌫˜ ϭ ⌫ ͑͑n ϩ ͒⌬p ͑c ϩ ϩ c ͒͒ 2 , 2 n 1 n (A3)
where n is the largest integer not larger than p/⌬p (thus n⌬p Յ p Ͻ (n ϩ 1)⌬p). The N N ⌺N ϭ N N densities cn , cnϩ1 are leaves of a northern patched surface {..., cnϪ1, cn , N cnϩ1, . . .} which satisfy (A2); they also are taken to satisfy
N Ϫ ͑ ͒ ͑ ͒ Ϫ N cnϩ1 B p B p cn ϭ ⌫˜ N ϭ , (A4) ͑n ϩ 1͒⌬p Ϫ p p Ϫ n⌬p
which determine them. (The second equality of (A4) follows from the first and from (A2).) ⌺N In this way, p and B( p) generate all the leaves of the patched isopycnal surface ( p) that most nearly joins to the bridge at pressure p. Similarly, a southern patched surface ⌺S ϭ cS cS cS n⌬p Յ p Ͻ n ϩ 1 ⌬p {..., nϪ1, n, nϩ1, . . .} can be generated. For ( 2) , we take N S cn , cn to be the almost-matching density leaves crossing the inter-regional boundary (the 90 Journal of Marine Research [63, 1
Table 3. Potential density matching at the equator for modified patched isopycnals, with ⌬p ϭ 1000 db
N S N S N S Ϫ3 p/db c0 c0 c1 c1 c2 c2/kg m 200 26.904 26.881 300 26.999 26.961 400 27.094 27.039 500 27.183 27.111 31.622 31.694 600 31.725 31.783 700 31.819 31.862 800 31.905 31.934 900 31.984 31.997 1000 32.052 32.052 1100 32.125 32.114 1200 32.187 32.169 1300 32.241 32.218 1400 32.286 32.261 1500 32.323 32.299 36.808 36.832 1600 36.851 36.866 1700 36.885 36.895 1800 36.913 36.920 1900 36.936 36.939 2000 36.954 36.954
ϭ ⌬ N ϭ S ϭ equator). (Only if p n p are they the same, and the same as B( p): cn cn n⌬p ϵB n ϩ 1 ⌬p Յ p Ͻ n ϩ ⌬p cN cS B( ) n .) For ( 2) ( 1) , we take nϩ1, nϩ1 to be the almost matching density leaves. In any event, one obtains in this way two sets of densities as a function of pressure: cN( p), cS( p). This implies a cross-regional association of patched surfaces to make up a joint north-south patched surface. Subtracting (A4) from a similar equation for south branch parameters, one may see that
cN͑p͒ Ϫ cS͑p͒ ϭ ͑p Ϫ n⌬p͒͑⌫˜ S Ϫ ⌫˜ N͒, for n⌬p Յ p Ͻ ͑n ϩ 1͒⌬p, 2 (A5) ϭ Ϫ͑͑n ϩ ͒⌬p Ϫ p͒͑⌫˜ S Ϫ ⌫˜ N͒ ͑n ϩ 1͒⌬p Յ p Ͻ ͑n ϩ ͒⌬p 1 , for 2 1 .
Though ⌫˜ N, ⌫˜ S themselves depend on cN( p), cS( p), the difference, as one can see from Figure 9 at fixed p, does not vary strongly with density. Thus the difference cN( p) Ϫ cS( p) p ϭ n⌬p 1 ⌬p ⌫˜ N Ϫ ⌫˜ S p ϭ n ϩ 1 ⌬p ranges from zero at ,upto2 ( )at ( 2) , changes sign abruptly, then back to zero at p ϭ (n ϩ 1)⌬p. This shows that the regionally matched Ϯ1 ⌬p ⌫˜ N Ϫ ⌫˜ S surfaces have discontinuities of density at the equator as large as 2 ( ). These discontinuities vanish in the continuous limit, ⌬p 3 0. For coarsely resolved potential density surfaces, ⌬p ϭ 1 hb, with differences of ⌫˜ S Ϫ ⌫˜ N Ϸ 1.4 ϫ 10Ϫ8 mϪ2 s2 at pressures lower than 2000 db (Fig. 9), they may be as large as 0.07 kg mϪ3. Table 3 shows 2005] de Szoeke & Springer: All-Atlantic temperature-salinity-pressure relation 91
Figure 18. Density difference at the inter-regional boundary vs. southern density for modified regionally differentiated patched potential density (from Table 3).
some north-south correspondences of cN( p), cS( p) calculated by solving (A3), (A4); the difference cN Ϫ cS is shown as a function of cS on Figure 18. Thus we would modify Reid’s (1994) method of patching isopycnals to take account of the properties of the bridge in matching isopycnals across regions with markedly different ϪS relations. Here is how the example considered in the main text, of isopycnals around ϭ 1 31.678 crossing 500 db from greater pressure from the north at the interregional ϭ boundary (taken to be the equator), would be modified. If 1 31.678 crosses the equator ϭ at slightly higher pressure than 500 db, one would join it to 1 31.742 south of the N S equator, obtained by interpolating the c1 , c1 correspondences in Table 3. Following this ϭ isopycnal to the south, supposing it crosses 500 db, one would link it to 0 27.149, ϭ obtained from interpolating Table 2 (or Table 1). On the other hand, if 1 31.678 crosses ϭ 500 db slightly north of the equator, one might wish to join it to 0 27.229 (Table 2), though this isopycnal is found far deeper than 500 db. Nevertheless interpolation of Table ϭ 3 indicates that when this crosses the equator it is to be joined to 0 27.147 (and thence ϭ to 1 31.741, from Table 2), which one may take to be virtually identical, given ϭ truncation and rounding errors in the calculations, with 27.149 ( 1 31.742), arrived at via the alternative path. This illustrates how the modified method eliminates the difficulty noted in the main text, of ranges of isopycnals that cannot be properly matched across 92 Journal of Marine Research [63, 1 regional boundaries, because infinitesimally close isopycnals follow widely divergent paths. On the other hand, the gaps in the isopycnals across regional boundaries so created ϭ ϭ (e.g., between 1 31.678 and 1 31.742 on Fig. 15—a gap of order 100 db) provide a locus for “diapycnal” flow, i.e., flow across the joint surface formed by the linking of the northern and southern leaves. This is the analogue of the reversible diapycnal flow across continuous regionally differentiated orthobaric isopycnals associated with anomalies of compressibility from the respective virtual compressibilities of the north and south branches.
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Received: 22 July 2004; revised: 21 December, 2004.