2830 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
Orthobaric Density: A Thermodynamic Variable for Ocean Circulation Studies
ROLAND A. DE SZOEKE,SCOTT R. SPRINGER, AND DAVID M. OXILIA College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon
(Manuscript received 21 October 1998, in ®nal form 30 December 1999)
ABSTRACT A new density variable, empirically corrected for pressure, is constructed. This is done by ®rst ®tting com- pressibility (or sound speed) computed from global ocean datasets to an empirical function of pressure and in situ density (or speci®c volume). Then, by replacing true compressibility by this best-®t virtual compressibility in the thermodynamic density equation, an exact integral of a Pfaf®an differential form can be found; this is called orthobaric density. The compressibility anomaly (true minus best-®t) is not neglected, but used to develop a gain factor on the irreversible processes that contribute to the density equation and drive diapycnal motion. The complement of the gain factor, Ϫ 1, multiplies the reversible motion of orthobaric density surfaces to make a second contribution to diapycnal motion. The gain factor is a diagnostic of the materiality of orthobaric density: gain factors of 1 would indicate that orthobaric density surfaces are as material as potential density surfaces. Calculations of the gain factor for extensive north±south ocean sections in the Atlantic and Paci®c show that it generally lies between 0.8 and 1.2. Orthobaric density in the ocean possesses advantages over potential density that commend its use as a vertical coordinate for both descriptive and modeling purposes. A geostrophic streamfunction exists for the momentum equations transformed to orthobaric density coordinates so that the gradients of orthobaric density surfaces give precisely the geostrophic shear. A form of Ertel's potential vorticity can be de®ned whose evolution equation contains no contribution from the baroclinicity vector. Orthobaric density surfaces are invariant to the choice of reference pressure. All of these are properties that potential density lacks. In the continuous limit, patched potential density surfaces, which are formed by joining segments of locally referenced potential density surfaces in various pressure ranges and are extensively used in descriptive physical oceanography, become a particular form of orthobaric density surfaces. The method of selecting the segments is equivalent to choosing the virtual compressibility function. This correspondence is a useful aid in interpreting patched potential density surfaces. In particular, there is a material ¯ow across such surfaces that is analogous to the reversible ¯ow across orthobaric isopycnals.
1. Introduction that is determined by the speci®cation of the coordinate variable. As this coef®cient is the chemical potential In a companion paper (de Szoeke 2000) the equations when entropy is used as coordinate, it is termed the of motion were set down in a form in which an arbitrary chemical potential analogue. If conventional potential thermodynamical variable was used as vertical coordi- density is used as vertical coordinate, these salinity- nate instead of depth. The variable was expressed as a gradient terms are by no means negligible in pressure 1 function of pressure, speci®c volume, and salinity ; the ranges more than a few hundred decibars from the ref- only restriction on it was that it be monotonic with erence pressure. respect to depth. It was shown that, in general, using Potential density of seawater was originally put for- such a variable as a coordinate introduces into the mo- ward as a generalization of the notion of potential tem- mentum equations terms explicitly involving salinity perature for a single-component ¯uid like dry air (WuÈst gradients, multiplied by a thermodynamic coef®cient 1933; Montgomery 1938). In the case of dry air, po- tential temperature possesses dual thermodynamic and dynamic properties that make it uniquely useful. It is an alias of entropy, so it can change only because of 1 The third variable, salinity, could be replaced by any variable independent of the other two. The ensuing discussion would still irreversible processes, such as molecular diffusion, tur- follow, with appropriate modi®cations. bulent mixing, diabatic heating, cabbeling, etc.; we may call such a variable quasi-conservative or quasi-mate- rial. It is also the sole determinant in the ¯uid of the Corresponding author address: Dr. Roland A. de Szoeke, College of Oceanic and Atmospheric Sciences, Oregon State University, 104 buoyant force due to density variations. Potential den- Oceanography Admin. Building, Corvallis, OR 97331-5503. sity of seawater, on the other hand, though it may be E-mail: [email protected] quasi-conservative, is severely lacking in its dynamical
᭧ 2000 American Meteorological Society NOVEMBER 2000 DE SZOEKE ET AL. 2831 properties. It may exhibit inversions even in regions faces (Reid and Lynn 1971), will be examined (section where the ocean is stably strati®ed (Ekman 1934; Lynn 4). An interesting result will be established; namely, that and Reid 1968). It is not the sole determinant of buoyant in the limit of arbitrarily closely spaced locally refer- force, for salinity gradients on potential density surfaces enced potential density segments, a form of orthobaric may still generate signi®cant contributions to the mo- density is obtained. The speci®cation of the matching mentum balance (de Szoeke 2000). This behavior is a of potential densities at the interfaces between their consequence of the thermobaric character of seawater, pressure ranges is equivalent to selecting the ``best-®t,'' namely, that the thermal expansion coef®cient of sea- or virtual, compressibility. Just as regionally differen- water depends strongly on pressure (Ekman 1934; Mc- tiated methods of patching potential density are em- Dougall 1987a). ployed (Reid 1994), geographically variable extensions McDougall (1987b; also Jackett and McDougall of the orthobaric density idea may likewise be contem- 1997) introduced the notion of neutral density to over- plated. The properties of the latter may be used to gain come some of the dif®culties associated with potential insight into the former. density, including the problem of monotonicity when true strati®cation is stable. However, neutral density is 2. Toward a quasi-conservative thermodynamic not a thermodynamic function; it depends not only on variable the thermodynamic state of a water sample, character- ized by the triplet of pressure, temperature, and salinity, A principal aim of this paper is to determine a variable but also on the geographical location where it was col- ϭ (p, ␣), (2.1) lectedÐa dependence determined by reference to a global hydrographic dataset. This hybrid character a function only of pressure p and speci®c volume ␣ ϭ makes it dif®cult to work with, both theoretically and Ϫ1, that is nearly quasi-conservative2; that is, its sur- practically. faces are nearly normal to the dianeutral vector. We start In this paper we shall consider thermodynamic var- from the density equation, averaged over small-scale iables that are pycnotropicÐfunctions of pressure, p, processes, and speci®c volume, ␣, only. Such a variable, when used D Dp as vertical coordinate, produces no extra gradient terms, Ϫ⌫ ϭq. (2.2) besides the Montgomery function gradient, in the mo- Dt Dt mentum balance equations (de Szoeke 2000). This pyc- Here q speci®es the sources and sinks of density due to notropic property is what commends such variables for (i) irreversible molecular ¯uxes of heat and salt, which use in de®ning Ertel's (1942) potential vorticity. How- are usually negligible; (ii) recti®cation due to nonli- ever, in general, pycnotropic variables lack the quasi- nearities in the equation of state, which produce cab- conservative property that potential density possesses: beling and thermobaric effects; and (iii) small-scale tur- that the variable changes only because of irreversible bulent eddy transport divergences (McDougall 1987a; processes that transport or create heat and salt. McDougall and Garrett 1992; Davis 1994). The second The aim of this paper is to construct a variable, , term on the left represents the reversible adiabatic com- which we call empirical orthobaric speci®c volume, that pression in which ⌫ is the coef®cient of adiabatic com- is a function of p and ␣, monotonic in depth, and ap- pressibility,3 and may be considered a function of p, ␣,S: proximately quasi-conservative. By ``approximate,'' we 1/c 2(p, ,S), (2.3) shall mean that adiabatic changes in due to com- ⌫ ϭ ␣ pressibility are as small as can be arranged for such a c being the speed of sound. Suppose we write variable. We shall construct this variable by empirically ⌫ ϭ⌫ ϩ⌬⌫, (2.4a) ®tting adiabatic compressibility (or sound speed) to a 0 function of p and ␣, and then using this best-®t com- where pressibility to construct an integral of the Pfaf®an dif- 1 ferential form for density. ⌫ϭ , (2.4b) 0 2 The signi®cance of orthobaric speci®c volume (or its c0 (p, ␣) reciprocal, orthobaric density) is twofold. It may be used 11 as an alternative to potential density, conventional or ⌬⌫ ϭ Ϫ , (2.4c) 22 patched, for purposes of descriptive display of ocean- c (p, ␣, S) c0 (p, ␣) ographic data. Because of its attractive dynamical and 2 andc0 (p, ␣) shall be suitably chosen. We call ⌫ 0 the thermodynamic properties, it may be useful for theo- retical manipulations and for numerical modeling of ocean circulation. Its dual usefulness for description and modeling derives from the same attributes. 2 Exact conservation of any such variable is impossible (not only because of irreversible processes). The estimation of the degree of The relation of orthobaric density surfaces to patched ``nearness'' to conservation is part of the aim of this paper. potential density surfaces, the latter constructed by join- 3 Usually ␣⌫ is called the compressibility. For our purposes, the ing segments of locally referenced potential density sur- unconventional usage is convenient. 2832 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 virtual or best-®t compressibility. The goal is to choose 2 c0 to make the compressibility anomaly ⌬⌫ as small as practicable. This will be taken up in the following sec- tions. Equation (2.2) may be written D␣␣2 Dp Dp ϩϭϪ␣2⌬⌫ Ϫ ␣2q. (2.5) 2 Dt c0 Dt Dt The left side of this is an exact Pfaf®an differential; it may be written D /Dt, where (an integrating factor) and are pycnotropic functions determined as follows (Sneddon 1957). De®ne the virtual speci®c volume A(pЈ | p, ␣) by solving A ϪA2ץ ϭ (2.6a) 2 (pЈ c0 (pЈ, Aץ subject to
A(p | p, ␣) ϭ ␣. (2.6b) FIG. 1. Schematic showing a few members of the family ␣ ϭ (p, ), each for ®xed orthobaric speci®c volume . This is the potential speci®c volume relative to reference ␣ pressure pЈ, obtained as though the adiabatic compress- ibility of seawater were 1/c2 (p, ␣). At a ®xed reference 2␣ץץ 0 pressure, say pЈϭ0, A(0 | p, ␣) de®nes a function ln ϭϪ . (2.10) 2 c0␣ץ pץ (p, ␣) ϭ A(0 | p, ␣). (2.7) Given the well-behaved character of the right side of We call the orthobaric (meaning pressure-corrected) this equation, it is scarcely conceivable that can attain speci®c volume, conditioned on the choice of c 0. zero. Hence we conclude that Ͼ 0 strictly. The virtual speci®c volume may also be used to obtain De Szoeke (2000) showed two extreme examples of the inverse function of (2.7), namely ␣(p, ). Replacing forms of orthobaric speci®c volume, conditioned re- (2.6b) by spectively on the choices c 0 ϭϱand c 0 ϭ 0. The former A(0 | 0, ) ϭ (2.6c) corresponds to in situ speci®c volume itself, while the latter can be thought of as pressure. While neither of and integrating (2.6a) to pЈϭp, the original in situ these is particularly conservative, either may serve as speci®c volume ␣ must be recovered; namely, a useful coordinate, especially pressure. ␣(p, ) ϭ A(p |0,). (2.8) a. Standard salinity; compressibility anomaly A few members of this family, ␣ ϭ ␣(p, ), for ®xed , that is, isopleths of , are shown schematically on The virtual compressibility (2.4b) can be taken to Fig. 1. (Actual examples will be shown later, in Fig. 7). de®ne a standard salinity function S(p, ␣) by means of The assignation of a numerical label to the curves in the implicit relation Fig. 1 corresponds to the speci®c volume that a hypo- ⌫ (p, ␣,S( p, ␣)) ϭ⌫( p, ␣). (2.11) thetical water parcel evolving according to (2.6a) would 0 have at the reference pressure (zero in this case). Choos- Hence the compressibility anomaly (2.4c) is ing a different reference pressure in (2.7) and (2.8) ⌬⌫ ϭ ⌫ (p, ␣, S) Ϫ⌫[p, ␣, S(p, ␣)] would alter the label of a particular curve in Fig. 1 though not the form of the curve itself. The curves in ⌫ץ Fig. 1 are invariant in this sense, and selection of a ഡ [S Ϫ S(p, ␣)]. (2.12) Sץ reference pressure is arbitrary. p,␣ Ϫ2 p, with ,Sheld constant, a changeץ/␣ץ ␣The integration factor is given by Since ⌫ ϭϪ of independent variables shows that ␣ץ ␣ ץ ⌫ץ ␣⌫ץ ⌫ץ ( ϭ . (2.9 S Ϫ2 . p ϭϪ ϭ␣␣S lnץ p ␣Sץ p,SץS p, ␣ ץS p, ץ Because ␣(0, ) ϭ A(0 | 0, ) ϭ , it follows that ϭ ␣ (2.13) /(A(pЈ |0,ץ 1atp ϭ 0. By differentiating Eq. (2.6a) for -pЈ with respect to , we may show The expression after the ®rst equality of (2.13) is strongץ/(pЈ, )␣ץpЈϭץ that ly dominated by the second term. The expression after NOVEMBER 2000 DE SZOEKE ET AL. 2833
Ϫ1 2 Ϫ1 the second equality is proportional to McDougall's ϭ (1 ϩ ⌬⌫␣ p ) , (2.21) (1987a) thermobaric parameter Tb. It is also inversely proportional to the pressure (or depth) scale H dis- which will be called the buoyancy gain factor associated ␣ . This may be written, using (2.17), played by Akitomo (1999; see that author's Fig. 1): with 2 2 g g ( ϭϭ , (2.22 ץ ␣␣ ץ ⌫ץ ␣ 222 ϭ ln ϵ Tb ഡ ln␣ ␣nzϪ␣ pn pץ p Ϫ␣␣ ץ ,S pץSS␣ ␣ or, upon eliminating p between (2.17) and (2.22), Ϫ1 ϵ H␣ . (2.14) g2⌬⌫ This shows that compressibility anomalies occur be- ϭ 1 ϩ (2.23) n2 cause of deviations of water mass properties, speci®cally of salinity from the standard salinity S(p, ␣), coupled [cf. McDougall (1989)]. with the thermobaric character of seawater (Tb ± 0). In remarks following Eq. (2.10) we concluded that Ͼ 0. Hence the condition for the validity of the trans- formation to as vertical coordinate, namely that b. Buoyancy frequency ϭ 1/z Ͼ 0, (2.24) The average buoyancy frequency n is de®ned by z requires, from (2.22), that g2 ␣ץ g n2 ϭϪ, (2.15) (zc2 Ͼ 0. (2.25ץ ␣ and is calculated from the microstructure-averaged spe- (Actually, either sign of is acceptable; only a change ci®c volume pro®le and the average compressibility of sign is intolerable. By convention, we settle on the (2.3). By using the hydrostatic relation, positive sign.) The gain factor is an index of the quasi-conser- pp vativeness of the scalar . The irreversible sources andץ ϭϭϪg␣Ϫ1, (2.16) -zz sinks of density (including small-scale turbulent diffuץ sion), q, are multiplied by to give the sinks and sources one may write (2.15) as of in Eq. (2.20). The other contribution toÇ , con- taining the factor ( Ϫ 1), is due to a remnant of the pggץ222␣␣ץ g n2 ϭϩ ϩϪ, reversible compressibility effect. Large is an indica- 222 zcc tion of predominating reversible contributions toÇ . [Forץzc00ץ␣ the ®rst term of which, multiplied by ␣gϪ1 Ϫ1,isthe example, if were in situ speci®c volume, could be 2 z. Hence, using the second equal- O(10) (de Szoeke 2000).] A choice ofc0 that rendersץ/ץ exact differential ity of (2.16), and (2.4c) close to 1 will be sought. g2 n2 ϭϪ Ϫg2⌬⌫. (2.17) d. The transformed equations of motion 2 ␣ p The averaged primitive equations of motion can be put into a form that uses as vertical coordinate the or- c. Gain factor thobaric speci®c volume variable described above by Equation (2.5) may be written Eq. (2.7), provided only that condition (2.24) or (2.25) is met. These equations are D Dp Ç ϵϭϪ␣Ϫ12⌬⌫ Ϫ ␣Ϫ12q. (2.18) Du M ١ Dt Dt ϭϪf k ϫ u Ϫ Dt I The substantial derivative of pressure, written in terms of as independent variable, anticipating a result to be ϩ F (2.26) Mץ established below [Eq. (2.29)], is ϭ⌸ (2.27) ץ Dp (p ϩ Çp, (2.19( ١ ´ ϩtIuץ) ϭ ץץ Dt (١I ´(pu) ϩ (p Ç) ϭ 0; (2.28 p ϩ ץ tץ ١I is the gradient operator on surfaces of constant where . So Eq. (2.18) becomes y) is the gradient operator alongץ ,xץ) ١I ϵ where Ϫ1 2 -١I)p, -isopleths, u ϭ (u, , 0), and F is the horizontal fric ´ t ϩ uץ)(p Ç ϭϪp ␣ q ϩ ( Ϫ 1 (2.20) tional force. Equation (2.20) should be substituted for p Ç in (2.28). The substantial rate of change operator where is de®ned by 2834 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
Ç ; (2.29) The angle  between e and e 0 is given byץ١I ϩ ´ t| ϩ uץD/Dt ϵ the special functions appearing in (2.26)±(2.28) are |e ϫ e| sin ϭ 0 . (2.39) p |e0||e| H ϭ ͵ ␣(pЈ, ) dpЈ, (2.30) Substituting, one obtains 0 ١p ١p ϭ ␣2⌬⌫e ϫ ϫ ␣١⌫⌬e ϫ e ϭ ␣2 pЈ, ) 0 0)␣ץ p ⌸ϭ dpЈ (Exner function), (2.31) 2 (١p. (2.40 ϭ ␣ ⌬⌫e ϫץ ͵ 0 Both e and e 0 are dominated by their vertical compo- M ϭ H ϩ gz (Montgomery function). (2.32) nents, so that (2.15) and (2.22) show that Details may be found in de Szoeke (2000). Notice par- ␣␣ 2 2 ( terms from (2.26) |e| ഡ n ,|e0| ഡ n . (2.41ץ/SץS, ticularly the absence of ١ I gg and (2.27). This is a consequence of the pycnotropic character of . Because of this also, an Ertelian potential These are adequate approximations for the denominator vorticity can be de®ned, of (2.39). Hence, using the second equality of (2.40), and (2.23), f ϩ ( xyϪ u )| ١p Q ϭ , (2.33) e0 ϫ p sin ഡ ( Ϫ 1) . (2.42) ΗΗn2 which would be conserved if were conservative (Ç ϵ D /Dt ϭ 0) and if friction were negligible (F ϭ 0). Using the ®rst equality of (2.38), and (2.22), The function ␣(p, ) to be used in (2.30), (2.31) is ١p ١ ϫ ␣ ١p e ϫ de®ned by Eq. (2.8); it is the inverse function of (2.7). 0 ϭ . 2 From (2.9) and (2.31), the integrating factor is related ngz to by ⌸ Transforming the horizontal partial derivatives into de- rivatives along constant- surfaces, one obtains ⌸ ץ ϭ . (2.34) ␣ pץ sin ഡ | Ϫ 1| [p2222(1 ϩ z ) ϩ p (1 ϩ z ) Ϫ 2ppzz]. 1/2 g x y y x xyxy Making the geostrophic approximation in (2.26), and
١Iz| K 1, this is very well approximated by| using the hydrostatic balance (2.27), one may eliminate Because M from these equations to obtain the thermal wind bal- ␣ (p|, (2.43 ١| |ance, sin ഡ | Ϫ 1 g I ( u ϭϪ١I⌸. (2.35ץf k ϫ -١Ip is the gradient of pressure in constant- sur where ١I to (2.30) and (2.32) and substituting Hence gradients of the Exner function on orthobaric faces. Applying speci®c volume surfaces give the geostrophic ``shear,'' in (2.43),
u. On account of (2.34), Ϫ1ץ (١IM|. (2.44 ١Iz Ϫ g| |sin ഡ | Ϫ 1
(١Ip, (2.36 ١I⌸ϭ ١Iz is the slope of orthobaric speci®c ,In this expression
-١Ip is practically the same as the slope of volume surfaces, which one may take to change by hun in which Ϫ1 ١IM is constant surfaces. There are no additional terms in dreds of meters through the oceans, while g (2.35), as there would be if the coordinate were potential the gradient of dynamic height, which varies on the density, for example (de Szoeke 2000). order of meters at most. Hence, (2.44) may be approx- imated by
(١Iz|. (2.45| |e. Neutrality of orthobaric speci®c volume surfaces  ഡ | Ϫ 1 Water parcels are often assumed to follow neutral This means that the slope of a neutral trajectory at a trajectories that are everywhere perpendicular to the lo- point differs from the slope of the orthobaric speci®c cal dianeutral vector volume surface through the same point by a proportion | Ϫ 1|. 2 ϩ ␣ ⌫١p (2.37) If the diffusivity tensor for the concentration of a ␣١ e ϭ (McDougall 1987b; Davis 1994; Eden and Willebrand scalar is diagonal with respect to principal axes oriented 1999). One may ask how closely neutral trajectories to the dianeutral vector e at a point, follow orthobaric speci®c volume surfaces, ϭ const, K ϭ diag(KH, KH, KD), (2.46) whose local normal is then the rotated tensor, with respect to axes oriented to 2 ϩ ␣ ⌫ 0١p. (2.38) e0,is ␣١ ١ ϭ e0 ϭ NOVEMBER 2000 DE SZOEKE ET AL. 2835
Ϫ1 FIG. 2. Average adjusted sound speedc*0 (in m s ) as a function of pressure and adjusted speci®c volume, ␣*, computed from the global 1Њϫ1Њ yearly averaged NODC hydrographic dataset.
1 Ϫ  surface to 500 dbar at depth; each ␦␣ bin was chozen 12 1 to be 0.005 ϫ 10Ϫ5 m3 kgϪ1 (corresponding to ␦ ϭ KЈ ഡ KH Ϫ121  2 , (2.47) 0.05 kg mϪ3). The average sound speed, standard de- K /K ϩ 2 12DH viation, and number of points within each bin are shown
١Iz (Redi 1982; de Szoeke in Figs. 2, 3, and 4, respectively. In each of these ®gures(where (1,  2) ϭ ( Ϫ 1 2 and Bennett 1993). The size of the term KH enhancing the abscissa is adjusted speci®c volume, the ``true'' dianeutral diffusivity K in the third diagonal D ␣* ϭ ␣ ϩ ␥p, (3.1) element is a matter of particular concern. It is worth stressing that the orientation of the principal axes of K where ␥ ϭ 3.941 ϫ 10Ϫ9 m3 kgϪ1 dbarϪ1. This adjust- in (2.46) is an assumption, however plausible. ment is done purely for convenience to reduce the range of speci®c volume caused by pressure dependence, per- mitting a ®ner scale for *. 3. Orthobaric speci®c volume: An empirical ␣ The colored region in Fig. 4 represents the population pycnotropic variable density of water in the world's oceans with respect to In this section we calculate a mean estimate of pressure and speci®c volume. This region is relatively c0(p, ␣) from the global distribution of temperature and narrow in the deep ocean, re¯ecting the fact that there salinity in the ocean and use it to construct an empirical is little variation of speci®c volume there. The range of orthobaric speci®c volume variable. We examine its speci®c volume at a given pressure increases shallower suitability as a coordinate for use in modeling the World than 1000 dbar, and becomes relatively large shallower Ocean and analyzing oceanographic data. than 500 dbar. Most of the region is contiguous, except From the global degree-square averaged National on the left-hand side of the ®gure. Although there is no Oceanographic Data Center (NODC) hydrographic da- explicit geographic information contained in the ®gure, taset of T and S at standard depths (Boyer and Levitus some parts of it can be identi®ed with speci®c ocean 1994), pressure, speci®c volume, and sound speed were basins because of their distinctive pressure-speci®c vol- calculated at every point on a three-dimensional spatial ume relationships. The detached region at the left, for grid. Sound speeds were sorted into ␦p ϫ ␦␣ bins. The example, corresponds to the Mediterranean and Red size of the ␦p bins was dictated by the standard depths Seas. Because of the unusual water properties of these of the NODC data and ranged from 10 dbar near the seas, their physical isolation from the rest of the oceans, 2836 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
FIG. 3. Sound speed standard deviation (in m sϪ1) about the average shown in Fig. 2.
FIG. 4. Number of data points per bin corresponding to the calculations of Figs. 2 and 3. NOVEMBER 2000 DE SZOEKE ET AL. 2837
FIG. 5. Data used in the WOCE estimate of average sound speed functionc*0 ( p, ␣*). The two sections A16 and P16 are displayed in Figs. 8±13. as well as the poor coverage of these regions in the inset), the standard deviation is 8 m sϪ1 or smaller. Var- NODC data, we have decided to exclude these waters iability in the sound speed estimate comes from two from further consideration in this calculation. Exami- sources. The ®rst is the intrinsic error of projecting a nation of the number of points in each ␦p ϫ ␦␣* bin function of three independent variables (p, ␣*, S) onto (Fig. 4) shows that most of the middepth water falls a function of just two independent variables (p, ␣*). along one of two branches within the domain. The left The second is the discretization error, that arises from branch represents the relatively denser (at a given pres- gridding the data in the ®rst stage of our analysis. In sure) waters of the Atlantic Ocean, and the right branch addition, the NODC data themselves have been subject represents the less dense waters of the Paci®c Ocean. to considerable smoothing and binning, which may also The bin-averaged sound speed, c 0, has a fairly simple affect the calculation. dependence on pressure and speci®c volume. Figure 2 To assess the possible effects of the analysis proce- displays averaged adjusted sound speed dure, which is applied in producing the gridded data,
c*0 ϭ c 0 Ϫ p (3.2) we independently calculated an empirical sound speed function directly from hydrographic data available from Ϫ1 Ϫ1 with ϭ 0.018 m s dbar . The parameter is ar- WOCE cruises and other sources. The geographic dis- bitrarily chosen to reduce the linear trend in the depen- tribution of these data are shown in Fig. 5. Though dence of sound speed on pressure that would otherwise extensive horizontal coverage is plainly lacking, the dominate the ®gure. Usually, at a given pressure, sound very high vertical resolution makes possible a much speed increases with increasing speci®c volume. This ®ner bin size in pressure (2 dbar). The difference be- is expected, as both sound speed and speci®c volume tween the WOCE-derived empirical sound speed func- increase with temperature. (An exception to this rule tion and the NODC function (Fig. 2) is shown in Fig. occurs on the far left-hand side of the domain where there are low speci®c volumes but high sound speeds 6. In the areas of the p, ␣* plane where the datasets owing to the warm but very salty waters found in the overlap, the differences are small. They are less than 20 Ϫ1 Mediterranean and Red Seas.) The average adjusted ms within the top 500 dbar of the ocean, and less than 2 m sϪ1 deeper than 1000 dbar (Fig. 6 inset). Since sound speedc*0 in Fig. 2 may be considered an empirical function relating sound speed to pressure and adjusted these differences are smaller than the standard deviation speci®c volume ␣*. of the NODC data, we conclude that the preprocessing Standard deviation, shown in Fig. 3, quanti®es the applied to the NODC data has not compromised it for variability of the true sound speed about the bin-aver- this purpose. aged sound speed. It is generally smaller than 10 m sϪ1 Favoring the distributed horizontal coverage of the except in the upper ocean at pressures less than 300 NODC data, we used the average adjusted sound speed dbar, where it may exceed 40 m sϪ1. In the mid and function of Fig. 2 to calculate orthobaric speci®c volume deep ocean at pressures greater than 1000 dbar (Fig. 3 . The adjusted speci®c volume is related to the virtual 2838 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
Ϫ1 FIG. 6. Difference (in m s ) betweenc*0 (p, ␣*) computed from WOCE and NODC (Fig. 2) datasets.
Ϫ ␥p)2 *␣)*␣ץ ,(speci®c volume function of section 2 by Eqs. (2.8 ϭ ␥ Ϫ , (3.4a) (3.1), 2 [p [c0*(p, ␣*) ϩ pץ ␣*(p, ) ϭ A(p |0,) ϩ ␥p. (3.3) subject to In terms of this, the differential equation (2.6a) becomes ␣* ϭ at p ϭ 0. (3.4b) A Runge±Kutta procedure was used to integrate Eq. (3.4a). The resulting trajectories are shown in Fig. 7. They are contours of constant , when the latter is thought of as a function of p and ␣*. Such contours exist throughout the populous regions of the domain. Away from the sea surface, the value of at a given pressure is greater than ␣*, re¯ecting the effect of pres- sure on speci®c volume. (Recall that much of the linear effect of pressure has already been removed in the trans- formation to ␣*; without the transformation, these con- tours would extend much farther to the left of Fig. 7.) The contours of also have curvature owing to the nonlinear dependence of compressibility on pressure. Furthermore, the contours of diverge with increasing
pressure because of the dependence ofc*0 on ␣*onthe right-hand side of Eq. (3.4a). The spacing of the con- tours at great pressure is more than twice that at zero pressure. This re¯ects the approximate doubling of [Eqs. (2.9), (2.34)] between surface and great pressureÐ the origin of which lies in the thermobaric effect (Mc- FIG. 7. Isopleths of (p, ␣*) computed by solving Eq. (2.6) using Dougall 1987a; de Szoeke 2000).
the average sound speed,c*0 (p, ␣*), of Fig. 2. Various ®elds relevant to the calculation of orthobaric NOVEMBER 2000 DE SZOEKE ET AL. 2839
Ϫ1 FIG. 8. Sound speed anomaly, ⌬c ϭ c0 Ϫ c (in m s ), for WOCE sections A16 (top) and P16 (bottom).
speci®c volume are displayed for two north±south trend p removed from c 0 by (3.2) has been restored.] WOCE sections in the Paci®c and Atlantic (Tsuchiya et Note that compressibility anomaly ⌬⌫, given by (2.4c), al. 1992, 1994), nominally along 150ЊW and 25ЊW (Fig. may be obtained from
5). Sound speed anomaly, Ϫ3 ⌬⌫ Ӎ 2c0 ⌬c (3.6) ⌬c ϭ c Ϫ c, (3.5) 0 to good approximation. Sound speed anomalies are large between average sound speed function c0(p, ␣) (Fig. 2) and negative in the upper North Atlantic, while ⌬c ϳ and actual sound speed is shown in Fig. 8. [The linear Ϫ15 m sϪ1 for depths shallower than 1500 m. Anomalies 2840 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
FIG. 9. Salinity, S (in psu), for WOCE sections A16 (top) and P16 (bottom). Different color ranges were used in the two oceans. are moderately large and positive in the upper South Sound speed anomaly is invariably associated with sa- Atlantic, ⌬c ϳ 5msϪ1 shallower than 1500 m, though linity (Fig. 9): negative anomalies with high salinity with a lens of negative anomaly, ⌬c ϳϪ5msϪ1 shal- anomalies, and vice versa. lower than 600 m, in the subtropical South Atlantic. In Sound speed anomalies are generally smaller in mag- the deep Atlantic small negative anomalies are found, nitude in the Paci®c, and mostly positive, while in the between 0 and Ϫ5msϪ1, with somewhat larger mag- Atlantic they are mostly negative. This merely re¯ects nitudes in the deep North Atlantic. Small positive values the fresher waters of the Paci®c (Fig. 9). Values are occur along the bottom associated with Antarctic waters. large in the upper North Paci®c, with ⌬c of order 10 m NOVEMBER 2000 DE SZOEKE ET AL. 2841
FIG. 10. Buoyancy frequency n, in cycles per hour, for WOCE sections A16 (top) and P16 (bottom). The 0.4 cph contour is shown dashed. sϪ1 shallower than 1000 m, somewhat smaller in the section of this variable. It is included here because of South Paci®c at the same depths, the largest values oc- the role it plays in the calculation of the buoyancy gain curring in the Antarctic Circumpolar Current. The only factor [Eq. (2.23)]. The 0.4 cph [ϭ7.0 ϫ 10Ϫ4 rad negative values are found shallower than 500 m in the sϪ1] contour is shown dashed. This value corresponds subtropical gyre of the South Paci®c, and between 500 to a density gradient of 0.01 kg mϪ3/200 m, the nu- m and 1500 m in the tropical band. merator corresponding to the limit of density difference Buoyancy frequency is shown in Fig. 10. This re- that can be discriminated (Fofonoff 1985) over the sembles what is seen, for example, in FlatteÂ's (1979) 200-m resolution sought in Fig. 10. Values smaller than 2842 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
FIG. 11. Gain factor for empirical orthobaric density on WOCE sections A16 (top) and P16 (bottom). The 0.4 cph buoyancy frequency contour from Fig. 10 is shown dashed. this we take to be not practically distinguishable from Atlantic, values as low as 0.6 are found on the section, zero. but only at the very edge of the bottom pycnostad, where The gain factor associated with orthobaric density the estimate of n 2 that goes into the calculation of is [Eq. (2.23)] is shown in Fig. 11. This factor lies mostly marginally reliable. Values of tend to be smaller than in the range 0.8 to 1.2 throughout both oceans outside 1 in the Atlantic, associated with more-saline water mas- the bottom layer with buoyancy frequency Ͻ0.4 cph, ses, except for the fresher Antarctic waters and their with occasional bull's-eyes in the upper ocean, presum- equatorward extensionsÐintermediate water and bot- ably associated with mixed layers. In the deep North tom water. By contrast, values tend to be larger than 1 NOVEMBER 2000 DE SZOEKE ET AL. 2843 in the Paci®c because waters are fresher, except in the altered, the forms of the orthobaric isopycnals in Fig. near-surface South Paci®c subtropics and in a middepth 13 would not change, although the numerical labels, , equatorial band. In the Paci®c, values of are somewhat of the surfaces would. Potential temperature also has nearer 1 than in the Atlantic. this reference-invariant property,4 though potential den- The gain factor is an index of the quality of or- sity in seawater does not. Second, the gradients of or- thobaric speci®c volume as a coordinate. It is nec- thobaric isopycnals in Fig. 13 give precisely the geo- essary for the transformation to that remain the strophic shear [Eq. (2.35)], which gradients of potential same sign everywhereÐa requirement that is easily sat- density surfaces do not (de Szoeke 2000). is®ed. The irreversible sources of in situ density q are The chief reservation about using orthobaric density multiplied by to give the irreversible sources of for descriptive and modeling purposes is the occurrence [Eq. (2.20)]. The closeness of this factor to 1 makes the of residual compressibility contributions to diapycnal correction practically inconsequential. The turbulent material ¯ow, though this is an irreducible concomitant mixing coef®cients and such, which constitute the cal- of the variable composition of seawater and thermobaric culation of q, are far more uncertain than order 20%. effects. This contribution is measured by the parameter The rate of change of pressure, calculated as though Ϫ 1, a nondimensional formulation of the compress- observed following orthobaric speci®c volume surfaces, ibility anomaly ⌬⌫. Orthobaric density sections like Fig. and multiplied by Ϫ 1, makes a thermodynamically 13 ought to be evaluated in conjunction with the cor- reversible contribution toÇ . Though the Ϫ 1 factor responding sections of like Fig. 11, which we dis- may be small, of order Ϯ0.1, the effect of this contri- cussed extensively above. In the following section we bution should not be neglected. In the next section, we will show that a similar contribution to diapycnal mass shall demonstrate a link between patched potential den- ¯ow must occur through patched potential density sur- sity (Reid and Lynn 1971) and a form of orthobaric faces. density, and show that, through the rifts between the Interested readers may obtain datasets of the empirical leaves constituting the patched surfaces, there is a ma- functionc*0 (p, ␣*) from the authors. Using these, Eqs. terial ¯ow that is the discrete analog of the reversible (3.4) may be solved by standard methods to obtain tra- contribution to Ç . jectories of constant on the p, ␣* plane. In addition, Meridional sections of the index of neutrality  2, cal- a Matlab toolbox is available to compute orthobaric den- culated from Eq. (2.45), are displayed in Fig. 12 on a sity, , from hydrographic data for arbitrarily dimen- logarithmic scale. The largest values, տO(10Ϫ8), are sioned (p, S, T) arrays. found along the ocean bottom, almost invariably within the band where buoyancy frequency is less than 0.4 cph, and where the estimates of Ϫ 1, hence  2, are most 4. Patched potential density 2 uncertain. Outside this band  is typically orders of To overcome some of the dif®culties with the use of magnitude smaller. The criterion for the neglect of the potential density in pressure ranges far from its refer- 2 KH term in the third diagonal element of KЈ, the dif- ence pressure, Reid and Lynn (1971) calculated poten- fusivity tensor with respect to orthobaric density co- tial densities over restricted pressure ranges (typically ordinates, Eq. (2.47), is that of order 1000 dbar) referenced to a pressure in the center 2 Ϫ8 of the range. We shall call a constant potential density  Շ KD/KH ഠ 10 . (3.7) surface within such a restricted pressure range a leaf. To estimate the right side of this inequality, we have Patched isopycnal ``surfaces'' are formed by joining ad- Ϫ5 2 Ϫ1 adopted the nominal values KD ϭ 10 m s , KH ϭ jacent leaves across the transitions between their pres- 3 2 1 10 m sϪ . This criterion is fairly well satis®ed through- sure ranges (Fig. 14a). The word ``surfaces'' is put in out the sections of Fig. 12. We conclude that for the quotes here because of the inevitable discordances be- purposes of calculating turbulent diffusion with respect tween the leaves, which must occur at their joints and to orthobaric density surfaces, that is, the term q ap- which arise from the variation of compressibility with pearing in (2.2) or (2.20), the approximation of the di- temperature (the thermobaric effect) and the geograph- agonal elements of KЈ [Eq. (2.47)] as KH, KH, KD is ical variation of T±S properties. acceptable. In this section we shall study the Reid and Lynn Finally, sections of orthobaric density, shown as (1971) method of patching potential density leaves. In Ϫ1 particular, we shall examine the discordances between ϭ Ϫ 1000, (3.8) the leaves, showing how they permit exchange of water are displayed in Fig. 13. These sections appear quite through the resulting patched surfaces, even if the in- conventional, exhibiting the features expected in ``den- dividual leaves are considered impermeable (i.e., when sity'' sections. However, the use of orthobaric density has certain advantages over potential density. Surfaces of constant orthobaric density (called orthobaric iso- pycnals, for short) are invariant with respect to reference 4 Although only approximately. The slight dependence of adiabatic pressure. If the reference pressure of Eq. (3.4b) were lapse rate of seawater on salinity vitiates this useful property. 2844 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
FIG. 12. Logarithm (base 10) of index of neutrality 2, de®ned in the text, for empirical orthobaric density on the WOCE sections A16 (top) and P16 (bottom). The 0.4 cph buoyancy frequency contour from Fig. 10 is shown dashed. no mixing or cabbeling is held to occur across potential joints, as described above, is equivalent to replacement density surfaces), and calculate the rate of exchange of true seawater compressibility by an approximate across the discordances. It will be shown that patched form, dependent only on pressure and speci®c volume, potential density surfaces tend, in the limit of ever ®ner just as was done to obtain the empirical orthobaric den- individual leaves (®ner divisions of pressure intervals), sity of sections 2 and 3. The rate of mass exchange to constant surfaces of a form of orthobaric density. This across the discordances between leaves is the discrete means that the procedure of matching leaves at the analog of the reversible ¯ux, obtained above, of water NOVEMBER 2000 DE SZOEKE ET AL. 2845
FIG. 13. Empirical orthobaric density for WOCE sections A16 (top) and P16 (bottom). across the corresponding form of orthobaric density sur- mass exchange through the discordances may be reduced faces. in the individual subregions, additional horizontal mass Potential density patching as it is usually practiced al- exchanges through the patched potential density surfaces lows for regional variation of the patching procedure. For may arise at the discontinuous horizontal joints between example, Reid (1994) patches the 1 ϭ 31.938 leaf to the the subregions. However, to begin, we shall put aside for 0 ϭ 27.44 leaf in the North Atlantic and to the 0 ϭ later this complication of regionally differentiated patching 27.30 leaf in the South Atlantic. This is done to allow for and assume that a given 1 leaf, for example, is uniquely regional differences in the T±S property distributions of associated with a single 0 leaf everywhere. We may call the two oceans. We shall see in this situation that, while this a universal form of patched potential density. 2846 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
TABLE 1. Speci®cations of patched isopycnal surfaces.a /
0 1 2 3 4 North Atlantic 26.750 31.090 35.337d 39.587d 43.837d 27.440 31.938 36.288d 40.538d 44.788d 27.630 32.200 36.550d 40.800d 45.050d 27.824 32.456 36.980 41.395 45.735d 27.846 32.485 37.017 41.440 45.780d 27.874 32.523 37.067 41.500 45.840b 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 South Atlantic 26.750 31.090 35.337d 39.587d 43.837d 27.300 31.938 36.288d 40.538d 44.788d 27.630 32.200 36.550d 40.800d 45.050d 27.755c 32.425 36.980 41.400 45.735d 27.770 32.445 37.013 41.440 45.780d 27.787 32.476 37.041 41.500 45.840 27.800 32.487 37.057 41.538 45.880 27.804 32.498 37.074 41.547 45.907 27.815 32.502 37.080 41.564 45.920
a Excerpted from Reid (1994). Boldface-type numbers indicate Reid's (1994) identi®cation labels for the patched surfaces. b Changed from 45.838. c Corrected (J. L. Reid 2000, personal communication). d Extrapolated.
surface never extends to greater than 3500 dbar, in which
case the m value of its deepest occurring leaf is used (Table 1). We will not follow in this exception but al-
ways use a 4 label, obtained by extrapolation if nec- essary. This extrapolation is arbitrary but of no con- FIG. 14. (a) Patched potential density schematic. A constant 0 sequence except in furnishing a unique 4 label for each surface is patched to a surface at 500 dbar, and so on, so as to 1 surface. minimize the discordance between the two surfaces or leaves. (b) ±S diagram for the North Atlantic on 500 dbar (from NODC archive). The entries from Table 1 (denoted by pluses) have
Pairs of 0, 1 surfaces were so chosen by Reid (1994) that they been plotted on Fig. 15 as intersect in the main ±S line on this level surface. Ϫ1 ␣* ϭ (m ϩ 1000) ϩ ␥p (4.1)
[cf. Eq. (3.1)] versus N (speci®cally 4, i.e., N ϭ 4), a. Patched isopycnals: A formal description and joined by straight lines for each reference pressure
We show in Table 1 excerpts of similar tables from pm ϭ 0, 1, 2, 3, 4 hbar (1 hbar ϭ 1000 dbar). The curves Reid's (1994) analysis of circulation in the North and in Fig. 15 may be read as de®ning a relation between
South Atlantic. The rows in Table 1 identify the m ϭ N and m Ϫ1 ␣m Ϫ 1000 values, with respect to reference pressures m ϭ Ã (pmN, ) (4.2) pm, for m ϭ 0, 1, 2, 3, 4, of certain potential density leaves de®ned, respectively, over the ranges p m Ͻ p Ͻ at the discrete pressures pm. Given any in situ density 1 p mϩ1 [where p m ϭ 2 (pm ϩ pmϪ1) for m Ͼ 0, and p 0 ϭ and pressure, m and pm, one can identify the corre- 0]. The concatenation of these leaves from each row in sponding reference pressure level and look up the cor-
Table 1 gives a patched potential density surface [called responding isopycnal value ( 4 in the case of Fig. 15). simply an isopycnal surface by Reid and Lynn (1971), [Note that trivially à (pNN, ) ϭ N.] The function (4.2) though we shall call it a patched isopycnal surface, to furnishes the chain of m values whose leaves make up distinguish it from other kinds of isopycnal surfaces]. a patched potential density surface characterized by the
For example, in the North Atlantic, a leaf with density density label N. of 27.44 between the surface and 500 dbar is connected Suppose a water sample from pressure pm, with (in to the 1 ϭ 31.938 leaf from 500 to 1500 dbar, and so situ) density m ϭ Ã (pmN, ) and salinity S, is taken on (Fig. 14a). It is convenient to label such a patched adiabatically and with ®xed composition to pressure isopycnal surface by the numerical value of its 4 leaf. pmϩ1. There it will have a density pot(pmϩ1 | pm, m, S), Reid (1994) follows this same practice, except when a where the function pot(pЈ | p, ,S) gives the potential NOVEMBER 2000 DE SZOEKE ET AL. 2847
of the chosen 0, 1 surfaces in the ±S diagram of Fig.
14b de®nes the salinity value S(0, 1000 dbar, 0) used in Eq. (4.4) above. These methods readily generalize to
give a value of mϩ1 corresponding to a preselected m.
In this way the set of leaves {m : p m Ͻ p Ͻ p mϩ1} that make up a patched surface may be constructed.
b. Coarse orthobaric density approximations to patched density Before considering the theoretical limit of taking ever
®ner spacings of the set of reference pressures pm,itis useful to review some interesting interpretations of the information in Table 1 and Fig. 15. The latter shows how coarsely resolved are the particular patched iso- FIG. 15. Orthobaric density function corresponding to Reid's pycnals displayed by Reid (1994): ®rst, the pressure (1994) patched potential density surfaces. Ordinate is in situ density, intervals over which the constituent leaves are de®ned adjusted by ␥ p [de®ned in the text]; abscissa is orthobaric density 5 with respect to 4 hbar. Contours at ®xed pressure p are shown; are quite wide; second, only a small number (nine) of South Atlantic (solid), North Atlantic (dashed); pluses indicate the en- isopycnal patchings are speci®ed to represent the entire tries from Table 1. density continuum. (We will call these the standard patched isopycnals.) On the second point, one might readily specify a con- density, with respect to reference pressure pЈ, of a water tinuum of intermediate patched isopycnals merely by sample with in situ pressure p, density , and salinity linearly interpolating (for example) between the stan- S. This density matches mϩ1 ϭ à (pmϩ1, N) only for a dard patched isopycnals. On the ®rst point, the function special value of salinity given by the solution of à (pm, 4) gives the in situ densities, at the discrete
pressures pm ϭ 0, 1, 2, 3, 4 hbar, that are associated pot(pmϩ1 | pm, m, S) ϭ à (pmϩ1, N). (4.3) with a given 4 label. Although the patched isopycnals This special salinity value is designated by appear to admit no interpretation ofà at pressures in- S ϭ S(p , p , ), (4.4) termediate between these discrete values, each of which m mϩ1 m stands for a ®nite range, one might interpret an inter- indicating its dependence on the pressure levels pm, pmϩ1 polation between these discrete pressures as follows. and on the density value m of the upper isopycnal sur- Consider two simple choices of interpolation indicated face. [The dependence on N can be eliminated by in- in Fig. 16. This shows ␣* ϭ (à ϩ 1000)Ϫ1 ϩ ␥p, in verting (4.2); listing the dependence on mϩ1 would be situ speci®c volume corrected by the offset ␥p, as a similarly super¯uous.] If this value of salinity occurs at function of p for 4 ϭ 45.88. The discrete data points, a point along the intersection of the m ϭϭà (pmN, ) denoted by asterisks, have been joined in two ways: by const leaf with the transition pressure level p mϩ1, then a sequence of centered stair steps and by linear seg- the mϩ1 ϭϭà (pmϩ1, N) const leaf will pass through ments. For either choice of interpolation (or any other) the same point. (Figure 14a furnishes a schematic il- Fig. 16 may be read as specifying the in situ density lustration of such an occurrence between a surface 0 à at pressure p that is to be associated with the given and a surface at 500 dbar.) 1 label. Conversely, the numerical label of a water The Reid and Lynn (1971) method for selecting the 4 4 parcel with in situ densityà at pressure p, converted to set { } of density leaves that constitute a patched is- m ␣*, may be read off Fig. 15, and identi®es a surface of opycnal surface is illustrated in the ±S diagram of Fig. water parcels that share this label. This surface is an 14b for m ϭ 0, that is, at the transition from to 0 1 isopleth of a form of orthobaric density, whose value at the p ϭ 500 dbar pressure level. The surface to 0 1 coincides with in situ density at 4 hbar. The two inter- be associated with is chosen to pass centrally through 0 polation choices give two forms of orthobaric density. the range of the observed data points. (The details of The stair-step orthobaric density isopleths correspond this selection procedure need not concern us.) The im- fairly closely to the potential density leaves, patched portant point is that there is a ®nite, even if narrow, together at the transition pressures: within each pressure range of observed ,Svalues along any surface at 0 interval the in situ density, after correction for pressure, 500 dbar (say). This range differs from the ,Svalues along the 1 surface chosen to correspond. This means that in physical space, illustrated schematically in Fig.
14a, the 0 leaf intersecting 500 dbar cannot join per- 5 fectly onto the corresponding 1 leaf. The intersection Reid (1994) shows one additional isopycnal. 2848 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
sion of Fig. 2 could be constructed by differencing the rows of Table 1.
c. Patched potential density: The continuous limit We turn now to the question of taking the limit of ever ®ner spacing of the potential density leaves in
patched surfaces, that is, taking each ⌬p ϭ pmϩ1 Ϫ pm → 0. It will ®rst be shown how the salinity function (4.4) generates a virtual compressibility function, which
de®nes a kind of orthobaric density. Subtract m ϭ pot(pm | pm, m, S) ϭ Ã (pmN, ) from both sides of Eq. (4.3), where S is given by (4.4), and divide by ⌬p:
pot(pmϩ1 | pmm, , S ) Ϫ pot(pmm| p , m, S ) ⌬p FIG. 16. Adjusted in situ density vs pressure for orthobaric density à (pmϩ1, NmN) Ϫ à (p , ) 4 ϭ 45.88; stair step and linearly interpolated versions; South At- ϭ . (4.5) lantic (solid), North Atlantic (dashed). Asterisks indicate the entries ⌬p from Table 1. Orthobaric density based on globally averaged com- → pressibility of section 2 (dot-dashed), with error bars. In the limit ⌬p 0, the left side becomes ( (pЈ | p , , Sץ pot mm ϭ⌫(p , , S ). (4.6) pЈ mmץ is constant.6 Figure 16 also shows the orthobaric density ΗpЈϭpm isopleth from Fig. 7, based on globally averaged com- pressibility, that goes through à ϭ 45.88 at p ϭ 4 hbar. This is the partial rate of change of potential density Error bars (Ϯ2 std dev, encompassing 95% con®dence) with respect to reference pressure, and is therefore the are shown on the global orthobaric isopycnal. These are adiabatic compressibility evaluated at p m , m , computed by summing variance estimates of sound S(pm, pmϩ1, m) (Phillips 1966). In the limit this may speed from the ␦p ϫ ␦␣ bins of Fig. 3 from the starting be written point of integration of (2.6)ÐpЈϭ4 hbarÐto pressure ⌫ (p, ,S(p, )) ϵ⌫(p, ), (4.7) p, namely, 0 where 4hb 2␣2⌬p 2 var (p) ϭ var (pЈ), ϭ à (p, ) (4.8) ␣ 3 c N pЈϭp c0 and as though sound speed ¯uctuations were uncorrelated among bins. These error bars may be interpreted as de- S(p, ) ϭ lim S(pmm, p ϩ1, m) (4.9) → ®ning an envelope likely to contain at 95% con®dence pmϩ1 pm the statistically expected orthobaric isopycnal based on are the continuous limits of (4.2) and (4.4). (The sub- the expectation value of the (p, ␣)-sorted sound speed. scripts m, super¯uous in the limit, have been dropped.) The linearly interpolated versions of Reid's (1994) The right side of (4.5), in the limit ⌬p → 0, becomes pЈ. Putting all this together, one sees thatץ/( à (pЈ, ץ patched isopycnals for the South and North Atlantic are quite close to the global-average orthobaric isopycnal, N the functionà (pЈ, N) is given by the solution of the lying mostly within the error bars. differential equation There is no fundamental reason to favor the stair- Ãץ step interpolation over linear interpolation, each of ϭ⌫(pЈ, à ), (4.10a) pЈ 0ץ .which gives a different form of orthobaric density Note that the virtual compressibility function, ⌫ ϭ 0 subject to p)N , is in®nite at the transition pressures for theץ/ Ãץ) stair-step orthobaric density, but remains ®nite for the à ϭ N at pЈϭpN. (4.10b) linearly interpolated orthobaric density. A coarse ver- Equation (4.7) de®nes a virtual compressibility function,
⌫ 0, dependent only on pressure, p, and in situ density, , and conditioned on S(p, ), the continuous limit of 6 The correspondence to the patched density of Reid (1994) can (4.4). Comparison of (4.7) with (2.11) con®rms that be improved by elaborating the correction ␥p in Eq. (3.1) to as com- S(p, ) is identical to the standard salinity function de- plicated a function s(p) as necessary to remove as much of the pres- sure dependence of in situ density as possible in each pressure range. ®ned in section 2. Equation (4.8) describes the in situ The correction may even be made discontinuous at the transition density at pressure p that is to be used in constructing pressures to optimize this removal. the continuous limit of the patched isopycnal surface NOVEMBER 2000 DE SZOEKE ET AL. 2849
generated from the reference density N. Comparing stantaneously be made to rise or drop to reach the new (4.10) to Eqs. (2.6a,c), (2.8) one sees, by making the potential density leaf. Equation (4.15) resembles an es- identi®cations timate made by McDougall (1987b) of potential density change along a neutral trajectory [though with S Ϫ S Ϫ1 à ϭ ␣ Ϫ 1000 replaced by Ϫ(␣/␣S)( Ϫ )]. 2 ϭ Ϫ1 Ϫ 1000 (4.11) Divided by n g/␣, where n is the local buoyancy fre- N quency, (4.15) gives an estimate of the vertical dis- 2 ⌫ϭ001/c (p, ␣) placement the ¯oat undergoes (positive if upward) in that the equations are identical. making the transition, namely, Ϫ2 zm Ϫ zmϩ1 ഡ g␣⌬⌫n (pmϩ1 Ϫ pm) d. Quasi-conservative property Ϫ1 ഡ g ␣( Ϫ 1)(pmϩ1 Ϫ pm), (4.16) In the discrete version of patched density, the poten- where (2.12) has been used in the ®rst equality and tial densities of individual leaves are quasi-conservative. (2.23) in the second. Suppose both sides of (4.16) are In the continuous limit a form of orthobaric density is divided by ⌬t, the time taken by the ¯oat to move from recovered, which is not in general quasi-conservative, one central reference pressure, pm, to the next, pmϩ1.As as we saw in section 2. This apparent paradox is resolved → → pmϩ1 pm and ⌬t 0, the quotient (pmϩ1 Ϫ pm)/⌬t by observing that there is concentrated reversible ma- tends to the rate of change of pressure following hori- terial ¯ow through the joints between the individual zontal motion along what has been identi®ed as a lim- leaves of the patched isopycnals. This becomes contin- iting patched isopycnal, namely N ϭ const, now con- uously distributed in the limit and is synonymous with tinuous: the reversible material ¯ow, given by the second term → (١I)p. (4.17 ´ t ϩ uץ) of Eq. (2.20), across the limiting orthobaric isopycnals. (pmϩ1 Ϫ pm)/⌬t The recognition of this material ¯ow across patched Hence surfaces (as across orthobaric isopycnals) is important → Ϫ1 (١I)p. (4.18 ´ t ϩ uץ)(to the assessment of the meaning of such surfaces in (zm Ϫ zmϩ1)/⌬t g ␣( Ϫ 1 practical oceanographic contexts. For this reason we offer the following thought experiment that illustrates The right side of (4.18) invites comparison to the second the inevitability of the cross-surface ¯ow at the tran- term of (2.20), the reversible material ¯ow across or- sitions between potential density leaves. thobaric isopycnals. Thus we see that the material that Suppose a subsurface ballasted ¯oat, which otherwise ¯ows past the hypothetical ballasted ¯oat as it travels is advected by horizontal ¯uid motion, is programmed from one potential density leaf to the next in the vicinity to adjust its buoyancy so as to follow patched potential of a transition pressure is the discrete analogue of the reversible material ¯ow through the corresponding form density surfaces. Suppose the ¯oat is following a m leaf in the downward sense and approaching the tran- of orthobaric isopycnal. sition pressure p mϩ1 at which it will switch to follow a mϩ1 leaf. The particular mϩ1 leaf to which it will switch e. Regional differentiation has the valueà (pmϩ1, N) , preassigned by the Reid and Lynn (1971) procedure, and given by (4.3) at the salinity As already noted, Reid's (1994) patching procedure S speci®ed by (4.4). Before the transition, the ther- makes allowance for regional variation of T±S proper- modynamic state of the water parcel surrounding the ties. That is, different sets of numerical values for the m ϭ à (pmN, ) leaves are selected for one ocean basin ¯oat is completely described by m, pm, and its actual salinity S. The potential density referenced to p of compared to another, or for one hemispheric basin com- mϩ1 pared to the other. For example, Table 1 and Figs. 15, this water parcel is pot( pmϩ1 | pm, m, S). The difference of this value from the target potential density is therefore 16 show different patched isopycnal speci®cations for the North and South Atlantic. Even within a hemispheric ⌬ ϭ pot(pmϩ1 | pm, m, S) basin such as the North Atlantic, Reid (1994) allows for some variation between western and eastern regions Ϫ pot[pmϩ1 | pm, m, S(pm, pmϩ1, m)]. (4.14) (not shown in Table 1). This undoubtedly ensures small- By carrying out successive Taylor expansions, ®rst of er variations of salinity from the regional standard, S Ϫ pmϩ1 about pm at ®xed S (or S) and m, then of S about S R (in the mean-square sense, say), than from the S, and using (4.6), one obtains that (4.14) is approxi- lumped global standard salinity S Ϫ S G. Hence the re- mately gional density jumps at transitions, given by (4.15), to which the cross-surface material ¯ow is proportional, is ⌫ץ ⌬ ഡ [S Ϫ S(pmm, p ϩ1, nm)](p ϩ1 Ϫ pm), (4.15) similarly reduced. Likewise the continuously distributed Sץ p,␣ material ¯ow across orthobaric isopycnals could be re- where the coef®cient is furnished by (2.13). This is the duced by using regional averages to de®ne the virtual in situ density change through which the ¯oat must in- compressibility rather than lumped global averages as 2850 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30
FIG. 17. Regionally differentiated orthobaric density section for WOCE Cruise A16 along 25ЊW, based on linear interpolation between Reid's (1994) density surfaces, de®ned differently for North and South Atlantic. Note the discontinuities at the equator. (Bold isopycnals correspond to the standard Reid surfaces.) in section 3. However, this reduction comes at a price: density of a water parcel taken to 4 hbar as though its
P) 4 . The latter may beץ/ Ãץ) there will be horizontal nonconformities in such re- compressibility were ⌫ 0 ϭ gionally differentiated patched surfaces (or orthobaric computed by differencing, in effect, Fig. 15.] We stress density surfaces) at the boundaries between the regions. that such discontinuities must occur for patched poten- Material can leak across these nonconformities just as tial density surfaces interpolated between the surfaces it does at the vertical transitions between the potential 0 ϭ 26.750 and 1 ϭ 31.938 [these are two of Reid's density leaves. It can be shown that the reduction, by (1994) standard isopycnals, see Table 1; they correspond using regionally differentiated speci®cations, of the lo- to 4 ϭ 43.837 and 44.788]. Such discontinuities are a cal material ¯ow across vertical transitions between concomitant of the regional differentiation employed in leaves of a patched surface is compensated by the in- de®ning the patched potential density surfaces. crease of material exchange at the resulting nonconfor- These discontinuities constitute gaps through which mities among regions. This is a consequence merely of water may pass, even if the continuous portions of the mass conservation, not of any property of the surfaces. surfaces could be regarded as material. The necessity To illustrate these remarks, an example is shown in of accounting for this exchange of material is a disad- Fig. 17 of a density section computed for the tropical vantage of employing regional differentiation of water portion of WOCE Cruise A16 in the Atlantic at 25ЊW. properties in the de®nition of patched potential density
What is displayed in Fig. 17 is 4 for each pressure and surfaces. An additional disadvantage of regional differ- in situ density pair reported from the station, calculated entiation is the forfeiture, in the vicinity of interregional by bilinear interpolation of the data in Table 1. As we borders, of the property of the Montgomery function of have noted, this form of orthobaric density gives a de- being an acceleration potential (sometimes called a geo- fensible approximation to the patched potential density strophic streamfunction), as it otherwise is in orthobaric surfaces of Reid (1994), permitting a continuum of sur- density surfaces, and would be approximately for faces to be constructed between the standard ones spec- patched potential density surfaces constructed without i®ed. Because different rules are used for calculating 4 regional differentiation (de Szoeke 2000). There are dis- in the North and South Atlantic (Table 1), there are continuities in acceleration potential at the interregional inevitable discontinuities in Fig. 17 at the equator, borders. equivalent to vertical displacements of order 50 m. [To An unexceptionable example of the use of regional forestall possible confusion, we emphasize that 4 is not differentiation is Reid's (1994) subdivision of the denser potential density referenced to 4 hbar, but the virtual waters of the North Atlantic into western and eastern NOVEMBER 2000 DE SZOEKE ET AL. 2851 subbasins separated by the Reykjanes Ridge. In this case of this material ¯ow occurs at the discordances between the differentiated waters are not in direct contact because the segmented leaves of potential density at the tran- of the intervening ridge so that no horizontal discon- sition pressures. For the regionally variable form of tinuity occurs between the differently speci®ed water patched surfaces, additional horizontal material ¯ows masses. (A good case could be made for dealing sim- may occur at the borders between the constituent re- ilarly with enclosed marginal seas like the Mediterra- gions, as we saw above. Presumably, for empirical neu- nean and Red Seas.) Formally this regionality might be tral density, cross-surface material ¯ows of both types speci®ed as follows: occur, although now continuously distributed by the Jackett±McDougall algorithm. ⌫0(p, ␣) ϭ⌫ deep(p, ␣), p Ͼ p max
ϭ⌫AB(p, ␣)or⌫ (p, ␣), p Ͻ pmax, (4.19) 5. Summary depending on whether one is in region A or B (e.g., An empirically based, pycnotropic (i.e., function only western or eastern Atlantic). of pressure, p, and in situ speci®c volume, ␣), variable was devised. It was constructed by calculating a global f. Neutral density ®t of adiabatic compressibility to p and ␣ and then using this best-®t compressibility in the density equation to Empirical neutral density variables can be constructed obtain an exact integral of the density tendency and the that minimize everywhere the squared difference of the best-®t compressibility. This exact integralÐthe desired slope of surfaces of the variable from neutral tangent new variableÐmay be termed the pressure-corrected or
-١ Ϫ⌫١p. The orthobaric speci®c volume, . Orthobaric speci®c vol planes, which are perpendicular to calculation may be based on the global hydrographic ume possesses the same useful dynamical properties po- dataset (Jackett and McDougall 1997), or focused on a tential temperature has in a single-component ¯uid. single ocean basin like the North Atlantic (Eden and First, it possesses a geostrophic streamfunction, the Willebrand 1999). While this idea seems intuitively sim- Montgomery function. This means that under circum- ilar to orthobaric speci®c volume, which may be thought stances where geostrophy prevails, vertical shear is pro- to be an approximate integral of the left side of Eq. portional to the gradient of orthobaric density surfaces (2.2), an important feature of Jackett and McDougall's (which is the thermal wind relation). Second, to the (1997) neutral density, which distinguishes it from the extent that the tendency of is negligible, a near-con- universal forms of orthobaric density and patched po- servative potential vorticity satz may be devised. Nei- tential density, is that it depends not only on the ther- ther of these properties pertains for potential density (de modynamic state of a water sample, given by the triplet Szoeke 2000). p, T, S but also on the longitude and latitude at which The tendency equation for was derived. It consists it was observed. (Although regionally differentiated of two parts. One part is given by the same irreversible forms of orthobaric density and patched potential den- sources and sinks (including turbulent transport diver- sity would also possess elements of geographical de- gence) that contribute to density tendency, multiplied pendence.) While the complexity of the neutral density by the buoyancy gain factor , the ratio of apparent -z) to true stability (buoyץ/ץ algorithm makes it dif®cult to analyze, we may rely on stability (proportional to the empirical demonstration by Jackett and McDougall ancy frequency squared). This factor is quite close to (1997) that neutral density surfaces closely follow the 1, as a diagnostic calculation shows for meridional regionally differentiated form of Reid's (1994) patched ocean sections through the Atlantic and Paci®c. So this potential density surfaces. Indeed, these authors aver contribution to the orthobaric speci®c volume tendency that neutral density surfaces are the continuous analogue Ç is about the same as the potential speci®c volume of discretely de®ned patched potential density. They tendency␣Ç P due to irreversible contributions. presumably mean by this the continuous limit of the The angle between orthobaric density surfaces and discrete spacing of vertical pressure intervals, but also locally de®ned neutral tangent planes, or trajectories, the horizontally continuous analogue of the regionally was calculated for two meridional ocean sections. This varying form of the Reid and Lynn (1971) potential angle was suf®ciently small that, for the purposes of density patching method. As we have shown above, both computing a turbulent diffusivity tensor (assumed to be of these procedures (vertical and horizontal patching) diagonal with respect to neutral tangent planes) with imply reversible exchanges of mass across the patches, respect to orthobaric coordinates, it may be neglected even if potential density itself could be regarded as per- except possibly near the ocean bottom. fectly conservative. The continuous limit merely dis- The other part of the tendency comes from the tributes the cross-surface mass exchange at the patches compressibility anomaly, the difference between true continuously; it does not eliminate it. For orthobaric compressibility and the best-®t compressibility, multi- density surfaces this cross-surface material ¯ow is ex- plied by the apparent vertical motion of orthobaric den- plicitly given by the second term of Eq. (2.20). For sity surfaces. Its magnitude is measured by the smallness patched potential density surfaces an analogous version of Ϫ 1, which is proportional to the compressibility 2852 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 30 anomaly. This term seems to be the inevitable price for Acknowledgments. T. McDougall and an anonymous banishing salinity gradient contributions from the hor- reviewer provided valuable criticism of an earlier draft. izontal momentum balance and hydrostatic balance. De- This research was supported by National Science Foun- spite the closeness of the gain factor to 1, its contribution dation Grants 9319892, 9402891; the Jet Propulsion toÇ may not be negligible. Laboratory under the TOPEX/Poseidon Announcement For these reasons, we suggest the empirical orthobaric of Opportunity, Contract 958127; and by NASA Grant speci®c volume (or density) variable as an alternative NAGS-4947. to potential density both for descriptive uses in dis- playing oceanographic data and for theoretical uses in REFERENCES modeling ocean circulation. It would be premature to Akitomo, K., 1999: Open-ocean deep convection due to thermobar- claim the ``best-®t'' compressibility we have devised as icity. 1. Scaling argument. J. Geophys. Res., 104, 5225±5234. the last word. The version based on the global ocean- Boyer, T., and S. 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