A Neutral Density Variable for the World's Oceans

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DAVID R. JACKETT AND TREVOR J. MCDOUGALL Antarctic CRC, University of Tasmania and Division of Oceanography, CSIRO, Hobart, Tasmania, Australia (Manuscript received 20 June 1995, in ®nal form 13 March 1996)

ABSTRACT The use of density surfaces in the analysis of oceanographic data and in models of the ocean circulation is widespread. The present best method of ®tting these isopycnal surfaces to hydrographic data is based on a linked sequence of potential density surfaces referred to a discrete set of reference pressures. This method is both time consuming and cumbersome in its implementation. In this paper the authors introduce a new density variable, neutral density ␥n, which is a continuous analog of these discretely referenced potential density surfaces. The level surfaces of ␥n form neutral surfaces, which are the most appropriate surfaces within which an ocean model's calculations should be performed or analyzed. The authors have developed a computational algorithm for evaluating ␥n from a given hydrographic observation so that the formation of neutral density surfaces requires a simple call to a computational function. Neutral density is of necessity not only a function of the three state variables: salinity, temperature, and pressure, but also of longitude and latitude. The spatial dependence of ␥n is achieved by accurately labeling a global hydrographic dataset with neutral density. Arbitrary hydrographic data can then be labeled with reference to this global ␥n ®eld. The global dataset is derived from the Levitus climatology of the world's oceans, with minor modi®cations made to ensure static stability and an adequate representation of the densest seawater. An initial ®eld of ␥n is obtained by solving, using a combination of numerical techniques, a system of differential equations that describe the fundamental neutral surface property. This global ®eld of ␥n values is further iterated in the characteristic coordinate system of the neutral surfaces to reduce any errors incurred during this solution procedure and to distribute the inherent path-dependent error associated with the de®nition of neutral surfaces over the entire globe. Comparisons are made between neutral surfaces calculated from ␥n and the present best isopycnal surfaces along independent sections of hydrographic data. The development of this neutral density variable increases the accuracy of the best-practice isopycnal surfaces currently in use but, more importantly, provides oceanographers with a much easier method of ®tting such surfaces to hydrographic data.

1. Introduction inverse models are faced with the problem of ®nding the actual positions of the surfaces in the model domain. The role of ``density'' surfaces in models of the ocean Even in the case of the prognostic level models, vali- is fundamental. These models range from the analysis dation of these models can only be made by comparison of hydrographic data on the most appropriate mixing surfaces (Reid 1986, 1989, 1994; McCartney 1982) to with data from the real ocean. It must be borne in mind the inversion of hydrographic data to obtain ocean cir- that the strong lateral mixing of these prognostic models culation and mixing (Schott and Stommel 1978; Wunsch occurs along neutral surfaces. As such, maps of prop- 1978; Killworth 1986; Zhang and Hogg 1992) to prog- erties along neutral surfaces and the heights of neutral nostic ``level'' or ``layer'' models (Bleck et al. 1992; surfaces are the most relevant variables from the real Hirst and Wenju 1994), which integrate forward in time ocean to be compared with the model output. the basic momentum and tracer conservation equations. Historically, the ®rst surfaces used for oceanic models All of these models have as their goal a description of were in situ density surfaces. Wust (1933) and Mont- the oceanic circulation and all use the concept of mixing gomery (1938) ®rst realized the inadequacy of these surfaces, between or along which the underlying physics surfaces for describing the circulation even at shallow governing oceanic ¯ow is analyzed. While prognostic depth, and they proposed a new variable, potential den- level models can adequately cope with these mixing sity, for quantitatively describing the isopycnal spread- surfaces (they only require the direction of the gradient ing of water masses in the ocean. Despite the widespread of the surfaces in order to align the mixing tensor), acceptance of this isopycnal mixing, de®ciencies in potential density surfaces for accurately describing precise isopycnal ¯ow have been known for some time. For example, McDougall (1987a) shows a surface across the Corresponding author address: Dr. David R. Jackett, Division of Oceanography, CSIRO, GPO Box 1538, Hobart, Tasmania 7001, Aus- North Atlantic Ocean where potential density and neu- tralia. tral surfaces diverge by 1500 m. Lynn and Reid (1968),

᭧1997 American Meteorological Society

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Reid and Lynn (1971), and Ivers (1975) have all used while McDougall (1987a) made a detailed analysis of a series of potential density surfaces, referred to three the properties of these surfaces. These surfaces were different reference pressures, namely, 0, 2000, and 4000 de®ned by McDougall in terms of gradients of salinity db. In more recent work, Reid (1994) has found it nec- S and potential temperature ␪ in the neutral surface, essary when describing the geostrophic circulation of while in McDougall and Jackett (1988) it was shown the Atlantic Ocean to further re®ne the de®nition of that the normal to these surfaces is in the direction of ١␪, where ␣ and ␤ are the thermal expansion␣ ١S Ϫ␤ isopycnal surfaces by using potential density surfaces referred to a range of reference pressures, with incre- and saline contraction coef®cients, respectively. It is this ments in some regions of as little as 500 db. latter property that motivates the de®nition of neutral The reduction of the errors associated with a single surfaces that is most useful for our purposes, namely, reference pressure by the introduction of a range of that they are the surfaces everywhere perpendicular to ١␪). It is important to note that␣ ١S Ϫ␤)reference pressures makes the actual computation of the the vector ␳ ,١␪) is simply ϪgϪ1␳N2␣ ١S Ϫ␤)isopycnal surfaces dif®cult. Reid (1989, 1994), for ex- the z component of ␳ ample, has de®ned 10 isopycnal surfaces in the North where g is the acceleration due to gravity and N is the and South Atlantic Oceans with 88 different ␴n values, buoyancy or Brunt±VaÈisaÈllaÈ frequency. -١␪) in the real ocean is a well␣ ١S Ϫ␤)where n ϭ 0, 0.5, 1, 1.5, 2, 3, 4, and 5 (corresponding Since ␳ to reference pressures up to 5000 db), depending on the de®ned function of three-dimensional space, the exis- spatial location of the water involved (there being 11 tence of neutral tangent planes as the planes perpen- ,١␪) is guaranteed. Unfortunately␣ ١S Ϫ␤)different spatial regions). The choice of the number of dicular to ␳ these ␴n values and the de®nitions of the corresponding the envelope of all such tangent planes is not a math- regions is subjective. The assignment of ␴n values is ematically well-de®ned surface. For such a surface to ١␪) must satisfy the condition␣ ١S Ϫ␤)based on a match between differing ␴n values as one exist, A ϭ ␳ progresses from one spatial region to the next. When of integrability (Phillips 1956), that is, its helicity (Lilly following such an ``isopycnal'' surface from one ref- 1986) H must be zero; erence pressure to another (say from a given ␴ value (ϫ A ϭ 0. (1 ١´H ϭ A 1 to a desired ␴2 value), data at the mid pressure (1500 db in this case) from all the casts in the region is plotted This condition is satis®ed exactly when there is an in- on a ␴1 versus ␴2 diagram and a straight line is ®tted tegrating factor b ϭ b(x, y, z) with the property that ١␪) is irrotational [see also Eqs. (10) and␣ ١S Ϫ␤)by least squares. The desired ␴2 value is then found as b␳ the value on the straight line at the speci®ed ␴1 value. (11) of McDougall and Jackett 1988]. Under these con- The addition of further isopycnal surfaces requires the ditions, a scalar potential ␥n exists that satis®es choice of additional ␴ values and regions based on this (١␪). (2␣ ١S Ϫ␤)n ϭ b␳␥١ n matching procedure. The computation of these isopycnal surfaces for a range of reference pressures is both Formally, it can be shown that (i) the existence of neutral ١␪), (ii) zero helicity␣ ١S Ϫ␤)time consuming and far from straightforward. Also, surfaces orthogonal to ␳ there are nearly always discontinuities in the slopes of [as de®ned in (1)], and (iii) the existence of the scalar these isopycnal surfaces across the de®ning regions. potential ␥n and integrating factor b satisfying (2) are In this paper we de®ne a new density variable, neutral all equivalent. density, denoted by ␥n, which is a function of salinity Another way of looking at the above de®nition of S (psu), in situ temperature T (ЊC), pressure p (db), neutral density is to apply the classical result from vector longitude, and latitude. Surfaces of constant ␥n de®ne analysis known as Helmholtz's theorem (see Phillips the neutral surfaces, which provide the proper frame- 1956). This states that an arbitrary vector, V, can be work for an ocean model's calculations and analysis. expressed as the sum of two vectors, one of which is These neutral density surfaces are essentially a contin- irrotational, the other solenoidal. That is, for any vector ϫ ١ ١␾ and uous analog of the discrete potential density surfaces ®eld V there are two other vector ®elds ϫ W. With our de®nition ١ ١␾ ϩ referred to various pressures (Reid's isopycnals), which W satisfying V ϭ is the current best practice method of quantitatively de- of ␥n we are attempting to ®nd a scalar function, b, of scribing isopycnal mixing. space for which the Helmholtz decomposition of V ϭ ١␪) consists of only the irrotational part␣ ١S Ϫ␤)The computation of ␥n for particular hydrographic b␳ -n. The degree to which this is achievable is deter␥١ data uses an accurately prelabeled global dataset, and much of the present paper is concerned with the con- mined by the (small) size of the helicity H. struction of this labeled dataset. Labeling of arbitrary In the real ocean H does not generally satisfy (1) hydrographic data is achieved by making basic neutral exactly, despite that its magnitude is small. In Fig. 1 we surface calculations from the observation to the four have plotted the frequency distribution of helicity H for nearest neighbor casts (in terms of latitude and longi- the global dataset developed in section 3 derived from tude) in the dataset. the Levitus (1982) oceanographic atlas of the World Early work on neutral surfaces was done by Pingree Ocean. From this ®gure it is clear that 95% of the he- (1972), Ivers (1975), and Foster and Carmack (1976), licity values of the ocean lie between Ϯ1.8 ϫ 10Ϫ17 kg2

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In section 2 we present an accurate method for ®nding the point where a neutral surface passing through a bottle {S, T, p} intersects a neighboring cast of hydrographic data, and we show that the errors involved in this calculation are quadratic in the differences in pressure and potential temperature on the neutral surface, being proportional to ⅙∫ pd␪. Section 3 introduces a global dataset that facilitates the de®nition of neutral density ␥n. This dataset is derived from the Levitus climatology but has vertical stability constraints imposed and mod- i®cations made to the Antarctic shelf waters. In section 4 we solve the simultaneous system of differential equations de®ned by (2) using a variety of numerical techniques and so obtain an initial ®eld of ␥n values for our FIG. 1. Probability density function (dotted line) and cumulative distribution function (solid line) of the helicity values, H, found in global dataset. The distribution of the errors accumu- the global dataset of section 3. The scale of H is in units of 10Ϫ18 lated during this solution procedure are then smoothed kg2 mϪ9, and 95% of the ocean H values lie between Ϯ1.8 ϫ 10Ϫ17 out over the entire ocean through an iterative improve- kg2 mϪ9. ment to the initial ␥n ®eld. The global dataset therefore has had a ®eld of ␥n values assigned to it, and the labeling of arbitrary hydrographic data can now be mϪ9, with extreme values being of the order Ϯ7.0 ϫ achieved by making neutral excursions to the four clos- 10Ϫ16 kg2 mϪ9. We show in appendix A that a uniform est casts of the labeled dataset, where an accurate es- helicity of 10Ϫ17 kg2 mϪ9 over a square area of ocean timate of its ␥n value is available. Section 5 details the 100 km on a side would give an ambiguity in density actual method of labeling arbitrary hydrographic data of 0.003 kg mϪ3. This ambiguity is just below the present with ␥n, while section 6 compares the resulting neutral instrumentation error in density. density surfaces with the current isopycnal surfaces. The ``path dependent'' nature of neutral surfacesÐ that is, the phenomenon of tracing a neutral trajectory around an ocean basin and returning to the original lon- 2. The basic neutral surface calculation gitude and latitude at a depth different to that of the The basic neutral surface calculation determines the starting pointÐhas been recognized for some time (see point where a neutral surface passing through a given Reid and Lynn 1971; McDougall 1987a; McDougall and bottle (SÄ,TÄ,pÄ) intersects a neighboring cast of hydro- Jackett 1988; and Theodorou 1991). McDougall (1987a) graphic data. The cast is de®ned by n bottles {(S , T , found this error to be only about 10 m for a neutral k k pk), k ϭ 1, 2, ´´´, n} but is a continuous function of p ``helix'' around the main gyre in the North Atlantic or z. In order to develop an accurate method for ®nding Ocean, while Theodorou (1991) found it to be only 4 this point, we introduce the neutral surface gradient op- m around a much smaller closed trajectory in the Ionian ١n, which is de®ned by taking derivatives of ,erator Sea. In McDougall and Jackett (1988) the pitch of this variables in neutral surfaces. More speci®cally, the com- neutral helix has been quanti®ed in terms of the hydro- /␾ץ ١n␾ for a scalar function, ␾, such as ponents of xͦ , are de®ned asץ graphic properties of the water masses on the closed trajectories, with the four examples of neutral surfaces n in the North Atlantic all having depth errors of less than ␦␾ lim , 10 m. In sections 3 and 4 of this paper we ®nd that the ␦x→0 ␦x size of this error over local scales is of the order of meters and over global scales of the order of tens of where ␦␾ is evaluated in the neutral surface and the meters. The small size of this helical-path-dependent distance ␦x is evaluated in the horizontal geopotential error, coupled with the small size of the helicity values plane. The conventional three-dimensional gradient op- -١n (e.g., see Gill 1982 or Mc is related to ١ H for the ocean in Fig. 1, provides the motivation for erator searching for a well-de®ned neutral density variable, ␥n, Dougall and Jackett 1988) by which locally nearly possesses the neutral surface prop- ץ (ϩ m, (3 ١ ϭ ١ .(erty (2 n z Η x,y ץ Throughout this paper we use the word ``cast'' to denote a vertical pro®le of hydrographic data and ``bot- where tle'' to signify data taken from a particular depth. Al- ץ -though this nomenclature is not usually applied to av z Ηץ -eraged data such as Levitus (1982) or the dataset de veloped in this paper, it does serve as a succinct way x,y of describing hydrographic data. is the usual Cartesian derivative and m is a vector nor-

Unauthenticated | Downloaded 09/28/21 11:32 AM UTC 240 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 mal to the neutral surface with unit k component. It follows immediately from the application of (3) that the neutral surface property (2) is equivalent to

n (␪), (4 ١␣ S Ϫ ١␤)١nnn␥ ϭ b␳

n ١n␥ ϭ 0 is identical to the solution so the solution to ١n␪) ϭ 0. This observation is simply␣ ١nS Ϫ␤)of ␳ that the neutral surface is tangent to the locally referenced potential density surface (see McDougall 1987a), and it forms the basis of our basic neutral surface calculation. Given no knowledge of the water mass properties Ä Ä between (S,T,pÄ) and the cast {(Sk, Tk, pk), k ϭ 1, 2, ´´´, n}, discretization of the locally referenced potential density about a point midway between the bottle and cast provides us with the following algorithm: the point

(S, T, p) on the cast {(Sk, Tk, pk), k ϭ 1, 2, ´´´, n}on the same neutral surface as the bottle (SÄ,TÄ,pÄ) is that point which satis®es E(p) ϭ ␳(S, ␪, p) Ϫ ␳(SÄ, ␪Ä, p) ϭ 0, (5) where ␪ and␪Ä are potential temperatures referred to the sea surface, and p ϭ (p ϩ pÄ)/2. That is, the two parcels (S, T, p) and (SÄ,TÄ,pÄ) are deemed to obey the neutral property if they have the same potential density when referred to their average pressure. Note that we have chosen to write (5) in terms of the equation of state expressed as a function of salinity, potential temperature, and pressure. Equation (5) can equally well be FIG. 2. Salinity±potential temperature diagram (a), and E(p) func- written using the International Equation of State,␳Ã (S, tion (b), for a bottle and Levitus cast of hydrographic data in the T, p), as Southern Ocean for which a triple crossing has occurred. The solid, dotted, and dashed lines in (a) show the three local potential density E(p) ϭ ␳Ã(S, ␪[S, T, p, p], p) surfaces through the bottle giving rise to the three solutions, while (b) indicates how reversals in the sign of E can result in these triple ÄÄÄ p Ϫ␳Ã(S, ␪[S, T, pÄ,p], p) ϭ 0, (6) crossings of E(p) with the p axis. and it is this functional form of E(p) that we solve for the cast values (S, T, p).

The method then consists of forming values of Ek ϭ where one of these triple crossings has occurred. The E(pk) for each of the bottles (Sk, Tk, pk), k ϭ 1, 2, ´´´, n, solid, dotted, and dashed lines in Fig. 2a show the three on the cast, and identifying the intervals [pk, pkϩ1] over potential density surfaces through the bottle correspond- which the sign of E(p) changes. In each of these inter- ing to the three solutions of (6) at 192.4 db, 229.4 db, vals a Newton±Raphson technique is employed to ac- and 522.6 db. The dotted line has a potential density of curately ®nd the positions of (S, T, p) on the cast that 34.62 kg mϪ3 corresponding to a midpressure value of satisfy E(p) ϭ 0 to within prescribed tolerances (5 ϫ 1471.2 db, while the solid and dashed lines have cor- 10Ϫ3 m in depth and 5 ϫ 10Ϫ5 kg mϪ3 in E). Should the responding values (34.7 kg mϪ3, 1489.7 db) and (35.4 Newton±Raphson scheme fail to converge for a partic- kg mϪ3, 1636.3 db). The choice of which of these so- ular interval (as in the case of a double zero), a simple lutions we take as the solution of E(p) ϭ 0 is a dif®cult interval halving technique is adopted with the same error one to make and is the subject of work in progress. In tolerances. Fig. 2b we have shown the E(p) function for this bottle/ In the vast majority of cases, a single point (S, T, p) cast combination. It is clear that the multiple solutions is identi®ed on the cast as being neutrally connected to of E(p) ϭ 0 can only occur when there are reversals in Ä Ä (S,T,pÄ). However, in some cases we are faced with the signs of Ep. multiple solutions of E(p) due to ``triple crossings'' of A Taylor series expansion of Ep about the point (S, the E(p) function with the p axis. These occurrences are T, p) reveals a leading term proportional to N2 at the real and predominantly occur in the Southern Ocean. cast, the constant of proportionality being positive. The Figure 2 shows the salinity±potential temperature dia- derivation of this Taylor series can be found in appendix gram and the E(p) function for the bottle (34.4 psu, A. Thus, the occurrence of triple crossings of E(p), a

Ϫ1.0ЊC, 2750 db) and a cast in the Southern Ocean direct consequence of a reversal in the sign of Ep,is

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{(Sk, Tk, pk), k ϭ 1, 2, ´´´, n}. Anticipating the global trajectory puts ␤(S,␪, p)⌬S Ϫ ␣(S, ␪, p)⌬␪ equal to zero dataset we develop in section 3, we have completely (see appendix A), instead of putting all of the right-hand circumvented these triple crossing problems in all the side of (8) to zero. Thus, we incur a relative error of ∫ basic neutral surface calculations we have performed in ϪTb n (p Ϫ p) d␪, or an absolute density error of about 3 ∫ Ϫ3 this paper by imposing realistic stability constraints on 10 Tb n (p Ϫ p) d␪ kg m . For a linear variation of p ∫ ∫ the cast data making up our global dataset. We still, with ␪, n (p Ϫ p) d␪ is zero, and so n (p Ϫ p) d␪ is however, detect the existence of these triple crossings equal to the area between the data on a neutral trajectory for arbitrary hydrographic data, and when such solutions and the straight line between the end points on a p±␪ occur, we do not attempt (at this stage) to make an diagram (see Fig. A1). The thermobaric parameter Tb is objective choice between them: the neutral surface cal- approximately 2.7 ϫ 10Ϫ8 KϪ1 (db)Ϫ1 [see Fig. 9 of culation returns with a special pressure value indicating McDougall 1987b] and the maximum triangular error this situation. from this effect is 2.7 ϫ 10Ϫ5 ϫ (1/2)⌬p⌬␪. A more The de®nition that we have used of the neutral re- realistic error estimate is one-third of this triangular er- lationship between two bottles, E ϭ ␳(S, ␪, p) Ϫ ␳(SÄ, ror, say 5 ϫ 10Ϫ6⌬p⌬␪, where ⌬p is measured in de- ␪Ä, p) ϭ 0, is a ®nite difference approximation to the cibars. For the realistic maximum ocean values of ⌬p exact relationship ϭ 1000 db and ⌬␪ ϭ 1.5ЊC, this error estimate is 7.5 ϫ 10Ϫ3 kg mϪ3. (␪)´dl ϭ 0, (7 ١␣ S Ϫ ١␤)␳ ͵ nn n 3. The dataset along a neutral trajectory. The accuracy of this ®nite difference expression, E, is also examined in appendix Reid and Lynn (1971) have shown that, as an iso- A, where it is found that when the variations of pressure pycnal surface rises toward the sea surface in the North- and potential temperature on a neutral surface are pro- ern and Southern Hemispheres, it coincides with dif- portional, the error in E is cubic in the differences ⌬S ferent potential density surfaces (referenced to sea level) ϭ S Ϫ SÄ, ⌬T ϭ T Ϫ TÄ, and ⌬p ϭ p Ϫ pÄ. For the values in the two hemispheres. In subsequent work, Reid (1986, of ⌬S, ⌬T, and ⌬p that are encountered over a 4Њ extent 1989, 1994) has taken this isopycnal mixing analysis of longitude or latitude (say 0.3 psu, 1.5ЊC, 1000 db) further by de®ning isopycnal surfaces in the South Pa- these cubic errors are quite small (typically 2 ϫ 10Ϫ4 ci®c, and the South and North Atlantic Oceans, in an kg mϪ3). attempt to describe the total geostrophic circulation of In general, potential temperature and pressure will not these basins. In these papers Reid has shown that not vary linearly along a neutral trajectory. In this case the only is it between hemispheres that these surfaces differ error involved with the ®nite difference approximation, signi®cantly in potential density, but even within the E, to a neutral trajectory is quadratic rather than cubic, same ocean basin they must be de®ned with different potential density labels in different regions (sometimes being proportional to Tb⌬p⌬␪, where Tb is the thermobaric parameter of McDougall (1987b). This qua- referred to the same pressure, but mostly referred to dratic error is much more serious than the cubic error different pressure levels). For example, in the most re- discussed in appendix A and can be quanti®ed as fol- cent speci®cations of the isopycnal surfaces in the North lows. If p does not vary linearly with ␪, then the leading and South Atlantic Oceans, the layers can differ by as terms in much as 0.14 in ␴0 between the hemispheres, while 6 ∫ of the 10 surfaces in the north and 5 of the 10 in the -١n␪) ´dl are south have been de®ned with multiple values (on oc␣ ١n S Ϫ␤) n casions up to three of four different values) of potential ␪)´dl ١␣ S Ϫ ١␤) ͵ nn density, all referred to the same pressure. n By analogy with Reid's work, it follows that ␥n, being a continuous approximation to these discretely refer- ϭ͵͵␤(S, ␪,p)dS Ϫ ␣(S, ␪,p)d␪ enced potential density surfaces, cannot be a function nn of the three state variables S, T, and p but must also ϭ␤(S,␪,p)⌬SϪ␣(S,␪,p)⌬␪ depend on geographical location. Thus, despite the existence of a well-de®ned equation of state in terms of S, T, and p, any attempt to describe ␥n in terms of just ϩ␤ (pϪp)dS Ϫ ␣ (p Ϫ p) d␪ p ͵ p ͵ these three variables will prove fruitless. Given this spa- nn tial dependence of ␥n, it is clear that any representation ϭ␤(S,␪,p)⌬SϪ␣(S,␪,p)⌬␪of ␥n must incorporate a knowledge of the spatial distribution of temperature and salinity. One method for ϪT(pϪp)d␪.accommodating this is to describe the entire ocean by b͵ n a comprehensive dataset that possesses an accurate rep- (8) resentation of ␥n. An arbitrary {S, T, p, lat, long} ob-

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servation from the ocean will not normally have the densest bottom bottle (in terms of ␴4) from the 16 casts same combination of S, T, and p as the labeled reference surrounding the point in question, thereby attempting dataset at that longitude and latitude, but it will not be to capture an ocean with the widest possible density very distant from it since the reference dataset is an range. In the Antarctic Circumpolar Current this pro- average of actual observations taken from the same lo- cedure sometimes resulted in there being no cast that cation. A label can be found for this observation by was representative of the center of the ACC, and values ∫ ®nding the depths on the surrounding four casts in the of ␳0Tb⅙pd␪ [see McDougall and Jackett (1988) or (A10) reference data where the hydrographic properties are of appendix A] of about 0.03 kg mϪ3 were found around such that it and the {S, T, p} observation lie on the local closed neutral circuits on the 4Њϫ4Њgrid. This same neutral surface. The ␥n label of an arbitrary {S, was recti®ed by ensuring that casts from the central T, p, lat, long} observation is then taken as a weighted ACC were used rather than those based on the maximum n average of these ␥ values found in the labeled dataset. ␴4 criterion. At the surface, all casts have also been It is important to realize that the use of the prelabeled complemented with the corresponding seasonal Levitus global dataset in the de®nition of neutral density ␥n does data. Speci®cally, in the top 200 m of the ocean, the not diminish its use as the most natural density variable annual data was linearly interpolated toward the season in the ocean. This follows from the fact that the rela- that had the least dense surface water, with no change tionship of lying on the same neutral density surface is, at 200 m and the seasonal data completely replacing the apart from the path-dependent errors (which are quan- annually averaged data at the surface. Data points in the ti®able), an equivalence relation. Consider the process Arctic Ocean, the Mediterranean, Baltic, Black, Caspian of labeling the data from two adjacent casts from a and Red Seas, the Persian Gulf, and Hudson Bay were modern hydrographic section. On each of these two all excluded from the subsampled dataset. casts there is a point that communicates neutrally with To justify the 4Њ resolution of our global dataset and a point on one of our prelabeled casts. The equivalence to further quantify the local path-dependent error as- relation of the neutral property implies that all three sociated with the ill-de®ned nature of neutral surfaces, points lie on the same neutral surface and, in particular, we have conducted an experiment on small closed neu- that the two points on the observed casts that receive tral trajectories in the Levitus (1982) atlas. For each the same ␥n label also possess the neutral property. The bottle on each cast in our subsampled global dataset, only caveat here is the presence of path dependence, we have made two different types of neutral excursions and this can be shown to be a very small effect (see to each of the casts to the immediate east and north. below). We conclude that even though the prelabeled The ®rst applies the basic neutral surface calculation global dataset that we use may be much older than ob- once with the neighboring cast, while the second inserts servational data requiring labeling, the equivalence re- the four casts from the original 1Њϫ1ЊLevitus data lationship formed by our neutral density variable en- between the bottle and the cast in question and then sures that the labeling procedure achieves our aim of makes ®ve neutral surface calculations. This provides labeling observational data with a neutral density vari- us with estimates of the absolute depth errors at the two able. neighboring casts due to our dataset being stored every At the present time the only datasets encompassing 4Њ rather than every 1Њ. Averaging over all bottles on the entire ocean are the climatological atlas of Levitus the original cast, we can now ®nd mean and maximum (1982) and the global dataset of Reid (1986, 1989, absolute depth errors associated with 4Њ eastward and 1994). The former consists of averaged values of S and northward closed neutral excursions for each cast in the T on a regular grid at standard depths, while the latter subsampled dataset. In Fig. 3a we have plotted the cu- data consists of observed values of S, T, and p at ir- mulative distributions for the entire ocean of these cast regular spatial locations. For our purposes, the Levitus mean and maximum absolute depth errors, the solid line data possesses two signi®cant advantages over the Reid representing the mean errors and the dashed line the data: the regular nature of the data in terms of longitude, maximum errors. It is evident that 95% of the Levitus latitude, and depth (corresponding to our x, y, and z (1982) ocean have cast-mean absolute depth errors of coordinates) and that the dataset does not contain ad- less than 4.2 m and maximum absolute depth errors of jacent casts from different seasons. For these reasons less than 14.1 m. The small sizes of these errors vali- we have chosen the climatology of Levitus (1982) as dates the resolution we have adopted for our global the dataset for describing the world's oceans. dataset, albeit this resolution having been chosen for the The Levitus data consists of measurements of S and reasons of computational expediency. Figure 3b shows T at 33 standard depth levels at a 1Њ resolution, located these same errors, but now expressed in terms of the at each 1/2Њ of longitude and latitude. Solution of (2) difference in locally referenced potential density at the for such a large dataset is computationally a very large cast 4Њ to the east or north. These errors have been 3 ∫ problem, so we have subsampled the original data onto compared with the estimate 10 Tb⅙pd␪, derived in ap- a4Њϫ4Њgrid, located on every fourth degree of lon- pendix A, and are almost exactly equal to this estimate, gitude and latitude commencing at (88ЊS, 0Њ). The data con®rming our Taylor series analysis results that this at each of these locations is taken as the cast with the quadratic error term due to the path-dependent nature

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lem, but high quality observational data such as Reid's dataset (Reid 1986, 1989, 1994) also contains equally numerous occurrences of these static instabilities, almost certainly due to instrumentation errors. The effect of statically unstable water in our dataset would be disastrous for our application, since negative N2 values would cause overturning of neutral surfaces and vertical inversions of ␥n in the labeled dataset. This n can be seen by recalling that ␥z ϭ b␳(␤Sz Ϫ ␣␪z). Mc- Dougall (1988) has shown that b can be written as an exponential of the sum of two integrals of water mass properties along neutral surfaces and, as such, must al- n ways be positive. Thus, the sign of␥z is the opposite to the sign of N2, which means that when N2 Ͻ 0, ␥n is not monotonic with depth. A technique must therefore be employed that eliminates these buoyancy frequency problems from the subsampled 4Њ data. Recent work by the present authors (Jackett and Mc- Dougall 1995) has addressed this problem with the development of an algorithm that minimally adjusts hydrographic data so that the resulting buoyancy frequency pro®les are larger than a speci®ed lower bound. Using this algorithm, the entire reference dataset has been sta- bilized with quite mild minimal buoyancy frequency pro®les. The resulting modi®cations made to the Levitus dataset were all reasonable [and well within the error tolerances found in Levitus (1982)], graphically not vis- ible in the vast majority of cases. The formation of Antarctic Bottom Water by salt re- jection during ice formation in winter causes additional problems with the Antarctic shelf water casts taken from FIG. 3. Cumulative distribution function over the entire ocean of the Levitus atlas. Figure 4a shows two casts in our sub- cast mean (solid line) and maximum (dashed line) absolute depth sampled database along the 140ЊE meridian, one at 64ЊS errors associated with making the basic neutral surface calculation over 4Њ of longitude or latitude rather than using the 1Њϫ1Њdata. with a depth of 4000 m and the other on the shelf at In panel (a) we have plotted depth differences (m) on the horizontal 68ЊS with a depth of 400 m. Also shown are two bottles, axis, while in panel (b) we have plotted the differences in terms of marked A and B, at depths of 300 and 950 m, respec- the locally referenced potential density (kg mϪ3). Ninety-®ve percent tively, taken from the summer, WOCE line SR3 at of the Levitus ocean have mean absolute errors less than 4.2 m and maximum absolute errors of less than 14.1 m. 66.4ЊS, 140.2ЊE in February 1994. The dashed line represents a neutral excursion between bottle A and the more northern cast, such an excursion to the southern of neutral surfaces far outweighs the cubic error terms cast on the shelf not being possible. Figure 4a clearly inherent in our ®nite-difference approximations. demonstrates that the shelf water contained in our sub- Although the Levitus atlas possesses the aforemen- sampled dataset does not have the ability to neutrally tioned advantages, it was constructed using objective communicate with even relatively shallow observational analysis performed on geopotential surfaces, rather than data taken in its immediate vicinity. The averaging na- the more accurate locally referenced potential density ture of the objective analysis in smoothing out extreme surfaces. Also, despite the careful elimination of stati- density values has resulted in even the densest cast in cally unstable water prior to the objective analysis, no the 4Њϫ4Њbox of Levitus data on the shelf being unable stability criterion was actually implemented during or to make neutral excursions to summer observational after the averaging procedure. The consequence of these data. The situation with winter data, were it obtainable, shortcomings is that nearly half of the 4Њϫ4Њcast data would be worse. chosen from the Levitus climatology possess vertical Accordingly, we have systematically replaced all Ant- inversions of locally referenced potential density or, arctic shelf water in our subsampled dataset with deeper equivalently, have buoyancy frequency pro®les with and denser data from neighboring waters. All data along negative values. It should be pointed out that these sta- 72Њ and 76ЊS has been replaced with the denser data at bility problems only affect some 5% of the bottle data, 68ЊS at the same latitude. For shelves north of this, the but when projected to cast data, this ®gure increases to replacement has been selective. For instance, Fig. 5 47%. Not only is it averaged data that suffers this prob- shows the replacement strategy to the south of Drake

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exception to this was that we have retained the Ker- guelen Plateau and the intersection of the Southeast In- dian Ridge with the Paci®c Antarctic Ridge and the Antarctic continent since deep water (below 3000 m) on either side of these two bathymetric features does not mix in the real ocean. Returning to Fig. 4a, it is clear from the slope in the S±␪ diagram of the neutral surface through A to the denser cast [now also the cast at (68ЊS, 140ЊE) in our dataset] that this reference cast will not neutrally communicate with the deeper observational bottle B. To facilitate such connection we have found it necessary to extend all water south of 64ЊS to much denser water. The dashed line in Fig. 4b shows this extension for the cast in question, and the dotted line indicates the neutral excursion from bottle B to this extended cast. The (S, T, p) point adopted for the extension of the southern casts in the dataset is taken as (34.67 psu, Ϫ2.18ЊC, 5500 db), the salinity value being the salinity of the densest water found in the real ocean (Sverdrup et al. 1942), the pressure being that of the deepest level in the Levitus and our reference dataset, and the temperature being the in situ temperature at (34.67 psu, 5500 db), which is 0.25ЊC below the freezing line at (34.67 psu, 500 db). This pressure of 500 db was motivated by the maximum depth of icebergs. The extension in the global dataset is made for all bottles below min (4500 db, depth of the cast) by linearly interpolating the data at this depth on the cast to the frozen bottom water at 5500 db. FIG. 4. Two casts in the subsampled 4Њϫ4ЊLevitus dataset on Finally we return to the question of the validity of the Antarctic shelf, and two bottles of observational data A and B using a prelabeled global dataset to de®ne our neutral taken from the summer WOCE line SR3 at (66.4ЊS, 140.2ЊE). The density variable. Apart from this question there is the dotted lines depict the neutral surfaces passing through A and B to the deeper of the two casts, and the dashed line represents the ex- further issue of the accuracy of the Levitus data itself. tension of this deeper cast to frozen bottom water. For example, Lozier et al. (1994) have pointed out that the Cartesian averaging that was used in forming the Levitus dataset results in spurious water masses where Passage. The numbers at the center of each of the boxes density surfaces slope signi®cantly and where the S±␪ in the ®gure represent the depth of the cast in the dataset diagram of vertical casts is curved. They have shown at that location (expressed in meters), and the thick solid that along an isopycnal horizon the changes in S and ␪ line depicts the delineation we have used between the are as large as 0.2 psu and 1ЊC, while along an isobaric South Paci®c and South Atlantic Oceans (see later). The horizon, the change in potential density is as large as Antarctic continent is shown by the gray line, and the 0.1 kg mϪ3. The question then arises as to whether our arrows indicate which casts are replaced by others in method of labeling data with neutral density (based on the dataset. All replacement is made sequentially from Levitus data) can potentially be in error by 0.1 kg mϪ3 the east and then from the north so that, for example, in these regions. The answer to this question is no, and the cast at 68ЊS, 312ЊE replaces all casts southwest of the reason is that apart from the path-dependent errors it in the Weddell Sea. The box at 72ЊS, 288ЊE is without the basic neutral calculation forms an equivalence re- a number in Fig. 5 since the original 1Њϫ1ЊLevitus lation. It follows from this that, if observations A and data contains no observations in the 4Њϫ4Њbox, so we B both individually communicate neutrally with a point have duplicated the observation from the north. This L in our labeled Levitus dataset (hence, points A and was the only occurrence of such data paucity in the B get the same neutral density label by our algorithm), entire ocean. then point A will also communicate neutrally with point The replacement of Antarctic shelf water over the B. That point L is, say, 1ЊC warmer than it ought to be entire longitudinal range of the ocean was made in a is only important in that it can introduce some small similar manner, where shelf water was replaced by dense path-dependent errors. If we assume that L is too warm water from the appropriate neighboring ocean. The only by 1ЊC and too deep by 100 db, then the maximum

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FIG. 5. Replacement strategy for the Antarctic shelf waters on the southern side of Drake Passage. The solid line depicts the delineation used between the Paci®c and Atlantic Oceans and the arrows indicate which casts are replaced by others in the global dataset. possible path-dependent error in neutral density is [from order hyperbolic partial differential equations. In gen- Eq. (A10)] eral, the most accurate method of solution of these equations is with the method of characteristics (see Carrier 1 103Tb⅜pd␪ഠ2.7 ϫ 10Ϫ5 ϫϫ100 ϫ 1 and Pearson 1976 or Smith 1965). This technique iden- b ͵ 2 n ti®es characteristic curves or surfaces in the problem n Ϫ3 Ϫ3 domain, along which values for ␥ can be found by ϭ 1.3 ϫ 10 kg m , integrating initial data speci®ed on some appropriate where we have allowed for the full triangular area in (i.e., noncharacteristic) curve (or surface). These char- the p±␪ diagram. It is concluded that while the Levitus acteristics also de®ne a region of dependence of the data does contain the anomalies as described by Lozier initial data, and values of ␥n outside this region cannot, et al. (1994), these anomalies do not signi®cantly affect in theory, be accurately determined. our use of the Levitus dataset in the neutral density Not surprisingly, the characteristic surfaces of (9) turn algorithm. The same argument also shows that the mean out to be neutral surfaces, along which ␥n is constant. temporal change of ocean properties due to, say, global The method of characteristics for our problem therefore warming does not lead to signi®cant errors in the neutral reduces to ®tting neutral surfaces to our three-dimen- density label. sional dataset, with the values of ␥n on these surfaces being determined by a knowledge of ␥n down just one cast. 4. Labeling the reference dataset with n ␥ An individual neutral surface can be ®tted to our Having now constructed a stable global hydrographic three-dimensional hydrographic data {S,T, p, lat, long} dataset, we need to accurately label this dataset with ␥n. by starting from a given cast and then growing laterally This labeled dataset will subsequently be used to assign in space using the basic neutral calculation for each step. ␥n values to arbitrary hydrographic data. The coupled During this procedure the set of locations are partitioned system of differential equations de®ned by (2) succinct- into two disjoint subsets: those that have (have not) been ly describes the neutral property we desire for neutral assigned a pressure value for the particular neutral sur- density ␥n. As an initial ®eld of ␥n values for our ref- face. Points from the latter are moved to the former by erenced global dataset, we therefore seek the solution keeping track of the perimeter set of locations of the to the differential equations currently assigned surface. This perimeter set is de®ned to be the set of all locations where an actual pressure (١␪), (9␣ ١S Ϫ␤)n ϭ b␳␥١ value on the last iteration of the process exists, with it where b is an unknown scalar function of space. initially being de®ned as a single point (the main ref- These equations are a simultaneous system of ®rst- erence cast). Neutral excursions are then made from this

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FIG. 6. A map of the iteration number at which each location in the global dataset was assigned a neutral surface pressure value. The surface shown is the shallowest of the 100 neutral surfaces ®tted to the dataset that did not pass through Indonesian waters above 1200 m. perimeter set to all unallocated points that are at a dis- regions. It also shows that when the surfaces reach 0Њ tance of 4Њ of longitude or latitude. In this way the size from the east, they are continued westward into the of the perimeter set increases from 1, rises to a maxi- western Atlantic Ocean. mum, and then decays to 0, when the complete surface The single cast down which we provide the initial ␥n has been ®tted to the data. When the surface either out- data was chosen to be the cast at 16ЊS, 188ЊE, this being crops or undercrops, the pressure value is ¯agged, and the central Paci®c Ocean cast that captured a maximum the surface is pursued no further in this direction. After 80% of the bottle data of our global dataset. By this we termination of the iterative procedure, those points re- mean that 80% of the total number of bottles in the maining to be assigned have either outcropped on one dataset lie between adjacent neutral surfaces that em- or more occasions, or are more than one horizontal grid anate from this reference cast. The values for ␥n down point from the currently de®ned neutral surface, and in this cast can be chosen arbitrarily, but we have made it both cases are left unassigned. A major advantage of coincide with the ``local potential density,'' a variable generating the surfaces in the two lateral directions us- ®rst proposed by Veronis (1972). Speci®cally, we de®ne ing this method is that the resulting surfaces have the 0 ability to iterate around bathymetric features. ␥n (z)ϭ␴(0) Ϫ ␳gϪ12 N dz, (10) During this procedure, various water masses are in- ref 0 ͵ z hibited from communicating with each other. For example, below 1200 m the Paci®c and Indian Oceans are which is a simple integration of (9) in the vertical with not permitted to be connected through Indonesia, but a b pro®le of unity. It should be noted that although above this, neutral excursions are allowed, thereby mod- this density variable can be uniquely de®ned for any eling the Indonesian Through¯ow (Godfrey and Gold- cast, it does not constitute a well-de®ned function of ing 1981). In contrast, the Atlantic and Paci®c Oceans three-dimensional space (as was pointed out by Veronis n are never joined through the Panama Canal. Also, for 1972). However, extending the de®nition of␥ref from reasons which will be subsequently discussed, the Pa- the one single cast at 16ЊS, 188ЊE to the global dataset ci®c and Atlantic Oceans are not allowed to commu- using characteristic surfaces does generate a mathe- nicate through Drake Passage at this stage of the pro- matically well-de®ned variable. This is achieved by al- cess. lowing b to vary in space [see Eq. (2)]. The values of One hundred neutral surfaces were ®tted to the global ␥n at the reference cast in the Paci®c Ocean ranged from dataset, all emanating from a single cast in the central 22.3797 kg mϪ3 at the surface, equal to the in situ and Paci®c Ocean. Figure 6 shows one of these surfaces that potential densities there, to 28.2211 kg mϪ3 at the ocean commenced on the Paci®c Ocean cast at mid depth, this ¯oor. being the shallowest of the neutral surfaces that was This method of ®tting neutral surfaces to the global blocked through Indonesia. The contoured number is dataset does result in errors in ␥n, which become ap- the iteration number at which each particular location parent when examining points on adjacent casts on the is assigned a neutral surface pressure value. The ®gure same neutral surface that have been generated with en- clearly demonstrates the circuitous nature of the surface tirely different trajectories from the reference cast. This generation and the closures of the various bathymetric depth error, which can be quanti®ed by making one ®nal

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immediately above and below each bottle. Using the path (in x±y space) taken by either one of these surfaces (usually the closest, unless it outcrops) from the bottle in question back to the Paci®c Ocean reference cast, the basic neutral surface calculation can again be employed to ®nd a neutral trajectory from the arbitrary bottle back to the reference cast. The value of ␥n for the particular bottle can then be found by accurately computing

0 ␴(0) Ϫ ␳gNdzϪ12 0 ͵ zint

up the main reference cast, where zint is the height of FIG. 7. A plot of the real, dz࿞real, against the estimated, dz࿞est intersection of this neutral trajectory with the reference from Eq. (11), of the depth error (in meters) along the arti®cial barrier- imposed at Drake Passage of global-scale closed neutral circuits. The cast. Some of the bottles at the very top and bottom of correlation coef®cient of the two depth errors is 0.995. the ocean that do not lie between adjacent neutral surfaces can use the paths of the shallowest and deepest surfaces respectively, to generate paths back to the Pa- neutral excursion from one of these casts to its neighbor, ci®c Ocean reference cast. (The majority of such routes is the inherent path-dependent error associated with the either outcrop or intersect the sea bed along these paths, de®nition of ␥n. To minimize the occurrence of this error and in this case the bottles cannot be labeled using this in our dataset we have stopped communication between method.) When this was done, an additional 4% of the the Paci®c and Atlantic Oceans at this stage of the pro- ocean was assigned ␥n values, taking the proportion of cess by the inclusion of an arti®cial barrier near Drake ocean captured by the 100 neutral surfaces to 84%. Passage. This minimizes the number of points where This leaves 16% of the ocean, distributed almost such different paths meet and so simpli®es the subse- equally between bottles near the top and those near the quent task of spatially smoothing this error. bottom of the global dataset, which remain to be as- Previous work in McDougall and Jackett (1988) has signed values of ␥n. We return to the numerical solution quanti®ed this helical error in neutral density in terms of the system of differential equations (9) for these val- of the hydrographic properties of the data on the now ues. Although the method of characteristics is the nor- closed path. There it was shown that the error in depth, mal solution method of hyperbolic partial differential ⌬z, associated with such a closed contour C is equations, it is by no means the only process for solving these equations (see Smith 1965). Characteristic grids T ⌬z ഠ Ϫ b ⅜ pd␪, (11) are certainly the most appropriate when the initial data gNϪ12͵ C contain discontinuities, but in the absence of such continuity problems, alternate grids and numerical methods where the thermobaric parameter T ഠ 2.7 ϫ 10Ϫ8 dbϪ1 b can be considered. The continuity (smoothness) of the KϪ1 is approximately a constant (see also appendix A). hydrographic data contained in our global dataset and The seam down our arti®cial Paci®c/Atlantic Ocean bar- n rier allows us to examine the accuracy of this formula the exponential form for b implies that ␥ should be a over global scales. In Fig. 7 the estimate (11) of ⌬z continuous three-dimensional variable. We are therefore made using the closed neutral surface contours for all at liberty to choose an alternate grid and solution strat- 100 ®tted surfaces at the Drake Passage barrier is plotted egy for the fundamental system (9). against the actual error. The correlation of these two ⌬z The most natural choice of a grid for (9) is a rect- errors is remarkably good (a correlation coef®cient of angular Cartesian coordinate system, coinciding with 0.995), indicating (11) is indeed a very good estimate the positions at which we have our regular hydrographic of the helical error in ␥. The mean absolute value of data. Finite difference approximations can be written ⌬z, the difference between the two heights on the casts for the system of equations (9), and the solution ob- at the Drake Passage barrier over these global scales is tained, using standard techniques, from boundary data only 29.5 m, which further justi®es our search for a at the now labeled 84% of the global dataset. Elimi- well-de®ned neutral density variable that almost pos- nation of b between pairs of equations in (9) results in sesses the neutral surface property. the coupled system [since ␳x ϭ ␳(␤Sx Ϫ ␣␪x), ´´´, etc.] The ␥n values of the 100 ®tted neutral surfaces could ␳(␤SϪ␣␪ )␥ nnϭ ␳␥ now be suitably interpolated to provide initial ␥n values zzxxz for each of the bottles in the 80% of the captured ocean; nn ␳(␤SzzyyzϪ␣␪ )␥ ϭ ␳␥, (12) however, this would incur an unnecessary interpolation error. A more accurate evaluation for ␥n can be made where the subscripts denote partial differentiation. by using trajectory information contained in the surfaces These equations are discretized on a variety of grids in

Unauthenticated | Downloaded 09/28/21 11:32 AM UTC 248 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 order to obtain enough independent algebraic equations for iteratively improving the ␥n ®eld. This is based on for the determination of the unknown values of ␥n. making additional computations in the problem's natural The system of algebraic equations resulting from the characteristic coordinate system, the neutral surfaces. discretizations of (12) is overdetermined, roughly in the This relaxation procedure has the effect of laterally ratio of 6 to 1. Traditional ®nite difference techniques smoothing errors in ␥n along the neutral surfaces while nearly always choose discretizations and boundary data achieving continuous and smooth vertical pro®les of b. in such a way that the resulting algebraic system is Details of this relaxation technique can be found in ap- evenly determined, thereby enabling the solution to be pendix B. obtained from a simple matrix inversion. In the present The effect of the relaxation technique for laterally problem we have used the singular value decomposition smoothing the accumulated error in the ␥n ®eld can be to invert the overdetermined system. In practice, the seen in Fig. B3a, where we have plotted an estimate of number of equations (ϳ 65 100) and unknowns the maximum error associated with ␥n down each cast (ϳ11 350) made the inversion of (12) using one matrix in our global dataset. The vast majority of the ocean computationally expensive. Rather, the region of un- has this maximum error below the present instrument known ␥n values was split into smaller sized blocks, error of 0.005 kg mϪ3. The map for the average error each of roughly 500 unknowns. The blocks were chosen associated with each cast is very similar to Fig. B3a, to overlap, by two horizontal grid points, so that a max- only with a ␥n scale an order of magnitude less. imum number of equations was written for each block In conclusion, the ␥n ®eld resulting from the solution and so that the results from a previous inversion were of the differential system (9) using the characteristic coupled to the inversion of the next block. and ®nite difference techniques, followed by the itera- The ␥n values obtained by this procedure are located tive relaxation of the ␥n ®eld, produces a ␥n ®eld that outside the region of dependence (roughly, that space satis®es the differential system (9) an order of magni- between ␥n ϭ 22.38 and ␥n ϭ 28.22) of the reference tude better, on average, than the present instrumentation data at the Paci®c Ocean reference cast. As such, there error in density of 0.005 kg mϪ3. is a question as to the degree of accuracy of these values. A quantitative check can be made to determine whether solving (12) with an overdetermined system outside the 5. Labeling arbitrary hydrographic data with ␥n region of dependence de®ned by the problem's characteristics (as opposed to the more traditional matrix Having developed a global dataset that has been ac- inversion technique based on an evenly determined sys- curately assigned a ®eld of ␥n values, we are now in a tem) is, in fact, a valid numerical technique. By taking position to label external hydrographic data. An arbitrary {S, T, p, lat, long} observation from the real ocean ␳x, ␳y, and ␳(␤Sz Ϫ ␣␪z) as the derivatives of potential lies inside a box consisting of the four nearest casts in density, ␴␪ for example, the numerical procedure above should yield the corresponding potential density ®eld. our labeled dataset. Since our global dataset is based on a climatological atlas of the world's oceans, each of When this was done, the ␴␪ ®eld was inverted to the accuracy of the computations, that is, to order 10Ϫ4 kg these casts should possess bottles that are not very dis- mϪ3. This also suggests that potential dif®culties with tant, in terms of hydrography, from the unlabeled bottle. the convergence of the ®nite difference schemes is not Using the basic neutral surface calculation, four neutral a problem for this particular system of differential equa- trajectories are calculated between the arbitrary bottle tions or for the grids we are using. and the four neighboring casts. The immediate result of The solution of the differential system (9) contains this procedure is the four heights on the four neigh- errors in the ␥n ®eld due to several sources. The ®rst is boring casts, and these are then used, in the manner the inherent path-dependent error associated with the described below, to provide a ␥n value for the original ill-de®ned nature of neutral surfaces. This type of error {S, T, p, lat, long} observation. is most apparent along the arti®cial barrier, which we To increase the ef®ciency of the interpolation of the have so far imposed at Drake Passage. For the remainder ␥n values found on each of the four neighboring casts, of this paper this barrier is removed, allowing water to we have computed ␥n on each cast as a piecewise qua- neutrally mix through Drake Passage as it does in the dratic function in the vertical. This involves storing one real ocean. A second potential error source arises from extra variable, namely, the quadratic coef®cients, for the ®nite difference technique being applied to 16% of each cast in the global dataset. These parameters were the ocean, which lies outside the region of dependence evaluated by ®tting by least squares piecewise quadrat- of the initial data de®ned at the Paci®c Ocean reference ics to the ®nal ␥n ®eld. In appendix A we show that the cast. These two potential sources of error are additional error involved in using this piecewise quadratic repre- to the usual truncation errors involved in the numerical sentation of ␥n in the vertical induces an error of at most procedures we have adopted for the generation of the 10Ϫ4 kg mϪ3 in ␥n, well below the mean absolute error ␥n ®eld. level in the ␥n ®eld over the entire global dataset. Using To eliminate these errors, or at least distribute them this parameterization of ␥n, the interpolation of ␥n on out over the entire globe, we have developed a technique each cast in the global dataset can be achieved with a

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Ϫ y) and (1 Ϫ x)y. If one or more of the reference casts does not exist (because of the presence of continents) or if a neutral density estimate is unavailable because of a triple crossing, then the weighted average is made using three or less estimates of ␥n. We restrict the use of the vertical extrapolation procedure, that is used at the bottom of labeled data, when the datum is denser than the deepest labeled bottle of that cast. We do not accept a neutral density estimate if it requires neutral density to be extrapolated beyond the bottom bottle by more than 0.3 kg mϪ3. This re- striction is implemented smoothly in the following way. If one of the reference casts is extrapolated in order to FIG. 8. A cast of hydrographic data from our reference dataset label the datum, the weight of this cast is reduced lin- (solid line) and two bottles which require extension of this cast at early to zero as the amount of extrapolation, ⌬␥, in- the top and bottom (dashed lines) in order to successfully assign ␥n creases to 0.3 kg mϪ3. Of course, in each case, nor- values. The dotted lines show the two neutral surfaces through the bottles to the extended cast. malization is ensured by dividing by the sum of the ®nal weights. This smooth vertical interpolation procedure is used to ensure that if a series of very dense data are single multiplication, rather than with the many required encountered in the vicinity of a seamount or a conti- by alternate numerical integration techniques. nental shelf, the labeled values of neutral density will Despite the fact that we have added a seasonal mixed vary smoothly in the vertical as the number of casts layer and extended the Southern Ocean waters south of involved in its labeling vary from 4 to 3, ´´´, etc. 64ЊS to extremely dense water, the real ocean and model Once an unlabeled cast of hydrographic data has been output (in particular) will possess bottles with neutral assigned ␥n values by the above method, these values density values outside the range of ␥n in our global can be interpolated to ®nd the positions where speci®ed dataset. To successfully label these observations we ex- neutral density surfaces intersect the cast. This could be tend the casts involved on a needs basis at the top or done using simple linear interpolation; however, for ac- bottom of our global dataset. This extension is made curacy reasons we have implemented a straightforward just to a point where the extended cast neutrally com- quadratic technique that incorporates the nonlinear na- municates with the unlabeled bottle. At the surface, this ture of the equation of state of seawater. For the now- extension is made toward warmer water at the salinity n labeled cast of hydrographic data {(Sk, Tk, pk,),␥k kϭ of the top bottle on the cast, while at the ocean ¯oor it 1, 2, ´´´, ncast} and a given neutral density surface ␥n is in the direction given by R␳ ϭ ␣␪z/␤Sz ϭϪ1 (equal n n ϭ ␥␥Ã passing between bottles k and k ϩ 1 (i.e., kՅ contributions to the density gradient). Figure 8 shows ␥␥Ã nՅn ), ␥n in this interval is given by a typical cast that has required extension at the top in kϩ1 one case and at the bottom in another. A ␥n value is z now assigned to the unlabeled bottle by using the b value ␥nn(z)ϭ␥ϩb␳(␤SϪ␣␪ ) dz, (13) k ͵ zz from the very top or bottom of the unextended reference zk n cast, and the vertical integral of the relation ␥z ϭ b␳(␤Sz Ϫ ␣␪z) applied to the extension. This is straightforward where z is the depth level in the kth interval. A quadratic to do because the pressure of the new point on the cast structure can be given to ␥n(z) in this interval by taking is known (set equal to p ϭ 0 for a surface extension, b, Sz, and ␪z as constant over the interval, determined and equal to the bottom pressure of the cast for a bottom by the cast's ␥n and hydrographic values, and ␣ and ␤ extension), so that the vertical integral of ␳(␤Sz Ϫ ␣␪z) as known linear functions between the bottles. Equation is simply the difference in in situ density between the (14) then reduces to a quadratic new bottle on the cast and either the existing top or n 2 bottom bottle. ␥ (z) ϭ akz ϩ bkz ϩ ck, (14) The labeling strategy for an individual (S, T, p, lat, long) datum is as follows. First, the surrounding four where ak, bk, and ck are found in terms of the bottle casts of our reference dataset are located, and the basic hydrographic values by trivial algebra. The coef®cients n nn neutral calculation is made to each of these casts. Usu- are chosen to ensure that ␥ (z) is exactly␥␥kk andϩ1 at ally this gives four estimates of neutral density. If x and zkand zkϩ1 respectively. Equation (14) can then be sim- y represent the distances from the southwest reference ply and accurately solved to yield the point of inter- cast to the datum, normalized to the range [0, 1] cor- section of the ␥n ϭ ␥Ã n neutral surface with the arbitrary responding to the 4Њ longitude and latitude range, then cast. Corresponding values of S and T are found by the four weights for the four estimates of ␥n are initially linearly interpolating S and ␪ between the appropriate chosen as the linear interpolants, xy, x(1 Ϫ y), (1 Ϫ x)(1 bottle values.

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FIG. 9. Panel (a) shows a meridional section in the Indian Ocean FIG. 10. A meridional section in the Paci®c Ocean (a) along ap- along approximately 62ЊE taken from the Reid global hydrographic proximately 190ЊE taken from the Reid global hydrographic dataset. dataset. The solid lines represent the 26.8, 27.5, 27.95, and 28.1 The dashed lines represent the ␴0 ϭ 27.0, 27.5, 27.72, and 27.8 neutral surfaces obtained by interpolating the cast ␥n values after they potential density surfaces (referred to the sea surface), while the solid were labeled with neutral density, while the dashed lines represent lines represent ␥n surfaces corresponding to the average values of ␥n surfaces generated with the basic neutral surface calculation of section found on the ␴0 surfaces. Panel (b) shows the errors (in meters) 2 emanating from points on the ␥n surfaces near 23ЊS. Panel (b) shows between these two sets of surfaces. The two bottom surfaces have the errors (in meters) between these two sets of surfaces, the mean diverged to be 1438.9 m apart in the north. and maximum absolute differences being 5.9 and 61.1 m respectively.

a section from the Reid dataset using the accurate basic 6. Comparison of ␥n surfaces with present neutral surface calculation of section 2. Figure 9a shows isopycnal surfaces a meridional section near 62ЊE in the Indian Ocean, the To demonstrate the effectiveness of neutral density solid lines representing the ␥n surfaces for ␥n ϭ 26.8, in accurately representing neutral surfaces in the real 27.5, 27.95, and 28.1, and the dashed lines the surfaces ocean, we have sampled the Reid global dataset (Reid generated using the basic neutral surface calculation 1986, 1989, 1994) to obtain sections of hydrographic starting from initial points coincident with the ␥n sur- data that are different from the global dataset underlying faces near 23ЊS and which then proceed both north and the de®nition of ␥n. Since the Reid dataset is scattered south one cast at a time. In Fig. 9b we show the errors in spatial location, we have chosen these sections to be between these two sets of surfaces. The mean absolute as close as possible to a particular meridian of longitude, deviation between the two sets of surfaces over all sur- with a zonal resolution of about 4Њ of latitude. We have faces and all latitudes is 5.9 m, and the maximum ab- taken sections in each of the major ocean basins along solute deviation is 61.1 m, which to graphical accuracy which we compare neutral surfaces obtained by verti- makes them almost indistinguishable in the a panel. It cally interpolating ␥n, in the manner described in the should be noted that the ␥n surfaces are more likely to previous section, with surfaces calculated using several be the more accurate of the two sets of surfaces since alternative methods, including the present best isopycnal the use of the basic neutral surface calculation in this surfaces. Reid section is in a direction that is generally across the The ®rst comparison we make is between the ␥n neu- general circulation of the ocean. The ␥n ®eld has been tral surfaces (i.e., the neutral surfaces obtained by in- generated by averaging the initial ␥n ®eld over all lateral terpolating the cast ␥n values following labeling, as in directions, thereby reducing any path-dependent error the previous section) and the neutral surfaces ®tted to in the de®nition of a neutral surface. It is more important

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FIG. 11. A meridional section in the Atlantic Ocean (a) along ap- FIG. 12. A meridional section in the Atlantic Ocean (a) along approximately 330ЊE taken from the Reid global hydrographic dataset. proximately 330ЊE taken from the Reid global hydrographic dataset. The dashed lines represent the ®rst, third, ®fth, and ninth ``isopycnal'' The solid lines represent the ␥n surfaces of Fig. 11a, while the dashed surfaces de®ned in Reid (1994), while the solid lines represent the lines depict the neutral surfaces generated using the basic neutral closest ␥n surfaces corresponding to these isopycnals. Again, (b) surface calculation, emanating from a point near 40ЊS. Again, (b) shows the errors between the two sets of surfaces. The mean absolute shows the errors between the two sets of surfaces. The mean absolute deviation in depth is 15.4 m, while the maximum absolute deviation deviation in depth is 8.4 m, while the maximum absolute deviation is 179.8 m. is 42.6 m. to minimize the path-dependent error in the direction of the maximum absolute deviation in depth is 1438.9 m, the mean circulation than in the direction normal to the indicating the inadequacy of using ␴0 surfaces as ap- mean circulation. The vertical excursion between the proximating isopycnal surfaces. If potential density is two types of surfaces in Fig. 9 is then larger than that referred to a different pressure, say 4000 m, the situation applying in the physically more meaningful direction of is reversed, in the sense that these surfaces ®t neutral the mean circulation. Interpolation of ␥n using the sim- surfaces well at the bottom of the ocean but are inap- ple quadratic technique of the previous section is esti- propriate in surface waters. mated to cause an interpolation error of only 10Ϫ4 kg Figure 11 shows the most accurate method to date in mϪ3 and thus retains much of the accuracy of the basic ®tting isopycnal surfaces to hydrographic data. In the a neutral surface calculation. panel we have plotted surfaces 1, 3, 5, and 9 of Reid

In Fig. 10 we show how well ␴0 potential density (1994), based on potential density surfaces with a vary- surfaces perform in approximating the fundamental neu- ing reference pressure, on a meridional section near tral surface property (2) in the Paci®c Ocean. Here we 330ЊE in the Atlantic Ocean. These surfaces are depicted n have plotted the ␴0 ϭ 27.0, 27.5, 27.72, and 27.8 po- by the dashed lines, with the corresponding ␥ surfaces tential density surfaces, denoted by the dashed lines. (solid lines) being de®ned as those ␥n neutral surfaces For each of these surfaces we have found the mean value that minimize the mean absolute deviations (in terms of of ␥n on the surface, and plotted the corresponding neu- pressure) from the corresponding Reid isopycnals. In tral surface in the section (solid lines). It is clear that general, these isopycnal surfaces, although cumbersome n in surface waters the ␴0 surfaces approximate the ␥ to calculate, approximate the neutral surfaces well, al- surfaces quite well, but at depth there are considerable though in southern regions they can differ by nearly 200 discrepancies between the two. The mean absolute de- m. Again, we show the errors between the surfaces in viation between the two sets of surfaces is 72.3 m, while the b panel. The mean absolute deviation between the

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FIG. 13. Plots of the ®fth isopycnal from Reid (1994) in the North Atlantic Ocean. The distribution of ␥n (kg mϪ3)onthis isopycnal is shown in (a), while (b) shows the differences in pressure (db) between this isopycnal and a best ®t ␥n surface to this isopycnal. (c) The differences in pressure (db) between this best ®t ␥n surface and a neutral surface generated using the basic neutral surface calculation (as in section 4), emanating from a coincident point in the central Atlantic. The differences in the pressure scales between (b) and (c) indicate the improvement made by using ␥n surfaces over the isopycnal surfaces of Reid (1994) in approximating neutral surfaces. two sets of surfaces in Fig. 11 is 15.4 m with the max- present time these Reid isopycnal surfaces have only imum absolute deviation being 179.8 m. This cast was been de®ned for the North and South Atlantic and South typical of all the meridional sections in the Atlantic Paci®c Oceans. The ␥n surfaces are, on the other hand, Ocean. In Fig. 12a we show the neutral surfaces de®ned de®ned for the entire extent of the ocean excluding the using the basic neutral surface calculation of section 2 Arctic Ocean and the enclosed marginal seas. (dashed lines) for the four ␥n surfaces of Fig. 11a (solid In Fig. 13 we have concentrated on one particular lines) in the Atlantic. The mean absolute differences in isopycnal surface of Reid (1994), namely, his ®fth sur- depth between the two sets of surfaces is 8.4 m, while face (the third surface in Fig. 11a). Figure 13a shows the maximum absolute deviation is 42.6 m. Again, this the variation of ␥n on this isopycnal over the Atlantic ®gure demonstrates the accuracy of the ␥n surfaces in Ocean between 300ЊE and 0Њ, where a variation of 0.22 approximating the fundamental neutral surface property. kg mϪ3 in ␥n is evident. In Fig. 13b we show a map of While these deviations are much less than the best- the differences in pressure between the ``best ®t'' ␥n practice isopycnal approach of Reid (1994), the im- surface and this particular isopycnal. By best ®t we provement may well not be important for many ocean- again mean that ␥n surface that minimizes the mean ographic studies. Rather, we believe that our ␥n algo- absolute deviation from the Reid isopycnal. The mean rithm gives oceanographers a way of forming neutral absolute deviation of this ␥n neutral surface (where ␥n surfaces much more easily than is presently available ϭ 28.009) from the Reid isopycnal is 28.4 m over the by the isopycnal method of Reid. The extra accuracy region shown, with a maximum absolute deviation in (by a factor of about 5) is an added bonus. Also, at the the Weddell Sea of 328.8 m. To test which of these

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having triple crossings. This labeling exercise also allows us to quantify the speed of the assignment of ␥n values to hydrographic data. In the case of the Reid dataset, the average time taken for labeling each bottle of data was 12.7 msec, all computations being performed on a SUN SPARCstation 2. Finally, Brewer and Bradshaw (1975) have calculated the effect of variations of alkalinity, total carbon di- oxide, and silica content on the conductivity, salinity, and density of sea water. They estimate that the variations of these quantities can cause density differences of up to 10Ϫ2 kg mϪ3 between deep North Paci®c water and deep North Atlantic water compared with density values evaluated using existing algorithms. In the deep ocean this can amount to a vertical depth difference of 500 m. This issue obviously needs attention, and the international equation of state may need modi®cation. This would then directly affect the calculation of neutral density.

7. Summary In this paper we have de®ned a new density variable, neutral density, ␥n, which of necessity is a function of the ®ve variables S, T, p, latitude, and longitude. The level surfaces of ␥n form neutral surfaces, which are the widely accepted surfaces along which the strong lateral mixing occurs in the ocean, and across which the mixing is determined by the much smaller vertical (or dianeu- tral) diffusivity of small-scale mixing processes. As such, ␥n provides the most appropriate vertical coordinate for understanding the ocean circulation (e.g., see Hirst et al. 1996). Neutral density is de®ned in such a FIG.13.(Continued) way that it is the continuous analog of the discretely de®ned potential density surfaces that are the present surfaces is the more accurate in approximating the neu- best isopycnal surfaces. tral surface property, we have compared this ␥n ϭ The spatial dependence of neutral density is accom- 28.009 neutral surface with a neutral surface computed modated by linking the de®nition of ␥n with a global using the method of generating neutral surfaces in two hydrographic dataset. This dataset is based on the Levi- lateral directions (using the basic neutral surface cal- tus (1982) climatology of the world's oceans. An un- culation), as outlined in section 3. The surface was cho- fortunate aspect of the objective analysis that went into sen to emanate from the central location of (10ЊS, the making of the Levitus data is that nearly half of the 324ЊE) at the pressure of the ␥n ϭ 28.009 surface. Figure casts in the climatology contain vertical inversions of 13c shows the differences in pressure between these two local potential density at some point on the cast. To distinct neutral surfaces, the mean and maximum ab- rectify this situation we have minimally modi®ed these solute differences being 11.8 m and 63.3 m, respec- casts so that the resulting hydrography is statically sta- tively. The small sizes of these two errors, compared ble. In order to create a dataset with the widest possible with the magnitudes of the same errors associated with density range, we have also augmented the global da- Fig. 13b, again indicate the increased accuracy of using taset with summer water (taken from the seasonal Levi- ␥n surfaces in approximating neutral surfaces over the tus climatology) in the top 200 m of the water column present best isopycnal surfaces of Reid (1994). and have replaced Antarctic shelf waters with much With a view to testing the ability of the labeling pro- denser water taken from its near neighborhood. Further, cedure of section 5 to successfully assign ␥n values to we have extended all water south of 64ЊS to extremely arbitrary hydrographic data, we have labeled the global dense water, in order to ensure that we can label very dataset of Reid (1986, 1989, 1994), consisting of nearly dense Antarctic Bottom Water with neutral density. 6800 casts scattered worldwide. Of the dataset 99.29% The basic neutral surface calculation ®nds the point was successfully labeled with ␥n, those bottles not being of intersection of a neutral surface passing through a labeled having either extremely dense bottom water or particular bottle of data with a neighboring cast of hy-

Unauthenticated | Downloaded 09/28/21 11:32 AM UTC 254 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 27 drographic data. This point is found by solving a simple of state. However, once labeled, the cast can be used equation based on differences in local potential densities very ef®ciently for ®nding the depths of speci®c neu- using a Newton±Raphson technique. The errors asso- tral surfaces. ciated with this computation have been shown in ap- We have also made comparisons between the neutral n pendix A to be approximately (1/6)␳Tb⌬p⌬␪, where Tb surfaces formed from (i) the ␥ ®eld, (ii) neutral sur- is the thermobaric parameter 2.7 ϫ 10Ϫ8 (db)Ϫ1 KϪ1, and faces calculated from the basic neutral surface calcu- ⌬p and ⌬␪ are differences between the original bottle lation, and (iii) the present best isopycnal surfaces of and the point of intersection, the next highest error terms Reid (1994). Methods (i) and (ii) yield surfaces for being cubic in such differences. independent sections of hydrographic data that are al- We have de®ned neutral density via a simultaneous most indistinguishable when viewed graphically, dem- system of ®rst-order hyperbolic partial differential equa- onstrating the accuracy achieved by the ␥ n surfaces. tions. The solution of these equations provides us with The comparison between methods (i) and (iii) (the an initial ®eld of ␥n values for our global dataset. To mean absolute differences being of the order of tens solve these neutral surface partial differential equations of meters, and the maximum absolute deviations being we have used a combination of two numerical tech- of the order hundreds of meters) showed how well the niques: the method of characteristics and ®nite differ- Reid discretely referenced isopycnal surfaces perform ences. The ®rst of these yields the most accurate solution in satisfying the fundamental neutral surface property. for 84% of the global dataset, while the second inverts The use of the ␥ n variable for forming neutral surfaces for ␥n in the remaining 16% of the ocean. The ®nite improves on the accuracy of the present best method difference technique, based on three different grids, is of forming isopycnal surfaces, but more importantly it unusual in the sense that it produces an overdetermined does so in a manner that is very easily implemented. system of algebraic equations for the partial differential Software for labeling arbitrary hydrographic data equations. The ability of this procedure to successfully with ␥n and for ®nding the positions of speci®ed neutral invert potential density, ␴0, for our data validates the density surfaces within the water column is available accuracy of using this technique in the 16% of the ocean through the World Wide Web at http://www.ml.csiro.au/ requiring solution by a method different from the meth- ϳjackett/NeutralDensity. od of characteristics (␴0 was simply chosen because it is a well-de®ned function that has similar nonlinearities Acknowledgments. We wish to thank Drs. Nathan Bin- to the function we seek, ␥n). doff, Peter McIntosh, and Steve Rintoul for their con- To eliminate numerical errors associated with these structive comments on a draft of this paper. We thank solutions of the neutral surface differential equations, Dr Nathan Bindoff for supplying the WOCE data used and to distribute the path-dependent errors inherent in in Figure 4, and Dr Terry Joyce for providing the initial the de®nition of neutral surfaces, we have further it- impetus for the project. This work is part of the CSIRO erated the initial ␥ n ®eld. This was done globally and Climate Change Research Project. at all depths along the neutral surfaces, the problem's characteristic coordinate system. At the same time we APPENDIX A have imposed continuity and smoothness constraints on the vertical b pro®les, which we believe exist in a. Quanti®cation of helicity in terms of density the real ocean. In this way we have been able to gen- Substituting A ( S ) into (1), and noting ١␪␣ Ϫ ١␤ n ϭ ␳ erate a ␥ ®eld for our global dataset, which contains that both and are functions of (S, , p), helicity can n Ϫ4 Ϫ3 ␣ ␤ ␪ local ␥ errors of sizes 7 ϫ 10 kg m in the mean, be expressed as and 8 ϫ 10Ϫ3 kg mϪ3 maximum. These compare well 2 ١p ´ ١␪ with the present observational error in density of 5 ϫ H ϭ (␳0) ␤Tb١S ϫ Ϫ3 Ϫ3 ,␪ ١ p ϫ ١ ´ kg m . ϭ (␳ )2T ␣␪ k 10 Having successfully and accurately labeled our glob- 0 bz n n 3 Ϫ3 al dataset we are able to interpolate it, in terms of where ␳0 ϭ 10 kg m . From McDougall and Jackett spatial location and hydrography, to assign ␥ n values (1988), Eqs. (39) or (40), it is seen that the ambiguity to arbitrary hydrographic data. This we achieve by ®t- of a locally referenced potential density is given by the ting a local neutral surface between the datum in ques- areal integral in x±y space, on an approximately neutral ∫∫ ١npϫ١n␪´kdx dy. In terms of tion and the four nearest neighbors (in x±y space) in surface, of ␳0Tb the labeled dataset. These are then suitably combined helicity, this ambiguity in potential density is to form a single estimate of the bottle's ␥ n value. The ␥ n values produced by this procedure can then be in- H/(␳␣␪0 z)dx dy. terpolated using a simple quadratic technique to ®nd ͵͵ Ϫ4 Ϫ1 Ϫ3 Ϫ1 the positions of speci®ed neutral surfaces down an ar- Taking ␣ ϭ 2 ϫ 10 K and ␪z ϭ 3 ϫ 10 Km , bitrary cast of hydrographic data. The time taken to a uniform value of helicity of 10Ϫ17 kg2 mϪ9 over an label an external cast of hydrographic data can be ex- area of 100 km on a side would result in an ambiguity pensive, due to the many calls to the present equation in potential density of 0.003 kg mϪ3.

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(␪)´dl, (A2 ١␣ S Ϫ ١␤) ͵ nn n

where the integral is taken along a neutral trajectory between our two bottles.

Writing a Taylor series for both ␳(S1, ␪1, p1) and ␳(S2, ␪2, p2) about (S, ␪, p) and retaining terms up to third order, we ®nd

␳(S22, ␪ , p) Ϫ ␳(S 11, ␪ , p)

ϭ␳S⌬Sϩ␳␪⌬␪ 1 FIG. A1. Diagram illustrating the variation of ␪ and p along a neutral ϩ(␳(⌬␪)33ϩ␳(⌬S)) trajectory (the jagged line) when the variables are not proportional. 24 ␪␪␪ SSS The area on this p±␪ diagram between the data and the straight line is usually much less than the triangular area, 1/2⌬p⌬␪. 1 ϩ(␳⌬S(⌬␪)22ϩ␳(⌬S)⌬␪). (A3) 8 S␪␪ SS␪ b. Taylor series expansion for E Here all the partial derivatives are evaluated at (S, ␪, p p) Note that there are no quadratic terms in (A3). Di- Consider a ®xed bottle (SÄ, ␪Ä, pÄ) and a neighboring viding this equation by ␳(S, ␪,p ) we have cast de®ned by the continuous variables (S, ␪, p). The function E is now de®ned by [␳(S22, ␪ , p) Ϫ ␳(S 11, ␪ , p)]/␳(S, ␪, p) E ϭ ␳(S,␪, p) Ϫ ␳(SÄ, ␪Ä, p), (A1) ϭ␤(S,␪,p)⌬SϪ␣(S,␪,p)⌬␪ 1 where p ϭ (p ϩ pÄ)/2, and the aim of the algorithm is Ϫ(␣(⌬␪)33Ϫ␤(⌬S)) to ®nd positions on the cast where E ϭ 0. Differentiating 24 ␪␪ SS (A1) with respect to p down the cast we ®nd 1 22 Ϫ(␣S␪ ⌬S(⌬␪)ϩ␣SS(⌬S)⌬␪), (A4) Epppϭ ␳(S, ␪,p)[␤(S, ␪,p)S Ϫ␣(S, ␪,p)␪] 8 1 where it is readily shown that Ϫ␳ /␳ (ϵ ␣ Ϫ 2␣␣ ) ϩ[␳(S, ␪,p)␥(S, ␪,p)Ϫ␳(S,ÄÄ␪ÄÄ,p)␥(S, ␪,p)]. ␪␪␪ ␪␪ ␪ 2 is closely approximated by ␣␪␪ and similarly for other terms in (A4). Expanding ␳, ␣, ␤, and ␥ as a Taylor series about (S, Now we develop a corresponding series expression ␪,p), we then obtain for the exact integral (A2) along a neutral trajectory. This integral will depend on how S, ␪, and p vary along ␣ץ 1 Epppϭ ␳[␤S Ϫ ␣␪ ] ϩ ␳ (⌬p␪ pϪ⌬␪)ϩ´´´, the neutral trajectory from (S1, ␪1, p1)to(S2,␪2,p2). p Here we will initially assume that the variations of pץ2 where ␳, ␣, ␤ and all partial derivatives are evaluated and ␪ are proportional, while S varies in the manner at (S,␪, p), ⌬p ϭ p Ϫ⌬p,Ä ␪ϭ␪Ϫ␪Ä, and ⌬S ϭ S Ϫ that is required on a neutral trajectory. De®ning the SÄ, and only the largest two terms have been retained. perturbation quantities SЈϭSϪS,␪Јϭ␪Ϫ␪, and pЈ The leading term in the expansion is thus proportional ϭ p Ϫ p, we have to N2. ⌬p pЈϭ ␪Ј ⌬␪ c. Taylor series expansion of the neutral relationship E and Consider two bottles, (S1, ␪1, p1) and (S2, ␪2, p2)on 2 an exact neutral trajectory, separated by ⌬S ϭ S2 Ϫ S1, ⌬S ⌬␪ 2 ⌬␪ ϭ ␪2 Ϫ ␪1, and ⌬p ϭ p2 Ϫ p1. The separation in SЈϭ ␪ЈϪQ Ϫ(␪Ј) longitude and latitude is not relevant to the calculation. ⌬␪[]΂΃2 All Taylor series here will be expanded about the mean along the neutral trajectory, where Q is the quadratic values S ϭ (S ϩ S )/2, ␪ ϭ (␪ ϩ ␪ )/2, and p ϭ (p 1 2 1 2 1 coef®cient. Q is found by equating the slope dSЈ/d␪Јϭ ϩ p )/2. The neutral relationship 2 ⌬S/⌬␪ϩ2Q␪Јto be equal to the ratio of

E ϭ ␳(S2, ␪2, p) Ϫ ␳(S1, ␪1, p) ϭ 0 ␣(S, ␪,)ppϩ␣␪␪Јϩ␣SSЈϩ␣ppЈand ␤(S, ␪,)ϩ will be compared to the exact expression ␤␪␪Јϩ␤SSЈϩ␤ppЈat leading order. This gives

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␣␣2 ␪)´dl ١␣ ١nnS Ϫ␤) ␤2Q␤(S,␪,p)ϭ␣ϩ2␣Ϫ ␪ SS ͵ []␤␤΂΃ n ␣ ⌬p␣ ␪␪ 3 ϩ␣Ϫ␤ ϭ ␤(S, ␪, p)⌬S Ϫ ␣(S, ␪, p)⌬␪ Ϫ (⌬␪) ⌬␪␤[]pp 24 ⌬p ␤␣ ␣ p ␪ppp2 2 ϭCbbϩT, ϩQϪ(⌬␪)⌬pϪ⌬␪(⌬p) ⌬␪ ΂΃612 24 where the two brackets on the right-hand side are rec- ␣␣ Ϫ⌬S␪S(⌬␪)2 ϪSS (⌬S)2⌬␪ ognized as the caballing parameter, Cb, and the ther- 88 mobaric parameter, Tb, of McDougall (1991). The thermal expansion coef®cient is expanded about ␣␤SP Sp Ϫ⌬S⌬␪⌬pϩ(⌬S)2⌬p S, ␪,p ) as a Taylor series, retaining terms to second 612 order, giving ␤␤ ϩ⌬ppS(⌬p)2 ϩ SS (⌬S)3 . (A8) ␣(S, ␪,p)ϭ␣(S,␪,p)ϩ␣SSЈϩ␣␪␪ Јϩ␣ppЈ 24 24 111 2 2 2 We return to estimating the error in the ®nite-differ- ϩ␣SS(SЈ)ϩ␣␪␪(␪Ј)ϩ␣pp(pЈ) ∫ ١n␪)´dl ϭ 0. Since, by␣ ١nS Ϫ␤) ence form (A4) of n 222 ∫ -١n␪´dl is zero, (A8) can be sub␣ ١nS Ϫ␤) de®nition n ϩ␣S␪SЈ␪Јϩ␣Sp SЈpЈϩ␣␪␪p ЈpЈ. tracted from (A4) to give the error involved in using (A5) (A4) as the de®nition of a neutral trajectory. In this way, ∫ ⌬␪/2 ١n␪ ´dl is equal to ∫Ϫ⌬␪/2 ␣(S, ␪,p)d␪Ј, along an exact neutral trajectory, (A4) is not zero but is␣ The integral n and the only unusual term that occurs in performing this [␳(S , ␪ , p) Ϫ ␳(S , ␪ , p)]/␳(S, ␪, p) ⌬␪/2 22 11 integral is ␣S ∫Ϫ⌬␪/2 SЈ d␪Ј. This is performed using our quadratic expression for SЈ as a function of ␪Ј, obtaining ␣␤ 3 ␪pp 2 Ϫ1/6␣SQ(⌬␪) , noting that this is cubic in ⌬␪, not qua- ϭϪQ(⌬␪)⌬p dratic. The integral of (A5) is ΂΃12 6

␣␤Sp Sp ⌬␪/2 ␣␣ ϩ⌬S⌬␪⌬pϪ(⌬S)2⌬p ␣(S, ␪,p)d␪ϭ␣(S,␪,p)⌬␪ϩϪ␪␪ S Q(⌬␪)3 612 ͵ 24 6 Ϫ⌬␪/2 ΂΃ ␣␤ pp2 pp 2 ␣␣pp SS ϩ⌬␪(⌬p)ϩ⌬S(⌬p) . (A9) ϩ⌬␪(⌬p)2 ϩ(⌬S)2⌬␪ 24 24 24 24 ␣ Using our knowledge of ϩ ␪p (⌬␪)2⌬p 12 ⌬p ␣␣S Sp Q ϭ C ϩ T 2␤ , ϩ⌬␪S(⌬␪)2 ϩ⌬S⌬␪⌬p. bb⌬␪ 12 12 ΂[]΋΃ (A6) it is readily shown that this Q term in (A9) is only 10% of the (1/12)␣ (⌬␪)2⌬p term. The leading terms are the ␣ The S integral in (A2) is evaluated similarly (using dS ␪p ␪p and ␣pp terms. Using ⌬p ϭ 1000 db and ⌬␪ ϭ 1ЊC, (A9) ϭ⌬S/⌬␪d␪ϩ2Q␪Јd␪), giving Ϫ7 is approximately 2 ϫ 10 , implying an error in ␳(S2, ␪2, ⌬S/2 Ϫ4 Ϫ3 ␤ p) Ϫ ␳(S1, ␪1, p)of2ϫ10 kg m . This is a factor of ␤(S, ␪,p)ϭ␤(S,␪,p)⌬Sϩ ␪ Q(⌬␪)3 ͵ 6 10 less than the measurement precision of modern instru- Ϫ⌬S/2 mentation. It is also interesting to note that the cubic terms ␤␤ ϩ pQ(⌬␪)2⌬p ϩ⌬␪␪ S(⌬␪)2 in (A4) [all of which canceled with the same terms in (A8)] Ϫ7 3 624are also smaller than 10 [the largest is (1/24)␣␪␪(⌬␪) ␤␤ which is only 10Ϫ8 for ⌬␪ ϭ 1ЊC], so that the relationship ϩ⌬ppS(⌬p)2 ϩ SS (⌬S)3 24 24 ␣(S, ␪, p)⌬␪ Ϫ ␤(S, ␪, p)⌬S ϭ 0 is as accurate as the ␤␤ relation E ϭ 0 that we use to de®ne a neutral trajectory. ϩ⌬␪pSS⌬␪⌬pϩ␪(⌬S)2⌬␪ 12 12 ␤ d. Variation of potential density on a neutral ϩ Sp (⌬S)2⌬p. 12 trajectory

(A7) Potential density referred to any ®xed pressure, pr,is Taking the difference of (A6) and (A7), we ®nd (noting a function of only S and ␪ so that that ␤␪ ϭϪ␣S and ␤␪p ϭϪ␣Sp, ´´´, etc.) d␳␪ ϭ ␳␪[␤(S, ␪, pr) dS Ϫ ␣(S, ␪, pr)d␪].

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l 3 Along a neutral trajectory ␤(S, ␪, p)dS Ϫ ␣(S, ␪, p)d␪ is then 0.0128␣␪␪ b0␳ (⌬␪) , where ⌬␪ is the difference ϭ 0, so the variation of ␳␪ along a neutral trajectory is in potential density between the pair of bottles. The maximum value of ⌬␪ over all the data in our global ␣␣ dataset is 1.7ЊC, and taking ␣ 2 ϫ 10Ϫ7 KϪ3, this d␳ϭ␳␤(S, ␪,p)(S, ␪,p)Ϫ (S, ␪,p) d␪ ␪␪ ഠ ͵ ␪ ͵ ␪ rr␤␤ term gives a maximum error of 1.3 ϫ 10Ϫ5 kg mϪ3. The nn [] next term (the ␣␪ term) yields a maximum error of l 2 0.0256(b1⌬z)␳ (⌬␪) ␣␪. Now (b1⌬z) is the change in b 103T [p Ϫ p ] d␪ (kg mϪ3 ), ഠb ͵ r over this bottle pair, and this is taken to be 0.1, giving n a maximum cubic interpolation error of (using ␣␪ ഠ Ϫ5 Ϫ2 Ϫ5 Ϫ3 where we have recognized that the thermobaric param- 10 K ) 7.4 ϫ 10 kg m . The ␣p term is no larger eter Tb is equal to ␤(␣/␤)p, the ®rst term in the Taylor than this. expansion of the integrand has been retained, and ␳␪ has We conclude therefore that the cubic error in quadrati- been approximated by 103 kg mϪ3. cally interpolating ␥n between bottles is no larger than Following McDougall and Jackett (1988), this ex- 10Ϫ4 kg mϪ3. pression can be used to estimate the inherent ambiguity in ␥n around a closed circuit in x±y space. When tracking a neutral trajectory around a loop and back to the orig- APPENDIX B inal cast, the change in potential density along the loop Iterative improvement of the ␥n ®eld shows up as a vertical difference in potential density on this cast. This is equal to the vertical difference in ␥n In this appendix we take the initial ␥n ®eld, that is, divided by the local b value. From this the inherent the ␥n ®eld resulting from the solution of the differential ambiguity in ␥n along such a neutral trajectory is system (9), and iteratively smooth it in the neutral density coordinate system. This results in a ␥n ®eld in which the errors from all sources are signi®cantly better, on ⌬␥ n ഠ 103Tb⅜pd␪ (kg mϪ3 ), (A10) b ͵ average, than the present instrument error in density n 0.005 kg mϪ3. The basic idea is to average the ␥n ®eld where the integral is taken along the neutral trajectory, currently found at each bottle in the global dataset over which is closed by the excursion on the p±␪ diagram the values found locally on the neutral density surface up the cast between the ®rst and last point on the same through the particular bottle. The process is repeated cast. In terms of depth, (A10) becomes iteratively until it reaches a steady state. T For each bottle in the global dataset we form a ⌬z ഠ b ⅜ pd␪ (m). ``plate'' consisting of the bottle and the points of in- gNϪ12͵ n tersection of the neutral trajectories through that bottle with the casts immediately to the north, south, east, and west. The depths de®ning each plate need only be com- n e. Error in using a quadratic for ␥ between bottles puted once with the basic neutral surface calculation, Between the labeled bottles, neutral density is ac- since they depend on the hydrography of the data and n n curately given by vertically integrating ␥z ϭ b␳(␤Sz Ϫ not on the values of ␥ found on the neighboring casts. ␣␪z). Both Sz and ␪z are constant in this depth interval Since each plate locally de®nes the neutral surface pass- so that ing through the particular bottle, values of ␥n on the four ``satellite'' casts (in the north, south, east, and west zz directions) should be the same as the value of ␥n on the ␥nn(z)ϭ␥ϩ␳Sb␤dz Ϫ ␳␪ b␣ dz, kz͵z͵central cast. For the initial ␥n ®eld we have developed, zz kkthis is unfortunately found not to be the case. The errors where ␳ has been taken outside the integrals as its in ␥n on these plates are a manifestation of both the relative variation is small compared with those of b errors incurred during the numerical solutions of (9) and ␣. We concentrate on the second integral and ex- used to produce the ␥n ®eld, and the real path-dependent pand b as a linear function of z (b ϭ b0 ϩ b1zЈ where error inherent in the de®nition of a neutral surface. zЈϭzϪ[zk ϩ(1/2)⌬z]) and ␣ is a quadratic, ␣ ϭ ␣ In general, the satellite depths on the four neighboring 2 2 ϩ ␣␪␪zzЈϩ␣ppzzЈϩ(1/2)␣␪␪(␪z) (zЈ) ϩ ´´´. The terms casts de®ning each plate will fall between the standard n that are not captured by a quadratic expression for ␥ depths {zk, k ϭ 1, 2, ´´´, nz} of our global dataset. Here as a function of z are due to the vertical integrals nz ϭ 33 is the number of standard depths in the Levitus l 3 2 l 2 2 of Ϫ(1/2)␣␪␪b0␳ (␪z) (zЈ) , Ϫ␣␪b1␳ (␪z) (zЈ) and (1982) and the reference datasets. Consequently we need l 2 n Ϫ␣ppzb1␳ ␪z(zЈ) . When a linear trend is subtracted from to make estimates of the ␥ values at each of these plate 2 the vertical integral of (zЈ) between zЈϭϪ⌬z/2 and zЈ depths. This can be achieved by integrating b␳(␤Sz Ϫ n ϭϩ⌬z/2, it is found that the extreme values occur at ␣␪z) from a known ␥ value at one of the bottles on zЈϭϮ⌬z/12͙ , and these extreme values are each of the casts to the height of the plate on that cast. 3 Ϯ0.0256(⌬z) . The maximum error due to the ␣␪␪ term Considering a point of intersection of a plate with just

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cast n one of these satellite casts lying between standard depth where␥␥k,l is the cast estimate, { k,l,m , m ϭ 1, 2, ´´´, 4} n zk and zkϩ1, the value of ␥ we assign to this point is are the satellite estimates, r ⑀ [0, 1] is a suitable relax- z ation parameter, and nlk ϭ 4 for k Ͻ ncast and nlk ϭ 1 cast nn for k ϭ ncast. The␥k,l value for l ϭ 1 is the bottle ␥(z)ϭ␥k ϩb␳(␤SzzϪ␣␪ ) dz, (B1) ͵ value␥␥n , whilecast for l 1 and ␥n for m ϭ 1, 2, ´´´, zk k k,l,m± k,l,m 4 are all computed by evaluating (B3) on the appropriate n n where␥k are the bottle ␥ values. To complete this def- n cast using the current {(␥k , bk,0, bk,1), k ϭ 1, 2, ´´´, ncast} inition of ␥n(z) we require a knowledge of the vertical set of values. Notational convenience has necessitated pro®le of b down each cast. This is facilitated by letting the suppression of the n superscript in ␥n in several of b have a piecewise linear pro®le in the vertical, con- the equations in this section. strained to be continuous at the bottle depths. Speci®- n Once values of {(␥k , bk,0, bk,1), k ϭ 1, 2, ´´´, ncast} cally, if the value of z lies between zk and zkϩ1, k ⑀ {1, are known for every cast in the global dataset, the it- 2, ´´´, ncast} where ncast is the number of bottles on erative procedure consists of sequentially updating these the cast, we de®ne b as values, cast by cast, by minimizing

b(z) ϭ bk,0 ϩ bk,1(z Ϫ zk), ncast nlk ncastϪ1 n est 2 2 2 where b and b are constants and z ϭ (z ϩ z )/2. (␥ (zk,l) Ϫ ␥ k,l) ϩ 1.125 wb k k,1, (B5) k,0 k,1 k k kϩ1 ͸͸ ͸ The continuity constraints on the b pro®les are then kϭ1 lϭ1 kϭ1

bk,0 ϩ bk,1(zkϩ1 Ϫ zkk)/2 ϭ b ϩ1,0 Ϫ bkϩ1,1(zkϩ2 Ϫ zkϩ1)/2 where wk are weights described below and zk,l are the (B2) depths corresponding to the {k, l} index. This minimization was performed subject to the continuity con- for k ϭ 1, 2, ´´´, ncast Ϫ 2. straints (B2) on b, and For each cast in our global dataset we now have a parameterization of b that enables the vertical pro®le of zkϩ1 ␥n(z) to be completely evaluated down the cast. Ex- ␥nnϪ␥ϭ[bϩb(zϪzÅ)] kϩ1 k ͵ k,0 k,1 k plicitly, for z ⑀ [zk, zkϩ1], k ϭ 1, 2, ´´´, ncast Ϫ 1, this zk is given by ´ ␳(␤SzzϪ ␣␪ ) dz (B6) z ␥nn(z)ϭ␥ϩ[bϩb(zϪzÅ)] k ͵ k,0 k,1 k and zk

0.1 Յ bk,0 Ϯ bk,1(zkϩ1 Ϫ zk)/2 Յ 10, (B7) ´ ␳(␤SzzϪ ␣␪ ) dz. (B3) n where k ϭ 1, 2, ´´´, ncast Ϫ 1. The ␥n(z ) in (B5) are To update these bottle␥k values, and to ®nd values for k,l written in terms of the unknown coef®cients {(n , b , the constants {(bk,0, bk,1), k ϭ 1, 2, ´´´, ncast Ϫ 1}, we ␥k k,0 have added three additional neutral plates between each bk,1), k ϭ 1, 2, ´´´, ncast} using (B3), the integrals being bottle pair down the cast, making a total possible 129 (numerically) performed once for each {k, l} subinterval (4 ϫ 32 ϩ 1) plates for each cast. Again, the positions and stored for use on each iteration. Equation (B6) ex- n of these plates in the global dataset need only be com- presses a compatibility condition between the {␥k , k ϭ puted once and stored for subsequent use. It is worth 1, 2, ´´´, ncast} values and the piecewise linear pro®le noting that the degree of stabilization adopted in section of b, while (B7) ensures that b at either end of the kth 3 for ensuring our dataset was everywhere statically interval (and hence b over the entire interval since b is stable was more than adequate, in the sense that none linear over this range) is contained in the interval [0.1, of the plates on any cast in our dataset intersected each 10.0]. It should be noted that, although the constraints other, and there was no occurrence of triple crossings (B7) may have been hard constraints in the initial it- encountered during their generation. erations of the relaxation procedure, they were not active Each plate on any one cast can now have ®ve esti- in the ®nal b ®eld. mates made for the true value of ␥n on the plate: one The minimization of (B5) subject to (B2), (B6), and at the central cast and one at each of the satellite casts. (B7) can be achieved by the use of standard software These can be combined to form a single estimate of ␥n for the solution of constrained weighted least squares for the plate. Labeling these plates with indices {k, l}, problems (e.g., see IMSL 1991). As with the Gauss± where k ϭ 1, 2, ´´´, ncast is bottle number, and l ϭ 1, Seidel method of solution of Laplace's equation, con-

2, ´´´, nlk is an index (nlk is usually 4) for the internal vergence of the iterative procedure is greatly improved n plates corresponding to successively increasing depths if the new values of {(␥k , bk,0, bk,1), k ϭ 1, 2, ´´´, ncast} from (and including) bottle number k to the next deepest for a cast are used as soon as they become available for est bottle, we take as the plate estimate the evaluation of␥k,l in (B4) for any cast processed later

4 in the same iteration. An initial set of {(bk,0, bk,1), k ϭ ␥ estϭ (1 Ϫ r)␥ cast ϩ r ␥ n 4, (B4) 1, 2, ´´´, ncast Ϫ 1} values was found by running one k,l k,l͸ k,l,m ΂΃mϭ1 ΋ iteration of the procedure using only a single sum in

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nn FIG. B1. Map of local neutral surface differences of ␥␥maxϪ min after 1 iteration (a) and 50 iterations (b) of the relaxation procedure. All neutral density errors (kg mϪ3) are calculated on neutral surface 27.8, where neutral trajectories to each of the neighboring four casts have been ®tted to the surface at each point in the global dataset. the ®rst term of (B5) (the sum over bottle number k), the amount of vertical structure in the b ®eld. The est n with␥k,1 values being provided by the initial ␥ ®eld of weights, wk, for k ϭ 1, 2, ´´´, ncast Ϫ 1, are chosen to section 4, for k ϭ 1, 2, ´´´, ncast. be (1/8)⌬zk[␳(␤⌬S Ϫ ␣⌬␪)]k because for given values n n The objective function (B5) is essentially a trade-off of␥k at the ncast bottles, bk,1 affects the ␥ (z) at mid- n between (i) the sum of deviations of ␥ from an estimate depth (zk) between the bottles by the amount (1/8) n of the ␥ plate values at the 4 ϫ ncast Ϫ 3 neutral plates bk,1⌬zk[␳(␤⌬S Ϫ ␣⌬␪)]k.At¼and ¾ of the depths n down a cast and (ii) the sizes of the slopes of the piece- between the bottles, bk,1 affects the calculated ␥ (z)to wise linear pro®les of b. In this way a certain error is the extent (1/32)bk,1⌬zk[␳(␤⌬S Ϫ ␣⌬␪)]k, while of allowed in the neutral density ®eld in order to reduce course, the values of ␥n at the bottles are not affected

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∫ Ϫ3 FIG. B2. Map of ␳Tbbpd⅙ ␪(kg m ) from the hydrographic data (a), and the cumulative distribution nn functions (b), of this error (dashed line) and local neutral surface differences of␥␥maxϪ min (solid line). All neutral density errors (kg mϪ3) are calculated on neutral surface 27.8, where local plates have been ®tted to the surface at each point in the global dataset.

n at all. If bk,1 affected all the ␥ values equally, then the we have ®tted a plate as described above, and have used coef®cient outside the second term in (B5) would be 4 as an estimate of the ␥n error on this plate the value of n nn n as there are four ␥ terms for each bk,1 term. This factor ␥␥maxϪ min for the ®ve values of ␥ found there. The of 4 is downweighted by [0 ϩ (¼)2 ϩ 1 ϩ (¼)2]/4, grayscale in the ®gure varies from 0 to 0.019 kg mϪ3, becoming the 1.125 shown. This minimization proce- with large (though not the largest) values of this error dure has the effect of achieving continuous and smooth estimate being evident at the arti®cial barrier imposed vertical pro®les of b without incurring undue extra ␥n at Drake Passage in creating the initial ®eld. As the n errors. iteration procedure progresses, the values of␥k , k ϭ 1, Figure B1a shows a measure of the error in ␥n on the 2, ´´´, ncast on the casts settle down to nearly constant 27.8 neutral surface for each cast in our global dataset values, with the initial error on the plates being after the single iteration of the above procedure that smoothed out over the entire globe. Figure B1b shows determines the initial values of {(bk,0, bk,1), k ϭ 1, 2, ´´´, the same error estimates as those in Fig. B1a on the ncast}. At the depth of this neutral surface on each cast same neutral surface after 50 iterations of the relaxation

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nn FIG. B3. Map of the ␥␥maxϪ min errors on the plates (a), and the cumulative distribution functions ∫ (b), of this error (solid line) and ␳Tbbpd⅙ ␪errors (dashed line) associated with the neutral plates. All neutral density errors are for the entire ocean volume, where the map in (a) represents the maximum value down all plates of each cast in the dataset, and the distributions in (b) are over all plates in the ocean.

procedure, and it is clear the large initial errors at the l ϭ 1, 2, ´´´, nlk), for various choices of the relaxation Drake Passage barrier have dissipated. However, even parameter r, corresponds exactly with several of the though the ␥n ®eld has largely settled down after nearly numerical schemes [Jacobi, Gauss±Seidel, or successive 50 iterations, there are several regions in the global overrelaxation (SOR) techniques, see Dahlquist et al. dataset where sizeable plate errors occur on this partic- (1974)] used to iteratively ®nd the numerical solution ular neutral surface. of Laplace's equation. At the boundaries these schemes In order to reduce the sizes of the ␥n errors on the pay special attention to the boundary conditions of the plates, we have examined the numerical details of the underlying physical problem. In our case, these would n n iteration process for regions of possible production of be no ¯ux boundary conditions on ␥ , explicitly ␥x n n neutral density. One such source of ␥ generation can ϭϭ␥y 0 at the ocean edges. Accordingly, we have be identi®ed by noticing that the relaxation average (B4) introduced these boundary conditions into the plate es- est est used to ®nd the␥k,l estimates (k ϭ 1, 2, ´´´, ncast and timate (B4) by de®ning missing␥k,l,m values, for m ϭ

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1, 2, ´´´, 4 for any cast on the boundary as those found that 95% of the plates over the entire ocean have errors on the opposite cast on the plate, for k ϭ 1, 2, ´´´, ncast in the ␥n ®eld well below the present instrumentation Ϫ3 and l ϭ 1, 2, ´´´, nlk. It is worth noting that when r ± error (ϳ0.005 kg m ) associated with density. The nn 1, r is precisely the relaxation parameter in the SOR mean error given by ␥␥maxϪ min over all plates in the method above. However, unlike the signi®cant improve- global dataset is 9.1 ϫ 10Ϫ4. ment made in the convergence of the SOR method for appropriate choices of r Ͼ 1, all choices of r in this range led to unstable numerical results. This is almost REFERENCES certainly due to the interpolation error involved in computing the four ␥n values on the satellite casts on each Bleck, R., C. Rooth, D. Hu, and L. T. Smith, 1992: Salinity-driven thermohaline transients in a wind- and thermohaline-forced iso- plate using only a piecewise linear pro®le for b. pycnic coordinate model of the North Atlantic. J. Phys. Ocean- The effect of this no ¯ux boundary condition on the ogr., 22, 1486±1505. averaging scheme (B4) was to reduce the plate errors Brewer, P. G., and A. Bradshaw, 1975: The effect of the non-ideal on the neutral surfaces but, not to any great extent, errors composition of sea water on salinity and density. J. Mar. Res., 33, 157±175. still remaining on the plates in our global dataset. An- Carrier, G. F., and C. E. Pearson, 1976: Partial Differential Equa- other source of these plate errors is the inherent path- tionsÐTheory and Technique. Academic Press, 320 pp. dependent error associated with the neutral surface cal- Dahlquist, A., A. Bjorck, and N. Anderson, 1974: Numerical Meth- culations made in de®ning each plate. To quantify the ods. Prentice-Hall, 573 pp. extent of this error in the global dataset, we return to Foster, T. D., and E. C. Carmack, 1976: Frontal zone mixing and Antarctic Bottom Water formation in the southern Weddell Sea. the quite accurate estimates we have of this error in Deep-Sea Res., 23, 301±317. terms of the hydrography, previously discussed in sec- Gill, A. E., 1982: Atmosphere±Ocean Dynamics. Academic Press, tions 2 and 3. For each plate involving ®ve casts in our 662 pp. dataset, we ®nd the ⅙∫ pd␪area associated with a closed Godfrey, J. S., and T. J. Golding, 1981: The Sverdrup relation in the Indian Ocean, and the effect of Paci®c±Indian Ocean through- neutral circuit around the four satellite casts, com- ¯ow on Indian Ocean circulation and on the East Australian mencing at the (S, ␪, p) point found on the neutral sur- Current. J. Phys. Oceanogr., 11, 771±779. face on the eastern side of the plate. From appendix A Hirst, A. C., and C. Wenju, 1994: Sensitivity of a World Ocean GCM we know that for local values of b and ␳, and the ther- to changes in subsurface mixing parameterization. J. Phys. mobaric parameter T ϭ 2.7 ϫ 10Ϫ8 dbϪ1 KϪ1, bT ␳ ∫⅙ Oceanogr., 24, 1256±1279. b b ÐÐ, D. R. Jackett, and T. J. McDougall, 1996: The meridional pd␪corresponds to the error in neutral density as we overturning cells of a World Ocean model in neutral density follow the closed neutral trajectory around the edge of coordinates. J. Phys. Oceanogr., 26, 775±791. the plate. In Fig. B2a we have plotted this error from IMSL, 1991: Fortran Subroutines for Mathematical Applications. the hydrographic data for the 27.8 neutral surface over IMSL Inc., Houston, Texas, 1372 pp. Ivers, W. D., 1975: The deep circulation in the northern North At- the entire globe, where it is evident that there are regions lantic, with especial reference to the Labrador Sea. Ph.D. thesis, n of the ocean where the helical error in ␥ is of the same Scripps Institute of Oceanography, University of California, San order as the ␥n errors we are experiencing on the plates. Diego, 179 pp. Another way of looking at this is to form the cumulative Jackett, D. R., and T. J. McDougall, 1995: Minimal adjustment of frequency distributions of these two ␥n errors for all the hydrographic data to achieve static stability. J. Atmos. Oceanic Technol., 12, 381±389. data on the 27.8 neutral surface. Figure B2b shows these Killworth, P. D., 1986: A Bernoulli inverse method for determining two distributions, the dashed line representing the he- the ocean circulation. J. Phys. Oceanogr., 16, 2031±2051. lical errors of ␥n from the hydrographic data and the Levitus, S., 1982: Climatological Atlas of the World Ocean. NOAA solid line the ␥n errors associated with the ␥n ®eld on Prof. Paper No. 13, U.S. Govt. Printing Of®ce, 173 pp. Lilly, D. K., 1986: The structure, energetics and propagation of ro- the plates on this neutral surface after 50 iterations of tating convective storms. Part II: Helicity and storm stabilization. the procedure. In practice we ®nd that our worst values J. Atmos. Sci., 43, 126±140. nn of ␥␥maxϪ min are about one-half of the largest value Lozier, M. S., M. S. McCartney, and W. B. Owens, 1994: Anomalous ∫ anomalies in averaged hydrographic data. J. Phys. Oceanogr., of bTb␳ ⅙ pd␪on the plates. Figure B3 shows the ␥␥nnϪ errors over the entire 24, 2624±2638. max min Lynn, R. J., and J. L. Reid, 1968: Characteristics and circulation of ocean volume after 100 iterations of this relaxation tech- deep and abyssal waters. Deep-Sea Res., 15, 577±598. nique. Figure B3a shows these errors over all plates in McCartney, M. S., 1982: The subtropical recirculation of mode wa- our global dataset, the grayscale corresponding to the ters. J. Mar. Res., 40 (Suppl.), 427±464. maximum ␥n plate error over all (potentially 129) plates McDougall, T. J., 1987a: Neutral surfaces. J. Phys. Oceanogr., 17, 1950±1964. down each cast. In Fig. B3b we have plotted the two ÐÐ, 1987b: Thermobaricity, cabbeling and water mass conversion. n cumulative frequency distributions of the two ␥ errors J. Geophys. Res., 92, 5448±5464. nn ∫ (i.e., the ␥␥maxϪ min errors and the ␳bTb ⅙pd␪errors ÐÐ, 1988: Neutral surface potential vorticity. Progress in Ocean- on the plates) over all plates in our global dataset, the ography, Vol. 20, Pergamon, 185±221. two lines representing the same errors as in Fig. B2b. ÐÐ, 1991: Water mass analysis with three conservative variables. J. Geophys. Res., 96, 8687±8693. From this ®gure it is clear that we have achieved with ÐÐ, and D. R. Jackett, 1988: On the helical nature of neutral sur- n this iteration procedure the levels of error in ␥ that are faces. Progress in Oceanography, Vol. 20, Pergamon, 153±183. contained in the hydrographic data. Further, it is evident Montgomery, R. B., 1938: Circulation in the upper layers of the

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