## 1. Introduction

The role of “density” surfaces in models of the ocean is fundamental. These models range from the analysis of hydrographic data on the most appropriate mixing surfaces (Reid 1986, 1989, 1994; McCartney 1982) to the inversion of hydrographic data to obtain ocean circulation and mixing (Schott and Stommel 1978; Wunsch 1978; Killworth 1986; Zhang and Hogg 1992) to prognostic “level” or “layer” models (Bleck et al. 1992; Hirst and Wenju 1994), which integrate forward in time the basic momentum and tracer conservation equations. All of these models have as their goal a description of the oceanic circulation and all use the concept of mixing surfaces, between or along which the underlying physics governing oceanic flow is analyzed. While prognostic level models can adequately cope with these mixing surfaces (they only require the direction of the gradient of the surfaces in order to align the mixing tensor), inverse models are faced with the problem of finding the actual positions of the surfaces in the model domain. Even in the case of the prognostic level models, validation of these models can only be made by comparison with data from the real ocean. It must be borne in mind that the strong lateral mixing of these prognostic models occurs along neutral surfaces. As such, maps of properties along neutral surfaces and the heights of neutral surfaces are the most relevant variables from the real ocean to be compared with the model output.

Historically, the first surfaces used for oceanic models were in situ density surfaces. Wust (1933) and Montgomery (1938) first realized the inadequacy of these surfaces for describing the circulation even at shallow depth, and they proposed a new variable, potential density, for quantitatively describing the isopycnal spreading of water masses in the ocean. Despite the widespread acceptance of this isopycnal mixing, deficiencies in potential density surfaces for accurately describing precise isopycnal flow have been known for some time. For example, McDougall (1987a) shows a surface across the North Atlantic Ocean where potential density and neutral surfaces diverge by 1500 m. Lynn and Reid (1968), Reid and Lynn (1971), and Ivers (1975) have all used a series of potential density surfaces, referred to three different reference pressures, namely, 0, 2000, and 4000 db. In more recent work, Reid (1994) has found it necessary when describing the geostrophic circulation of the Atlantic Ocean to further refine the definition of isopycnal surfaces by using potential density surfaces referred to a range of reference pressures, with increments in some regions of as little as 500 db.

The reduction of the errors associated with a single reference pressure by the introduction of a range of reference pressures makes the actual computation of the isopycnal surfaces difficult. Reid (1989, 1994), for example, has defined 10 isopycnal surfaces in the North and South Atlantic Oceans with 88 different *σ*_{n} values, where *n* = 0, 0.5, 1, 1.5, 2, 3, 4, and 5 (corresponding to reference pressures up to 5000 db), depending on the spatial location of the water involved (there being 11 different spatial regions). The choice of the number of these *σ*_{n} values and the definitions of the corresponding regions is subjective. The assignment of *σ*_{n} values is based on a match between differing *σ*_{n} values as one progresses from one spatial region to the next. When following such an “isopycnal” surface from one reference pressure to another (say from a given *σ*_{1} value to a desired *σ*_{2} value), data at the mid pressure (1500 db in this case) from all the casts in the region is plotted on a *σ*_{1} versus *σ*_{2} diagram and a straight line is fitted by least squares. The desired *σ*_{2} value is then found as the value on the straight line at the specified *σ*_{1} value. The addition of further isopycnal surfaces requires the choice of additional *σ*_{n} values and regions based on this matching procedure. The computation of these isopycnal surfaces for a range of reference pressures is both time consuming and far from straightforward. Also, there are nearly always discontinuities in the slopes of these isopycnal surfaces across the defining regions.

In this paper we define a new density variable, neutral density, denoted by *γ*^{n}, which is a function of salinity *S* (psu), in situ temperature *T* (°C), pressure *p* (db), longitude, and latitude. Surfaces of constant *γ*^{n} define the neutral surfaces, which provide the proper framework for an ocean model’s calculations and analysis. These neutral density surfaces are essentially a continuous analog of the discrete potential density surfaces referred to various pressures (Reid’s isopycnals), which is the current best practice method of quantitatively describing isopycnal mixing.

The computation of *γ*^{n} for particular hydrographic data uses an accurately prelabeled global dataset, and much of the present paper is concerned with the construction of this labeled dataset. Labeling of arbitrary hydrographic data is achieved by making basic neutral surface calculations from the observation to the four nearest neighbor casts (in terms of latitude and longitude) in the dataset.

Early work on neutral surfaces was done by Pingree (1972), Ivers (1975), and Foster and Carmack (1976), while McDougall (1987a) made a detailed analysis of the properties of these surfaces. These surfaces were defined by McDougall in terms of gradients of salinity *S* and potential temperature *θ* in the neutral surface, while in McDougall and Jackett (1988) it was shown that the normal to these surfaces is in the direction of *β*∇*S* − *α*∇*θ*, where *α* and *β* are the thermal expansion and saline contraction coefficients, respectively. It is this latter property that motivates the definition of neutral surfaces that is most useful for our purposes, namely, that they are the surfaces everywhere perpendicular to the vector *ρ*(*β*∇*S* − *α*∇*θ*). It is important to note that the *z* component of *ρ*(*β*∇*S* − *α*∇*θ*) is simply −g^{−1}*ρ**N*^{2}, where *g* is the acceleration due to gravity and *N* is the buoyancy or Brunt–Väisällä frequency.

*ρ*(

*β*∇

*S*−

*α*∇

*θ*) in the real ocean is a well- defined function of three-dimensional space, the existence of neutral tangent planes as the planes perpendicular to

*ρ*(

*β*∇

*S*−

*α*∇

*θ*) is guaranteed. Unfortunately, the envelope of all such tangent planes is not a mathematically well-defined surface. For such a surface to exist,

**A**=

*ρ*(

*β*∇

*S*−

*α*∇

*θ*) must satisfy the condition of integrability (Phillips 1956), that is, its helicity (Lilly 1986)

*H*must be zero;

*H*

**A**

**A**

*b*=

*b*(

*x, y, z*) with the property that

*b*

*ρ*(

*β*∇

*S*−

*α*∇

*θ*) is irrotational [see also Eqs. (10) and (11) of McDougall and Jackett 1988]. Under these conditions, a scalar potential

*γ*

^{n}exists that satisfies

*γ*

^{n}

*b*

*ρ*

*β*

*S*

*α*

*θ*

*ρ*(

*β*∇

*S*−

*α*∇

*θ*), (ii) zero helicity [as defined in (1)], and (iii) the existence of the scalar potential

*γ*

^{n}and integrating factor

*b*satisfying (2) are all equivalent.

Another way of looking at the above definition of neutral density is to apply the classical result from vector analysis known as Helmholtz’s theorem (see Phillips 1956). This states that an arbitrary vector, **V,** can be expressed as the sum of two vectors, one of which is irrotational, the other solenoidal. That is, for any vector field **V** there are two other vector fields ∇*ϕ* and ∇ × **W** satisfying **V** = ∇*ϕ* + ∇ × **W**. With our definition of *γ*^{n} we are attempting to find a scalar function, *b,* of space for which the Helmholtz decomposition of **V** = *b**ρ*(*β*∇*S* − *α*∇*θ*) consists of only the irrotational part ∇*γ*^{n}. The degree to which this is achievable is determined by the (small) size of the helicity *H.*

In the real ocean *H* does not generally satisfy (1) exactly, despite that its magnitude is small. In Fig. 1 we have plotted the frequency distribution of helicity *H* for the global dataset developed in section 3 derived from the Levitus (1982) oceanographic atlas of the World Ocean. From this figure it is clear that 95% of the helicity values of the ocean lie between ±1.8 × 10^{−17} kg^{2} m^{−9}, with extreme values being of the order ±7.0 × 10^{−16} kg^{2} m^{−9}. We show in appendix A that a uniform helicity of 10^{−17} kg^{2} m^{−9} over a square area of ocean 100 km on a side would give an ambiguity in density of 0.003 kg m^{−3}. This ambiguity is just below the present instrumentation error in density.

The “path dependent” nature of neutral surfaces—that is, the phenomenon of tracing a neutral trajectory around an ocean basin and returning to the original longitude and latitude at a depth different to that of the starting point—has been recognized for some time (see Reid and Lynn 1971; McDougall 1987a; McDougall and Jackett 1988; and Theodorou 1991). McDougall (1987a) found this error to be only about 10 m for a neutral“helix” around the main gyre in the North Atlantic Ocean, while Theodorou (1991) found it to be only 4 m around a much smaller closed trajectory in the Ionian Sea. In McDougall and Jackett (1988) the pitch of this neutral helix has been quantified in terms of the hydrographic properties of the water masses on the closed trajectories, with the four examples of neutral surfaces in the North Atlantic all having depth errors of less than 10 m. In sections 3 and 4 of this paper we find that the size of this error over local scales is of the order of meters and over global scales of the order of tens of meters. The small size of this helical-path-dependent error, coupled with the small size of the helicity values *H* for the ocean in Fig. 1, provides the motivation for searching for a well-defined neutral density variable, *γ*^{n}, which locally nearly possesses the neutral surface property (2).

Throughout this paper we use the word “cast” to denote a vertical profile of hydrographic data and “bottle” to signify data taken from a particular depth. Although this nomenclature is not usually applied to averaged data such as Levitus (1982) or the dataset developed in this paper, it does serve as a succinct way of describing hydrographic data.

In section 2 we present an accurate method for finding 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 *p* *d**θ*. Section 3 introduces a global dataset that facilitates the definition of neutral density *γ*^{n}. This dataset is derived from the Levitus climatology but has vertical stability constraints imposed and modifications made to the Antarctic shelf waters. In section 4 we solve the simultaneous system of differential equations defined by (2) using a variety of numerical techniques and so obtain an initial field of *γ*^{n} values for our global dataset. The distribution of the errors accumulated during this solution procedure are then smoothed out over the entire ocean through an iterative improvement to the initial *γ*^{n} field. The global dataset therefore has had a field of *γ*^{n} values assigned to it, and the labeling of arbitrary hydrographic data can now be achieved by making neutral excursions to the four closest casts of the labeled dataset, where an accurate estimate of its *γ*^{n} value is available. Section 5 details the actual method of labeling arbitrary hydrographic data with *γ*^{n}, while section 6 compares the resulting neutral density surfaces with the current isopycnal surfaces.

## 2. The basic neutral surface calculation

*S̃, T̃, p̃*) intersects a neighboring cast of hydrographic data. The cast is defined by

*n*bottles {(

*S*

_{k},

*T*

_{k},

*p*

_{k}),

*k*= 1, 2, ···,

*n*} but is a continuous function of

*p*or

*z.*In order to develop an accurate method for finding this point, we introduce the neutral surface gradient operator, ∇

_{n}, which is defined by taking derivatives of variables in neutral surfaces. More specifically, the components of ∇

_{n}

*ϕ*for a scalar function,

*ϕ*, such as ∂

*ϕ*/∂

*x*|

_{n}, are defined as

*δ*

*ϕ*is evaluated in the neutral surface and the distance

*δx*is evaluated in the horizontal geopotential plane. The conventional three-dimensional gradient operator ∇ is related to ∇

_{n}(e.g., see Gill 1982 or McDougall and Jackett 1988) by

**m**is a vector normal 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}

*γ*

^{n}= 0 is identical to the solution of

*ρ*(

*β*∇

_{n}

*S*−

*α*∇

_{n}

*θ*) = 0. This observation is simply 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.

*S̃, T̃, p̃*) and the cast {(

*S*

_{k},

*T*

_{k},

*p*

_{k}),

*k*= 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 {(

*S*

_{k},

*T*

_{k},

*p*

_{k}),

*k*= 1, 2, ···,

*n*} on the same neutral surface as the bottle (

*S̃, T̃, p̃*) is that point which satisfies

*E*

*p*

*ρ*

*S,*

*θ*

*p*

*ρ*

*S̃,*

*θ̃*

*p*

*θ*and

*θ̃*

*p*

*p*+

*p̃*)/2. That is, the two parcels (

*S, T, p*) and (

*S̃, T̃, p̃*) 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 written using the International Equation of State,

*ρ̂*

*S, T, p*), as

*E*(

*p*) that we solve for the cast values (

*S, T, p*).

The method then consists of forming values of *E*_{k} = *E*(*p*_{k}) for each of the bottles (*S*_{k}, *T*_{k}, *p*_{k}), *k* = 1, 2, ···, *n,* on the cast, and identifying the intervals [*p*_{k}, *p*_{k+1}] over which the sign of *E*(*p*) changes. In each of these intervals a Newton–Raphson technique is employed to accurately find the positions of (*S, T, p*) on the cast that satisfy *E*(*p*) = 0 to within prescribed tolerances (5 × 10^{−3} m in depth and 5 × 10^{−5} kg m^{−3} in *E*). Should the Newton–Raphson scheme fail to converge for a particular interval (as in the case of a double zero), a simple interval halving technique is adopted with the same error tolerances.

In the vast majority of cases, a single point (*S, T, p*) is identified on the cast as being neutrally connected to (*S̃, T̃, p̃*). However, in some cases we are faced with multiple solutions of *E*(*p*) due to “triple crossings” of the *E*(*p*) function with the *p* axis. These occurrences are real and predominantly occur in the Southern Ocean. Figure 2 shows the salinity–potential temperature diagram and the *E*(*p*) function for the bottle (34.4 psu, −1.0°C, 2750 db) and a cast in the Southern Ocean where one of these triple crossings has occurred. The solid, dotted, and dashed lines in Fig. 2a show the three potential density surfaces through the bottle corresponding to the three solutions of (6) at 192.4 db, 229.4 db, and 522.6 db. The dotted line has a potential density of 34.62 kg m^{−3} corresponding to a midpressure value of 1471.2 db, while the solid and dashed lines have corresponding values (34.7 kg m^{−3}, 1489.7 db) and (35.4 kg m^{−3}, 1636.3 db). The choice of which of these solutions we take as the solution of *E*(*p*) = 0 is a difficult one to make and is the subject of work in progress. In Fig. 2b we have shown the *E*(*p*) function for this bottle/cast combination. It is clear that the multiple solutions of *E*(*p*) = 0 can only occur when there are reversals in the signs of *E*_{p}.

A Taylor series expansion of *E*_{p} about the point (*S, T, p*) reveals a leading term proportional to *N*^{2} at the cast, the constant of proportionality being positive. The derivation of this Taylor series can be found in appendix A. Thus, the occurrence of triple crossings of *E*(*p*), a direct consequence of a reversal in the sign of *E*_{p}, is very closely related to the static stability of the cast {(*S*_{k}, *T*_{k}, *p*_{k}), *k* = 1, 2, ···, *n*}. Anticipating the global dataset we develop in section 3, we have completely circumvented these triple crossing problems in all the basic neutral surface calculations we have performed in this paper by imposing realistic stability constraints on the cast data making up our global dataset. We still, however, detect the existence of these triple crossings for arbitrary hydrographic data, and when such solutions occur, we do not attempt (at this stage) to make an objective choice between them: the neutral surface calculation returns with a special pressure value indicating this situation.

*E*=

*ρ*(

*S,*

*θ*,

*p*

*ρ*(

*S̃,*

*θ̃*

*p*

*E,*is also examined in appendix A, where it is found that when the variations of pressure and potential temperature on a neutral surface are proportional, the error in

*E*is cubic in the differences Δ

*S*=

*S*−

*S̃,*Δ

*T*=

*T*−

*T̃,*and Δ

*p*=

*p*−

*p̃.*For the values of Δ

*S,*Δ

*T,*and Δ

*p*that are encountered over a 4° extent of longitude or latitude (say 0.3 psu, 1.5°C, 1000 db) these cubic errors are quite small (typically 2 × 10

^{−4}kg m

^{−3}).

*E,*to a neutral trajectory is quadratic rather than cubic, being proportional to

*T*Δ

_{b}*p*Δ

*θ,*where

*T*is the thermobaric parameter of McDougall (1987b). This quadratic error is much more serious than the cubic error discussed in appendix A and can be quantified as follows. If

_{b}*p*does not vary linearly with

*θ,*then the leading terms in

*β*(

*S*

*θ*

*p*

*S*−

*α*(

*S*

*θ*

*p*

*θ*equal to zero (see appendix A), instead of putting all of the right-hand side of (8) to zero. Thus, we incur a relative error of −

*T*

_{b}∫

_{n}(

*p*−

*p*

*d*

*θ*, or an absolute density error of about 10

^{3}

*T*

_{b}∫

_{n}(

*p*−

*p*

*d*

*θ*kg m

^{−3}. For a linear variation of

*p*with

*θ*, ∫

_{n}(

*p*−

*p*

*d*

*θ*is zero, and so ∫

_{n}(

*p*−

*p*

*d*

*θ*is equal to the area between the data on a neutral trajectory and the straight line between the end points on a

*p*–

*θ*diagram (see Fig. A1). The thermobaric parameter

*T*

_{b}is approximately 2.7 × 10

^{−8}K

^{−1}(db)

^{−1}[see Fig. 9 of McDougall 1987b] and the maximum triangular error from this effect is 2.7 × 10

^{−5}× (1/2)Δ

*p*Δ

*θ*. A more realistic error estimate is one-third of this triangular error, say 5 × 10

^{−6}Δ

*p*Δ

*θ*, where Δ

*p*is measured in decibars. For the realistic maximum ocean values of Δ

*p*= 1000 db and Δ

*θ*= 1.5°C, this error estimate is 7.5 × 10

^{−3}kg m

^{−3}.

## 3. The dataset

Reid and Lynn (1971) have shown that, as an isopycnal surface rises toward the sea surface in the Northern and Southern Hemispheres, it coincides with different potential density surfaces (referenced to sea level) in the two hemispheres. In subsequent work, Reid (1986, 1989, 1994) has taken this isopycnal mixing analysis further by defining isopycnal surfaces in the South Pacific, and the South and North Atlantic Oceans, in an attempt to describe the total geostrophic circulation of these basins. In these papers Reid has shown that not only is it between hemispheres that these surfaces differ significantly in potential density, but even within the same ocean basin they must be defined with different potential density labels in different regions (sometimes referred to the same pressure, but mostly referred to different pressure levels). For example, in the most recent specifications of the isopycnal surfaces in the North and South Atlantic Oceans, the layers can differ by as much as 0.14 in *σ*_{0} between the hemispheres, while 6 of the 10 surfaces in the north and 5 of the 10 in the south have been defined with multiple values (on occasions up to three of four different values) of potential density, all referred to the same pressure.

By analogy with Reid’s work, it follows that *γ*^{n}, being a continuous approximation to these discretely referenced potential density surfaces, cannot be a function of the three state variables *S, T,* and *p* but must also depend on geographical location. Thus, despite the existence of a well-defined equation of state in terms of *S, T,* and *p,* any attempt to describe *γ*^{n} in terms of just these three variables will prove fruitless. Given this spatial dependence of *γ*^{n}, it is clear that any representation of *γ*^{n} must incorporate a knowledge of the spatial distribution of temperature and salinity. One method for accommodating this is to describe the entire ocean by a comprehensive dataset that possesses an accurate representation of *γ*^{n}. An arbitrary {*S, T, p,* lat, long} observation from the ocean will not normally have the same combination of *S, T,* and *p* as the labeled reference dataset at that longitude and latitude, but it will not be very distant from it since the reference dataset is an average of actual observations taken from the same location. A label can be found for this observation by finding the depths on the surrounding four casts in the reference data where the hydrographic properties are such that it and the {*S, T, p*} observation lie on the same neutral surface. The *γ*^{n} label of an arbitrary {*S, T, p,* lat, long} observation is then taken as a weighted average of these *γ*^{n} values found in the labeled dataset.

It is important to realize that the use of the prelabeled global dataset in the definition of neutral density *γ*^{n} does not diminish its use as the most natural density variable in the ocean. This follows from the fact that the relationship of lying on the same neutral density surface is, apart from the path-dependent errors (which are quantifiable), an equivalence relation. Consider the process of labeling the data from two adjacent casts from a modern hydrographic section. On each of these two casts there is a point that communicates neutrally with a point on one of our prelabeled casts. The equivalence relation of the neutral property implies that all three points lie on the same neutral surface and, in particular, that the two points on the observed casts that receive the same *γ*^{n} label also possess the neutral property. The only caveat here is the presence of path dependence, and this can be shown to be a very small effect (see below). We conclude that even though the prelabeled global dataset that we use may be much older than observational data requiring labeling, the equivalence relationship formed by our neutral density variable ensures that the labeling procedure achieves our aim of labeling observational data with a neutral density variable.

At the present time the only datasets encompassing the entire ocean are the climatological atlas of Levitus (1982) and the global dataset of Reid (1986, 1989, 1994). The former consists of averaged values of *S* and *T* on a regular grid at standard depths, while the latter data consists of observed values of *S, T,* and *p* at irregular spatial locations. For our purposes, the Levitus data possesses two significant advantages over the Reid data: the regular nature of the data in terms of longitude, latitude, and depth (corresponding to our *x, y,* and *z* coordinates) and that the dataset does not contain adjacent casts from different seasons. For these reasons we have chosen the climatology of Levitus (1982) as the dataset for describing the world’s oceans.

The Levitus data consists of measurements of *S* and *T* at 33 standard depth levels at a 1° resolution, located at each 1/2° of longitude and latitude. Solution of (2) for such a large dataset is computationally a very large problem, so we have subsampled the original data onto a 4° × 4° grid, located on every fourth degree of longitude and latitude commencing at (88°S, 0°). The data at each of these locations is taken as the cast with the densest bottom bottle (in terms of *σ*_{4}) from the 16 casts surrounding the point in question, thereby attempting to capture an ocean with the widest possible density range. In the Antarctic Circumpolar Current this procedure sometimes resulted in there being no cast that was representative of the center of the ACC, and values of *ρ*_{0}*T*_{b}*pd**θ* [see McDougall and Jackett (1988) or (A10) of appendix A] of about 0.03 kg m^{−3} were found around local closed neutral circuits on the 4° × 4° grid. This was rectified by ensuring that casts from the central ACC were used rather than those based on the maximum *σ*_{4} criterion. At the surface, all casts have also been complemented with the corresponding seasonal Levitus data. Specifically, in the top 200 m of the ocean, the annual data was linearly interpolated toward the season that had the least dense surface water, with no change at 200 m and the seasonal data completely replacing the annually averaged data at the surface. Data points in the Arctic Ocean, the Mediterranean, Baltic, Black, Caspian and Red Seas, the Persian Gulf, and Hudson Bay were all excluded from the subsampled dataset.

To justify the 4° resolution of our global dataset and to further quantify the local path-dependent error associated with the ill-defined nature of neutral surfaces, we have conducted an experiment on small closed neutral trajectories in the Levitus (1982) atlas. For each bottle on each cast in our subsampled global dataset, we have made two different types of neutral excursions to each of the casts to the immediate east and north. The first applies the basic neutral surface calculation once with the neighboring cast, while the second inserts the four casts from the original 1° × 1° Levitus data between the bottle and the cast in question and then makes five neutral surface calculations. This provides us with estimates of the absolute depth errors at the two neighboring casts due to our dataset being stored every 4° rather than every 1°. Averaging over all bottles on the original cast, we can now find mean and maximum absolute depth errors associated with 4° eastward and northward closed neutral excursions for each cast in the subsampled dataset. In Fig. 3a we have plotted the cumulative distributions for the entire ocean of these cast mean and maximum absolute depth errors, the solid line representing the mean errors and the dashed line the maximum errors. It is evident that 95% of the Levitus (1982) ocean have cast-mean absolute depth errors of less than 4.2 m and maximum absolute depth errors of less than 14.1 m. The small sizes of these errors validates the resolution we have adopted for our global dataset, albeit this resolution having been chosen for the reasons of computational expediency. Figure 3b shows these same errors, but now expressed in terms of the difference in locally referenced potential density at the cast 4° to the east or north. These errors have been compared with the estimate 10^{3}*T*_{b}*pd**θ*, derived in appendix A, and are almost exactly equal to this estimate, confirming our Taylor series analysis results that this quadratic error term due to the path-dependent nature of neutral surfaces far outweighs the cubic error terms inherent in our finite-difference approximations.

Although the Levitus atlas possesses the aforementioned advantages, it was constructed using objective analysis performed on geopotential surfaces, rather than the more accurate locally referenced potential density surfaces. Also, despite the careful elimination of statically unstable water prior to the objective analysis, no stability criterion was actually implemented during or after the averaging procedure. The consequence of these shortcomings is that nearly half of the 4° × 4° cast data chosen from the Levitus climatology possess vertical inversions of locally referenced potential density or, equivalently, have buoyancy frequency profiles with negative values. It should be pointed out that these stability problems only affect some 5% of the bottle data, but when projected to cast data, this figure increases to 47%. Not only is it averaged data that suffers this problem, 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 *N*^{2} values would cause overturning of neutral surfaces and vertical inversions of *γ*^{n} in the labeled dataset. This can be seen by recalling that *γ*^{n}_{z}*b**ρ*(*β**S*_{z} − *α**θ*_{z}). McDougall (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 always be positive. Thus, the sign of *γ*^{n}_{z}*N*^{2}, which means that when *N*^{2} < 0, *γ*^{n} is not monotonic with depth. A technique must therefore be employed that eliminates these buoyancy frequency problems from the subsampled 4° data.

Recent work by the present authors (Jackett and McDougall 1995) has addressed this problem with the development of an algorithm that minimally adjusts hydrographic data so that the resulting buoyancy frequency profiles are larger than a specified lower bound. Using this algorithm, the entire reference dataset has been stabilized with quite mild minimal buoyancy frequency profiles. The resulting modifications made to the Levitus dataset were all reasonable [and well within the error tolerances found in Levitus (1982)], graphically not visible in the vast majority of cases.

The formation of Antarctic Bottom Water by salt rejection during ice formation in winter causes additional problems with the Antarctic shelf water casts taken from the Levitus atlas. Figure 4a shows two casts in our subsampled database along the 140°E meridian, one at 64°S with a depth of 4000 m and the other on the shelf at 68°S with a depth of 400 m. Also shown are two bottles, marked A and B, at depths of 300 and 950 m, respectively, taken from the summer, WOCE line SR3 at 66.4°S, 140.2°E in February 1994. The dashed line represents a neutral excursion between bottle A and the more northern cast, such an excursion to the southern cast on the shelf not being possible. Figure 4a clearly demonstrates that the shelf water contained in our subsampled dataset does not have the ability to neutrally communicate with even relatively shallow observational data taken in its immediate vicinity. The averaging nature of the objective analysis in smoothing out extreme density values has resulted in even the densest cast in the 4° × 4° box of Levitus data on the shelf being unable to make neutral excursions to summer observational data. The situation with winter data, were it obtainable, would be worse.

Accordingly, we have systematically replaced all Antarctic shelf water in our subsampled dataset with deeper and denser data from neighboring waters. All data along 72° and 76°S has been replaced with the denser data at 68°S at the same latitude. For shelves north of this, the replacement has been selective. For instance, Fig. 5 shows the replacement strategy to the south of Drake Passage. The numbers at the center of each of the boxes in the figure represent the depth of the cast in the dataset at that location (expressed in meters), and the thick solid line depicts the delineation we have used between the South Pacific and South Atlantic Oceans (see later). The Antarctic continent is shown by the gray line, and the arrows indicate which casts are replaced by others in the dataset. All replacement is made sequentially from the east and then from the north so that, for example, the cast at 68°S, 312°E replaces all casts southwest of it in the Weddell Sea. The box at 72°S, 288°E is without a number in Fig. 5 since the original 1° × 1° Levitus data contains no observations in the 4° × 4° box, so we have duplicated the observation from the north. This was the only occurrence of such data paucity in the entire ocean.

The replacement of Antarctic shelf water over the entire longitudinal range of the ocean was made in a similar manner, where shelf water was replaced by dense water from the appropriate neighboring ocean. The only exception to this was that we have retained the Kerguelen Plateau and the intersection of the Southeast Indian Ridge with the Pacific 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*–*θ* diagram of the neutral surface through A to the denser cast [now also the cast at (68°S, 140°E) 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 64°S 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, −2.18°C, 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 0.25°C 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.

*S*–

*θ*diagram of vertical casts is curved. They have shown that along an isopycnal horizon the changes in

*S*and

*θ*are as large as 0.2 psu and 1°C, while along an isobaric horizon, the change in potential density is as large as 0.1 kg m

^{−3}. The question then arises as to whether our method of labeling data with neutral density (based on Levitus data) can potentially be in error by 0.1 kg m

^{−3}in these regions. The answer to this question is no, and the reason is that apart from the path-dependent errors the basic neutral calculation forms an equivalence relation. It follows from this that, if observations A and B both individually communicate neutrally with a point L in our labeled Levitus dataset (hence, points A and B get the same neutral density label by our algorithm), then point A will also communicate neutrally with point B. That point L is, say, 1°C warmer than it ought to be is only important in that it can introduce some small path-dependent errors. If we assume that L is too warm by 1°C and too deep by 100 db, then the maximum possible path-dependent error in neutral density is [from Eq. (A10)]

*p*–

*θ*diagram. It is concluded that while the Levitus data does contain the anomalies as described by Lozier et al. (1994), these anomalies do not significantly affect our use of the Levitus dataset in the neutral density algorithm. The same argument also shows that the mean temporal change of ocean properties due to, say, global warming does not lead to significant errors in the neutral density label.

## 4. Labeling the reference dataset with *γ*^{n}

*γ*

^{n}. This labeled dataset will subsequently be used to assign

*γ*

^{n}values to arbitrary hydrographic data. The coupled system of differential equations defined by (2) succinctly describes the neutral property we desire for neutral density

*γ*

^{n}. As an initial field of

*γ*

^{n}values for our referenced global dataset, we therefore seek the solution to the differential equations

*γ*

^{n}

*bρ*

*β*

*S*

*α*

*θ*

*b*is an unknown scalar function of space.

These equations are a simultaneous system of first- order hyperbolic partial differential equations. In general, the most accurate method of solution of these equations is with the method of characteristics (see Carrier and Pearson 1976 or Smith 1965). This technique identifies characteristic curves or surfaces in the problem domain, along which values for *γ*^{n} can be found by integrating initial data specified on some appropriate (i.e., noncharacteristic) curve (or surface). These characteristics also define a region of dependence of the initial data, and values of *γ*^{n} outside this region cannot, in theory, be accurately determined.

Not surprisingly, the characteristic surfaces of (9) turn out to be neutral surfaces, along which *γ*^{n} is constant. The method of characteristics for our problem therefore reduces to fitting neutral surfaces to our three-dimensional dataset, with the values of *γ*^{n} on these surfaces being determined by a knowledge of *γ*^{n} down just one cast.

An individual neutral surface can be fitted to our three-dimensional hydrographic data {*S,T, p,* lat, long} by starting from a given cast and then growing laterally in space using the basic neutral calculation for each step. During this procedure the set of locations are partitioned into two disjoint subsets: those that have (have not) been assigned a pressure value for the particular neutral surface. Points from the latter are moved to the former by keeping track of the perimeter set of locations of the currently assigned surface. This perimeter set is defined to be the set of all locations where an actual pressure value on the last iteration of the process exists, with it initially being defined as a single point (the main reference cast). Neutral excursions are then made from this perimeter set to all unallocated points that are at a distance of 4° of longitude or latitude. In this way the size of the perimeter set increases from 1, rises to a maximum, and then decays to 0, when the complete surface has been fitted to the data. When the surface either outcrops or undercrops, the pressure value is flagged, and the surface is pursued no further in this direction. After termination of the iterative procedure, those points remaining to be assigned have either outcropped on one or more occasions, or are more than one horizontal grid point from the currently defined neutral surface, and in both cases are left unassigned. A major advantage of generating the surfaces in the two lateral directions using this method is that the resulting surfaces have the ability to iterate around bathymetric features.

During this procedure, various water masses are inhibited from communicating with each other. For example, below 1200 m the Pacific and Indian Oceans are not permitted to be connected through Indonesia, but above this, neutral excursions are allowed, thereby modeling the Indonesian Throughflow (Godfrey and Golding 1981). In contrast, the Atlantic and Pacific Oceans are never joined through the Panama Canal. Also, for reasons which will be subsequently discussed, the Pacific and Atlantic Oceans are not allowed to communicate through Drake Passage at this stage of the process.

One hundred neutral surfaces were fitted to the global dataset, all emanating from a single cast in the central Pacific Ocean. Figure 6 shows one of these surfaces that commenced on the Pacific Ocean cast at mid depth, this being the shallowest of the neutral surfaces that was blocked through Indonesia. The contoured number is the iteration number at which each particular location is assigned a neutral surface pressure value. The figure clearly demonstrates the circuitous nature of the surface generation and the closures of the various bathymetric regions. It also shows that when the surfaces reach 0° from the east, they are continued westward into the western Atlantic Ocean.

*γ*

^{n}data was chosen to be the cast at 16°S, 188°E, this being the central Pacific Ocean cast that captured a maximum 80% of the bottle data of our global dataset. By this we mean that 80% of the total number of bottles in the dataset lie between adjacent neutral surfaces that emanate from this reference cast. The values for

*γ*

^{n}down this cast can be chosen arbitrarily, but we have made it coincide with the “local potential density,” a variable first proposed by Veronis (1972). Specifically, we define

*b*profile of unity. It should be noted that although this density variable can be uniquely defined for any cast, it does not constitute a well-defined function of three-dimensional space (as was pointed out by Veronis 1972). However, extending the definition of

*γ*

^{n}

_{ref}

*b*to vary in space [see Eq. (2)]. The values of

*γ*

^{n}at the reference cast in the Pacific Ocean ranged from 22.3797 kg m

^{−3}at the surface, equal to the in situ and potential densities there, to 28.2211 kg m

^{−3}at the ocean floor.

This method of fitting neutral surfaces to the global dataset does result in errors in *γ*^{n}, which become apparent when examining points on adjacent casts on the same neutral surface that have been generated with entirely different trajectories from the reference cast. This depth error, which can be quantified by making one final neutral excursion from one of these casts to its neighbor, is the inherent path-dependent error associated with the definition of *γ*^{n}. To minimize the occurrence of this error in our dataset we have stopped communication between the Pacific and Atlantic Oceans at this stage of the process by the inclusion of an artificial barrier near Drake Passage. This minimizes the number of points where such different paths meet and so simplifies the subsequent task of spatially smoothing this error.

*z,*associated with such a closed contour

*C*is

*T*

_{b}≈ 2.7 × 10

^{−8}db

^{−1}K

^{−1}is approximately a constant (see also appendix A). The seam down our artificial Pacific/Atlantic Ocean barrier allows us to examine the accuracy of this formula over global scales. In Fig. 7 the estimate (11) of Δ

*z*made using the closed neutral surface contours for all 100 fitted surfaces at the Drake Passage barrier is plotted against the actual error. The correlation of these two Δ

*z*errors is remarkably good (a correlation coefficient of 0.995), indicating (11) is indeed a very good estimate of the helical error in

*γ*. The mean absolute value of Δ

*z,*the difference between the two heights on the casts at the Drake Passage barrier over these global scales is only 29.5 m, which further justifies our search for a well-defined neutral density variable that almost possesses the neutral surface property.

*γ*

^{n}values of the 100 fitted neutral surfaces could now be suitably interpolated to provide initial

*γ*

^{n}values for each of the bottles in the 80% of the captured ocean;however, this would incur an unnecessary interpolation error. A more accurate evaluation for

*γ*

^{n}can be made by using trajectory information contained in the surfaces 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 Pacific Ocean reference cast, the basic neutral surface calculation can again be employed to find a neutral trajectory from the arbitrary bottle back to the reference cast. The value of

*γ*

^{n}for the particular bottle can then be found by accurately computing

*z*

_{int}is the height of intersection of this neutral trajectory with the reference cast. Some of the bottles at the very top and bottom of 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 Pacific Ocean reference cast. (The majority of such routes either outcrop or intersect the sea bed along these paths, and in this case the bottles cannot be labeled using this method.) When this was done, an additional 4% of the ocean was assigned

*γ*

^{n}values, taking the proportion of ocean captured by the 100 neutral surfaces to 84%.

This leaves 16% of the ocean, distributed almost equally between bottles near the top and those near the bottom of the global dataset, which remain to be assigned values of *γ*^{n}. We return to the numerical solution of the system of differential equations (9) for these values. Although the method of characteristics is the normal solution method of hyperbolic partial differential equations, it is by no means the only process for solving these equations (see Smith 1965). Characteristic grids are certainly the most appropriate when the initial data contain discontinuities, but in the absence of such continuity problems, alternate grids and numerical methods can be considered. The continuity (smoothness) of the hydrographic data contained in our global dataset and the exponential form for *b* implies that *γ*^{n} should be a continuous three-dimensional variable. We are therefore at liberty to choose an alternate grid and solution strategy for the fundamental system (9).

*b*between pairs of equations in (9) results in the coupled system [since

*ρ*

_{x}=

*ρ*(

*β*

*S*

_{x}−

*α*

*θ*

_{x}), ···, etc.]

*γ*

^{n}.

The system of algebraic equations resulting from the discretizations of (12) is overdetermined, roughly in the ratio of 6 to 1. Traditional finite difference techniques nearly always choose discretizations and boundary data in such a way that the resulting algebraic system is evenly determined, thereby enabling the solution to be obtained from a simple matrix inversion. In the present problem we have used the singular value decomposition to invert the overdetermined system. In practice, the number of equations (∼ 65100) and unknowns (∼11350) made the inversion of (12) using one matrix computationally expensive. Rather, the region of unknown *γ*^{n} values was split into smaller sized blocks, each of roughly 500 unknowns. The blocks were chosen to overlap, by two horizontal grid points, so that a maximum number of equations was written for each block and so that the results from a previous inversion were coupled to the inversion of the next block.

The *γ*^{n} values obtained by this procedure are located outside the region of dependence (roughly, that space between *γ*^{n} = 22.38 and *γ*^{n} = 28.22) of the reference data at the Pacific Ocean reference cast. As such, there 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 region of dependence defined by the problem’s characteristics (as opposed to the more traditional matrix inversion technique based on an evenly determined system) is, in fact, a valid numerical technique. By taking *ρ*_{x}, *ρ*_{y}, and *ρ*(*β**S*_{z} − *α**θ*_{z}) as the derivatives of potential density, *σ*_{θ} for example, the numerical procedure above should yield the corresponding potential density field. When this was done, the *σ*_{θ} field was inverted to the accuracy of the computations, that is, to order 10^{−4} kg m^{−3}. This also suggests that potential difficulties with the convergence of the finite difference schemes is not a problem for this particular system of differential equations or for the grids we are using.

The solution of the differential system (9) contains errors in the *γ*^{n} field due to several sources. The first is the inherent path-dependent error associated with the ill-defined nature of neutral surfaces. This type of error is most apparent along the artificial barrier, which we have so far imposed at Drake Passage. For the remainder of this paper this barrier is removed, allowing water to neutrally mix through Drake Passage as it does in the real ocean. A second potential error source arises from the finite difference technique being applied to 16% of the ocean, which lies outside the region of dependence of the initial data defined at the Pacific Ocean reference cast. These two potential sources of error are additional to the usual truncation errors involved in the numerical procedures we have adopted for the generation of the *γ*^{n} field.

To eliminate these errors, or at least distribute them out over the entire globe, we have developed a technique for iteratively improving the *γ*^{n} field. This is based on making additional computations in the problem’s natural characteristic coordinate system, the neutral surfaces. This relaxation procedure has the effect of laterally smoothing errors in *γ*^{n} along the neutral surfaces while achieving continuous and smooth vertical profiles of *b.* Details of this relaxation technique can be found in appendix B.

The effect of the relaxation technique for laterally smoothing the accumulated error in the *γ*^{n} field can be seen in Fig. B3a, where we have plotted an estimate of the maximum error associated with *γ*^{n} down each cast in our global dataset. The vast majority of the ocean has this maximum error below the present instrument error of 0.005 kg m^{−3}. The map for the average error associated with each cast is very similar to Fig. B3a, only with a *γ*^{n} scale an order of magnitude less.

In conclusion, the *γ*^{n} field resulting from the solution of the differential system (9) using the characteristic and finite difference techniques, followed by the iterative relaxation of the *γ*^{n} field, produces a *γ*^{n} field that satisfies the differential system (9) an order of magnitude better, on average, than the present instrumentation error in density of 0.005 kg m^{−3}.

## 5. Labeling arbitrary hydrographic data with *γ*^{n}

Having developed a global dataset that has been accurately assigned a field of *γ*^{n} values, we are now in a position to label external hydrographic data. An arbitrary {*S, T, p,* lat, long} observation from the real ocean lies inside a box consisting of the four nearest casts in our labeled dataset. Since our global dataset is based on a climatological atlas of the world’s oceans, each of these casts should possess bottles that are not very distant, in terms of hydrography, from the unlabeled bottle. Using the basic neutral surface calculation, four neutral trajectories are calculated between the arbitrary bottle and the four neighboring casts. The immediate result of this procedure is the four heights on the four neighboring casts, and these are then used, in the manner described below, to provide a *γ*^{n} value for the original {*S, T, p,* lat, long} observation.

To increase the efficiency of the interpolation of the *γ*^{n} values found on each of the four neighboring casts, we have computed *γ*^{n} on each cast as a piecewise quadratic function in the vertical. This involves storing one extra variable, namely, the quadratic coefficients, for each cast in the global dataset. These parameters were evaluated by fitting by least squares piecewise quadratics to the final *γ*^{n} field. In appendix A we show that the error involved in using this piecewise quadratic representation of *γ*^{n} in the vertical induces an error of at most 10^{−4} kg m^{−3} in *γ*^{n}, well below the mean absolute error level in the *γ*^{n} field over the entire global dataset. Using this parameterization of *γ*^{n}, the interpolation of *γ*^{n} on each cast in the global dataset can be achieved with a single multiplication, rather than with the many required by alternate numerical integration techniques.

Despite the fact that we have added a seasonal mixed layer and extended the Southern Ocean waters south of 64°S to extremely dense water, the real ocean and model output (in particular) will possess bottles with neutral density values outside the range of *γ*^{n} in our global dataset. To successfully label these observations we extend the casts involved on a needs basis at the top or bottom of our global dataset. This extension is made just to a point where the extended cast neutrally communicates with the unlabeled bottle. At the surface, this extension is made toward warmer water at the salinity of the top bottle on the cast, while at the ocean floor it is in the direction given by *R*_{ρ} = *α**θ*_{z}/*β**S*_{z} = −1 (equal contributions to the density gradient). Figure 8 shows a typical cast that has required extension at the top in one case and at the bottom in another. A *γ*^{n} value is now assigned to the unlabeled bottle by using the *b* value from the very top or bottom of the unextended reference cast, and the vertical integral of the relation *γ*^{n}_{z}*b**ρ*(*β**S*_{z} − *α**θ*_{z}) applied to the extension. This is straightforward to do because the pressure of the new point on the cast is known (set equal to *p* = 0 for a surface extension, and equal to the bottom pressure of the cast for a bottom extension), so that the vertical integral of *ρ*(*β**S*_{z} − *α**θ*_{z}) is simply the difference in in situ density between the new bottle on the cast and either the existing top or bottom bottle.

The labeling strategy for an individual (*S, T, p,* lat, long) datum is as follows. First, the surrounding four casts of our reference dataset are located, and the basic neutral calculation is made to each of these casts. Usually this gives four estimates of neutral density. If *x* and *y* represent the distances from the southwest reference cast to the datum, normalized to the range [0, 1] corresponding to the 4° longitude and latitude range, then the four weights for the four estimates of *γ*^{n} are initially chosen as the linear interpolants, *xy, x*(1 − *y*), (1 − *x*)(1 − *y*) and (1 − *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 *γ*^{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 m^{−3}. This restriction is implemented smoothly in the following way. If one of the reference casts is extrapolated in order to label the datum, the weight of this cast is reduced linearly to zero as the amount of extrapolation, Δ*γ*, increases to 0.3 kg m^{−3}. Of course, in each case, normalization is ensured by dividing by the sum of the final weights. This smooth vertical interpolation procedure is used to ensure that if a series of very dense data are encountered in the vicinity of a seamount or a continental shelf, the labeled values of neutral density will vary smoothly in the vertical as the number of casts involved in its labeling vary from 4 to 3, ···, etc.

*γ*

^{n}values by the above method, these values can be interpolated to find the positions where specified neutral density surfaces intersect the cast. This could be done using simple linear interpolation; however, for accuracy reasons we have implemented a straightforward quadratic technique that incorporates the nonlinear nature of the equation of state of seawater. For the now- labeled cast of hydrographic data {(

*S*

_{k},

*T*

_{k},

*p*

_{k},

*γ*

^{n}

_{k}

*k*= 1, 2, ···,

*n*cast} and a given neutral density surface

*γ*

^{n}=

*γ̂*

^{n}passing between bottles

*k*and

*k*+ 1 (i.e.,

*γ*

^{n}

_{k}

*γ̂*

^{n}≤

*γ*

^{n}

_{k+1}

*γ*

^{n}in this interval is given by

*z*is the depth level in the

*k*th

*γ*

^{n}(

*z*) in this interval by taking

*b, S*

_{z}, and

*θ*

_{z}

*γ*

^{n}and hydrographic values, and

*α*and

*β*as known linear functions between the bottles. Equation (14) then reduces to a quadratic

*γ*

^{n}

*z*

*a*

_{k}

*z*

^{2}

*b*

_{k}

*z*

*c*

_{k}

*a*

_{k},

*b*

_{k}, and

*c*

_{k}are found in terms of the bottle hydrographic values by trivial algebra. The coefficients are chosen to ensure that

*γ*

^{n}(

*z*) is exactly

*γ*

^{n}

_{k}

*γ*

^{n}

_{k+1}

*z*

_{k}and

*z*

_{k+1}respectively. Equation (14) can then be simply and accurately solved to yield the point of intersection of the

*γ*

^{n}=

*γ̂*

^{n}neutral surface with the arbitrary cast. Corresponding values of

*S*and

*T*are found by linearly interpolating

*S*and

*θ*between the appropriate bottle values.

## 6. Comparison of *γ*^{n} surfaces with present isopycnal surfaces

To demonstrate the effectiveness of neutral density in accurately representing neutral surfaces in the real ocean, we have sampled the Reid global dataset (Reid 1986, 1989, 1994) to obtain sections of hydrographic data that are different from the global dataset underlying the definition of *γ*^{n}. Since the Reid dataset is scattered in spatial location, we have chosen these sections to be as close as possible to a particular meridian of longitude, with a zonal resolution of about 4° of latitude. We have taken sections in each of the major ocean basins along which we compare neutral surfaces obtained by vertically interpolating *γ*^{n}, in the manner described in the previous section, with surfaces calculated using several alternative methods, including the present best isopycnal surfaces.

The first comparison we make is between the *γ*^{n} neutral surfaces (i.e., the neutral surfaces obtained by interpolating the cast *γ*^{n} values following labeling, as in the previous section) and the neutral surfaces fitted to a section from the Reid dataset using the accurate basic neutral surface calculation of section 2. Figure 9a shows a meridional section near 62°E in the Indian Ocean, the solid lines representing the *γ*^{n} surfaces for *γ*^{n} = 26.8, 27.5, 27.95, and 28.1, and the dashed lines the surfaces generated using the basic neutral surface calculation starting from initial points coincident with the *γ*^{n} surfaces near 23°S and which then proceed both north and south one cast at a time. In Fig. 9b we show the errors between these two sets of surfaces. The mean absolute deviation between the two sets of surfaces over all surfaces and all latitudes is 5.9 m, and the maximum absolute deviation is 61.1 m, which to graphical accuracy makes them almost indistinguishable in the a panel. It should be noted that the *γ*^{n} surfaces are more likely to be the more accurate of the two sets of surfaces since the use of the basic neutral surface calculation in this Reid section is in a direction that is generally across the general circulation of the ocean. The *γ*^{n} field has been generated by averaging the initial *γ*^{n} field over all lateral directions, thereby reducing any path-dependent error in the definition of a neutral surface. It is more important to minimize the path-dependent error in the direction of the mean circulation than in the direction normal to the mean circulation. The vertical excursion between the two types of surfaces in Fig. 9 is then larger than that applying in the physically more meaningful direction of the mean circulation. Interpolation of *γ*^{n} using the simple quadratic technique of the previous section is estimated to cause an interpolation error of only 10^{−4} kg m^{−3} and thus retains much of the accuracy of the basic neutral surface calculation.

In Fig. 10 we show how well *σ*_{0} potential density surfaces perform in approximating the fundamental neutral surface property (2) in the Pacific Ocean. Here we have plotted the *σ*_{0} = 27.0, 27.5, 27.72, and 27.8 potential density surfaces, denoted by the dashed lines. For each of these surfaces we have found the mean value of *γ*^{n} on the surface, and plotted the corresponding neutral surface in the section (solid lines). It is clear that in surface waters the *σ*_{0} surfaces approximate the *γ*^{n} surfaces quite well, but at depth there are considerable discrepancies between the two. The mean absolute deviation between the two sets of surfaces is 72.3 m, while the maximum absolute deviation in depth is 1438.9 m, indicating the inadequacy of using *σ*_{0} surfaces as approximating isopycnal surfaces. If potential density is referred to a different pressure, say 4000 m, the situation is reversed, in the sense that these surfaces fit neutral surfaces well at the bottom of the ocean but are inappropriate in surface waters.

Figure 11 shows the most accurate method to date in fitting isopycnal surfaces to hydrographic data. In the a panel we have plotted surfaces 1, 3, 5, and 9 of Reid (1994), based on potential density surfaces with a varying reference pressure, on a meridional section near 330°E in the Atlantic Ocean. These surfaces are depicted by the dashed lines, with the corresponding *γ*^{n} surfaces (solid lines) being defined as those *γ*^{n} neutral surfaces that minimize the mean absolute deviations (in terms of pressure) from the corresponding Reid isopycnals. In general, these isopycnal surfaces, although cumbersome to calculate, approximate the neutral surfaces well, although in southern regions they can differ by nearly 200 m. Again, we show the errors between the surfaces in the b panel. The mean absolute deviation between the two sets of surfaces in Fig. 11 is 15.4 m with the maximum absolute deviation being 179.8 m. This cast was typical of all the meridional sections in the Atlantic Ocean. In Fig. 12a we show the neutral surfaces defined using the basic neutral surface calculation of section 2 (dashed lines) for the four *γ*^{n} surfaces of Fig. 11a (solid lines) in the Atlantic. The mean absolute differences in depth between the two sets of surfaces is 8.4 m, while the maximum absolute deviation is 42.6 m. Again, this figure demonstrates the accuracy of the *γ*^{n} surfaces in approximating the fundamental neutral surface property.

While these deviations are much less than the best- practice isopycnal approach of Reid (1994), the improvement may well not be important for many oceanographic studies. Rather, we believe that our *γ*^{n} algorithm gives oceanographers a way of forming neutral surfaces much more easily than is presently available by the isopycnal method of Reid. The extra accuracy (by a factor of about 5) is an added bonus. Also, at the present time these Reid isopycnal surfaces have only been defined for the North and South Atlantic and South Pacific Oceans. The *γ*^{n} surfaces are, on the other hand, defined for the entire extent of the ocean excluding the Arctic Ocean and the enclosed marginal seas.

In Fig. 13 we have concentrated on one particular isopycnal surface of Reid (1994), namely, his fifth surface (the third surface in Fig. 11a). Figure 13a shows the variation of *γ*^{n} on this isopycnal over the Atlantic Ocean between 300°E and 0°, where a variation of 0.22 kg m^{−3} in *γ*^{n} is evident. In Fig. 13b we show a map of the differences in pressure between the “best fit” *γ*^{n} surface and this particular isopycnal. By best fit we again mean that *γ*^{n} surface that minimizes the mean absolute deviation from the Reid isopycnal. The mean absolute deviation of this *γ*^{n} neutral surface (where *γ*^{n} = 28.009) from the Reid isopycnal is 28.4 m over the region shown, with a maximum absolute deviation in the Weddell Sea of 328.8 m. To test which of these surfaces is the more accurate in approximating the neutral surface property, we have compared this *γ*^{n} = 28.009 neutral surface with a neutral surface computed using the method of generating neutral surfaces in two lateral directions (using the basic neutral surface calculation), as outlined in section 3. The surface was chosen to emanate from the central location of (10°S, 324°E) at the pressure of the *γ*^{n} = 28.009 surface. Figure 13c shows the differences in pressure between these two distinct neutral surfaces, the mean and maximum absolute differences being 11.8 m and 63.3 m, respectively. The small sizes of these two errors, compared with the magnitudes of the same errors associated with Fig. 13b, again indicate the increased accuracy of using *γ*^{n} surfaces in approximating neutral surfaces over the present best isopycnal surfaces of Reid (1994).

With a view to testing the ability of the labeling procedure of section 5 to successfully assign *γ*^{n} values to arbitrary hydrographic data, we have labeled the global dataset of Reid (1986, 1989, 1994), consisting of nearly 6800 casts scattered worldwide. Of the dataset 99.29% was successfully labeled with *γ*^{n}, those bottles not being labeled having either extremely dense bottom water or having triple crossings. This labeling exercise also allows us to quantify the speed of the assignment of *γ*^{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 dioxide, 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 10^{−2} kg m^{−3} between deep North Pacific 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 modification. This would then directly affect the calculation of neutral density.

## 7. Summary

In this paper we have defined a new density variable, neutral density, *γ*^{n}, which of necessity is a function of the five variables *S, T, p,* latitude, and longitude. The level surfaces of *γ*^{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 dianeutral) diffusivity of small-scale mixing processes. As such, *γ*^{n} provides the most appropriate vertical coordinate for understanding the ocean circulation (e.g., see Hirst et al. 1996). Neutral density is defined in such a way that it is the continuous analog of the discretely defined potential density surfaces that are the present best isopycnal surfaces.

The spatial dependence of neutral density is accommodated by linking the definition of *γ*^{n} with a global hydrographic dataset. This dataset is based on the Levitus (1982) climatology of the world’s oceans. An unfortunate aspect of the objective analysis that went into the making of the Levitus data is that nearly half of the casts in the climatology contain vertical inversions of local potential density at some point on the cast. To rectify this situation we have minimally modified these casts so that the resulting hydrography is statically stable. In order to create a dataset with the widest possible density range, we have also augmented the global dataset with summer water (taken from the seasonal Levitus climatology) in the top 200 m of the water column and have replaced Antarctic shelf waters with much denser water taken from its near neighborhood. Further, we have extended all water south of 64°S to extremely dense water, in order to ensure that we can label very dense Antarctic Bottom Water with neutral density.

The basic neutral surface calculation finds the point of intersection of a neutral surface passing through a particular bottle of data with a neighboring cast of hydrographic data. This point is found by solving a simple equation based on differences in local potential densities using a Newton–Raphson technique. The errors associated with this computation have been shown in appendix A to be approximately ^{1}/_{6}*ρ**T*_{b}Δ*p*Δ*θ*, where *T*_{b} is the thermobaric parameter 2.7 × 10^{−8} (db)^{−1} K^{−1}, and Δ*p* and Δ*θ* are differences between the original bottle and the point of intersection, the next highest error terms being cubic in such differences.

We have defined neutral density via a simultaneous system of first-order hyperbolic partial differential equations. The solution of these equations provides us with an initial field of *γ*^{n} values for our global dataset. To solve these neutral surface partial differential equations we have used a combination of two numerical techniques: the method of characteristics and finite differences. The first of these yields the most accurate solution for 84% of the global dataset, while the second inverts for *γ*^{n} in the remaining 16% of the ocean. The finite difference technique, based on three different grids, is unusual in the sense that it produces an overdetermined system of algebraic equations for the partial differential equations. The ability of this procedure to successfully invert potential density, *σ*_{0}, for our data validates the accuracy of using this technique in the 16% of the ocean requiring solution by a method different from the method of characteristics (*σ*_{0} was simply chosen because it is a well-defined function that has similar nonlinearities to the function we seek, *γ*^{n}).

To eliminate numerical errors associated with these solutions of the neutral surface differential equations, and to distribute the path-dependent errors inherent in the definition of neutral surfaces, we have further iterated the initial *γ*^{n} field. This was done globally and at all depths along the neutral surfaces, the problem’s characteristic coordinate system. At the same time we have imposed continuity and smoothness constraints on the vertical *b* profiles, which we believe exist in the real ocean. In this way we have been able to generate a *γ*^{n} field for our global dataset, which contains local *γ*^{n} errors of sizes 7 × 10^{−4} kg m^{−3} in the mean, and 8 × 10^{−3} kg m^{−3} maximum. These compare well with the present observational error in density of 5 × 10^{−3} kg m^{−3}.

Having successfully and accurately labeled our global dataset we are able to interpolate it, in terms of spatial location and hydrography, to assign *γ*^{n} values to arbitrary hydrographic data. This we achieve by fitting a local neutral surface between the datum in question and the four nearest neighbors (in *x*–*y* space) in the labeled dataset. These are then suitably combined to form a single estimate of the bottle’s *γ*^{n} value. The *γ*^{n} values produced by this procedure can then be interpolated using a simple quadratic technique to find the positions of specified neutral surfaces down an arbitrary cast of hydrographic data. The time taken to label an external cast of hydrographic data can be expensive, due to the many calls to the present equation of state. However, once labeled, the cast can be used very efficiently for finding the depths of specific neutral surfaces.

We have also made comparisons between the neutral surfaces formed from (i) the *γ*^{n} field, (ii) neutral surfaces calculated from the basic neutral surface calculation, and (iii) the present best isopycnal surfaces of Reid (1994). Methods (i) and (ii) yield surfaces for independent sections of hydrographic data that are almost indistinguishable when viewed graphically, demonstrating the accuracy achieved by the *γ*^{n} surfaces. The comparison between methods (i) and (iii) (the mean absolute differences being of the order of tens of meters, and the maximum absolute deviations being of the order hundreds of meters) showed how well the Reid discretely referenced isopycnal surfaces perform in satisfying the fundamental neutral surface property. The use of the *γ*^{n} variable for forming neutral surfaces improves on the accuracy of the present best method of forming isopycnal surfaces, but more importantly it does so in a manner that is very easily implemented.

Software for labeling arbitrary hydrographic data with *γ*^{n} and for finding the positions of specified neutral density surfaces within the water column is available through the World Wide Web at http://www.ml.csiro.au/∼jackett/NeutralDensity.

## Acknowledgments

We wish to thank Drs. Nathan Bindoff, Peter McIntosh, and Steve Rintoul for their constructive comments on a draft of this paper. We thank Dr Nathan Bindoff for supplying the WOCE data used in Figure 4, and Dr Terry Joyce for providing the initial impetus for the project. This work is part of the CSIRO Climate Change Research Project.

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## APPENDIX A

#### Quantification of helicity in terms of density

**A**=

*ρ*(

*β*∇

*S*−

*α*∇

*θ*) into (1), and noting that both

*α*and

*β*are functions of (

*S,*

*θ*,

*p*), helicity can be expressed as

*ρ*

_{0}= 10

^{3}kg m

^{−3}. From McDougall and Jackett (1988), Eqs. (39) or (40), it is seen that the ambiguity of a locally referenced potential density is given by the areal integral in

*x*–

*y*space, on an approximately neutral surface, of

*ρ*

_{0}

*T*

_{b}∫ ∫ ∇

_{n}

*p*× ∇

_{n}

*θ*·

**k**

*dx dy.*In terms of helicity, this ambiguity in potential density is

*α*= 2 × 10

^{−4}K

^{−1}and

*θ*

_{z}= 3 × 10

^{−3}K m

^{−1}, a uniform value of helicity of 10

^{−17}kg

^{2}m

^{−9}over an area of 100 km on a side would result in an ambiguity in potential density of 0.003 kg m

^{−3}.

#### Taylor series expansion for E_{p}

*S̃,*

*θ̃*

*p̃*) and a neighboring cast defined by the continuous variables (

*S,*

*θ*,

*p*). The function

*E*is now defined by

*E*

*ρ*

*S,*

*θ*

*p*

*ρ*

*S̃,*

*θ̃*

*p*

*p*

*p*+

*p̃*)/2, and the aim of the algorithm is to find positions on the cast where

*E*= 0. Differentiating (A1) with respect to

*p*down the cast we find

*ρ, α, β,*and

*γ*as a Taylor series about (

*S, θ, p*), we then obtain

*ρ*,

*α*,

*β*and all partial derivatives are evaluated at (

*S,*

*θ*,

*p*), Δ

*p*=

*p*−

*p̃,*

*θ*=

*θ*−

*θ̃*

*S*=

*S*−

*S̃,*and only the largest two terms have been retained. The leading term in the expansion is thus proportional to

*N*

^{2}.

#### Taylor series expansion of the neutral relationship E

*S*

_{1},

*θ*

_{1},

*p*

_{1}) and (

*S*

_{2},

*θ*

_{2},

*p*

_{2}) on an exact neutral trajectory, separated by Δ

*S*=

*S*

_{2}−

*S*

_{1}, Δ

*θ*=

*θ*

_{2}−

*θ*

_{1}, and Δ

*p*=

*p*

_{2}−

*p*

_{1}. The separation in longitude and latitude is not relevant to the calculation. All Taylor series here will be expanded about the mean values

*S*

*S*

_{1}+

*S*

_{2})/2,

*θ*

*θ*

_{1}+

*θ*

_{2})/2, and

*p*

*p*

_{1}+

*p*

_{2})/2. The neutral relationship

*E*

*ρ*

*S*

_{2}

*θ*

_{2}

*p*

*ρ*

*S*

_{1}

*θ*

_{1}

*p*

*ρ*(

*S*

_{1},

*θ*

_{1},

*p*

_{1}) and

*ρ*(

*S*

_{2},

*θ*

_{2},

*p*

_{2}) about (

*S*

*θ*

*p*

*S*

*θ*

*p̄*) Note that there are no quadratic terms in (A3). Dividing this equation by

*ρ*(

*S*

*θ*

*p̄*) we have

*ρ*/

_{θθθ}*ρ*(≡

*α*− 2

_{θθ}*αα*) is closely approximated by

_{θ}*α*and similarly for other terms in (A4).

_{θθ}*S, θ,*and

*p*vary along the neutral trajectory from (

*S*

_{1},

*θ*

_{1},

*p*

_{1}) to (

*S*

_{2},

*θ*

_{2},

*p*

_{2}). Here we will initially assume that the variations of

*p*and

*θ*are proportional, while

*S*varies in the manner that is required on a neutral trajectory. Defining the perturbation quantities

*S*′ =

*S*−

*S*

*θ*′ =

*θ*−

*θ*

*p*′ =

*p*−

*p̄*, we have

*Q*is the quadratic coefficient.

*Q*is found by equating the slope

*dS*′/

*d*

*θ*′ = Δ

*S*/Δ

*θ*+ 2

*Q*

*θ*′ to be equal to the ratio of

*α*(

*S*

*θ*

*p̄*) +

*α*′ +

_{θ}θ*α*′ +

_{S}S*α*′ and

_{p}p*β*(

*S*

*θ*

*p̄*) +

*β*′ +

_{θ}θ*β*′ +

_{S}S*β*′ at leading order. This gives

_{p}p*C*, and the thermobaric parameter,

_{b}*T*, of McDougall (1991).

_{b}*S*

*θ*

*p̄*) as a Taylor series, retaining terms to second order, giving

*∇*

_{n}α*·*

_{n}θ

*dl*^{Δθ/2}

_{−Δθ/2}

*α*(

*S, θ, p*)

*dθ*′, and the only unusual term that occurs in performing this integral is

*α*

_{S}^{Δθ/2}

_{−Δθ/2}

*S*′

*dθ*′. This is performed using our quadratic expression for

*S*′ as a function of

*θ*′, obtaining −1/6

*α*(Δ

_{S}Q*θ*)

^{3}, noting that this is cubic in Δ

*θ*, not quadratic. The integral of (A5) is

*S*integral in (A2) is evaluated similarly (using

*dS*= Δ

*S*/Δ

*θ*

*dθ*+ 2

*Qθ*′

*dθ*), giving

*β*

_{θ}= −

*α*

_{S}and

*β*

_{θp}= −

*α*

_{Sp}, ···, etc.)

_{n}(

*β*∇

_{n}

*S*−

*α*∇

_{n}

*θ*)·

*dl*_{n}(

*β*∇

_{n}

*S*−

*α*∇

_{n}

*θ*·

*dl**Q*term in (A9) is only 10% of the (1/12)

*α*

_{θp}(Δ

*θ*)

^{2}Δ

*p*term. The leading terms are the

*α*

_{θp}and

*α*

_{pp}terms. Using Δ

*p*= 1000 db and Δ

*θ*= 1°C, (A9) is approximately 2 × 10

^{−7}, implying an error in

*ρ*(

*S*

_{2},

*θ*

_{2},

*p*

*ρ*(

*S*

_{1},

*θ*

_{1},

*p*

^{−4}kg m

^{−3}. This is a factor of 10 less than the measurement precision of modern instrumentation. It is also interesting to note that the cubic terms in (A4) [all of which canceled with the same terms in (A8)] are also smaller than 10

^{−7}[the largest is (1/24)

*α*

_{θθ}(Δ

*θ*)

^{3}which is only 10

^{−8}for Δ

*θ*= 1°C], so that the relationship

*α*(

*S*

*θ*

*p*

*θ*−

*β*(

*S*

*θ*

*p*

*S*= 0 is as accurate as the relation

*E*= 0 that we use to define a neutral trajectory.

#### Variation of potential density on a neutral trajectory

*p*

_{r}, is a function of only

*S*and

*θ*so that

*d*

*ρ*

_{θ}

*ρ*

_{θ}

*β*

*S,*

*θ*

*p*

_{r}

*dS*

*α*

*S,*

*θ*

*p*

_{r}

*d*

*θ*

*β*(

*S,*

*θ*,

*p*)

*dS*−

*α*(

*S,*

*θ*,

*p*)

*d*

*θ*= 0, so the variation of

*ρ*

_{θ}along a neutral trajectory is

*T*

_{b}is equal to

*β*(

*α*/

*β*)

_{p}, the first term in the Taylor expansion of the integrand has been retained, and

*ρ*

_{θ}has been approximated by 10

^{3}kg m

^{−3}.

*γ*

^{n}around a closed circuit in

*x*–

*y*space. When tracking a neutral trajectory around a loop and back to the original cast, the change in potential density along the loop shows up as a vertical difference in potential density on this cast. This is equal to the vertical difference in

*γ*

^{n}divided by the local

*b*value. From this the inherent ambiguity in

*γ*

^{n}along such a neutral trajectory is

*p*–

*θ*diagram up the cast between the first and last point on the same cast. In terms of depth, (A10) becomes

#### Error in using a quadratic for γ^{n} between bottles

*γ*

^{n}

_{z}

*b*

*ρ*(

*β*

*S*

_{z}−

*α*

*θ*

_{z}). Both

*S*

_{z}and

*θ*

_{z}are constant in this depth interval so that

*ρ*has been taken outside the integrals as its relative variation is small compared with those of

*b*and

*α*. We concentrate on the second integral and expand

*b*as a linear function of

*z*(

*b*=

*b*

_{0}+

*b*

_{1}

*z*′ where

*z*′ =

*z*− [

*z*

_{k}+ (1/2)Δ

*z*]) and

*α*is a quadratic,

*α*=

*α*

*α*

_{θ}

*θ*

_{z}

*z*′ +

*α*

_{p}

*p*

_{z}

*z*′ + (1/2)

*α*

_{θθ}(

*θ*

_{z})

^{2}(

*z*′)

^{2}+ ···. The terms that are not captured by a quadratic expression for

*γ*

^{n}as a function of

*z*are due to the vertical integrals of −(1/2)

*α*

_{θθ}

*b*

_{0}

*ρ*

^{l}(

*θ*

_{z})

^{3}(

*z*′)

^{2}, −

*α*

_{θ}

*b*

_{1}

*ρ*

^{l}(

*θ*

_{z})

^{2}(

*z*′)

^{2}and −

*α*

_{p}

*p*

_{z}

*b*

_{1}

*ρ*

^{l}

*θ*

_{z}(

*z*′)

^{2}. When a linear trend is subtracted from the vertical integral of (

*z*′)

^{2}between

*z*′ = −Δ

*z*/2 and

*z*′ = +Δ

*z*/2, it is found that the extreme values occur at

*z*′ = ±Δ

*z*/

*z*)

^{3}. The maximum error due to the

*α*

_{θθ}term is then 0.0128

*α*

_{θθ}

*b*

_{0}

*ρ*

^{l}(Δ

*θ*)

^{3}, where Δ

*θ*is the difference in potential density between the pair of bottles. The maximum value of Δ

*θ*over all the data in our global dataset is 1.7°C, and taking

*α*

_{θθ}≈ 2 × 10

^{−7}K

^{−3}, this term gives a maximum error of 1.3 × 10

^{−5}kg m

^{−3}. The next term (the

*α*

_{θ}term) yields a maximum error of 0.0256(

*b*

_{1}Δ

*z*)

*ρ*

^{l}(Δ

*θ*)

^{2}

*α*

_{θ}. Now (

*b*

_{1}Δ

*z*) is the change in

*b*over this bottle pair, and this is taken to be 0.1, giving a maximum cubic interpolation error of (using

*α*

_{θ}≈ 10

^{−5}K

^{−2}) 7.4 × 10

^{−5}kg m

^{−3}. The

*α*

_{p}term is no larger than this.

We conclude therefore that the cubic error in quadratically interpolating *γ*^{n} between bottles is no larger than 10^{−4} kg m^{−3}.

## APPENDIX B

### Iterative improvement of the γ^{n} field

In this appendix we take the initial *γ*^{n} field, that is, the *γ*^{n} field resulting from the solution of the differential system (9), and iteratively smooth it in the neutral density coordinate system. This results in a *γ*^{n} field in which the errors from all sources are significantly better, on average, than the present instrument error in density 0.005 kg m^{−3}. The basic idea is to average the *γ*^{n} field currently found at each bottle in the global dataset over the values found locally on the neutral density surface through the particular bottle. The process is repeated iteratively until it reaches a steady state.

For each bottle in the global dataset we form a“plate” consisting of the bottle and the points of intersection of the neutral trajectories through that bottle with the casts immediately to the north, south, east, and west. The depths defining each plate need only be computed once with the basic neutral surface calculation, since they depend on the hydrography of the data and not on the values of *γ*^{n} found on the neighboring casts. Since each plate locally defines the neutral surface passing through the particular bottle, values of *γ*^{n} on the four “satellite” casts (in the north, south, east, and west directions) should be the same as the value of *γ*^{n} on the central cast. For the initial *γ*^{n} field we have developed, this is unfortunately found not to be the case. The errors in *γ*^{n} on these plates are a manifestation of both the errors incurred during the numerical solutions of (9) used to produce the *γ*^{n} field, and the real path-dependent error inherent in the definition of a neutral surface.

*z*

_{k},

*k*= 1, 2, ···,

*nz*} of our global dataset. Here

*nz*= 33 is the number of standard depths in the Levitus (1982) and the reference datasets. Consequently we need to make estimates of the

*γ*

^{n}values at each of these plate depths. This can be achieved by integrating

*b*

*ρ*(

*β*

*S*

_{z}−

*α*

*θ*

_{z}) from a known

*γ*

^{n}value at one of the bottles on each of the casts to the height of the plate on that cast. Considering a point of intersection of a plate with just one of these satellite casts lying between standard depth

*z*

_{k}and

*z*

_{k+1}, the value of

*γ*

^{n}we assign to this point is

*γ*

^{n}

_{k}

*γ*

^{n}values. To complete this definition of

*γ*

^{n}(

*z*) we require a knowledge of the vertical profile of

*b*down each cast. This is facilitated by letting

*b*have a piecewise linear profile in the vertical, constrained to be continuous at the bottle depths. Specifically, if the value of

*z*lies between

*z*

_{k}and

*z*

_{k+1},

*k*

*ϵ*{1, 2, ···,

*n*cast} where

*n*cast is the number of bottles on the cast, we define

*b*as

*b*

*z*

*b*

_{k,0}

*b*

_{k,1}

*z*

*z*

_{k}

*b*

_{k,0}and

*b*

_{k,1}are constants and

*z*

_{k}= (

*z*

_{k}+

*z*

_{k+1})/2. The continuity constraints on the

*b*profiles are then

*k*= 1, 2, ···,

*n*cast − 2.

*b*that enables the vertical profile of

*γ*

^{n}(

*z*) to be completely evaluated down the cast. Explicitly, for

*z*

*ϵ*[

*z*

_{k},

*z*

_{k+1}],

*k*= 1, 2, ···,

*n*cast − 1, this is given by

*γ*

^{n}

_{k}

*b*

_{k,0},

*b*

_{k,1}),

*k*= 1, 2, ···,

*n*cast − 1}, we have added three additional neutral plates between each bottle pair down the cast, making a total possible 129 (4 × 32 + 1) plates for each cast. Again, the positions of these plates in the global dataset need only be computed once and stored for subsequent use. It is worth noting that the degree of stabilization adopted in section 3 for ensuring our dataset was everywhere statically stable was more than adequate, in the sense that none of the plates on any cast in our dataset intersected each other, and there was no occurrence of triple crossings encountered during their generation.

*γ*

^{n}on the plate: one at the central cast and one at each of the satellite casts. These can be combined to form a single estimate of

*γ*

^{n}for the plate. Labeling these plates with indices {

*k, l*}, where

*k*= 1, 2, ···,

*n*cast is bottle number, and

*l*= 1, 2, ···,

*nl*

_{k}is an index (

*nl*

_{k}is usually 4) for the internal plates corresponding to successively increasing depths from (and including) bottle number

*k*to the next deepest bottle, we take as the plate estimate

*γ*

^{cast}

_{k,l}

*γ*

^{n}

_{k,l,m}

*m*= 1, 2, ···, 4} are the satellite estimates,

*r*

*ϵ*[0, 1] is a suitable relaxation parameter, and

*nl*

_{k}= 4 for

*k*

*n*cast and

*nl*

_{k}= 1 for k =

*n*cast. The

*γ*

^{cast}

_{k,l}

*l*= 1 is the bottle value

*γ*

^{n}

_{k}

*γ*

^{cast}

_{k,l,m}

*l*≠ 1 and

*γ*

^{n}

_{k,l,m}

*m*= 1, 2, ···, 4 are all computed by evaluating (B3) on the appropriate cast using the current {(

*γ*

^{n}

_{k}

*b*

_{k,0},

*b*

_{k,1}),

*k*= 1, 2, ···,

*n*cast} set of values. Notational convenience has necessitated the suppression of the

*n*superscript in

*γ*

^{n}in several of the equations in this section.

*γ*

^{n}

_{k}

*b*

_{k,0},

*b*

_{k,1}),

*k*= 1, 2, ···,

*n*cast} are known for every cast in the global dataset, the iterative procedure consists of sequentially updating these values, cast by cast, by minimizing

*w*are weights described below and

_{k}*z*are the depths corresponding to the {

_{k,l}*k, l*} index. This minimization was performed subject to the continuity constraints (B2) on

*b,*and

*k*= 1, 2, ···,

*n*cast − 1. The

*γ*

^{n}(

*z*

_{k,l}) in (B5) are written in terms of the unknown coefficients {(

*γ*

^{n}

_{k}

*b*

_{k,0},

*b*

_{k,1}),

*k*= 1, 2, ···,

*n*cast} using (B3), the integrals being (numerically) performed once for each {

*k, l*} subinterval and stored for use on each iteration. Equation (B6) expresses a compatibility condition between the {

*γ*

^{n}

_{k}

*k*= 1, 2, ···,

*n*cast} values and the piecewise linear profile of

*b,*while (B7) ensures that

*b*at either end of the

*k*th interval (and hence

*b*over the entire interval since

*b*is linear over this range) is contained in the interval [0.1, 10.0]. It should be noted that, although the constraints (B7) may have been hard constraints in the initial iterations of the relaxation procedure, they were not active in the final

*b*field.

The minimization of (B5) subject to (B2), (B6), and (B7) can be achieved by the use of standard software for the solution of constrained weighted least squares problems (e.g., see IMSL 1991). As with the Gauss–Seidel method of solution of Laplace’s equation, convergence of the iterative procedure is greatly improved if the new values of {(*γ*^{n}_{k}*b*_{k,0}, *b*_{k,1}), *k* = 1, 2, ···, *n*cast} for a cast are used as soon as they become available for the evaluation of *γ*<+><+><+>^{est}_{k,l}*b*_{k,0}, *b*_{k,1}), *k* = 1, 2, ···, *n*cast − 1} values was found by running one iteration of the procedure using only a single sum in the first term of (B5) (the sum over bottle number *k*), with *γ*<+><+><+>^{est}_{k,1}*γ*^{n} field of section 4, for *k* = 1, 2, ···, *n*cast.

The objective function (B5) is essentially a trade-off between (i) the sum of deviations of *γ*^{n} from an estimate of the *γ*^{n} plate values at the 4 × *n*cast − 3 neutral plates down a cast and (ii) the sizes of the slopes of the piecewise linear profiles of *b.* In this way a certain error is allowed in the neutral density field in order to reduce the amount of vertical structure in the *b* field. The weights, *w*_{k}, for *k* = 1, 2, ···, *n*cast − 1, are chosen to be (1/8)Δ*z*_{k}[*ρ*(*β*Δ*S* − *α*Δ*θ*)]_{k} because for given values of *γ*^{n}_{k}*n*cast bottles, *b*_{k,1} affects the *γ*^{n}(*z*) at middepth (_{k}) between the bottles by the amount (1/8)*b*_{k,1}Δ*z*_{k}[*ρ*(*β*Δ*S* − *α*Δ*θ*)]_{k}. At ¼ and ;q3 of the depthsbetween the bottles, *b*_{k,1} affects the calculated *γ*^{n}(*z*) to the extent (1/32)*b*_{k,1}Δ*z*_{k}[*ρ*(*β*Δ*S* − *α*Δ*θ*)]_{k}, while of course, the values of *γ*^{n} at the bottles are not affected at all. If *b*_{k,1} affected all the *γ*^{n} values equally, then the coefficient outside the second term in (B5) would be 4 as there are four *γ*^{n} terms for each *b*_{k,1} term. This factor of 4 is downweighted by [0 + (¼)^{2} + 1 + (¼)^{2}]/4, becoming the 1.125 shown. This minimization procedure has the effect of achieving continuous and smooth vertical profiles of *b* without incurring undue extra *γ*^{n} errors.

Figure B1a shows a measure of the error in *γ*^{n} on the 27.8 neutral surface for each cast in our global dataset after the single iteration of the above procedure that determines the initial values of {(*b*_{k,0}, *b*_{k,1}), *k* = 1, 2, ···, *n*cast}. At the depth of this neutral surface on each cast we have fitted a plate as described above, and have used as an estimate of the *γ*^{n} error on this plate the value of *γ*^{n}_{max}*γ*^{n}_{min}*γ*^{n} found there. The grayscale in the figure varies from 0 to 0.019 kg m^{−3}, with large (though not the largest) values of this error estimate being evident at the artificial barrier imposed at Drake Passage in creating the initial field. As the iteration procedure progresses, the values of *γ*^{n}_{k}*k* = 1, 2, ···, *n*cast on the casts settle down to nearly constant values, with the initial error on the plates being smoothed out over the entire globe. Figure B1b shows the same error estimates as those in Fig. B1a on the same neutral surface after 50 iterations of the relaxation procedure, and it is clear the large initial errors at the Drake Passage barrier have dissipated. However, even though the *γ*^{n} field has largely settled down after nearly 50 iterations, there are several regions in the global dataset where sizeable plate errors occur on this particular neutral surface.

In order to reduce the sizes of the *γ*^{n} errors on the plates, we have examined the numerical details of the iteration process for regions of possible production of neutral density. One such source of *γ*^{n} generation can be identified by noticing that the relaxation average (B4) used to find the *γ*^{est}_{k,l}*k* = 1, 2, ···, *n*cast and *l* = 1, 2, ···, *nl*_{k}), for various choices of the relaxation parameter *r,* corresponds exactly with several of the numerical schemes [Jacobi, Gauss–Seidel, or successive overrelaxation (SOR) techniques, see Dahlquist et al. (1974)] used to iteratively find the numerical solution of Laplace’s equation. At the boundaries these schemes pay special attention to the boundary conditions of the underlying physical problem. In our case, these would be no flux boundary conditions on *γ*^{n}, explicitly *γ*^{n}_{x}*γ*^{n}_{y}*γ*^{est}_{k,l,m}*m* = 1, 2, ···, 4 for any cast on the boundary as those found on the opposite cast on the plate, for *k* = 1, 2, ···, *n*cast and *l* = 1, 2, ···, *nl*_{k}. It is worth noting that when *r* ≠ 1, *r* is precisely the relaxation parameter in the SOR method above. However, unlike the significant improvement made in the convergence of the SOR method for appropriate choices of *r* > 1, all choices of *r* in this range led to unstable numerical results. This is almost certainly due to the interpolation error involved in computing the four *γ*^{n} values on the satellite casts on each plate using only a piecewise linear profile for *b.*

The effect of this no flux boundary condition on the averaging scheme (B4) was to reduce the plate errors on the neutral surfaces but, not to any great extent, errors still remaining on the plates in our global dataset. Another source of these plate errors is the inherent path- dependent error associated with the neutral surface calculations made in defining each plate. To quantify the extent of this error in the global dataset, we return to the quite accurate estimates we have of this error in terms of the hydrography, previously discussed in sec~tions 2 and 3. For each plate involving five casts in our dataset, we find the *p d**θ* area associated with a closed neutral circuit around the four satellite casts, commencing at the (*S,* *θ*, *p*) point found on the neutral surface on the eastern side of the plate. From appendix A we know that for local values of *b* and *ρ*, and the thermobaric parameter *T*_{b} = 2.7 × 10^{−8} db^{−1} K^{−1}, *bT*_{b}*ρ* ^{;oc}∫*p d**θ* corresponds to the error in neutral density as we follow the closed neutral trajectory around the edge of the plate. In Fig. B2a we have plotted this error from the hydrographic data for the 27.8 neutral surface over the entire globe, where it is evident that there are regions of the ocean where the helical error in *γ*^{n} is of the same order as the *γ*^{n} errors we are experiencing on the plates. Another way of looking at this is to form the cumulative frequency distributions of these two *γ*^{n} errors for all the data on the 27.8 neutral surface. Figure B2b shows these two distributions, the dashed line representing the helical errors of *γ*^{n} from the hydrographic data and the solid line the *γ*^{n} errors associated with the *γ*^{n} field on the plates on this neutral surface after 50 iterations of the procedure. In practice we find that our worst values of *γ*^{n}_{max}*γ*^{n}_{min}*bT*_{b}*ρ* *p d**θ* on the plates.

Figure B3 shows the *γ*^{n}_{max}*γ*^{n}_{min}*γ*^{n} plate error over all (potentially 129) plates down each cast. In Fig. B3b we have plotted the two cumulative frequency distributions of the two *γ*^{n} errors (i.e., the *γ*^{n}_{max}*γ*^{n}_{min}*ρ**bT*_{b} *p d**θ* errors on the plates) over all plates in our global dataset, the two lines representing the same errors as in Fig. B2b. From this figure it is clear that we have achieved with this iteration procedure the levels of error in *γ*^{n} that are contained in the hydrographic data. Further, it is evident that 95% of the plates over the entire ocean have errors in the *γ*^{n} field well below the present instrumentation error (∼0.005 kg m^{−3}) associated with density. The mean error given by *γ*^{n}_{max}*γ*^{n}_{min}^{−4}.