The Materiality and Neutrality of Neutral Density and Orthobaric Density

AUGUST 2009 D E S Z O E K E A N D S P R I N G E R 1779

ROLAND A. DE SZOEKE AND SCOTT R. SPRINGER* College of Oceanic and Atmospheric Sciences, Oregon State University, Corvallis, Oregon

(Manuscript received 1 May 2008, in ﬁnal form 2 February 2009)

ABSTRACT

The materiality and neutrality of neutral density and several forms of orthobaric density are calculated and compared using a simple idealization of the warm-sphere water mass properties of the Atlantic Ocean. Materiality is the value of the material derivative, expressed as a quasi-vertical velocity, following the motion of each of the variables: zero materiality denotes perfect conservation. Neutrality is the difference between the dip in the isopleth surfaces of the respective variables and the dip in the neutral planes. The materiality and neutrality of the neutral density of a water sample are composed of contributions from the following: (I) how closely the sample’s temperature and salinity lie in relation to the local reference u–S relation, (II) the spatial variation of the reference u–S relation, (III) the neutrality of the underlying reference neutral density surfaces, and (IV) irreversible exchanges of heat and salinity. Type II contributions dominate but have been neglected in previous assessments of neutral density properties. The materiality and neutrality of surfaces of simple orthobaric density, deﬁned using a spatially uniform u–S relation, have contributions analogous to types I and IV, but lack any of types II or III. Extending the concept of orthobaric density to permit spatial variation of the u–S relation diminishes the type I contributions, but the effect is counterbalanced by the emergence of type II contributions. Discrete analogs of extended orthobaric density, based on regionally averaged u–S relations matched at interregional boundaries, reveal a close analogy between the extended orthobaric density and the practical neutral density. Neutral density is not superior, even to simple orthobaric density, in terms of materiality or neutrality.

1. Introduction For a long time, potential density was accepted as the correct standard for comparison with water property It has been a century since the demonstration of the core layers, but Lynn and Reid (1968) showed a sen- relation of ocean currents to density gradients (Helland- sitive and anomalous dependence of the topography Hansen and Nansen 1909). The data from the Meteor of potential density surfaces on the choice of their expedition (Wu¨ st 1935) showed the close alignment of reference pressure. The distortion of these surfaces core layers in the ocean, on which water properties like complicates the relationship between potential density salinity, dissolved oxygen, etc., assumed extremal values, gradients and dynamically important pressure gradients with strata of density corrected, however crudely, for (de Szoeke 2000). This is most dramatically illustrated adiabatic compression. [Usually, s was used, which is t by the well-known reversal of the vertical gradient of merely the in situ density with its pressure dependence the potential density (referenced to zero pressure) in a neglected; occasionally, the specific volume anomaly has stably stratified region of the deep Atlantic Ocean. The been employed (Reid 1965).] Montgomery (1938) gave a cause can be traced to the fact that the effective thermal cogent, qualitative explanation of why water properties expansion coefficient of potential density is that of the should spread out along (corrected) density surfaces. in situ density at the potential density’s reference pressure, thereby incorrectly representing thermal effects on buoyancy at pressures far from the reference pressure * Current affiliation: Earth and Space Research, Seattle, Washington. (McDougall 1987b). To mitigate this shortcoming, potential density has been computed over narrow pressure ranges (e.g., 0–500, 500–1500, 1500–2500 dbar, etc.), Corresponding author address: Roland A. de Szoeke, 104A COAS Administration Bldg., Oregon State University, Corvallis, referenced to pressures within the respective ranges OR 97331. (s0, s1, s2, etc.) and matched at the transition pres- E-mail: [email protected] sures to form composite surfaces (Reid and Lynn 1971).

DOI: 10.1175/2009JPO4042.1

Ó 2009 American Meteorological Society 1780 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39

A further elaboration allows for geographical variation This is not to imply that irreversible contributions to the by matching different values of potential density across balance of x are negligible, only that reversible contri- the transition pressures in different regions. This pro- butions to strict materiality are under consideration. cedure [sometimes called the patched potential density Potential temperature and potential density (with a method (de Szoeke and Springer 2005)] has been used single reference pressure) are material variables. How- successfully (e.g., Reid 1994) to create global-scale ever, in situ density r governed by the balance maps of potential temperature and salinity, radio tracers Dr Dp r ›r such as tritium corrected for its radio decay, anthropo- 5 c2 1 q_ , where c2 5 , (1.1) genic tracers like chlorofluorocarbons, or nutrients like Dt Dt ›p u,S nitrate and phosphate corrected for oxygen concentra- tion [so-called NO and PO (Broecker 1981)]. is not material because of reversible adiabatic com- r The formulation of the patched potential density is pression, apart from irreversible effects q_ . not entirely satisfactory (de Szoeke and Springer 2005). Though not material, density (or specific volume 21 A significant limitation is that it is defined discretely in V 5 r ) has the following crucial substitution property both the vertical and horizontal directions. The vertical for the combination that occurs in the equations of discretization was refined by de Szoeke and Springer motion of pressure gradient force and gravitational ac- (2005), who showed that decreasing the vertical pressure celeration: interval of the patches from the typical 1000 to 200 dbar, r1$p gk 5 $ M(V) k9(V)[› M(V) p], say, may render the matching procedure ill-defined V V because there is no longer a one-to-one correspon- (1.2) dence between potential density surfaces across re- where M(V) 5 Vp 1 gz is the Montgomery function gional boundaries. Two major definitions of continuous (closely related to dynamic height) of density surfaces corrected density variables have been proposed: neutral and k9(V) 5 (k 2 $ z)/› z.2 The exact replacement, as density (Jackett and McDougall 1997) and simple or- V V it were, of 2r21$p by the three-dimensional gradient thobaric density (de Szoeke et al. 2000). These two var- of M(V), expressed relative to nonorthogonal slanting iables seem, at least superficially, to be very different. nonunit basis vectors i, j, and k9(V), and of 2gk by pk9(V) Part of the goal in this paper is to derive them within a in the vertical component, is of crucial consequence to common framework so that their relationship can be the balance of forces regarded from isopycnal surfaces understood more easily. A second goal is to compare (de Szoeke 2000). The effective buoyant force acts in their performance in idealized settings that are analyt- the direction of k9(V) 5 $V, that is, normal to isopycnals. ically tractable. We propose three quantitative stan- When the left side of (1.2) is set equal to the Coriolis dards for comparison: 1) How material are the surfaces force, this substitution becomes the basis of the thermal of the variable? 2) How closely do the gradients of the wind calculation of the dynamic height and geostrophic variable correspond to dynamically important density Ð currents: M(V) 5 pdV (Sverdrup et al. 1942). Taking gradients? 3) How neutral are surfaces of the variable? the vertical component of the curl of the momentum The first two standards are motivated directly by early balance relative to isopycnal coordinates furnishes a discoveries about the relationship between thermo- potential vorticity based on the spacing of in situ iso- dynamic variables and motion, and the third is moti- pycnals. The balance of this potential vorticity is vated by considerations of the preferred directions of marred, however, by very strong vortex stretching due mixing. These concepts will now be defined precisely. to motions across in situ density surfaces arising from Strict materiality is the quality of being material, or 1 adiabatic compression. conservative: a fluid variable is material (conservative) if it does not change following the motion, or Dx/Dt 5 0. Quasi-materiality is the property of a variable changing only in response to irreversible, diabatic effects; that is, 2 Notation conventions: $p 5 $ p 1 k› p and $ p 5 i› p 1 Dx=Dt 5 q_x [ $ (K$x), where K is a diffusivity tensor z z z xjz j›yjz p,where›xjz p 5 pxjz, ›yjz p 5 pyjz, etc., are partial derivatives for the variable. In this paper, by materiality we shall at fixed z; ›zp 5 pz, etc., are partial derivatives with respect to z; mean (at very slight risk of confusion) quasi-materiality. i, j, k are orthogonal unit basis vectors, horizontal and vertical; $Vz 5 i›xjVz 1 j›yjVz is the gradient of isopycnal surfaces; ›xjVz 5 zxjV, ›yjVz 5 zyjV, etc., are partial derivatives at constant V; ›Vz 5 zV,etc., 1 (V) (V) It has been traditional in physical oceanography to denote as are partial derivatives with respect to V;and$VM 5 i›xjVM 1 (V) conservative what we call here material. Because the former has j›yjVM is the two-dimensional vector gradient at constant V of the recently been used in a rather different sense (McDougall 2003), Montgomery function M(V). Similar usages with independent vari- we have adopted the latter term. ables other than V will be employed. AUGUST 2009 D E S Z O E K E A N D S P R I N G E R 1781

The density of seawater is a function of potential weighted, between slopes of candidate surfaces and the temperature, salinity, and pressure. If there were a neutral plane slope. A variable that labels such candi- temperature–salinity relation unvarying in space or time date surfaces may be called a neutral density of the first in the oceans, density could be regarded as a function kind. 4 only of pressure and potential temperature u: r21 5 Simple orthobaric density s —a function only of V(p, u). In that case the pressure gradient plus gravi- density and pressure, and called simple to distinguish it tational acceleration combination occurring in the from extended orthobaric density, considered below momentum balance may be replaced by the following: (section 5)—is calculated by assuming a virtual com- 21 22 pressibility r c0(r, p) —also a function only of pres- 1 (u) (u) (u) (u) r $p gk 5 $uM [›uM P ]k9 , sure and density—to correct the in situ density for 4 22 (1.3) compression: u ds 5 dr 2 c0(r, p) dp, where u(r, p) is an integrating factor. De Szoeke et al. (2000) fitted where M(u) 5 gz 1 H(u) is the Montgomery function, the a function of p and r to the global field of sound speed in the ocean to furnish c (r, p). This was shown to be sum of the gravitationalÐ potential and heat content, the 0 (u) p (u) (u) equivalent to fitting a global average u–S relation. Be- latter defined by H 5 0 V(p, u)dp; P 5 [›H /›u]p (u) cause it depends only on the in situ density and pres- is the Exner function; and k9 5 (k 2 $uz)/›uz 5 $u. sure, the simple orthobaric density has the dynamical This has obvious parallels to (1.2), with similar inter- 4 u property (1.3), with s substituted for u. The material pretations. The potential vorticity based on -surface 4 derivative of s may be calculated (de Szoeke and (isentrope) spacing ›uz is far more attractive than the r-surface potential vorticity because the potential tem- Springer 2005). The simple orthobaric density is not perature is material and the dientropic vortex stretching material unless the true sound speed c is identical with is much reduced or negligible (Eliassen and Kleinschmidt the hypothetical c0 everywhere; that is, unless the as- 1954). Also, the potential temperature gradient $u is sumed u–S relation actually applies everywhere. The parallel to the dianeutral vector, difference c0 2 c is a measure of the materiality of the orthobaric density. A normalized index c of material- d 5 r1($r 1 c2$p) (1.4) ity, called the buoyancy gain factor, was devised for simple orthobaric density. This factor is so called be- (about which more will be developed below), when cause it gives the ratio of the apparent buoyancy fre- there is an unvarying temperature–salinity relation. quency squared, based on the orthobaric density, to the Isentropes in this situation are neutral surfaces. This true buoyancy frequency squared. Its difference from property is closely linked to the material property of 1 is proportional to c0 2 c and is, thereby, a measure of potential temperature. the ocean’s deviation from the hypothetical universal The nexus among the three properties of potential u–S relation. One of the tasks of this paper is to quan- temperature—dynamical substitution (1.3), materiality, tify the neutrality of the simple orthobaric density and neutrality—is broken when the temperature–salinity (section 4). Another task is to explore how using spa- relation is not unvarying but dependent on geographic tially variable u–S relations to define the extended or- location. In such a case the dianeutral vector d cannot thobaric density affects materiality and neutrality be parallel everywhere to any scalar gradient, let alone (section 5). $u, though it may be taken to define at any point a Jackett and McDougall (1997) described how to neutral plane to which it is perpendicular. An early compute a variable g, which they called neutral density, hope was that the totality of neutral planes would de- constructed in such a way that its isopleth surfaces3 are fine a generalization of neutral surfaces, but this does closely tangent to neutral planes everywhere. Given, not happen (McDougall 1987a; Davis 1994). The for- say, a synoptic hydrographic section, the method cal- mation of well-defined surfaces from neutral planes culates apparent displacements of water samples on the requires that their helicity vanish everywhere; that is, section from a standard global reference dataset, such as d $ 3 d 5 0. This amounts to demanding an unvary- the Levitus climatology, and associates neutral density ing temperature–salinity relation, which brings us back values preassigned to the reference data to the synoptic to isentropes, closing the logical circle! Failing this, samples. One of the goals of this paper is to clarify the global measures of the closeness to neutral planes of the isopleth surfaces of a given variable can be devised— which could be called the surfaces’ neutrality. For ex- 3 Neutral density and its isopleth surfaces are well defined, even ample, Eden and Willebrand (1999) considered several when neutral surfaces do not exist (i.e., nonzero helicity). We will global indices of the mean-square difference, variously be careful to refer to neutral density surfaces, not neutral surfaces. 1782 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39 definition of this neutral density, designated here as the 2. Neutral density second kind, and, in particular, to distinguish it from a. Preliminaries neutral density of the first kind. McDougall and Jackett (2005a) proposed measures of materiality and neutral- In what follows it is useful to have the following ity for neutral density (section 2), modeled on de definitions and concepts at hand. The equation of state Szoeke et al.’s (2000) work with orthobaric density. r21 5 V(p, u, S) is a function of pressure, potential tem- However, we will show that in their calculation of perature, and salinity (Jackett et al. 2006). The dianeutral neutral-density materiality a consequential term was vector defined by (1.4) may also be written unjustifiably neglected, and their conclusions about the 1 relative materiality of neutral density and orthobaric d 5 a$u b$S, where a 5 V (›V/›u)p,S and density (McDougall and Jackett 2005b) are suspect. In 1 b 5 V (›V/›S)p,u (2.1) addition, we quantify the neutrality of the neutral density for the first time and derive an expression for the (McDougall 1987a). Its vertical component is k d 5 N2/g additional forces that arise on neutral density surfaces, (where k is a unit vertical vector, g is the earth’s gravi- which do not possess the substitution property (1.3) (see tational acceleration, and N is the buoyancy frequency) the appendix). and a, b are the thermal expansion and haline contraction The plan of this paper is as follows. In section 2 coefficients. The dip of a neutral plane, or neutral dip, is we briefly review how the neutral density of a water the two-dimensional vector whose direction in the hori- sample is calculated relative to a reference global hy- zontal gives the orientation of the plane, and whose drographic dataset. In the appendix, we calculate the magnitude is the tangent of the angle between the neutral material derivative of neutral density. We also derive a plane and the horizontal, namely, formula for the difference between the slopes and ori- (d 3 k) 3 k d 1 (k d)k a$ u 1 b$ S entations of neutral density surfaces and neutral h 5 5 5 z z . planes (neutrality), and a generalization of the dynam- k d k d a›zu b›zS ical substitution property (1.3) for neutral density. (2.2) In section 3 we take as the reference dataset for neutral density a simple model of the Atlantic that represents [This usage is common in describing bedding planes in the ocean’s meridional variation of the u–S rela- physical geology (Holmes 1945).] The dianeutral vector tion, though constructed in such a way that in situ iso- is integrable, that is, d 5 u$n, so that h 52$zn/›zn 5 pycnals are level and the reference ocean is motionless. $nz, if and only if the helicity vanishes: Supposing the ‘‘real’’ ocean to have been vertically d $ 3 d 5 m*$p a$u 3 b$S 5 0, (2.3) displaced from the reference ocean by meridionally varying amounts, we calculate, by methods analogous to where m* 5 ›ln (a/b)/›p. The latter has typical recipro- m2 1 ’ Jackett and McDougall’s (1997), the distribution of cal values for seawater of * 5000 dbar (McDougall neutral density and its material derivative. In section 4 1987b; Akitomo 1999; Eden and Willebrand 1999). we calculate the simple orthobaric density, based on Condition (2.3) is satisfied if S 5 S(u, p). In that case, a mean u–S relation of the reference dataset, of the surfaces of constant n are everywhere tangential to displaced real ocean, as well as its material derivative neutral planes (perpendicular to d) and may be properly and neutrality, which we compare to those of neutral called neutral surfaces. The dynamical substitution (1.3) density. We also calculate the dual-hemisphere ortho- is valid, with n substituted for u. (Under the slightly baric density (de Szoeke and Springer 2005), where more restrictive condition S 5 S(u), n may be taken to different mean northern and southern u–S relations be identical to u, as was assumed in the introduction.) representative of the respective hemispheres are em- Various practical kinds of neutral density have been ployed. We show how the two different simple or- devised even when (2.3) is not satisfied. But the tangent thobaric densities in the hemispheres may be patched planes of their surfaces cannot perforce be perpendic- together. In an extension of these ideas, by subdividing ular everywhere to the dianeutral vector, so estimates of the ocean into more and more provinces, each with its their closeness to this ‘‘ideal’’ ought to be formed. This distinctive mean u–S relation, and eventually passing to is one of the tasks undertaken in the following. the obvious continuous limit, we obtain what we have b. Neutral density of the first kind called the extended orthobaric density (section 5), whose properties can be compared to those of neutral Suppose we call $gz the dip of a surface z 5 z(g, x), as density. A discussion and summary conclude the paper labeled by the constant parameter g. The vector dif- (section 6). ference between this dip and the neutral dip, AUGUST 2009 D E S Z O E K E A N D S P R I N G E R 1783

Levitus climatology, from which averaged vertical $gz h, (2.4) reference profiles of potential temperature and sa- 5 is called the neutrality of the surface. Its magnitude linity, uref(p, x), Sref(p, x), at any position x in the j$gz 2 hj gives the angle between the surface and the world’s oceans have been compiled. (One might call neutral plane. When the neutrality is zero, surfaces of this the reference ocean.) The reference surfaces constant g are said to be neutral. Even if not, we shall are specified by the function call these surfaces neutral density surfaces of the first 4 p 5 p (g, x) [or its inverse, g 5 g (p, x)]. kind and call g the neutral density, when $gz ’ h,to ref ref some acceptable degree of approximation. Several ways (2.7) of approximating this equality have been described. One way is to integrate (ii) Next, suppose a water sample has been collected with the values u, S at ambient pressure p, position ›z[g, x(s)]/›s 5 h x9(s)/jjx9(s) , (2.5) x, and time t, from the present ocean (so called to distinguish it from the reference ocean). The target where s is the distance along a horizontal path x(s), to pressure level pr is the pressure at which the refer- form a three-dimensional neutral trajectory x(s), z(g, s). ence ocean’s potential density matches the present By repeating this over a network of such paths chosen to sample’s potential density, each referenced to the cover the entire ocean, and ‘‘connecting’’ paths sharing weighted-average p 5 ~lp 1 lpr [with ~l 1 l 5 1; the same value of g, McDougall (1987a) suggested that Jackett and McDougall (1997) invariably use l 5 ‘‘approximate surfaces’’ may be formed. There are un- ~l 5 0.5 for the weights]. This prescription may be avoidable mathematical difficulties with this approach written as when the helicity of the dianeutral vector does not r r vanish (Davis 1994). However, a realizable approach is V[p, uref(p , x), Sref(p , x)] 5 V(p, u, S) (2.8) to choose z(g, x) to minimize the neutrality globally, that is, to minimize and is called the neutral property (Jackett and r ð McDougall 1997). It furnishes p as a function of 2 u, S, p, and position x. The last step is to obtain jW ($gz h)j dV, (2.6) V the neutral density of the second kind of the water sample by substituting pr into the predetermined r where V is the total volume of the ocean and W is a function gref(p , x). suitable weight function. Eden and Willebrand (1999) Graphically, this identifies the reference neutral considered several similar minimization principles. Neu- density surface that goes through the target level pr at tral density surfaces of the first kind exist even when the position x in the reference ocean, and associates its neutrality $ z 2 h differs from zero. They must be re- g neutral density (first kind) with the present water sam- garded, however, as one step removed from neutral ple. This second kind of neutral density is two steps planes. removed from neutral planes: first by the association the c. Neutral density of the second kind neutral property [Eq. (2.8)] makes between the present ocean and the reference ocean, and second by the re- It is impractical to recalculate the global neutral lation of the reference neutral density to neutral planes. density field whenever a new hydrographic section becomes available. Accordingly, Jackett and McDougall d. Material rate of change: Cross-neutral mass flow (1997) described a two-stage procedure for assigning the Animportantpropertyofneutraldensityofthe value of a second kind of practical neutral density to a second kind can be obtained by taking the material sample of water collected at a certain horizontal posi- derivatives6 of both sides of Eqs. (2.8) and (2.7) and tion and ambient pressure. eliminating Dpr/Dt to furnish an equation for its material (i) The first stage is to prepare beforehand a set of reference neutral density surfaces of the first kind 5 We prefer to use pressure p rather than height z as a vertical from an archive of hydrographic data, such as the coordinate. 6 The material derivative operator is given by any of the fol-

lowing: D/Dt 5 ›tjz 1 u $z 1 w›z 5 ›tjp 1 u $p 1 p_›p 5 ›tjg 1 u $g 1 g›_ g, etc., where w 5 z_ 5 Dz/Dt, p_ 5 Dp/Dt, g_ 5 Dg/Dt, 4 Surfaces may be labeled alternatively by any G(g). It is con- etc.; ›tjz, etc., are partial time derivatives with z, etc., held constant. venient to resolve this ambiguity by setting g to the potential This invariance to reference frame is demonstrated by de Szoeke density at a central location (e.g., Jackett and McDougall 1997). and Samelson (2002). 1784 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39 rate of change, g_ [ Dg/Dt, in the present ocean. (De- tails are given in the appendix.) We call the value of g_ at a point in space and time the materiality; if the materiality were zero everywhere, g would be perfectly material, or conservative. Materiality gives the rate of flow eg (m s21, positive when upward) across neutral density surfaces. This is obtained in the appendix by Eq. (A.2), which is reproduced here: ›p rgeg [ g_ 5 (c^g 1)[~l(› 1 u $) p 1 lu $ pr] ›g |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}t g g I 1 ~lm(p pr)D1u $ q |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}g ref II N2 $ pr 1 D1 ref g h u g r g ref |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ref III FIG. 1. Neutral density schematic. Two neutral density surfaces, 1 ( p) g and g95g 1Ds, are shown in the (top) ‘‘present’’ dataset (at two 1 D|fflfflfflffl{zfflfflfflffl}q_ . (2.9) times: t and t 1Dt) and (bottom) in the reference dataset (fixed). IV The distinction between the present thickness p92p 5Ds›p/›g and the reference thickness p9r 2 pr 5Ds›pr/›g is illustrated. Ap- ^g r 1 In this equation, c 5 1 m[q qref(p , x)]D is the parent vertical motions in the present dataset, wa 5 (›t 1 u $)g p, r r neutral-density buoyancy gain factor;7 q, s are normal- and in the reference dataset, wa 5 u $g p [using a horizontal ized forms of u, S (slightly nonlinearly transformed in current from the present data at p, x, t (or g, x, t)], are indicated. the case of q; see the appendix); m 5 Vup/Vu is the thermobaric parameter (McDougall 1987b); D is a factor given by (A4) in the appendix; and Ds›p/›g is the (III) reference neutrality, that is, differences between r vertical thickness (in pressure units, i.e., decibars) of an the neutral density surface dip $gzref 52$g p / increment Ds of neutral density. The right side of (2.9) (rrefg) and the true neutral dip href (see the ap- uses a simplified equation of state given by Eqs. (A.1a) pendix) in the reference dataset; and and (A.1b) in the appendix. (IV) the irreversible rates of change of temperature One may distinguish four types of contributions to the and salinity that constitute the buoyancy rate of ( p) material rate of change, or the diapycnal material flow, change q_ [see Eq. (A5) in the appendix]. arising from the following sources: e. Neutrality (I) the sum of the apparent vertical motion in the Another result established in the appendix is an present ocean, wa 5 (›t 1 u $)g p, as though the constant-g surface were material, and in the ref- equation for the neutrality, the difference in the present r r ocean between the dip $gz of the neutral density (of the erence ocean, wa 5 u $g p (Fig. 1), each weighted by ~l or l, respectively, and attenuated by c^g 1, second kind) and the neutral dip h. This difference is which is a measure of the departure of the water given by Eq. (A.9): sample’s q, s from the local reference u–S relation; N2 (II) differences between ambient pressure in the pres- ($ z h) 5 D(c^g 1)$ (~lp 1 lpr) ent data and target pressure in the reference da- g g |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}g taset, p 2 pr, combined with advection of the I 1 m(p pr)$ (~lq 1 lq) reference temperature on neutral density surfaces |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}g ref (if the reference temperature changes on the ref- II erence neutral density surfaces, it is because of 2 r N $g p the spatial variation of the reference ocean’s u–S 1 ref h . (2.10) g r g ref relation); |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ref III

7 McDougall and Jackett (2005a) deﬁne cg 1 5 l(c^g 1), One may identify contributions analogous to types I, II, which is convenient when l 5 ~l 5 0.5 is used. and III for materiality. Note the distinction between the AUGUST 2009 D E S Z O E K E A N D S P R I N G E R 1785 neutralities of the present ocean (left-hand side) and the reference ocean (term III). McDougall and Jackett (2005b) derived an expression similar to (2.9). Jackett and McDougall (1997) argued that, given their method of preparing the reference dataset, the type III term in (2.9) [and by in- ference in (2.10)] is small. McDougall and Jackett (2005a) conducted a global census of c^g from the world hydrographic dataset showing that it is very close to

1(q 2 qref ’ 0) and concluded that the type I contribution to the materiality (and, one may infer, neutrality) of neutral density due to deviations from the local u–S relation is negligibly small. Further, following from a r r claim that m(q 2 qref)u $g p (type I) and m(p 2 p )u $gqref (type II) were typically of the same order, they asserted that diapycnal flow contributions from spatial gradients of the reference u–S relation were like- FIG. 2. The u–S relation of the Atlantic on the 500-dbar pressure wise negligible. This claim is very dubious. In section 3 surface. The ordinate is q: shifted, transformed, and normalized potential temperature. The abscissa is normalized and shifted sa- we shall present an illustration clarifying that type II linity. The grayscale clouds represent data points taken from the contributions to neutrality and materiality cannot be World Ocean Circulation Experiment (WOCE). The coefficients neglected. and parameters a1, a2, b, u0, S0, given in the appendix, are chosen so that unit changes in q or s give unit changes in the normalized f. Neutral density in the displaced reference ocean specific volume V/V0. The solid lines are analytic representations of the main southern and northern u–S lines; their slopes are R 5 The treatment in the appendix is quite general with dq/ds 5 2.0. The dashed line is a constant-density line linking the regard to the reference dataset and the present ocean. A main lines at 500 dbar. special case is of particular interest. Suppose the present ocean is obtained from the reference ocean by displac- tions as the cause of nonzero helicity and the nonexis- ing the latter vertically by an increment Dp, possibly tence of neutral surfaces. We will suppose the present itself a function of p, x, and t: ocean to be a vertically displaced form of the reference ocean from which we can calculate the neutral density q 5 q (p 1Dp, x), and s 5 s (p 1Dp, x). (2.11) ref ref distribution and its materiality and neutrality.

When this is substituted into the right side of (2.8), an a. A simple model of the warm-water obvious solution for target pressure is Atlantic Ocean r p 5 p 1Dp. (2.12) The u–S relation of the Atlantic on the horizontal 500-dbar pressure surface is shown in Fig. 2 in terms of r ^g Hence, q 2 qref(p , x) 5 0 and c 5 1 exactly, so that the modified variables q, s [Eq. (A1b) in the appendix]. the type I terms in the materiality and neutrality of The diagram shows the main southern and northern neutral density, Eqs. (2.9) and (2.10), vanish. But the r lines of the tropical to temperate upper ocean, with a type II terms, proportional to p 2 p 52Dp 6¼ 0, do not. constant-density bridge linking the main lines. The nonlinear transformation of potential temperature u has 3. Properties of neutral density in an straightened a slight curvature of the lines. On any idealized ocean pressure surface shallower than 1500 dbar [only 500 dbar is shown here, but see de Szoeke and Springer To evaluate the performance of neutral density (and, (2005) for more pressure surfaces], the main lines are in section 4, orthobaric density) with regard to materi- practically identical, while the bridge lies at different ality and neutrality, it is useful to consider a simplified densities. The two fitted lines in Fig. 2, model of the Atlantic Ocean’s meridional temperature 1 and salinity structure that illustrates its salient hydro- s 5 R q 7 s1, with R ’ 2.0 (density ratio) and graphic features. This model will be taken as the ref- s ’ 0.23 3 103, (3.1) erence dataset for the neutral density algorithm. Its 1 motivation is the spatial dependence of the Atlantic’s may be considered idealized representations of the main u–S relation, which was identified in the preceding sec- u–S lines at temperatures above 48C(q $ 20.11 3 1023). 1786 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39

FIG. 3. Profiles of u vs S at fixed latitudes on a meridional section FIG. 4. A cross section of u (solid) and S (dashed) in a simple of the Atlantic (grayscale lines). (See Fig. 2 caption and the ap- model of the quiescent warm-water Atlantic. The in situ isopycnals pendix for coefficients and parameters.) The variation is greatest are level. between 158S and 158N. At higher latitudes, the northern and southern u–S relations are nearly constant. The idealized northern and southern u–S relations in Fig. 2 are shown. Atlantic bridge region. For y/L .11(y/L ,21), we replace y/L in (3.2a) by 11(21). We take 2L ’ 3000 km More elaborate representations of the u–S lines than and n 5 1 2 R21 ’ 0.5. (The origin, y 5 0, need not be

(3.1), taking account of the convergence of the northern the equator.) The qref and sref fields are chosen to give a and southern lines at colder temperatures, may readily horizontally uniform profile of the normalized specific be considered, though in the interests of brevity we have volume anomaly, foregone such a treatment here. It may be more con- d p 5 s9 p ventional to show the spatial variation of u–S relations as ref( ) B , (3.2c) in Fig. 3, where each line of data represents a meridional here assumed linear with depth with constant coefficient location. This illustrates the evolution along a meridional s9 . We call this the quiescent reference ocean (Fig. 4). section from the southern to the northern extremes ide- B One might as well assign the d values of level pressure alized in Eq. (3.1). The near-constant density bridge surfaces (or rather their negatives) as the neutral den- structure at each pressure, exemplified in Fig. 2, unfolds sity labels of the reference ocean, that is, as in Eq. (2.7):8 to form the meridionally dependent u–S profiles of Fig. 3. The fit of parallel, straight lines to the q–s relation at gref(p, x) 5 dref( p) 5 sB9 p. (3.2d) each latitude for waters warmer than 48C or shallower than 1500 dbar seems quite accurate. The lines are biased If dref is eliminated from (3.2a) and (3.2b), a meridio- to the saltier side on passing from 208Sto208N, but are nally variable reference q–s relation emerges: unvarying with latitude south and north, respectively, of 1 these limits (overlooking some anomalously salty water sref 5 R qref 1 (y/L)s1, where jjy/L # 1 (3.2e) from the Mediterranean at northern midlatitudes). A linear fit to this bias between 208Sand208N may be (cf. Fig. 3). The q–s relations at successive positions oversimplified but not unrepresentative. from y 52L to L are straight lines of slope R, shifting Thus, Figs. 2 and 3 motivate the following idealiza- to saltier water. The total shift is 2s1. South of y 52L tions of the temperature and salinity fields of the (or north of y 5 L) the q–s relation is taken to be warmer layers of the subtropical to tropical Atlantic: spatially uniform, as given by Eq. (3.1). The end-point northern and southern q–s relations, though very sim- d ( p) 1 ( y/L)s q (p, y) 5 ref 1 , and (3.2a) ple, represent the real ocean very well, as does the ref n 1 mp mapping of the intermediate q–s relations onto the

sref(p, y) 5 (1 1 mp)qref(p, y) dref(p), (3.2b) 8 More generally, if the in situ density surfaces of the reference for jy/Lj # 1, where y 56L are the northern and dataset were not level, the reference neutral density surfaces r southern geographical boundaries of the transequatorial would depend on position, g 5 gref(p , x) (see the appendix). AUGUST 2009 D E S Z O E K E A N D S P R I N G E R 1787 constant-density bridges (Fig. 2). More schematic than (for p , p0 ; 1000 dbar, say), suppose the displacement strictly accurate is the assumption of a meridionally is linear in x and y: linear transition between the end points (Fig. 3). D ’D « 1 1 A consequence of the density-compensated bridge p 0( x/L y/L 1)/2, (3.4) structure [Eq. (3.2)] is that in situ isopycnals and neutral with D 5 100 m. The dominant, type II, contribution to planes (McDougall 1987a) are horizontal in the refer- 0 Eq. (2.9) is, using (3.3), ence dataset. The dianeutral vector [Eq. (2.1)] is everywhere vertical, ~lm s ms eg ’ Dpy 1 ’ ~l 1 yD (y/L 1 1)/2. 1 2 II rgs9 nL rgns9L 0 dref 5 krref g[sB9 1 mqref(p, y)] 5 kg Nref(p, y); (3.2f) B B (3.5) hence, the reference helicity is trivially zero: dref $ 3 dref 5 0. The reference neutrality is also zero; the refer- The parameter « in (3.4) may be so small as to be neg- r ence state exactly (and trivially) satisfies 2$g p /(rrefg) 5 ligible in Dp in (3.5), but it is essential in providing a 21 href 5 0, so that the type III terms that occur in (2.9) nonzero meridional velocity y 5 f sB9(p 2 p0)D0«/2L and (2.10) are absent. Reference-neutral surfaces are to make (3.5) nonzero and nontrivial. Fundamentally, level. the zonal structure of (3.4) provides three-dimensional From Eq. (3.2) describing the quiescent reference structure in the displaced ocean to furnish nonzero state, one obtains helicity. The meridional dependence of this is shown in 8 Fig. 5a for ~l 5 0.5, normalized by a parameter including > 2 r > Nref ›p 1 the meridional velocity y. > 5 sB9 1 mqref ’ sB9, 5 , > r g2 ›g s9 The neutrality of neutral density is given by Eq. (2.10). > ref B <> Applying this formula to the displaced reference ocean, ›q s9 mq s9 ref 5 B ref ’ B , (3.3) in which only the type II term on the right survives, one > r r > ›p n 1 mp n obtains, using the approximations of (3.3), > > > s /L s : q 5 1 ’ 1 mDp s ms D $g ref j r j , 1 1 0 n 1 mp nL $gz h ’ j ’ j (y/L 1 1)/2. rgsB9 nL rgnsB9L where j is the meridional unit vector. [When approxi- (3.6) mations are given in (3.3), the contributions from the small parameter m have been neglected.] This difference, or mismatch, is shown in Fig. 5b. Cu- riously, even if « [ 0, despite the displaced ocean’s b. Idealized neutral density in the displaced being perfectly two-dimensional with zero helicity (so reference ocean that neutral surfaces are well defined), the neutrality of Let the place of the hydrographic archive used as the neutral density surfaces does not vanish. An estimate reference data for the neutral density algorithm (2.8) be of the dimensionless parameter appearing in both (3.5) taken by the two-dimensional quiescent reference state and (3.6) is [Eq. (3.2)]. Suppose that the present ocean is the reference ocean displaced by Dp(p, x), as in Eqs. (2.11). In m s D (5000 dbar)1 (0.23 3 103) 1 0 ’ general, then, helicity will not be zero and neutral sur- 1 nrg L sB9 (0.5)(1 dbar m ) (1500 km) faces will not exist. However, neutral density and its (100 dbar) 5 isopleth surfaces are well defined and can be obtained 3 ’ 1.1 3 10 . (0.6 3 106 dbar1) by inserting Eq. (2.11) on the left of (2.8). As remarked r p g 25 27 26 21 above, the target pressure is then given by (2.12), and This gives e ; 10 y ; 10 –10 ms for y ; the neutral density by 1022–1021 ms21. In comparison to typical Ekman pump- r ing magnitudes near the ocean surface (;1026 ms21), g 5 sB9p 5 sB9(p 1Dp). such dianeutral material flows are not inconsiderable. Neutral density surfaces are described by pr 5 const. Interesting properties of neutral density, such as mate- 4. Properties of orthobaric density in an riality and neutrality, may be estimated. The buoyancy idealized ocean gain factor for the displaced ocean is c^g 5 1 (because r q 2 qref(p , x) 5 0), and type I terms in (2.9) and (2.10) The idealized representation in section 3 of the vanish. In a pressure range centered around 500 dbar Atlantic Ocean illustrates well the central irremediable 1788 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39

g FIG. 5. (top) The type II contribution to diapycnal mass ﬂow eII [Eq. (2.9)] for neutral density 4 (dashed), and the type I contribution to diapycnal mass ﬂow eI [Eq. (4.2)] for simple (global) orthobaric density (solid), with both normalized by ms1/(nsB9 )yD0/(Lrg). (bottom) Neutrality for neutral density [difference (mismatch) between the dip of the surfaces and the neutral dip; see Eq. (2.10)] (dashed), and for simple (global) orthobaric density [Eq. (4.5)] (solid), nor-

malized by ms1/(nsB9 )D0/Lrg. feature of the ocean’s density field that gives nonzero a. Simple orthobaric density helicity: the spatial variation of the u–S relation. We An orthobaric density function for the entire ideal- now turn to calculating orthobaric density functions ized Atlantic may be defined in the following way. Take predicated on u–S relations from various locations in the the reference q–s relation (3.2e) at some central loca- reference ocean. These relations and density functions tion, say y 5 0, that is, s 5 R21q . By (3.2a), q j are summarized in Table 1. We also calculate their ref ref ref y50 is related to a normalized specific volume anomaly d buoyancy gain factors (Table 2), cross-isopleth mass [Eq. (A1a) in the appendix] and pressure p by flows (materiality), and neutrality (difference of the surfaces’ $z from neutral dip h), and compare them with d qrefjy50 5 . those for neutral density. n 1 mp

TABLE 1. Orthobaric density functions.

q–s (q–d) relation Orthobaric density d Global (y 5 0) s 5 R1q [q j 5 d(n 1 mp)1] s4 5 ref ref ref 0 1 1 mp/n d s Northern (y 5 L) s 5 R1q 1 s [q j 5 (d 1 s )(n 1 mp)1] sN 5 m(L)1 1 1 s ref ref 1 ref L 1 1 1 mp/n 1 d 1 s Southern (y 52L) s 5 R1q s [q j 5 (d s )(n 1 mp)1] sS 5 m(L)1 1 s ref ref 1 ref L 1 1 1 mp/n 1 d s y /L (y ) 1 1 j Arbitrary center (y ) s 5 R1q 1 s y /L s j 5 m(y ) 1 s y /L j ref ref 1 j j 1 1 mp/n 1 j [q j 5 (d 1 s y /L)(n 1 mp)1] ref yj 1 j ms y Note: m(y) 5 1 1 1 nsB9L AUGUST 2009 D E S Z O E K E A N D S P R I N G E R 1789

TABLE 2. Buoyancy gain factors in the displaced reference ocean for various orthobaric density functions. 0 1 1, y . L s m 4 1 Global c4 1 ’ 1 @ y/L, jjy , L A u~ 5 (1 1 mp/n) ns9 B 1, y , L s m 0, y . L Northern cN 1 ’ 1 u~N 5 (1 1 mp/n)1m(L)1 ns9 1 y/L,0, y , L B s m 1 y/L, L , y , 0 Southern cS 1 ’ 1 u~S 5 (1 1 mp/n)1m(L)1 0, y , L nsB9 y y s1m j 1 1 Arbitrary center (yj) cj 1 ’ , jy yjj , L/(N 1) u~j 5 (1 1 mp/n) m(yj) nsB9 L Extended cO 2 1 ’ 0 u~O 5 (1 1 mp/n)1m(y)1

By substituting this into the adiabatic compressibility is the buoyancy gain factor, in which anomaly (›d/›p) 5 mq, one obtains the virtual com- q,s 4 4 pressibility anomaly, which may be used to pose the qrefjy50(s ) 5 s /n and 4 4 following differential equation: u~ 5 ›s /›d 5 (1 1 mp/n)1. dd md 5 mq j 5 . Only at the location from which the reference q–s re- dp ref y50 n 1 mp lation has been taken (y 5 0) will q 5 qref in the dis- 4 This furnishes the integral 2d/(1 1 mp/n) 5 const. One placed ocean, and c 5 1, reducing term I in (4.2) to may take the value of this ‘‘constant,’’ deﬁning a surface zero. The remaining part of (4.2), labeled IV, is a term 4 ( p) in d, p space, at a reference pressure, say p 5 0, to be s . proportional to q_ , the sum of all diabatic, irreversible In this way one obtains a simple ‘‘global’’ orthobaric inﬂuences on the negative buoyancy [Eq. (A5), see the density9 as a function of d, p: appendix). We have assigned the terms in (4.2) type designations based on their appearance analogous to d s4 5 . (4.1) Eq. (2.9). Note that terms of types II or III do not appear. 1 1 mp/n The buoyancy gain factor of simple orthobaric density

4 for the displaced reference ocean is listed in Table 2 and The material rate of change of s may be calculated. It shown in Fig. 6. Note the linear decrease with y of 4 can be written as c 2 1. A similar decrease is evident in an Atlantic 4 4 4 4 4 meridional section of c 2 1 (de Szoeke et al. 2000, rge [ ›p/›s s_ 5 (c 1)(› 1 u $) 4 p |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}t s their Fig. 11). (Since the reference u–S relation in the I latter calculation was taken from southern waters, there 4 4 4 (p) 1 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}›p/›s u~ c q_ (4.2) is a positive bias in the real-world section compared to IV the idealized section.) Unlike the type I term in the neutral density materiality, Eq. (2.9), the contribution (de Szoeke et al. 2000; de Szoeke and Springer 2005). 4 of the term of which this is a factor to material flow Here, Ds›p/›s is the vertical thickness (dbar) of an 4 across orthobaric isopycnals does not vanish. Consider increment Ds of orthobaric density. In addition, e is 4 the steady-state estimate of the apparent vertical motion, the net flow (m s21) across a s isopycnal and consists (›t 1 u $)s4 p ’ y(›p/›y)s4 ’ yD0/2L. The contribution of two parts. The first is an adiabatic part—a residual of from the type I term of (4.2) to diapycnal flow is the compressibility effect—given by the apparent mo-

4 ms y tion (›t 1 u $)s p of orthobaric isopycnals, attenuated 4 1 4 eI ’ yD0 . (4.4) by c 2 1, where rgnsB9L 2L

4 4 4 4 1 c 5 f1 1 u~ m[q qrefjy50(s )]›p/›s g (4.3) The meridional dependence of this, normalized by

ms1D0/(rgnsB9L)y, is shown in Fig. 5a; also shown is the 4 meridional average value of jeI j (similarly normalized). 4 One may compare e to eg , given by (3.5). Though their 9 More systematically, de Szoeke et al. (2000) deﬁned a global I II orthobaric density function on the basis of a volumetrically aver- meridional dependences are offset one from the other, aged compressibility function—equivalent to a volumetrically av- their magnitudes and parameter dependences are sim- g eraged u–S relation. ilar. The latter, eII, was neglected by McDougall and 1790 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39

FIG. 6. (top) A dual-hemisphere orthobaric isopycnal for the displaced reference ocean. Note the discontinuity at y 5 0. Meridional motion will carry material through this discontinuity (leakage). The dotted line is a surface of simple orthobaric density based on a single global- average u–S relation. (bottom) Buoyancy gain factor c differences from 1 (Table 2, lines 2 and 3) (solid) for the joint isopycnal in the top panel. The factor does not account for the material ﬂow through the discontinuity at y 5 0. The dotted line is the buoyancy gain factor for the simple, global orthobaric isopycnal (Table 2, line 1).

g g Jackett (2005a). From (2.9) and (4.2), eI } c 2 1 and b. Regional orthobaric density functions 4 4 eI } c 2 1. McDougall and Jackett (2005a) show per- g 4 g 4 Orthobaric density functions can be defined specific suasively that c 2 1 c 2 1, so that indeed eI eI . g g to various regions, each predicated on a reference q–s In our illustration, in fact, eI [ 0, c 2 1 [ 0. But g relation typical of its region. To construct global cor- in compensation it must be recognized that e is not II rected density surfaces, we must specify how to match negligible. the various density functions across their interregional The dip of an orthobaric density surface $ 4 p/(rg) s boundaries (de Szoeke and Springer 2005). We will il- differs from the neutral plane dip h [Eq. (2.2)] by lustrate this difficulty with some examples. $ 4 p 4 $ 4 p 4 s h 5 (c 1) s ’ (c 1)h 1) NORTHERN ORTHOBARIC DENSITY rg rg ms D y An orthobaric density might be predicated on the q–s ’ j 1 0 (4.5) 21 ns9rgL 2L relation (3.2e) at y 5 L, sref 5 R qref 1 s1. This choice B is important because it characterizes the water mass (de Szoeke et al. 2000). The last approximate equality properties of much of the northern North Atlantic, and follows from the first row entry of Table 2, and has a the northern portion of the tropical bridge region. If such magnitude similar to the neutrality of neutral density a choice is pursued along the preceding lines, one arrives surfaces [Eq. (3.6)]. See Fig. 5b. at the specification of what might be called the northern The panels in Fig. 5 illustrate a major result of this orthobaric density function sN, given in the second row paper. They show typical estimates of diapycnal flow of Table 1. [Any function of (2d 2 s1)/(1 1 mp/n)might (materiality) and the neutrality of neutral density and be chosen for sN. We have chosen a particular linear orthobaric density. The dianeutral material flow eg must function, for a reason that will become clear presently.] be expected to have magnitudes similar to the ortho- The material derivative of s N gives an equation exactly 4 baric diapycnal flow e . Likewise, the neutralities of like (4.2), involving a buoyancy gain factor c N, defined neutral density and orthobaric density are similar. like (4.3), with 4 superscripts replaced by N superscripts, AUGUST 2009 D E S Z O E K E A N D S P R I N G E R 1791

N 2 21 and with qrefjy5L(s ) given in Table 1. The buoyancy (m s , positive when toward lighter water). Divided gain factor cN and the integrating factor u~N for this by 2L, this gives an equivalent cross-isopycnal mass northern orthobaric density are listed in Table 2 for the flow, averaged over the bridge region, of displaced reference ocean; cN is shown in Fig. 6. 2s m s mD Buoyancy gain factors on a meridional Atlantic section O 1 1 0 eII ’ (yDp)y50/2L ’ (y)y50. (4.8) based on a northern (and a southern) u–S relation are rgnsB9 2rgnsB9L shown in de Szoeke and Springer’s (2005) Fig. 13. This will be shown normalized below (see Fig. 8a). This 2) SOUTHERN ORTHOBARIC DENSITY form is convenient for comparison to (4.4) (it is twice 4 the average of jeI j), the flow across simple orthobaric All this might be repeated to develop another ortho- 4 baric density function sS based on the q–s relation in isopycnals (s 5 const), and to (3.5), the material flow (3.2e) at the southern end of the tropical bridge, y 52L; across neutral density surfaces. 21 that is, sref 5 R qref 2 s1. This will yield the entry in the third row of Table 1, a linear function of (2d 1 s1)/ 5. Extended orthobaric density (1 1 mp/n). Again, the material derivative of sS, analogous to (4.2), may be obtained. [See also de Szoeke The bridge region need not be split into only two parts, and Springer’s (2005) Fig. 13.] northern and southern, as for the dual-hemisphere density function. Suppose the bridge region is subdivided 3) DUAL-HEMISPHERE ORTHOBARIC DENSITY into a number N of intervals jy yjj , L/(N 1), where N S The constant m factors specifying s , s in Table 1 yj 5 L(2j N 1)/(N 1) for j 5 1, ... N. The treat- have been chosen so that when the undisplaced refer- ment outlined above for the dual-hemisphere density ence bridge specific volume anomaly (SVA) profile function may be applied, with obvious modifications, to applies, d 52sB9p, the two are numerically identical. each such subinterval, characterized by the u–S relation Thus, the overall dual-hemisphere quasi-orthobaric given by (3.2e) at its center, y 5 yj. This leads to the density function defined by specification in each subinterval of an orthobaric den- (y ) sity function s j listed in the fourth row of Table 1. sO(p, d, y) 5 sN(p, d), for y . 0, and Each such density function has an associated buoyancy 5 sS(p, d), for y , 0, (4.6) gain factor cj (Table 2). The boundaries between the subintervals correspond to y 5 0 in the dual-hemisphere will be continuous across y 5 0 if the said density profile orthobaric density. The cj 21 factors determining the pertains there. However, if the density profile differs quasi-materiality of a local orthobaric density, defined by reference to the local u–S relation, are reduced by a from this standard, say, d ’2sB9(p 1Dp), as in the displaced ocean of section 2, the difference between the factor of 1/(N21). The discontinuities, analogous to two expressions of (4.6) on either side of y 5 0 is ap- (4.7), then occurring at the edges of subintervals, 5 proximately (using entries from Table 1) yj11/2 L(2j N)/(N 1), are similarly reduced, O (y ) (y ) Ds 5 [s j11 (p, d) s j (p, d)] O O O d’sB9( p1Dp) Ds 5 s 5 1 s 5 y 0 y 0 1 N S ’ 2s1mn Dpjy /(N 1), 5 [s (p, d) s (p, d)] j11/2 d’sB9( p1Dp) 1 but are N21 times more numerous (Fig. 7). Given a ’ 2s1mn Dp. (4.7) meridional flow y, there will be a mass transport through (In the approximate equality, higher-order terms in m each such gap, which, averaged over the spacing be- are neglected.) This gives a discontinuous displacement tween gaps, 2L/(N21), may be thought of as an equiv- for a sO isopycnal across y 5 0of alent diapycnal flow,

O s1m O mD eII ’ (yDp)j , (5.1) Ds 2s1 p yj11/2 Dz 5 ’ LrgnsB9 rgsB9 rgnsB9 distributed over the interval jyyj11/2j,L/(N1). These (deeper on the north side, Dz , 0 when displacement flows are shown in Fig. 8 for N 5 2 (dual-hemisphere from the standard profile is upward, or Dp . 0; see orthobaric density), 4, and 10; also shown are the (me- O Fig. 6). This discontinuity is a potential site of material ridionally unvarying) averages of |eI | ; O[1/(N 2 1)] in O leakage. Given meridional velocity y, there is volume each subinterval. In the limit N / ‘, eII tends to the flux through this discontinuity of 2s1m/(rgnsB9)(vDp)y50 continuous line shown in Fig. 8. 1792 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39

FIG. 7. (top) Examples of multilinked orthobaric isopycnals based on u–S relations centered on multiple intervals for link numbers N 5 4, 10. (The latter has been offset deeper by 100 dbar.) Note the discontinuities at the joints. Meridional motion will carry material through these discontinuities. The dotted line is the extended orthobaric isopycnal (N / ‘). (bottom) Buoyancy gain factor differences from 1 or 1.1 for the multilinked isopycnals in the top panel. These factors do not account for the material ﬂow through the discontinuities. The buoyancy gain factor for the extended orthobaric isopycnal (dotted) is 1.

An alternative view is gained by starting from the a. Material derivative continuous limit of the arbitrarily centered extended The material rate of change of (5.2) may be written orthobaric density function (fourth row, Table 1):

O (y) s (d, p, y) 5 s (d, p) rgeO [ s_ O›p/›sO 5 (cO 1)(› 1 u $) p |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}t sO 1 d (y/L)s1 I 5 m(y) 1 (y/L)s1 . (5.2) 1 1 mp/n s m ›p sO v 1 u~OcO (1 1 mp/n) p nL ›sO s9 The reference temperature field (3.2a) may be expressed |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}B in terms of this entity as II O O O ( p) 1 |fflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflffl}›p/›s u~ c q_ , (5.4) O 1 O qref(s ; y) 5 n [m(y)s 1 (y/L)s1]. IV

If the ocean were in the quiescent reference state given where by (3.2), then, substituting in (5.2), ÈÉ O O O O 1 c 5 1 1 u~ m[q qref(s ; y)]›p/›s , s9p 1 1 sO 5 B . u~O 5 (1 1 mp/n) m(y) . 1 1 mp/n

Otherwise, in the state displaced by Dp from the qui- Equation (5.4) gives the rate at which matter crosses a O escent, s isopycnal. The terms labeled I and IV, a residual compressibility effect and a diabatic contribution, re- O 1 s sB9p(1 1 mp/n) ’ sB9Dp. (5.3) spectively, are analogous to similar terms in (4.2). For the displaced reference ocean, it can be easily shown O O This approximation will be used below. that q 2 qref(s ; y) 5 0, so that c 5 1, and the ﬁrst AUGUST 2009 D E S Z O E K E A N D S P R I N G E R 1793

O FIG. 8. Type II diapycnal flow eII [Eq. (5.1)] for discrete extended orthobaric density, equivalent to the meridional mass flow through the gaps shown in Fig. 7, and located at the heavy vertical lines, for several choices of numbers N of links: (top) N 5 2 (dual-hemispheric O orthobaric density), (middle) N 5 4, and (bottom) N 5 10. The continuous limit (N / ‘)ofeII neutral [Eq. (5.5)] is shown in all panels. Type II neutral density diapycnal flow eII [Eq. (2.9)] is also O shown in the bottom panel. Type I diapycnal flow eI for extended orthobaric density is shown for N 5 4, 10. All normalized by ms1/(nsB9)yD0/(Lrg). term in (5.4) vanishes. The contribution of term II may contributions to the EOB diapycnal flow remain, re- be written sembling strongly, other than the factor ~l, the diapycnal flow for neutral density. McDougall and Jackett (2005a), m s ms eO ’ 1 yDp ’ 1 yD ( y/L 1 1)/2. (5.5) while confirming the smallness of the type I contribu- II rgs9 nL rgs9nL 0 B B tions, completely neglected the type II contributions to neutral density materiality. Their conclusion that neutral It is proportional to the meridional motion y, the merid- density is superior to orthobaric density in its material ional temperature gradient on sO surfaces ›q (sO; y)/ ref properties rests on this fallacy. ›y ’ s1/nL, the thermobaric coefficient m, the reciprocal O O O 2 2 buoyancy frequency squared ›p/›s u~ c ’ rg /N , b. Neutrality and the displacement Dp from the quiescent state. It coincides with the continuous limit of (5.1), as shown in The difference between the dips of extended orthobaric Fig. 8. The neutral density diapycnal flow [Eq. (3.5); see density surfaces and neutral planes can be calculated. Fig. 8c] differs only in the factor ~l. Take the quasi-horizontal gradient of (5.2) on constant O Reviewing the progression from simple global ortho- s surfaces. This gives, after some manipulation, baric density to dual-hemisphere orthobaric density, to N-fold discrete extended orthobaric density (EOB), to O $sO z h 5 (c 1)($pz $sO z) continuous EOB, one sees that type I diapycnal flow, g ms the sole reversible contributor to diapycnal flow across 1 j 1 [sO(1 1 mp/n) 1 s9p]. N2 nLs9 B simple orthobaric density surfaces, is augmented by a B type II contribution when dual-hemisphere EOB is con- (5.6) sidered, of about the same magnitude. As the number N of subintervals is increased, the type II contributions For the displaced reference ocean, again cO 5 1, and supplant the type I contributions, which dwindle in- (5.3) pertains, so that the last factor in the second term is 2 versely with N. In the continuous limit, only type II 2sB9Dp; also N /g ’ rgsB9. Hence, 1794 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39 ms Dp ms D y motion) of the quiescent state, and computed distribu- $ z ’ 1 ’ 1 0 1 1 . sO h j j tions of materiality and neutrality for each candidate rgsB9nL rgsB9nL 2L L variable. While this test does not exhaust the range of (5.7) real-world possibilities, it throws light on the relative performances of the variables. This is indistinguishable from Eq. (3.6), the neutrality We compared, in Fig. 5, the material flow (material- of neutral density, shown in Fig. 5b. One may compare ity) across isopleths of neutral density and simple or- it to (4.5), the neutrality of simple global orthobaric thobaric density, and the mismatch (neutrality) of the density surfaces, also shown in Fig. 5b. While the me- dips of said surfaces from the neutral dip. In sections 2 ridional dependence is shifted and reversed, the amount and 4, we identified several contributions to each of of variation is identical. these indices: type I, due to divergence of the local u–S relation from the reference u–S relation, and type II, from the meridional gradient of the reference u–S re- 6. Summary and discussion lation. Because the meridional variation of the u–S Density is an important variable, not merely as a relation is built into the reference ocean, there is by tracer, but especially because of its link to motion ac- construction no type I contribution for neutral density, celeration through the buoyant force. As in situ density’s though there must be a type II contribution. For simple utility as a tracer is limited by its susceptibility to adia- orthobaric density, on the other hand, the reference u–S batic compression, efforts are made to correct for this relation is meridionally unvarying so there is no type II effect. This would be uncontroversial if the temperature– contribution, although, since the local u–S relation dif- salinity relation of the ocean were unvarying with po- fers from the reference relation, there must be a type I sition. As this is not so, however, only partial solutions contribution. Although their spatial distributions are to the difficulty are possible. This paper has considered different, the respective types of contribution to the two attempts at such solutions: neutral density and materiality and neutrality of the two variables are sim- orthobaric density. The results are judged by a num- ilar and comparable in magnitude. McDougall and ber of criteria: materiality, the degree to which reversible Jackett (2005a), who reached a different conclusion, contributions (apart from irreversible, diabatic con- which favored neutral density, did so by overlooking the tributions) to the substantial rate of change of the cor- type II contributions. rected density variable are minimal; neutrality, the A dual-hemisphere orthobaric density function was degree to which isopleth surfaces of the variable are devised (section 4). It takes advantage of the close everywhere normal to the dianeutral vector d [Eq. hemispheric validity of two different u–S relations: (1.4)]; and the degree to which buoyancy forces can be southern and northern. This permits the local reduction regarded as acting across isopleth surfaces, embodied in of type I materiality and neutrality contributions. But a the paradigm of Eq. (1.3) for isentropes when the correspondence between the two distinct orthobaric temperature–salinity relation is unvarying. These crite- density functions must be established in order to con- ria are not independent, as the resemblances of nect isopleth surfaces between hemispheres. This is Eq. (2.10) (neutrality) to (2.9) (materiality) show. achieved by supposing meridional continuity when the Our approach was analytic. We constructed an ide- ocean is in the reference state. When the ocean is not in alization of the temperature–salinity structure of the the reference state, discontinuities occur in the surfaces upper kilometer or more of the Atlantic Ocean, in connected across the interhemispheric boundary (the which the ocean transitions continuously with latitude equator). These are important because, even apart from a southern u–S relation to a northern one. A from the materiality of the surfaces in the respective simplified equation of state was assumed, though with a hemispheres, they are sites where matter may pass representative, nonzero, thermobaric parameter, Vup/Vu. through the gaps in the surfaces. This form of ortho- For the reference ocean required by Jackett and baric density is notable because it is a model of the two- McDougall’s (1997) neutral density method, for which hemisphere form of patched potential density (Reid the Levitus climatology is usually adopted, we supposed 1994) obtained in the limit when the vertical intervals a quiescent rest state, though with the said meridional (or patches, typically chosen to be 1000 dbar thick in u–S variations. On the other hand, for the reference u–S practice) are infinitesimally small. The patched poten- relation required for simple orthobaric density, we took tial density has been seminal in the development of an intermediate relation between the northern and both neutral density (Jackett and McDougall 1997) and southern extremes. For the present ocean, as a test, we orthobaric density (de Szoeke et al. 2000; de Szoeke assumed a vertically displaced form (with resulting and Springer 2005). AUGUST 2009 D E S Z O E K E A N D S P R I N G E R 1795

The dual-hemisphere idea can be extended to form of orthobaric density will, like neutral density, exhibit multiregional orthobaric density functions, each chosen extra virtual forces. For example, on dual-hemisphere with reference to a local u–S relation, and joined be- orthobaric density surfaces (based on different northern tween neighboring regions by the requirement of conti- and southern u–S relations in each hemisphere of the nuity when the ocean is in the reference state (section 5). globe), which may exhibit discontinuities at the equator, Thus, type I materiality can be reduced virtually to an impetus proportional to the discontinuity must be nothing, while the discontinuities among regions, and administered to a hypothetical water parcel crossing the the material flow across them, even though individually equator to move it to the designated continuation of the reduced, become more frequent. In the limit of a me- surface. This is analogous to the continuous virtual ridionally continuous u–S relation, the latter material forces on neutral density surfaces. Such forces must be flow is seen to be an analog of the type II cross-material applied to water parcels to keep them moving on con- flow of neutral density. This analogy is quantitative. tinuous, extended orthobaric density surfaces. Apart from a 50% reduction factor for neutral density An important question remains about the nature of {which can be traced to the choice of a half-and-half orthobaric density and neutral density. It concerns the reference pressure [Eq. (2.8)]}, the meridional structure sizes of the contributions to materiality that have been of the type II materiality is identical. the main subject of this paper—the reversible terms, Though the materiality of a variable is of obvious principally types I and II—relative to the irreversible, utility in interpreting its spatial variation, the funda- adiabatic terms (denoted as type IV in sections 2 and 4 mental significance of neutrality is not so clear. It is and in the appendix). The angle between simple often said that the neutral plane at a specific point (the orthobaric density surfaces and neutral planes has been plane perpendicular to d at that point) contains the estimated on long meridional hydrographic sections bundle of trajectories on which a water parcel can begin through the Atlantic and Pacific Oceans [see de Szoeke to move without change in heat or energy, or that et al.’s (2000) Fig. 12]. This angle is to very good ap- 21 2 2 r (2$p 1 c $r) 52c d is the buoyant restoring force proximation the neutrality magnitude j$s4 z hj calcu- of adiabatic test particle excursions. We cannot sub- lated in section 4. De Szoeke et al. (2000) and McDougall stantiate such statements. Rather, when, for example, a and Jackett (2005a) reached different conclusions about spatially unvarying u–S relation applies, so that neutral the significance of the errors incurred in assuming that surfaces exist (and are coincident with isentropes), the the diffusivity tensor’s principal axes are aligned with combination of the pressure gradient force and buoy- orthobaric density surfaces (or neutral density surfaces) ancy force with respect to isentropes may be replaced rather than along neutral planes. However, a recent by the dynamical relation (1.3), in which the effective critical examination of these various alignment hypoth- buoyant force is P(u)k9(u) 5P(u)$u, acting parallel to d. eses shows that they are superfluous and makes the di- This is indeed a fundamental property of neutral sur- vergence of views irrelevant and moot (de Szoeke 2009, faces when they exist. From it, familiar inferences may unpublished manuscript). be drawn about potential vorticity conservation and thermal wind balances. The extension or modification of this property to the case of the spatially varying u–S APPENDIX relation (so that no neutral surfaces exist) is the perti- nent dynamical question. We obtained in the appendix the generalization [Eq. (A.12)], with respect to neutral Neutral Density density surfaces, of the substitution (1.3), and showed a. Materiality that there are additional terms—virtual, nonconser- vative forces in effect—associated with the spatial The general definition of neutral density, Eq. (2.8), r variation of the u–S relations implicit in the reference gives the target pressure p in terms of a water sample’s datasets used in the construction of neutral density. values of potential temperature u and salinity S at am- These additional virtual forces are experienced by water bient pressure p and horizontal location x, using ar- parcels constrained to follow neutral density surfaces. A chived reference profiles uref and Sref at that location. r systematic study of the importance of these forces is yet Equation (2.7) with p substituted then gives g. It will be lacking. convenient, though not essential, to use the following 4 The simple orthobaric density s is specifically con- simplified equation of state: structed to exhibit under any circumstances the dynam- r1 5 V( p, u, S) 5 V( p) 1 V d and d 5 (1 1 mp)q s, ical substitution property (1.3) that isentropes possess 0 when the u–S relation is unvarying. Nonsimple variants (A.1a) 1796 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39 where N2 ›q ›pr [ ref ~lm r ref D 2 (p p ) r rrefg ›p ›g 1 2 r 1 q 5 a1(u u0) 1 a2(u u0) and s 5 b(S S0). ›p ›p ›p 2 ~ 1 m(q qref) l 1 l (A.1b) ›g ›g ›g N2 ›q 5 1 lm(p pr) . (A.4) Here, r is the density, V is the specific volume, d is the rg2 ›p

SVA normalized by V0, u is the potential temperature, S The first equality in (A.4) defines D, while the second is the salinity, V(p) 5 V(p,u0,S0), and m 5 (Vpu/Vu)p50 ’ 2 3 1024 dbar21 is the thermobaric coefficient follows from taking the partial g derivative, at constant (McDougall 1987b).A1 In (A.1b), we used the parameters x and t, of both sides of Eq. (2.8). Also, u 5 58C, S 5 34.63 psu,A2 a 5 1.13 3 1024 8C21, a 5 0 0 1 2 q_( p) [ (1 1 mp)q_ 1 s_ 1.35 3 1025 8C22, and b 5 0.799 3 1023 psu21, for the first and second thermal expansion coefficients, and 5 (11mp)[$ (K$q) 1 a2$u K $u] 1 $ (K$s). the haline contraction coefficient, respectively. The (A.5) potential density r with respect to a fixed reference r ( p) pressure r is given by Here, K is the turbulent diffusivity tensor, q_ represents the total effect on the negative buoyancy of irre- r1 5 V(r, u, S) 5 V(r) 1 V [(1 1 mr)q s]. (A1c) versible exchanges of heat and salt [the superscript ( p) r 0 emphasizes that the normalized thermal expansion coefficient V1›V/›q 5 1 1 mp, occurring in (A5), is When the material derivatives of both sides of (2.8) 0 calculated at pressure p], and a 5 V21›2V/›u2 is the and (2.7) are taken, and p_r [Dpr/Dt 5 u$ pr 1g›_ pr 2 0 g g second expansion coefficient, or cabbeling coefficient, is eliminated, one obtains the following relation among defined after Eq. (A.1b). In obtaining (A.2) use was the material derivative g_ of neutral density and q_ , s_, p_, made of identities and definitions such as _ [ and x u: 8 > p_ 5 (› 1 u $) p 1 g›_ p, > t g g g ›p g r > rge [ g_ 5 (c^ 1)[~l(› 1 u $) p 1 lu $ p ] <> k d N2 ›q ›s ›g |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}t g g ref 5 ref 5 (1 1 mpr) ref 1 ref , r g r g2 ›pr ›pr I > ref ref r 1 > r 1 ~lm(p p )D u $ q >(d 3 k) 3 k 5 (1 1 mp )$ r q 1 $ r s |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}g ref :> ref p ref p ref 5 (k d )h . II ref ref 2 $ pr 1 Nref g (A.6) 1 D href u g rrefg r |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl} Here, qref, sref, Nref, dref are functions of p and position r 21 III x {with rref 5 [V(p , qref, sref)] } in the reference ocean, 1 ( p) r 1 D|fflfflfflffl{zfflfflfflffl}q_ . (A.2) to be evaluated at p 5 pref(g, x), while q, s, and N are IV functions of p (or g), x, and t in the present ocean. The equalities in the second and third lines of (A.6) define

The first equality of (A.2) defines the material velocity the vertical and horizontal components of dref, the dia- eg across surfaces of constant neutral density g; the neutral vector in the reference dataset. The last equality Roman numerals identify terms referred to in the main defines the neutral dip href, a vector whose magnitude text; and c^g is the buoyancy gain factor for neutral and direction in the horizontal plane give the gradient density, defined by and orientation, respectively, of neutral planes. The derivation of (A.2) has as yet used no partic- ^g 1 ular property of the reference neutral density surfaces. c 1 5 m(q qref)D , (A.3) However, the vanishing, if possible, of the factor in parentheses of term III in (A.2) (which concerns only fields from the reference dataset) could be used to provide pr 5 A1 r More generally, what follows may be readily extended to pref(g, x) [or its inverse g 5 gref(p , x)]. The exact van- cover any equation of state of the form d 5 P(p)q(u, S) 2 s(u, S), ishing of term III is possible if and only if dref $ 3 dref 5 where u and S are the true potential temperature and salinity. A2 Here, psu refers to United Nations Educational, Scientific 0. If this condition is not fulfilled, one may settle for and Cultural Organization (UNESCO) Practical Salinity Scale of pref(g, x), which makes term III ’ 0 in some appropriate 1978 (PSS78) salinity units (UNESCO 1981). global minimal sense (Eden and Willebrand 1999). AUGUST 2009 D E S Z O E K E A N D S P R I N G E R 1797

Equation (A.2) is similar to an equation derived by One may compare (A.2) to (5.4) for the flow across McDougall and Jackett (2005a,b), except that in the first extended orthobaric isopycnals. Term I of Eq. (A.2) is expression in (A.4) for the factor D these authors in the sum of two terms weighted, respectively, by ~l and l, r ^g effect assume that Nref ’ N and ›p /›g ’ ›p/›g.We and both multiplied by c 2 1. The first term is the retain the generality of (A.4) because the buoyancy apparent vertical motion (›t 1 u $)g p of neutral frequency and spacing of neutral density surfaces may density surfaces p 5 p(g, x, t) (as though they were differ substantially from the reference ocean to the in- material) in the present instantaneous dataset. The r stantaneous present ocean. We have also retained second is the apparent vertical motion u $g p of neu- r generality in the choice of weights l and ~l in the ref- tral density surfaces p 5 pref(g, x) frozen in the refer- erence pressure p 5 ~lp 1 lpr of the potential densities ence dataset, as though the present instantaneous current in Eq. (2.8); this includes McDougall and Jackett’s u 5 u(g, x, t) was projected into the reference dataset on (2005a,b) standard setting of l 5~l 5 0.5. McDougall and the corresponding neutral density surface (Fig. 1). This Jackett’s cg factor is related to (A.3) by weighted sum of apparent vertical motions seems rather peculiar. Equation (5.4) contains a counterpart only of cg 1 ’ l(c^g 1). (A.7) the first of these two contributions. [If the weights ~l 5 1, l 5 0 were chosen, the correspondence of terms One may also see from (A.3) and (A.4) that between (A.2) and (5.4) would be very close.] Most important, the second term of (5.4) for the 1 c^g 1 N2 ›q 5 m 1 lm r material derivative of the extended orthobaric density 2 (p p ) , (A.8) r q qref rg ›p corresponds to the p 2 p term (II) of (A.2). For l 5 ~l 5 0.5, using Eq. (A.3) and pz 52rg, the first few which has a limiting value even when q / qref. terms of Eq. (A.2) become 8 9 <> => eg 5 0.5m (q q )(› p 1 u $ p 1 u $ pr) (p pr)u $ q (Dp )1 1 . (A.2)9 :>|fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ref tjg g g |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}g ref ;> z I II

McDougall and Jackett (2005a) assert that, ‘‘Since left side and in term III on the right of (A.9) are the r (p 2 p )u $gqref is typically of the same order as neutralities in the present and reference datasets, that is, r (q 2 qref)u $gp , all contributions to [Eq. (A.2)9]typi- the respective differences between the neutral density cally have the same magnitude.’’ Then, because q 2 qref } surface dip and the neutral plane dip. Even if the cg 2 1 ’ 0, it would follow that all shown terms, types I reference neutrality on the right of (A.9) were zero, the and II, are negligible. However, in the main text we other two terms, particularly the second, proportional to contest the quoted statement. m(p2pr), may significantly bias the neutrality in the present dataset. b. Neutrality c. Generalized Montgomery function By taking the gradient along neutral density surfaces Substituting the equation of state (A.1a) into the $ of both sides of Eq. (2.8), and making use of the g neutral property (2.8), one obtains definitions and identities (A.4) and (A.6), one arrives at r r (1 1 mp)qref(p , x) sref(p , x) 5 (1 1 mp)q s N2 ($ z h) 5 m(q q )$ (~lp 1 lpr) g g |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ref g or I r r r r d 5 d 1 m ~lq 1 lq 1 m(p p )$ (~lq 1 lq) ref(p , x) (p p )[ ref(p , x) ], (A.10a) |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}g ref II where 2 r N $gp 1 ref h . (A.9) r r r r g r g ref dref(p , x) 5 (1 1 mp )qref(p , x) sref(p , x), and |fflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflfflffl}ref pr 5 p (g, x). (A.10b) III ref

(The Roman numerals refer to terms discussed in the One may regard Eq. (A.10) as a metaequation of state, main text.) The two quantities in the parentheses on the furnishing density as a function of metastate variables g 1798 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 39 and x, in addition to p and q. Defining a generalized substitution property (1.3). To quantitatively consider heat content distributions of hydrographic variables on neutral density surfaces and the motion fields that might have ð p brought them about, one must take account of the extra (g) H (p, g, q, x) 5 Vdp forces that occur in the momentum balances with re- ðpr p spect to such surfaces. For this the generalized substi- r r 5 V( p)dp 1 V0dref(p , x)(p p ) tution (A.12) is relevant and indispensable. pr 1 r 2 ~ r 1 V0m(p p ) [lqref(p , x) 1 lq] 2 REFERENCES (A.11) Akitomo, K., 1999: Open-ocean deep convection due to thermobaricity. 1. Scaling argument. J. Geophys. Res., 104, 5225– and a generalized Montgomery function M(g) 5 H(g) 1 gz, 5234. one can show that Broecker, W. S., 1981: Geochemical tracers and ocean circulation. Evolution of Physical Oceanography, B. A. Warren and C. Wunsch, Eds., The MIT Press, 434–460. 1 (g) Davis, R. E., 1994: Diapycnal mixing in the ocean: Equations for r $p gk 5 $gM 1Q$gq 1 X large-scale budgets. J. Phys. Oceanogr., 24, 777–800. (g) (g) (g) 1 [›gM 1Q$gq 1P ]k9 , de Szoeke, R. A., 2000: Equations of motion using thermodynamic coordinates. J. Phys. Oceanogr., 30, 2814–2829. (A.12) ——, and R. M. Samelson, 2002: The duality between the Boussinesq and non-Boussinesq hydrostatic equations of motion. J. Phys. (g) Oceanogr., 32, 2194–2203. where k9 5 (k $gz)/›gz 5 $g, and ——, and S. R. Springer, 2005: The all-Atlantic temperature– ›H(g) ›pr ›H(g) salinity–pressure relation and patched potential density. J. Mar. P(g) 5 5 , (A.13a) Res., 63, 59–93. ›g ›g ›pr p,q,x x p,q,x ——, ——, and D. M. Oxilia, 2000: Orthobaric density: A thermodynamic variable for ocean circulation studies. J. Phys. Oceanogr., ›H(g) 1 30, 2830–2852. Q5 5 l m r 2 Eden, C., and J. Willebrand, 1999: Neutral density revisited. Deep- V0 (p p ) , (A.13b) ›q 2 Sea Res. II, 46, 33–54. p,g,x Eliassen, A., and E. Kleinschmidt, 1954: Dynamic meteorology. Geophysik II, Vol. 48, Handbuch der Physik, J. Bartels, Ed., ›H(g) Springer, 1–154. X5$ H(g) 5$ H(g) 1$ pr , Helland-Hansen, B., and F. Nansen, 1909: The Norwegian Sea. Its p,q,g p,pr ,q g ›pr p,q,x physical oceanography based upon the Norwegian researches 1900–1904. Report on Norwegian Fishery and Marine Inves- (A.13c) tigations, Vol. 2, 390 pp. and 28 plates. Holmes, A., 1945: Principles of Physical Geology. Ronald Press, ›H(g) 538 pp. 5 V(pr) V d (pr, x) ›pr 0 ref Jackett, D. R., and T. J. McDougall, 1997: A neutral density p,q,x variable for the world’s oceans. J. Phys. Oceanogr., 27, 237– V m(p pr)[~lq (pr, x) 1 lq] 263. 0 ref ——, ——, R. Feistel, D. G. Wright, and S. M. Griffies, 2006: Al- ›d (pr, x) 1 V (p pr) ref gorithms for density, potential temperature, conservative 0 ›pr temperature, and the freezing temperature of seawater. 1 ›q (pr, x) J. Atmos. Oceanic Technol., 23, 1709–1728. 1 ~l m r 2 ref Lynn, R. J., and J. L. Reid, 1968: Characteristics and circulation of V0 (p p ) r , and 2 ›p deep and abyssal waters. Deep-Sea Res., 15, 577–598. (A.13d) McDougall, T. J., 1987a: Neutral surfaces. J. Phys. Oceanogr., 17, 1950–1964. (g) r r ——, 1987b: Thermobaricity, cabbeling and water mass conver- $p, pr , qH 5 V0(p p )$pr dref(p , x) sion. J. Geophys. Res., 92, 5448–5464. 1 r 2 r ——, 2003: Potential enthalpy: A conservative oceanic variable for 1 ~lV m(p p ) $ r q (p , x). 2 0 p ref evaluating heat content and heat fluxes. J. Phys. Oceanogr., (A.13e) 33, 945–963. ——, and D. R. Jackett, 2005a: An assessment of orthobaric (g) density in the global ocean. J. Phys. Oceanogr., 35, 2054– Here P , Q, and X are generalized metathermody- 2075. (g) namic forces given by the respective gradients of H . ——, and ——, 2005b: The material derivative of neutral density. Equation (A.12) is the generalization of the dynamical J. Mar. Res., 63, 159–185. AUGUST 2009 D E S Z O E K E A N D S P R I N G E R 1799

Montgomery, R. B., 1938: Circulation in upper layers of southern Sverdrup, H. U., M. J. Johnson, and R. Fleming, 1942: The Oceans: North Atlantic deduced with use of isentropic analysis. Pa- Their Physics, Chemistry, and General Biology. Prentice Hall, pers in Physical Oceanography and Meteorolology, Vol. VI, 1060 pp. No. 2, Massachusetts Institute of Technology–Woods Hole UNESCO, 1981: Background papers and supporting data on the Oceanographic Institution, 55 pp. international equation of state of seawater 1980. UNESCO Reid, J. L., 1965: Intermediate waters of the Pacific Ocean. Johns Tech. Papers in Marine Science, No. 38, 192 pp. Hopkins Oceanographic Studies, No. 2, 85 pp. Wu¨ st, G., 1935: Schichtung und Zirkulation des Atlantischen ——, 1994: On the total geostrophic transport of the North Ozeans. Die Stratosphare (The Stratosphere of the Atlantic Atlantic Ocean: Flow patterns, tracers, and transports. Prog. Ocean). Wissenschaftliche Ergebnisse der Deutschen Atlan- Oceanogr., 33, 1–92. tischen Expedition auf dem Forschungs und Vermessungs- ——, and R. J. Lynn, 1971: On the influence of the Norwegian– schiff ‘‘Meteor’’ 1925–1927 (Scientific Results of the German Greenland and Weddell Seas upon the bottom waters of the Atlantic Expedition of the RV ‘‘Meteor’’ 1925–1927), Vol. 6, Indian and Pacific Oceans. Deep-Sea Res., 18, 1063–1088. Part 1, Instalment 2, De Gruyter, 288 pp.