An Assessment of Orthobaric Density in the Global Ocean

2054 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35

TREVOR J. MCDOUGALL AND DAVID R. JACKETT CSIRO Marine Research, Hobart, Australia

(Manuscript received 20 May 2004, in final form 18 April 2005)

ABSTRACT

Orthobaric density has recently been advanced as a new density variable for displaying ocean data and as a coordinate for ocean modeling. Here the extent to which orthobaric density surfaces are neutral is quantified and it is found that orthobaric density surfaces are less neutral in the World Ocean than are potential density surfaces referenced to 2000 dbar. Another property that is important for a vertical coordinate of a layered model is the quasi-material nature of the coordinate and it is shown that orthobaric density surfaces are significantly non-quasi-material. These limitations of orthobaric density arise because of its inability to accurately accommodate differences between water masses at fixed values of pressure and in situ density such as occur between the Northern and Southern Hemisphere portions of the World Ocean. It is shown that special forms of orthobaric density can be quite accurate if they are formed for an individual ocean basin and used only in that basin. While orthobaric density can be made to be approximately neutral in a single ocean basin, this is not possible in both the Northern and Southern Hemisphere portions of the Atlantic Ocean. While the helical nature of neutral trajectories (equivalently, the ill-defined nature of neutral surfaces) limits the neutrality of all types of density surface, the inability of orthobaric density surfaces to accurately accommodate more than one ocean basin is a much greater limitation.

1. Introduction The density of seawater is a significantly nonlinear Density is used in oceanography for a variety of pur- function of temperature, pressure, and salinity, and the poses. Its horizontal gradient is used to deduce the ver- nonlinearity that causes oceanographers the most tical shear of horizontal velocity (assuming geostrophic trouble is the so-called thermobaric nonlinearity, which balance) and, because small-scale turbulent mixing is so is primarily due to the pressure dependence of the ther- weak compared with the lateral mixing of mesoscale mal expansion coefficient (or equivalently, the tem- eddies, it is important to be able to calculate the sur- perature dependence of the sound speed). The idea faces along which this strong lateral diffusion occurs. that properties are mixed predominately along isopyc- Oceanographers assume that the strong mixing of heat nals has a long history in oceanography, and the ther- and salt that is achieved by mesoscale eddies occurs mobaric nature of seawater led Reid and Lynn (1971) along the locally referenced potential density surfaces. to form a series of potential density surfaces referenced Shrinking this argument to a point defines the so-called to a series of different pressures in order to better ap- neutral tangent plane and the smallness of the observed proximate the surfaces along which the ocean mixes dissipation of mechanical energy in the ocean gives tracers. Following de Szoeke et al. (2000) we call these strong support to this being the correct plane in which isopycnals of Reid and Lynn’s (1971) patched potential the strong mesoscale mixing occurs (McDougall and density surfaces. Jackett 2005b). Oceanographers would like to analyze McDougall (1987a) defined the concept of a neutral data and run layered ocean models with respect to sur- tangent plane along which fluid parcels could be ex- faces that everywhere coincide with neutral tangent changed without feeling any buoyant forces. While planes, but the equation of state dictates that this is not these neutral tangent planes are well defined, they do possible and various compromises must be made. not link up to form a well-defined neutral surface because of the thermobaric terms in the equation of state. Corresponding author address: Trevor McDougall, CSIRO Ma- The ill-defined nature of a neutral surface was dis- rine Research, Castray Esplanade, Hobart, TAS 7001, Australia. cussed by McDougall and Jackett (1988) who showed E-mail: [email protected] that individual neutral trajectories in the ocean are

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helical in nature. Nevertheless, McDougall and Jackett We “road test” orthobaric density in the global (1988) and later Jackett and McDougall (1997; the neu- ocean, finding that it is significantly less neutral than is tral density software was available online at http://www. potential density referenced to 2000 dbar. Also, ortho- ml.csiro.au/ϳjackett/NeutralDensity) showed that, baric density is significantly non–quasi material ␴ while the helical nature of neutral trajectories may al- whereas 2 is 100% quasi material. low a significant mass flux to move vertically through In this paper we regard in situ density ␳ ϭ ␳(S, ⌰, p) the ocean without the need for small-scale mixing ac- to be a function of salinity S (expressed on the practical tivity, the ill-defined nature of neutral surfaces is small salinity scale), conservative temperature ⌰, and pres- in the sense that a well-defined surface [such as can be sure p [p is absolute pressure minus 0.101 325 MPa ϭ calculated with the neutral density algorithm ␥ n of 10.1325 dbar; see Feistel (2003) and Jackett et al. (2005, Jackett and McDougall (1997)] can be found that is manuscript submitted to J. Atmos. Oceanic Technol.)]. almost neutral everywhere; this statement will be fur- Conservative temperature (McDougall 2003) is propor- ϭ ther quantified in this paper (see Figs. 4 and 5). tional to potential enthalpy (referenced to p 0 MPa) De Szoeke et al. (2000) sought a density variable, and is a factor of more than 100 closer to being proportional to the heat content of seawater than is poten- called orthobaric density ␳␷ that is as neutral as possible ␪ ⌰ ␪ but is restricted to be a function only of pressure and in tial temperature . The difference between and is not central to the present paper and we shall sometimes situ density, that is, ␳␷ ϭ ␳␷(p, ␳). Such a pycnotropic ⌰ variable has the desirable properties that (i) a geo- simply refer to as temperature. In sections 2–4, we quantify the nonneutral and non- strophic streamfunction can be found in each ␳␷ surface so that the horizontal pressure gradient can be ex- quasi-material aspects of orthobaric density surfaces and compare these attributes with those of other types pressed in the convenient and numerically accurate of density surfaces. The reason for the irreducible and form of a divergence along the orthobaric density sur- significant nonneutral behavior of orthobaric density is face and (ii) the potential vorticity equation written in isolated in sections 5 and 6. One of the main conclu- terms of gradients of orthobaric density does not con- Ϫ2 sions of de Szoeke et al. (2000) is that the non-quasi- ١p ١␳ ϫ · ١␳␷ tain the baroclinic production term ␳ material nature of orthobaric density also characterizes since this is zero. finely patched potential density surfaces. In sections 5 Orthobaric density is a clever combination of pres- and 6 we show that this is incorrect; the explanation sure and in situ density that has the property that, so involves the important distinction between local helic- long as water mass variations occur in a monotonic way ity and the interhemispheric water mass contrasts. The with pressure (see section 5 below), ␳␷ can be made to paper ends with a discussion section. be quite neutral. We will show that in practice this means that it is possible to tune the orthobaric density variable so that it is a good approximation to neutral 2. Orthobaric density and vertical stability density surfaces in a single ocean basin. However, a Orthobaric specific volume ␯ is defined by the fol- major point of the present paper is that ␳␷ cannot be made to be approximately neutral in the global ocean. lowing relationship between total differentials (de Here we seek quantitative answers to the following Szoeke et al. 2000): three questions: (i) how close is the vertical gradient of 1 orthobaric density to being proportional to N2, the ␾d␯ ϭ d͑1ր␳͒ ϩ dp, ͑1͒ ␳2 2 square of the buoyancy frequency, (ii) how neutral are c0 orthobaric density surfaces, and (iii) how quasi material where c0 can be regarded as an approximation to the is orthobaric density? A variable is said to be quasi sound speed that is restricted to be a function only of material if it only changes when irreversible mixing oc- ϭ ␳ pressure and density, that is, c0 c0 (p, ), and the curs. If a fluid parcel does not change its salinity and boundary condition ␯ ϭ 1/␳ is applied at p ϭ 0 dbar. conservative temperature, then the value of a quasi- Orthobaric specific volume ␯ and the integrating factor material variable of the fluid parcel will also be un- ␾ are also functions only of pressure and density, that changed. Examples of quasi-material variables are en- is, ␾ ϭ ␾(p, ␳) and ␯ ϭ ␯(p, ␳). tropy, salinity, potential temperature, potential enthal- Here we discuss orthobaric density, ␳␷ ϵ 1/␷, rather py, and conservative temperature (see McDougall than orthobaric specific volume. The corresponding to- 2003). Variables such as density that vary with pressure, tal differential form is even for fixed values of salinity and conservative tem- ⌽ ␳ ϭ ␳ Ϫ ⌫ ͑ ͒ perature, are said to be non–quasi material. d ␯ d 0dp, 2

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␳ ϭ ␳ ϭ Ϫ ϭ ␳ Ϫ ␳ ⌰ and the boundary condition ␷ is applied at p 0 c0 c c0(p, ) c(p, , ). The derivation of the last 2 dbar, the integrating factor ⌽ϭ␾(␳/␳␷) is also a func- expression in (5) followed the same route as that of de ␳ ␳Ϫ1⌫ tion of only p and , 0 is an approximation to the Szoeke et al. (2000) in arriving at their (2.13), the cor- ⌫ ϭ Ϫ2 ϭ⌫ ␳ compressibility of seawater, and 0 c0 0(p, ). responding relation here being We write the true compressibility of seawater as ␳Ϫ1⌫ϭ Ϫ1 ⌫p| ⌰, where ⌫ is taken to be the functional form Ѩץ/␳ץ ␳ S, ͯ ϭϪ␳ ͑ ͒ ⌫ϭ⌫(p, ␳, ⌰). Following de Szoeke et al. (2000) we Ѩ⌰ Tb 7 p,␳ will loosely talk of ⌫ as the compressibility of seawater Ϫ1 even though the compressibility is actually ␳ ⌫. The so that the property contrasts are related by integrating factor is given by ⌬⌫ Ϸ Ϫ3⌬ ϷϪ␳ ͓⌰ Ϫ ⌰ ͑ ␳͔͒ ͑ ͒ Ѩ␳ 2c0 c Tb 0 p, . 8 ⌽ ϭ ͯ ͑ ͒ Ѩ␳ 3 ␯ p ⌰ ␳ Here 0(p, ) is a reference temperature as a function ϭ ␳ ϵ ␳ and is equal to unity at p 0 where ␷ . That is, at of pressure and in situ density. the sea surface, orthobaric density is equal to the po- It is also instructive to write down the expression for ␴ tential density 0 The software provided by de Szoeke the vertical gradient of potential density, which is et al. (2000) takes the pressure and in situ density of a readily found from the definitional differential expres- datum to be labeled with orthobaric density and inte- ␳ ϭ ␤ ⌰ Ϫ ␣ ⌰ ⌰ sion d ln ⌰ (S, , pr)dS (S, , pr)d to be p|␳ (along which theץ/␳ץ grates along the characteristic v ϭ⌫Ϫ1 ϭ ␳ | ץ ␳ץ slope / p ␳␯ 0 is known) to p 0, and ␷ is as- g Ѩ␳ Ϫ ⌰ Ϸ 2 Ϫ ͑ Ϫ ͒⌰ signed to be the value of ␳ at p ϭ 0 along this charac- N gTb p pr z ␳⌰ Ѩz teristic. R T The vertical gradient of orthobaric density (at fixed ϭ 2ͫ Ϫ ␳ b ͑ Ϫ ͒ͬ ͑ ͒ N 1 p pr , 9 latitude and longitude) can be related to the square of ͑R␳ Ϫ 1͒ ␣ the buoyancy frequency where R␳ ϭ ␣⌰ /␤S is the stability ratio of the water 2 ϭϪ ␳Ϫ1͑␳ Ϫ ⌫ ͒ ͑ ͒ z z N g z pz 4 column and the buoyancy frequency has been used in 2 ϭ ␣⌰ Ϫ ␤ using (2) and the hydrostatic relation by the expressions the form N g( z Sz). To illustrate the problems with representing vertical g Ѩ␳␯ Ϫ⌽ ϭ N2 ϩ g2͑⌫ Ϫ ⌫ ͒ ϵ ␺N2 stability with both orthobaric density and potential ␳ Ѩz 0 density, consider the following cases. From Fig. 8 of de Ϫ Szoeke et al. (2000), we see that the sound speed Ϸ N2 ϩ 2g2c 3͑c Ϫ c͒ 0 0 anomaly in the North Atlantic Ocean at a depth of 1500 Ϸ 2 Ϫ 2␳ ͓⌰ Ϫ ⌰ ͑ ␳͔͒ ͑ ͒ Ϫ Ϫ1 N g Tb 0 p, , 5 m is about 15 m s {consistent with a water mass contrast [⌰Ϫ⌰(p, ␳)] of about 3°C} and using this ␺ 0 where the first part of (5) serves to define , while in value of ⌬c in (5) shows that, if N2 is less that 10Ϫ6 sϪ2, the last expression the thermobaric parameter must be then the vertical gradient of ␳␷ changes sign such that ϭ ␣ Ϫ written with respect to pressure in pascals as Tb p the water column would appear to be unstably stratified ␣ ␤ ␤ ϭ ␤ ␣ ␤ Ϸ ϫ Ϫ12 Ϫ1 Ϫ1 ( / ) p ( / )p 2.7 10 K Pa (McDougall whereas, in fact, the buoyancy frequency is real and 1987b). This thermobaric parameter encapsulates the positive. Consider now the example of a potential den- key nonlinearity of the equation of state and it can be sity that is being used 2000 dbar distant from its refer- approximately regarded as being due to the depen- ence pressure where R␳ Ϸ 3, then (9) shows that the dence of the thermal expansion coefficient on pressure, vertical gradient of potential density can also go to zero or equivalently, on the variation of the compressibility in this situation. of seawater with temperature. Notice the similarity be- We rewrite (5) in the form tween (5) and the expression in (42) of McDougall (1989) for the vertical gradient of specific volume g Ѩ␳␯ anomaly. The middle expression here used the approxi- Ϫ ϵ ␺⌽Ϫ1N2 ␳ Ѩz mation Ϸ ⌽Ϫ1 2 Ϫ ⌽Ϫ1 2␳ ͓⌰ Ϫ ⌰ ͑ ␳͔͒ ͑ ͒ ⌬⌫ Ϸ Ϫ3⌬ ͑ ͒ N g Tb 0 p, 10 2c0 c, 6 where, following the sign convention of de Szoeke et al. and compare this and (9) with the corresponding ex- ⌬⌫ϭ⌫Ϫ⌫ ϭ⌫ ␳ ⌰ Ϫ⌫ ␳ ⌬ ϭ (2000), 0 (p, , ) 0(p, ) and c pression for the vertical gradient of neutral density,

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g Ѩ␥ n Ϫ ϵ bN2 Ϸ N2 ␳ Ѩz

ͬ ͒ ١ ⌰ Ϫ ⌰ ϫ ͫϪ 2␳ ͵ Ϫ2͑١ exp g Tb N ␥ p ␥p · dl . ␥n

͑11͒ The first part of (11) defines b while the exponential expression for b was derived by McDougall [1988, his (47)] and is approximate because (i) the variation of the saline contraction coefficient with pressure has been neglected in comparison with the larger proportional change in the thermal expansion coefficient with pressure and (ii) the original expression in McDougall

(1988) was derived for gradients along a neutral trajec- FIG. 1. Percentages of the global ocean volume (dark) and of tory, whereas (11) is written in terms of spatial gradi- data pairs (light) in which the vertical gradient of the density ents in a neutral density surface. Because we have variable decreases with depth. shown in McDougall and Jackett (1988), Jackett and McDougall (1997), and the present paper that neutral its propensity for vertical instability by altering the ref- density surfaces are very close to being neutral, the erence compressibility or reference sound speed c (p, second approximation in writing (11) is negligible. In- 0 ␳) [see (5)], however doing so would further degrade p is simply the ␥١ ⌰Ϫ⌰␥١ terestingly the combination p the approach of orthobaric density to neutrality and ,⌰ ١ ,isobaric gradient of conservative temperature p also degrade its quasi-material behavior (these aspects which is almost the same as the horizontal gradient, .(are discussed in the following two sections ⌰ ١ H . The three equations of (9), (10), and (11) are handy expressions for the vertical gradients of the three types of density variable—namely, potential density, 3. Orthobaric density and neutrality orthobaric density, and neutral density. Since (2) is a relationship between three total differ- In Fig. 1 we present the percentage of data from the entials, it also applies in a neutral tangent plane (which World Ocean that has vertical reversals of various always exists and in which parcels can be exchanged forms of density. We have used the global atlas data of without encountering buoyant forces) so that Koltermann et al. (2004) and Gouretski and Kolter- ͒ ͑ ١ ⌫ ␳ Ϫ ١ ␳ ϭ ⌽١ mann (2004) and the right-hand-most pair of bars rep- n ␯ n 0 np, 12 ϭ⌰ ١␣ resent the percentage of bottle pairs in this atlas that 2 while in a neutral tangent plane we also have n have a negative value of N . The left-hand bar of each Ϫ S and therefore [where ␳ 1⌫ is the adiabatic (and ١␤ pair is for the percentage of level pairs while the right- n isohaline) compressibility of seawater] hand-most bar is for the percentage of the ocean vol- ͒ ͑ ١⌫ ␳ ϭ ١ ume, where we have weighted the atlas data by the n np. 13 cosine of latitude and by the pressure difference over Combining (12) and (13) we have which the reversal occurred. Apart from the right- ␳ ϭ ⌽Ϫ1͓⌫͑ ␳ ⌰͒ Ϫ ⌫ ͑ ␳͔͒١ ١ hand-most pair of bars we discarded every bottle pair n ␯ p, , 0 p, np for which the atlas had N2 Ͻ 0 and evaluated the per- ١ ⌬Ϸ ⌽Ϫ1 Ϫ3 centages on the remaining data. Potential density re- 2 c0 c np ferred to zero pressure has nearly 1.75% of the ocean Ϫ p, ͑14͒ ϷϪ⌽ 1␳T ͓⌰ Ϫ ⌰ ͑p, ␳͔͒١ volume ostensibly unstably stratified; using a reference b 0 n pressure of 1000 or 2000 dbar improves things. which demonstrates that when ⌰ in the ocean can be Orthobaric density has vertical reversals in 0.65% of approximated as a single-valued function of pressure the ocean volume while neutral density has this prob- and density, ⌰ϭ⌰(p, ␳), then it is possible to form a ⌫ ␳ lem in only 0.15% of the ocean volume. Apart from reference compressibility function 0(p, ) so that ␳ Ϸ ١ potential density referenced to the sea surface, n ␷ 0 and the orthobaric density surface will every- orthobaric density performs the worst under this crite- where osculate with neutral tangent planes. The differ- rion of vertical stability in the ocean’s volume. The defi- ential expression (14) for the epineutral gradient of nition of orthobaric density could be altered to reduce orthobaric density can be compared with the corre-

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sponding gradient of potential density (with reference ␤ ␳ ϭ ١ pressure pr), which is [from the relations n ln ⌰ (S, ␣ ϭ ١ ⌰ ␤ ⌰ ١ ⌰ ␣ Ϫ ١ ⌰ , pr) nS (S, , pr) n and (S, , p) nS (S, ⌰ ١ ⌰ , p) n ], ͒ ͑ ⌰ ١͒ ␳ Ϸ ␳ ͑ Ϫ ١ n ⌰ ⌰Tb p pr n . 15 In Fig. 2 we show the variation of three types of density on neutral density surfaces. The standard deviation and the range of orthobaric density are both less than the corresponding quantities for the potential densities referenced to 0 and 2000 dbar. On the basis of this comparison it would seem that orthobaric density is more ␴ ␴ neutral than both 0 and 2, but we will later show that this is not the case when quantified in terms of the slopes of these surfaces. A meridional cross section of various density surfaces in the Atlantic Ocean is shown in Fig. 3a. The ␴⌰ surface is a potential density surface with a reference pres- ␴ sure of 0 dbar, while the 2 surface is a potential density surface with a reference pressure of 2000 dbar. The neutral trajectory proceeds from one “cast” in the ocean atlas to the next, all the while obeying the neutral property that successive vertically interpolated data points on the same trajectory have the same potential density when referenced to the average pressure of the two points (Jackett and McDougall 1997). The neutral density (␥n) surface balances this neutral trajectory in- formation on this one meridional section with the need to be as neutral as possible at other longitudes, so the difference between the lines marked neutral density and neutral trajectory in this figure is a manifestation of FIG. 3. Cross sections of various density surfaces along 328°E the real path dependence (i.e., helicity) in the ocean. azimuth (32°W) in the Atlantic that coincide at (a) 58°S and (b) The orthobaric density surface rises above the neutral the equator. density surface and, when this orthobaric density surface eventually outcrops (not shown) in the Northern Hemisphere, it coincides with the ␴⌰ surface that is shown on this diagram. Figure 3b shows the same neutral density surface but now the neutral trajectory and other density surfaces emanate from the equator rather than from near the ␴ ␴ sea surface at 58°S. The spatial variations of 0, 2, and ␳␷ are the same in Figs. 3a,b, and the different vertical excursions in pressure simply reflect the fact that the vertical gradients of these densities are much weaker at depth than near the sea surface. We now use (14) and the definition of the gain factor, ␺, in (5) together with the hydrostatic equation to find the difference between the slope of an orthobaric density surface and the neutral tangent plane, finding that p ١ ␳␯ ͑␺ Ϫ 1͒ ١ ϭϪ n ١ Ϫ ١ Ϫ n ϭ ͑␳ ͒ ␯z nz ␺ ␯ z pz FIG. 2. The standard deviation and range in the global ocean of ͑␺ Ϫ 1͒ z. ͑16͒ ١ three different types of density on a selection of neutral density ϷϪ surfaces. ␺ n

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The last part of this equation has ignored the small World Ocean where the fictitious diffusivity for mixing slope between an isobaric surface and the geopotential along orthobaric density surfaces exceeds 10Ϫ5 m2 sϪ1, surface. This result was also derived (by a slightly dif- as a function of depth in the water column. ferent route) by de Szoeke et al. [2000, their (2.45)]. The percentage of data that has a fictitious diapycnal Equation (16) says that the angle between the ortho- diffusivity greater than 10Ϫ5 m2 sϪ1 is shown in Fig. 5 baric surface and the neutral tangent plane is Ϫ(1 Ϫ for the global ocean (Fig. 5a) and for the North Atlantic ١ ␺Ϫ1 ) times the slope of the neutral tangent plane, nz. (Fig. 5b). This shows that 14% of the global ocean vol- In terms of the sound speed anomaly, (16) may be writ- ume has a false diapycnal diffusivity greater than 10Ϫ5 ten as m2 sϪ1 arising from mixing in orthobaric density surfaces as compared with a figure of 20% for potential 2 Ϫ3⌬ 2g c0 c density and 1.6% for neutral density using data from z. ͑17͒ ١ z ϷϪ ١ ١␯ z Ϫ n ͑ 2 ϩ 2 Ϫ3⌬ ͒ n the Gouretski and Koltermann (2004) atlas and almost N 2g c0 c zero when the data are taken from the labeled data that When properties are diffused in ocean models along a are contained in the neutral density software. In the surface other than the neutral tangent plane, density is North Atlantic, the density variable of Eden and Wil- fluxed across this neutral tangent plane and the false or lebrand (1999) (which was defined only for the North fictitious diapycnal diffusivity of density is given by the Atlantic and is a function only of salinity and potential product of the lateral diffusivity K and the square of the temperature) does remarkably well, although not as angle between the surfaces. For orthobaric density sur- well as neutral density. For the neutrality criterion dis- ␴ faces this fictitious diapycnal diffusivity is played in Fig. 5a, 2 surfaces are significantly more neutral than are orthobaric density surfaces in the global ͒ ͑ 2| ١|2 ϭ ͑ Ϫ ␺Ϫ1͒2| ١ Ϫ ١| fictitious ϭ D K ␯z nz K 1 nz . 18 ocean volume. At first sight this seems to conflict with ␴ the rather large epineutral variations of 2 as shown in At each point in the global ocean atlas we have evalu- Fig. 2. These results are consistent because the slope ated the right-hand side of (18) using a lateral diffusiv- error of a density surface depends on not only the ity K of 1000 m2 sϪ1 and using the definition of the gain epineutral variation of the density but also on the ver- factor ␺ in (5) and the contrast in sound speed under- tical gradient of this density. The more neutral behavior ␴ ␳ lying orthobaric density, as obtained from the software of 2 as compared with ␷ is due to the more robust ␴ supplied by de Szoeke et al. (2000). The logarithm (to vertical gradient of 2. Incidentally, when we searched the base 10) was taken and the frequency distribution for the reference pressure of a potential density vari- taken for all the data in the global ocean atlas and this able that had the smallest volume in which the fictitious is displayed in Fig. 4a. The fictitious diapycnal diffusiv- dianeutral diffusivity exceeded 10Ϫ5 m2 sϪ1, we found ␥n ␴ ity was also evaluated for lateral mixing along , 2, this best reference pressure to be within 20 dbar of and ␴⌰ surfaces. It is seen that as many locations have 2000 dbar. the fictitious diapycnal diffusivity greater than 10Ϫ4 The potential vorticity equation written in terms of m2 sϪ1 for mixing along orthobaric density surfaces as gradients of a variable ␭ contains the baroclinic pro- Ϫ2 -١p (Pedlosky 1979; Mc ١␳ ϫ · ١␭ for mixing along ␴⌰ surfaces. This occurs in approxi- duction term ␳ mately 5% of the World Ocean. By contrast, neutral Dougall 1995). For the case in which ␭ is orthobaric density surfaces display a fictitious diapycnal diffusivity density this production term is zero since ␳␷ is a func- greater than 10Ϫ4 m2 sϪ1 in just 0.17% of the ocean tion of only p and ␳. When ␭ is potential density, the volume. Neutral density surfaces are by far the best baroclinic production term can be written in terms of ⌰ ١ ϫ ١ approximation to neutral tangent planes among this se- helicity, that is, in terms of n p n · k. This pro- lection of surfaces. Also shown in Fig. 4a is the fre- duction term is also proportional to the pressure differ- quency distribution of the false diapycnal mixing for ence between the in situ pressure and the reference neutral density when the dataset is the Levitus data pressure of the potential density [see Eq. (59) of Mc- underlying the definition of neutral density: we believe Dougall (1988)]. Here we derive an expression for this that this represents the best that can be achieved by any production term in terms of the difference between the global density variable. slope of the neutral tangent plane and the potential ١p with ١␳ and The spatial locations where the fictitious diapycnal density surface. We first write both ␳ ϩ ␳ ١ diffusivity [(18)] is large are indicated in Figs. 4b,c. The respect to the neutral tangent plane as n zm and ϩ ١ colors in Fig. 4b represent the largest values of the np pzm [see section 3 of McDougall and Jackett fictitious diapycnal diffusivity down each vertical cast, (1988)] where m is normal to the neutral tangent plane while Fig. 4c shows the total number of locations in the and is defined in terms of the slope of the neutral tan-

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FIG. 4. (a) Frequency distribution of the logarithm of the fictitious diapycnal diffusivity arising from mixing along various coordinate surfaces. (b) The largest value of the fictitious diapycnal diffusivity [(18)] down each vertical cast is shown. (c) The total number of locations in the World Ocean where the fictitious diffusivity [(18)] exceeds 10Ϫ5 m2 sϪ1 is shown as a function of depth.

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responding production term for isopycnal potential vorticity, (21). Figure 6 shows that this is indeed the case.

4. The non-quasi-material nature of orthobaric density The material rate of change of locally referenced potential density is given by d␳ dp Ϫ ⌫ ϭ ␳͑␤S˙ Ϫ ␣⌰˙ ͒ ϵ q, ͑23͒ dt dt where, as in de Szoeke et al. (2000), q represents all manner of irreversible mixing processes, noting that q is not the divergence of a flux because of the several non- FIG. 5. (a) Percentages of the global ocean data for which the linear effects of the equation of state. Combining the Ϫ5 2 Ϫ1 fictitious diapycnal diffusivity is greater than 10 m s . The bar definition of orthobaric density, (2), with (23) leads to labeled JoeReid is for the Atlantic Ocean only. (b) The data only from the North Atlantic. For each pair of bars, the right-hand ␳ d ␯ Ϫ1 dp Ϫ1 ones are for the percentages of the global ocean volume while the ϭ ⌽ ⌬⌫ ϩ ⌽ q. ͑24͒ dt dt left-hand bars are the percentages of data pairs in the atlas. The data being examined are from Koltermann et al. (2004) ocean The material derivative of ␳␷ can be written as the prod- atlas except for the right-hand-most bars labeled gamma_n for uct of the velocity through the orthobaric density sur- which the data used came from the internal atlas of the neutral ␷ density software. face e and the vertical gradient of orthobaric density z, and we write the material derivative of pressureץ/␳␷ץ with respect to isobaric density surfaces as ␳ ϭ ١ ϩ ϭϪ١ ١ gent plane, nz,bym nz k. Noting that n ␳ Ϫ1 2 ϭϪ␳ ϩ⌫ dp ␯ ١⌫ ١␯ p ϩ e p , ͑25͒ · np and g N z pz, we find that ϭ p |␯ ϩ V dt t z Ϫ p ϫ m. ͑19͒ ١p ϭ ␳g 1N2١ ١␳ ϫ n where V is the horizontal velocity, to find that

The gradient of any scalar variable can be written in ␯ ١␯ p͒րp · e ϭ ͑␺ Ϫ 1͒͑p |␯ ϩ V terms of the normal to the neutral tangent plane m and t z ١ Ϫ ١ the difference in slope, ( ␭z nz), between the sur- ϩ ͑␣⌰˙ Ϫ ␤˙ ͒ր͑␣⌰ Ϫ ␤ ͒ ͑ ͒ S z Sz , 26 face of constant scalar and the neutral tangent plane as ,(␭ ␭ where, from (5 ϭϪ١ ١ (using ␭z H / Z) ␺ Ϫ 1͒ ϭ g2⌬⌫րN2͑ ͒ ͑ ͒ ١ Ϫ ١͑ ١␭ ϭ ␭ Ϫ ␭ zm z ␭z nz . 20 Ϸ 2 Ϫ3⌬ ր 2 Using (19) and (20) the baroclinic production term for 2g c0 c N potential density becomes ϷϪ␳ 2 ͓⌰ Ϫ ⌰ ͑ ␳͔͒ր 2 ͑ ͒ g Tb 0 p, N . 27 ͒ ١ Ϫ ١͑͒ ϭϪ͑ ␳͒Ϫ1 2͑Ѩ␳ րѨ ١ ١␳ ϫ ␳Ϫ2١␳ ⌰ · p g N ⌰ z ␳⌰z nz The second term on the right-hand side of (26) represents the effects of irreversible mixing processes on the p · k, ͑21͒ ١ ϫ n flow though orthobaric density surfaces. Note that this ␷ while that for neutral density is contribution to e is exactly the same as the contribution of irreversible diffusion to the flow through neutral ͒ ١ Ϫ ϭϪ͑ ␳͒Ϫ1 2␥n͑١ ١ ١␳ ϫ n␥␳Ϫ2١ · p g N z ␥z nz tangent planes [see McDougall (1984, 1991)]. De (p · k. ͑22͒ Szoeke et al. (2000) write the same term in their (2.20 ١ ϫ ⌰␣ ˙␤ Ϫ1 ␣⌰˙ Ϫ ץ ␳ץ n ␺⌽Ϫ1 as q( ␷/ z) [which is equal to ( S)/( z Ϫ ␤ ␺ Since by construction, the slope of neutral density sur- Sz)] and talk of the unwanted factor relating the faces are designed to be minimally different to the slope effects of irreversible diffusion (q) to the material rate of the coincident neutral tangent planes, one would ex- of change of orthobaric density. However, in the con- pect that the baroclinic production term for potential text of a layered ocean model it is the flow through the vorticity defined in terms of the gradient of neutral interfaces that is the important consequence of irrevers- density, (22), would be considerably less than the cor- ible mixing, and in this regard no factor of ␺⌽Ϫ1 ap-

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Ϫ n ١p, plotted on ١␳ ϫ · ␥FIG. 6. Maps of the baroclinic production terms for (a) neutral density, ␳ 2١ n Ϫ2 ,١p ١␳ ϫ · ⌰١␳ the 27.62 ␥ surface, and (b) the baroclinic production term for potential density, ␳ plotted on the 27.46 ␳⌰ surface in the Atlantic. pears in (26). That is, in respect to this irreversible dif- where ␺␥ is defined by fusion, orthobaric density surfaces are actually more 1 1 accurate than was realized by de Szoeke et al. (2000). ͫN2 ϩ ␳g2T ͑⌰ Ϫ ⌰r͒ Ϫ ␳g2T ͑p Ϫ pr͒⌰r ͬ 2 b 2 b p The first term in (26) arises from the non-quasi- ␺␥ ϭ material nature of orthobaric density; (␺ Ϫ 1)(p |␯ ϩ 1 t ͫ 2 ϩ ␳ 2 ͑⌰ Ϫ ⌰r͒ Ϫ ␳ 2 ͑ Ϫ r͒⌰r ͬ ␳ N g Tb g Tb p p p ١ V · ␯ p)/pz is the vertical velocity through a ␷ surface 2 due to the seemingly innocuous sliding of fluid flow ͑29a͒ along the sloping ␳␷ surface and the vertical heaving of ␥ the ␳␷ surfaces. The instantaneous vertical velocity past so that (␺ Ϫ 1) is ١ ϩ | geopotentials, w, is dominated by (pt ␯ V · ␯ p)/pz and approximately the fraction (␺ Ϫ 1) of this vertical 1 2 r Ϫ ␳g T ͑⌰ Ϫ ⌰ ͒ velocity appears as flow through orthobaric density sur- 2 b ͑␺␥ Ϫ 1͒ ϭ . faces. 1 ͫN2 ϩ ␳g2T ͑⌰ Ϫ ⌰r͒ Ϫ ␳g2T ͑p Ϫ pr͒⌰r ͬ In McDougall and Jackett (2005b) we show that the b 2 b p vertical velocity through neutral density surfaces is ͑29b͒ ␥ ␥ Ϫ1 ͒ p͒͑p␥١ · e Ϸ ͑␺ Ϫ 1͒͑p |␥ ϩ V t z Hence (␺␥ Ϫ 1) is nonzero to the extent that there is a ͑p Ϫ pr͒ water mass contrast(⌰Ϫ⌰r) between the seawater par- rͬ͑ ͒Ϫ1⌰ ١ r Ϫ ١ ͫ͒ ϩ ͑␺␥ Ϫ 1 V · ␥p V · ␥ p ͑⌰ Ϫ ⌰r͒ z cel that is being labeled and the data on the labeled reference dataset that communicates neutrally with the ␥ ϩ ␺ V · s seawater sample. Here (⌰Ϫ⌰r) and (p Ϫ pr) are the differences in temperature and pressure between the ͓a͑p͒⌰˙ Ϫ ␤͑p͒S˙͔ ϩ ͓␺␥ ϩ ͑␺␥ Ϫ 1͔͒ , ͑28͒ parcel’s temperature and pressure and those of the la- ͓ ͑ ͒⌰r Ϫ ␤͑ ͒ r͔ a p z p Sz beled dataset that are neutrally related to the parcel;

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Fig 6 live 4/C NOVEMBER 2005 MCDOUGALL AND JACKETT 2063 that is, the parcel, (S, ⌰, p), and the datum (Sr, ⌰r, pr) in the labeled dataset have the same density when moved adiabatically and isentropically to the pressure 0.5(p ϩ pr). For reasonable values of (⌰Ϫ⌰r) and (p Ϫ pr) the denominator in (29a) and (29b) is close to N2 and ␺␥ is close to 1. The third line of (28)—namely, ␺␥V · s—is due to the scalar product of the horizontal velocity and the difference in slope between the neutral tangent plane and the neutral density surface in the labeled reference dataset, ͒ ͑ r ١ r Ϫ ١ ϵ s nz ␥z . 29c We have seen in Figs. 4 and 5 that this contribution to (28) is particularly small [see the curve “␥n (labeled Levitus)” in Fig. 4a and the right-hand-most pair of bars in Fig. 5]. FIG. 7. The cumulative histogram of the occurrence of various values of |␺ Ϫ 1| and of |␺␥ Ϫ 1| in the ocean volume. The so-called reversible terms in the first two lines of (28) are the contribution to the flow through the neutral density surface of both (i) the adiabatic temporal denominator and this term is usually small when com- heaving of the neutral density surface and (ii) the adia- pared with N2. The term proportional to (p Ϫ pr) is also batic sliding of fluid along the neutral density surface. usually small when compared with N2. ␥ Note that, as (⌰Ϫ⌰r) tends to zero, (␺ Ϫ 1) also tends In Fig. 7a we show the frequency distribution of ␺ Ϫ to zero so that the second line of (28) does not misbe- 1 and ␺y Ϫ 1, and in Fig. 7b the cumulative distribution have but instead becomes of |␺ Ϫ 1| and |␺␥ Ϫ 1| for data from the ocean atlas. |␺ Ϫ 1 1 Ϫ1 This shows that 2% of the global ocean data have Ϫ ͫ 2 ϩ ␳ 2 ͑⌰ Ϫ ⌰r͒ Ϫ ␳ 2 ͑ Ϫ r͒⌰r ͬ | Ͼ |␺ Ϫ | Ͼ gTb N g Tb g Tb p p p 1 1.0 and 11% of the data have 1 0.25. By 2 2 ␥ contrast only 1% of the ocean data have |␺ Ϫ 1| Ͼ 0.25. r r ␺ A meridional cross section of is shown in Fig. 8 with . ⌰␥١ · ϫ ͑p Ϫ p ͒V Ͻ ␺ Ͻ r much of the Southern Ocean having 1.5 2, while⌰ ١ Ϫ r Since (p p )V · ␥ is typically of the same order as ␺ Ͻ r r some regions in the deep North Atlantic have 0 p , all contributions to the first two lines ␥١ · Ϫ⌰)V⌰) where the vertical gradient of orthobaric density ap- of (28) typically have the same magnitude. pears to imply unstable stratification. The fact that An estimate of the deviation of neutral density sur- ␥ |␺ Ϫ 1| is other than zero reflects the small differences faces from quasi materiality as compared with that of between the Gouretski and Koltermann (2004) atlas orthobaric density surfaces can be found by taking the and the Levitus atlas that is the basis of the neutral ratio of the first terms on the right-hand sides of (28) density software. and (26): The most important point to make here is that while 1 orthobaric density is nonquasi-material to the extent of ͑⌰ Ϫ ⌰r͒ ͑␺␥ Ϫ 1͒ 2 a built-in temperature difference of ⌰Ϫ⌰(p, ␳) that ϭ 0 ͑␺ Ϫ ͒ ͓⌰ Ϫ ⌰ ͑ ␳͔͒ exists between the Northern and Southern Hemi- 1 0 p, spheres, neutral density is only non–quasi material to 2 N the extent that the data being labeled are different from ϫ . 1 those that the ocean normally has at the location of the ͫN2 ϩ ␳g2T ͑⌰ Ϫ ⌰r͒ Ϫ ␳g2T ͑p Ϫ pr͒⌰r ͬ b 2 b p observation. Taking the orthobarically relevant temperature difference ⌰Ϫ⌰(p, ␳)of3°C to be repre- ͑ ͒ 0 30 sentative of the values in the atlas of the present North This ratio represents the relative influence of adiabatic Atlantic and then imagining a change in the water mass heaving at a fixed horizontal location on the vertical S Ϫ⌰structure at a particular location such that the flow through neutral density surfaces as compared with temperature measured along isopycnals is 1°C warmer ⌰Ϫ⌰ ␳ flow through orthobaric density surfaces. The ratio of than in the ocean atlas, then 0(p, )is4°C while N2 to the square bracket in the denominator here can ⌰Ϫ⌰r is 1°C. In this way we might imagine that usually be ignored since an epineutral temperature dif- orthobaric density is a factor of 4 less quasi material ference of 1°C contributes only 2.7 ϫ 10Ϫ7 sϪ2 to the than is neutral density.

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FIG. 8. A cross section of ␺ along 328°E (32°W) in the Atlantic Ocean.

If the observed ocean data have the same properties layers in a layered model that is eddy resolving. The as the ocean atlas that is contained in the neutral den- fluid parcels undergo vertical excursions of typically sity algorithm, then (⌰Ϫ⌰r) and (p Ϫ pr) are zero, ␺␥ 100 m with a typical time scale of say three months, that ϭ 1, and the neutral density of the observed data is is, the vertical velocity is harmonic with a magnitude of totally quasi material except for the small unavoidable order 100 m/(8 ϫ 106 s). Taking a value of (␺ Ϫ 1) of 0.2 term induced by neutral helicity, ␺␥V · s. This conclu- gives an instantaneous value of the nonadiabatic un- sion is contrary to that of de Szoeke et al. (2000) who wanted interlayer velocity, e␷, of 3.125 ϫ 10Ϫ6 msϪ1. claimed that continuously formed surfaces (such as Now this unwanted interlayer entrainment velocity re- finely patched potential density surfaces, finely patched verses in sign every three months and, over the long orthobaric density surfaces, or neutral density surfaces) term, it behaves as an unwanted diapycnal diffusivity in are non–quasi material to the same extent as orthobaric a numerical model that has an upstream advection density surfaces. They claimed that the same amount of scheme. McDougall and Dewar (1998) show that equal non-quasi-material behavior as besets orthobaric den- and opposite entrainment velocities across an interface sity surfaces will be distributed over the area of these of a layered model behave like a diffusivity. From (16) finely patched surfaces. To understand why this is in- or (22) of McDougall and Dewar (1998, p. 1466) we correct and to quantify these issues it is critical to real- find that the false diapycnal diffusivity scales as the ize that the water mass difference between hemispheres reversible interlayer velocity multiplied by the layer that bedevils orthobaric density does not present a thickness of the layer of the layered model. Taking this problem for patched potential density or neutral den- layer thickness to be 300 m, we see that the non-quasi- sity since these large-scale changes in water masses are conservative nature of orthobaric density results in an included in the definition of both of these density vari- unwanted amount of diapycnal mixing to the extent of ables. This issue is discussed in more detail in the next a diapycnal diffusivity of order section. Ϫ Ϫ Imagine using orthobaric density surfaces to define Dnumerical ϭ 10 3 m2 s 1. ͑31͒

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Fig 8 live 4/C NOVEMBER 2005 MCDOUGALL AND JACKETT 2065

⌰١ ϫ ١ ١ ␤ ϭ Since this is two orders of magnitude larger than what H Tb p · S is now deemed acceptable for such a diffusivity in ocean ⌰ ١ ϫ ١ ͒ ϭ ͑ 2ր models, it is clear that using orthobaric density as the N g Tb np n · k ͒ ͑ ⌰ ١ ϫ ١ ␤ vertical coordinate in an eddy-permitting layered ϭ pz Tb pS p · k, 33 model is incompatible with the present upwind scheme used for diapycnal mixing in some modern layered is everywhere zero in the region of interest. When neu- models. One might also question the use of the word tral helicity is nonzero, fluid motions along neutral tra- reversible to describe this type of interfacial velocity jectories have a helical nature (McDougall and Jackett since present numerical techniques convert this revers- 1988) so that mean vertical advection occurs without ible advection into irreversible mixing of fluid between the need for any diapycnal mixing or dissipation of me- the discrete fluid layers. chanical energy. The nonzero helicity is also respon- Hence if orthobaric density were to be used as the sible for the discordances that occur along adjacent vertical coordinate of a layered ocean model, one leaves of Reid’s patched potential density surfaces, as would have to use a vertical advection scheme other illustrated in Fig. 14 of de Szoeke et al. (2000). This can than the upwind scheme of the Miami Isopycnic Coor- be understood from the third line of (33) where if ⌰ ١ ϫ ١ dinate Ocean Model (MICOM). One choice might be pS p · k is nonzero at the matching pressure then to advect the mean of the two layer properties across potential density surfaces referred to different refer- each interface in a similar manner to z-coordinate mod- ence pressures do not meet along a line at this pressure els and to the layered model of Schopf and Loughe [see also (40) below]. However helicity is surprisingly (1995). While this vertically centered scheme is inher- small in the ocean (McDougall and Jackett 1988; ently less diffusive than the upwind scheme, the diffi- MJ05a), and we will find that the difference between culty in a layered model context is that the ratio of the water masses in the North and South Atlantic is a successive layer depths varies in a less controlled man- much larger issue affecting the neutrality and quasi ma- ner than in a z-coordinate model and this exacerbates teriality of orthobaric density surfaces. It is convenient the false diapycnal mixing (Yin and Fung 1991; in this section to assume that helicity (33) is zero ev- Treguier et al. 1996). erywhere in the ocean and to thus concentrate on the De Szoeke et al. (2000) point out that when potential hemispheric differences in water masses. ١ ␳ ϷϪ⌽Ϫ1␳ ⌰Ϫ⌰ ␳ ١ vorticity (PV) is defined in terms of the gradient of Equation (14), n ␷ Tb[ 0(p, )] np, orthobaric density, the baroclinic production term holds the key to understanding the inevitable lack of Ϫ2 .١p is zero. While this is true, this benefit neutrality when ␳␷ is used as a global density variable ١␳ ϫ · ١␳␯ ␳ of orthobaric density is often offset by its nonquasi- Unfortunately, an ocean with zero helicity, that is with -١␳ ϭ 0, only guarantees that neutral sur ١p ϫ · ⌰١ material nature, which contributes to the production term in the PV conservation equation. That is, the con- faces exist; it does not guarantee that the oceanic ⌰(p, servation equation of potential vorticity [see, e.g., ␳) is single valued. When ⌰(p, ␳) is multivalued it is no ⌫ ␳ ⌰ ⌫ ␳ (2.5.7) of Pedlosky (1979)] contains the source term longer possible to approximate (p, , ) with 0(p, ) d␳␯ /dt), where ␻ is the relative vorticity [see (14)], so orthobaric density is forced to vary along)١ · ( ␻ ϩ kf) and f is the Coriolis parameter, and the reversible ad- a neutral trajectory. vection contributes to this production term the amount To illustrate the issues here, consider a neutral tra- [from (26)] jectory as in Fig. 9a that outcrops in both the Northern ր ͔ ͑ ͒ and Southern Hemispheres and attains its maximum͒ ١ ١͓͑␺ Ϫ ͒͑Ѩ␳ րѨ ͒͑ | ϩ ͒ ␻ ϩ͑ kf · 1 ␯ z pt ␯ V · ␯p pz , 32 depth near the equator. Along this neutral trajectory which serves to counter the apparent advantage of we take S and ⌰ to vary monotonically northward, as achieving a zero value for another term in the PV equa- occurs in the Atlantic [see Figs. 2 and 3 in You and tion, namely the baroclinic production term. Note that McDougall (1990)]. One way to ensure that our ocean ϭ0 everywhere is to assume that the⌰١ ١S ϫ · ١p the source term (32) is present in the PV conservation has equation even when the flow is totally adiabatic and ocean is independent of longitude. That is, one can as- isohaline. sume that a single meridional section characterizes the entire ocean. This is a rather extreme way of achieving 5. The hemispheric limitation of orthobaric density zero helicity and, in fact, all that is required is that the Neutral surfaces are well defined when the neutral contours of constant p and ⌰ on a neutral surface are ϭ ⌰ ١ ϫ ١ helicity, defined by Jackett and McDougall (1997) and parallel so that n p n · k 0. At the deepest point McDougall and Jackett (2005a, manuscript submitted (i.e., at an extremum of pressure on the neutral trajec- ␳ ١ ١ to J. Phys. Oceanogr., hereinafter MJ05a) as tory) np and n are both zero [see (13)] so that the

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can be illustrated using the classical S–⌰ diagram (Fig. 9b) for properties along the neutral trajectory of Fig. 9a. Near the sea surface at point A, the neutral trajec- ␴ tory runs tangent to the 0 potential density contour through point A and as the neutral trajectory descends (going northward) the pressure increases and the slope of the neutral trajectory on the S–⌰ diagram decreases. For example, if point B were at 1000 dbar, then the neutral trajectory at point B on Fig. 9b would be tan- ␴ gent to a 1 contour there. Proceeding beyond point B as the neutral trajectory shoals toward the Northern Hemisphere outcrop, the slope of the neutral trajectory ␴ on Fig. 9b increases until it is again tangent to a 0 surface at the northern outcrop point C. However, notice that the potential density of point C is greater than that of the southern outcrop, point A. This is because the pressure along the path from A to B to C is everywhere greater than zero so that all the way along this ␴ path the neutral trajectory crosses 0 contours toward ␴ larger values of 0. The different potential density at the northern and southern outcrops of neutral surfaces and of patched potential density surfaces illustrated in Fig. 9 is a well- known feature, for example, from the work of Reid (1994) and You and McDougall (1990). The neutral surface ABC in Fig. 9 is 100% neutral (since we have assumed that helicity is zero) and, because the temperature and salinity vary monotonically along the surface, it can also be represented as a function of salinity and/ or conservative temperature, ƒ(S, ⌰). Hence this surface is also 100% quasi material. Also indicated on Fig. 9b is point D, which lies on a ␴ deeper neutral trajectory beginning from the same 0 value as point C, but now emanating from the Southern Hemisphere outcrop at point D. This illustrates that in our zero-helicity ocean there are in fact two different water parcels that have the same values of p and ␳. Specifically, when p ϭ 0 the two distinct water parcels (SC, ⌰C, pC ϭ 0) and (SD, ⌰D, pD ϭ 0) both have the FIG. 9. (a) Sketch of the variation of conservative temperature ␳ ␴ same value of —namely, the same value of 0. The fact and pressure along a neutral trajectory that outcrops in both that ⌰ is not a single-valued function ⌰(p, ␳) is further Southern and Northern Hemispheres. (b) The solid line is the ␳ path of the neutral trajectory on the S Ϫ⌰diagram and the points illustrated in the –p diagram of Fig. 9c where points C A, B, and C correspond to the same marked points as in (a). (c) and D overly each other and yet they have very differ- The variation of in situ density and pressure along the neutral ent temperatures. trajectory. The path of the orthobaric density surface from point A is also sketched on the S–⌰ diagram of Fig. 9b. planes of constant p and of constant ␳ coincide (and Where this orthobaric density surface outcrops at the ١p ϫ surface in the Northern Hemisphere it has the same also coincide with the neutral tangent plane) and ,١␳ ϭ 0. We show that it is from this point that, in potential density as the Southern Hemisphere outcrop general, a different branch of the ⌰(p, ␳) function point A. opens up so that the northern function ⌰(p, ␳) is dif- Figure 9c also illustrates the nonneutrality of ortho- ferent to the southern branch. baric density surfaces because points A and C lie on the The multiple-valued nature of the ⌰(p, ␳) function same neutral trajectory but the contrast in orthobaric

Unauthenticated | Downloaded 09/30/21 02:37 PM UTC NOVEMBER 2005 MCDOUGALL AND JACKETT 2067 density between these points is equal to the contrast in ␳ ϭ ␳ ϭ ␴ ϩ Ϫ3 potential density (since ␷ 0 1000 kg m at the sea surface). Consider now different choices for the ⌫ ␳ function 0(p, ). The three panels of Fig. 10 show the same sketch of in situ density and pressure as Fig. 9c for the data along the neutral trajectory of Fig. 9a, but now we add contours of orthobaric density. Figure 10a illus- ␳ ␳ ⌫ ␳ trates the contours of ␷(p, ) when 0(p, ) is chosen so as to make the orthobaric surfaces neutral in the Southern Hemisphere [that is, for the Southern Hemi- sphere ⌰(p, ␳) function], while Fig. 10b illustrates ␳␷(p, ␳ ⌫ ␳ ) when 0(p, ) is chosen so as to make the orthobaric surfaces neutral in the Northern Hemisphere [that is for the Northern Hemisphere ⌰( p, ␳) function]. The orthobaric density of de Szoeke et al. (2000) was ⌫ ␳ formed using globally averaged data for 0(p, ) and its contours are sketched in Fig. 10c. No matter which ⌫ ␳ choice is made for the 0(p, ) function, the contrast in orthobaric density between the outcrops in the two hemispheres is independent of this choice. This con- ⌬␳ ϭ ␳C Ϫ ␳A ⌬␳ ϭ ␳C Ϫ ␳A trast, ␯ ␯ ␯ is equal to 0 0 0 . From (14) we see that the variation of orthobaric density along a neutral trajectory is given by the integral (along the neutral trajectory)

C ␳C Ϫ ␳A ϭ ⌬␳ ϭ ͵ ⌽Ϫ1͑⌫ Ϫ ⌫ ͒ ͑ ͒ ␯ ␯ ␯ 0 dp 34 A and in the three panels of Fig. 11 we illustrate this ⌫ ␳ integral for the three choices of 0(p, ) that were discussed in the previous paragraph in connection with FIG. 10. The solid line in these panels shows the in situ density Fig. 10. The area enclosed in the three different panels and pressure along the neutral trajectory of Fig. 9a, and the of Fig. 11 are identical, even though the nonneutrality is dashed lines show contours of orthobaric density for three differ- distributed quite differently between the hemispheres. ent choices of the reference compressibility function. The integral in the ⌰–p diagram along the neutral trajectory ABC can be used to find the difference in density at pressure pE between points E and F is pro- potential density between the two outcrops, which is portional to the area on this diagram, that is, equal to the contrast in the two outcrop values of orthobaric density, so that (see Fig. 12a) F ␳F Ϫ ␳E Ϸ 3 ͵ ͑ Ϫ E͒ ⌰ ͑ ͒ 10 Tb p p d . 36 E ␴C Ϫ ␴A ϭ ⌬␳ 0 0 ␯ The difference in orthobaric density between these two C ϭ ͵ ⌽Ϫ1͑⌫ Ϫ ⌫ ͒ points is then 0 dp A F E Ϫ1 F E ͑␳␯ Ϫ ␳␯ ͒ Ϸ ⌽ ͑␳ Ϫ ␳ ͒ C Ϸ 3 ͵ ⌰ ͑ ͒ F T pd Ϫ 10 b . 35 Ϸ 3⌽ 1 ͵ ͑ Ϫ E͒ ⌰ ͑ ͒ A 10 Tb p p d . 37 E The last integral comes from integrating (15) along the This relation proves that it is the existence of the pres- neutral trajectory from A to B to C. sure extremum on the neutral trajectory that opens up This same type of integral along the neutral trajec- the orthobaric surface to nonneutrality. The pressure tory can be done at a pressure closer to the maximum extremum allows a nonzero area on the ␪–p diagram pressure where (see Fig. 12b) the change in the in situ between the data on the neutral trajectory and a fixed

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FIG. 12. (a) Sketch of the integration of pressure with respect to temperature along the neutral trajectory ABC. (b) The two points E and F on the neutral trajectory straddle the place where the pressure is maximal.

pressure equal to the pressure of two distinct points on the neutral trajectory. This nonzero area means that the density ␳ of these two parcels are different. Hence we have two parcels on a neutral trajectory that share the same pressure but different densities. Since orthobaric density is a function of p and ␳ we conclude that orthobaric density ␳␷(p, ␳) cannot be constant along the neutral trajectory when there is area enclosed on the ␪–p diagram of Fig. 12b and in the integral in (37). The possibility of this area arises when a pressure extremum exists on a neutral trajectory. The same conclusion can be reached by a reductio ad absurdum proof regarding the same two points E and F in Fig. 12b. Assume that these two points have not only the same pressure but also the same in situ density and hence the same orthobaric density since ␳␷ ϭ ␳␷ (p, ␳). However, since the two parcels have different tempera-

←

⌽Ϫ1 ⌫Ϫ⌫ FIG. 11. Sketch of the integrals of ( 0) with respect to pressure along the neutral trajectory ABC for different choices of ⌫ ␳ the reference compressibility function 0(p, ) corresponding to the choice displayed in Fig. 10.

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⌫ ␳ ⌰ Ϫ⌫ ␳ ⌰ tures, (p, , ) 0(p, ) is different at E than at F so the same ( , p) combination, one in each hemisphere. that [from (14)] the epineutral gradient of orthobaric It is only the presence of the pressure extremum ␳ ١ density, n ␷, cannot be zero all the way from E to F (circled) that stops an orthobaric density variable being along the neutral trajectory. This contradicts the initial neutral on the whole trajectory. Note that in each case assumption that the two parcels E and F have the same in Fig. 13 there is zero helicity and the neutral surfaces values of in situ and orthobaric densities. Hence we are well defined, as evidenced by the fact that the left- conclude that it is the multivalued nature of ⌰(p)in hand panels are composed of a series of lines so that no Fig. 12b; that is, the fact that ⌰ is not a single-valued area can be enclosed on this diagram by closed integra- function of p along the neutral trajectory that has en- tion paths in space. sured that orthobaric density must vary along the neu- We have shown that the points that deny orthobaric tral trajectory. density the possibility of being neutral are the locations ϭ ١ These two different proofs based on Fig. 12b, one where np 0 because this allows the existence of involving (37) and the other reductio ad absurdum different branches of the ⌰(p) function on a single neu- proof have both shown that, when temperature is not a tral surface. The epineutral pressure gradient vanishes single-valued function of pressure along a neutral tra- when ␳ ϭ ١ jectory, it is not possible to achieve n ␷ 0,sowe Ϫ1 ١␳͒, ͑38͒ ١p Ϫ⌫͑ ١p ϫ ١S͒ ϭ 0 ϭ ␳␤ Ϫ ⌰١␣͑ ١p ϫ ␳ ⌫ conclude that it is not possible to construct a 0(p, )so ⌫ ␳ ⌰ Ϫ⌫ ␳ ١␳ ϭ 0 or equivalently ١p ϫ that (p, , ) 0(p, ) is zero along the neutral or, in other words, when trajectory. This implies that this situation of ⌰(p) being when multivalued on a neutral trajectory is equivalent to tem- S. ͑39͒ ١␤ Ϫ ⌰ ١␣ ␳ ϭ 0 ϭ ١ perature not being a function only of p and ␳, ⌰(p, ␳). p p p The fact that ⌰ƒ(p, ␳) can also be seen from Figs. This shows that the multivalued nature of the ⌰(p, ␳) 9b,c, where points C and D have the same values of p function arises at points where the thermal wind is zero Ϫ⌰ ١␣ ϭ ϫ ١ ␳ ⌰ ϭ ␳Ϫ1 Ϫ1 2 ϫ and but different values of . We conclude that a [since ƒVZ g N k np gk ( p ١ ␤ ϭ⌰١ ϫ ١ ١ zero-helicity ocean in which both p · S 0 p S); see (16) of McDougall (1988)]. In MJ05a we ١ ⌰ ١ ١␳ ϭ ⌰ϭ ϫ ١ ⌰١ and · p 0 does not ensure that showed that zero helicity requires that p and pS be ⌰(p, ␳) and hence does not ensure that orthobaric sur- parallel, and here we find that, if these horizontal vec- ϭ ١␤Ϫ⌰ ١␣ ␤ ␣ faces can be made to coincide with neutral surfaces. tors are in the ratio of / so that p pS 0, ⌰| Ϫ In Fig. 13 we present some generic examples of n then orthobaric density surfaces lose the ability of being | p n diagrams and corresponding vertical cross sections made neutral even in a zero-helicity ocean. In Fig. 14 to illustrate when orthobaric density surfaces could be we show data on the S–⌰ diagram from the North and made neutral and when they could not. Figures 13a,b South Atlantic from a constant pressure of 505 dbar. have a neutral trajectory on which temperature is not a The data are reasonably close to being described by a monotonic function of pressure, but ⌰(p) is a single- single convoluted line, and the “width” of this line is a ⌫ ␳ ⌰ Ϫ⌫ ␳ valued function and so (p, , ) 0(p, ) could be measure of the helicity of the Atlantic ocean, that is, a made to be zero and the orthobaric density surface measure of the ill-defined nature of neutral surfaces in could be made neutral. Figures 13c,d have three points the Atlantic. The Southern Hemisphere data are rela- where there are pressure extrema on the neutral trajec- tively fresh, and the Northern Hemisphere data are ␴ ␴ tory and these are three separate reasons why ortho- relatively salty. The two 0 surfaces and the 1 surface baric density cannot be neutral in this example. Figures are the values of these patched potential densities of 13e,f show a water mass that has an extra line of data Reid’s (1994) second isopycnal surface in the Atlantic. ␴ ϭ ␴ ϭ that has a variety of temperatures but all at constant The 0 27.3 and 1 31.938 isopycnals coincide in pressure, approximating the influence of Mediterra- the Southern Hemisphere at 500 dbar in the middle of ␴ ϭ ␴ ϭ nean Water in the North Atlantic. At the junction point the cloud of data, while the 0 27.44 and 1 31.938 the pressure gradient is not defined, but nevertheless it isopycnals coincide in the northern cloud of data. The would be possible to have an orthobaric density surface widths of these clouds of data is a measure of helicity being neutral. If the Mediterranean Water line had and of the discordances that occur in the patched po- pressure increasing with temperature along the line, tential density surface patching procedure, while the then orthobaric density would necessarily lose its neu- much larger contrast in salinity and temperature between trality. In Figs. 13g,h, we see a neutral trajectory dip- the hemispheres is a measure of the inevitable nonneu- ping from the outcrops in both hemispheres to a single trality of a single-valued function such as ␳␷(p, ␳). maximum pressure, but with a more interesting varia- The occurrence of points or regions of zero horizon- tion of temperature, such that there are two points with tal density gradient is inevitable when one considers

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FIG. 13. (left) Sketches of ⌰–p diagrams and the corresponding sections in physical space of several generic neutral trajectories. The points marked with circles are the pressure extrema on the neutral trajectories and it is from these points that the different branches of the ⌰ϭ⌰(p, ␳) function emanate. The existence of these pressure extremes limit the ability of orthobaric density to be neutral. The points marked with crosses are points of extreme temperature on the neutral trajectory, and it is from these points that the different branches of the function p(S, ⌰) emanate (see McDougall and Jackett 2005b). The existence of these points limits the ability of a function of salinity and temperature to be neutral.

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function ⌰(p, ␳), We conclude that it is from locations ١␳ ϭ 0 that the different branches of ١p ϫ where the ⌰ϭ⌰(p, ␳) function diverge. That is, different branches of this function emanate from locations where ⌰ ١ ϭ ١ np 0 and n 0.

6. Orthobaric density in comparison with Reid’s isopycnals In Table 1 we show the Reid (1994) definitions of 10 patched isopycnals for the Atlantic Ocean. The boldface numbers are the label used by Reid to describe these isopycnals, and we note that for each patched isopycnal the boldface potential densities are common to both the North and South Atlantic. That is, as one

FIG. 14. Data from the ocean atlas from the North and South crosses the equator, Reid uses a single definition of ␴ Atlantic at a fixed pressure of 505 dbar. Also shown are two 0 potential density on each of his patched isopycnals. ␴ contours and one 1 contour, composing Reid’s (1994) second This contrasts with the interpolation procedure used by isopycnal surface in the Atlantic. de Szoeke et al. (2000) where some of the interhemispheric differences appear at the equator (see their more than a single hemisphere. In terms of our ⌰–p Fig. 17). diagram of data on neutral density surfaces (Fig. 12a), The first thing we note from Table 1 is that the po- ␴ given that a neutral density surface outcrops at high tential density with respect to zero pressure, 0, that is latitude with different temperatures (and salinities) in used as the definition of these patched isopycnals for the two hemispheres, it is inevitable that orthobaric data in the upper 500 dbar is often different in the density will be nonneutral to the extent of the area South and North Atlantic; for example, layer 2 is denser ␴ Ϫ3 enclosed in this diagram up to the p ϭ 0 axis since this in 0 terms by 0.14 kg m in the North Atlantic than in area is proportional to the difference between the po- the South Atlantic. Since at zero pressure orthobaric ␴ tential densities at the two outcrop locations in the density is equal to 0, it is clear that the differences in ␴ Southern and Northern Hemispheres. 0 that appear in Table 1 for each layer are equal to the The cause of this inherent limitation of orthobaric differences in orthobaric density at the two outcrops of density can be discussed more generally by considering Reid’s patched isopycnals. Since these patched isopyc- ١␳ ϭ 0 nals are reasonably neutral (see Fig. 5; McDougall and ١p ϫ the region in space near a location where ␴ as shown in Fig. 15. The surfaces of constant pressure Jackett 2005b), the contrasts in 0 in this table are a and density will in general have different curvatures, convenient way of seeing the extent of the nonneutral- and we show two points, marked A and B, that have the ity of orthobaric density surfaces. ␳ ϭ same values of pressure and in situ density. In general, Consider now the orthobaric density surface ␷ Ϫ3 there will be a horizontal gradient of temperature and 1027.816 kg m , which outcrops in the South Atlantic salinity in this region so that parcels A and B will have poleward of Reid’s tenth layer but outcrops in the different values of ⌰. Hence the temperatures of these North Atlantic equatorward of the fifth layer (see ␴ two parcels cannot both be described by the same Table 1 and compare 27.816 with the underlined 0 numbers for layers 5 and 10). Reid’s patched potential density surfaces 5 and 10 differ in depth by 2000 m at the equator, so we see that points that would mix and move neutrally to places that are 2000 dbar apart in the water column actually lie on the same orthobaric density surface. ⌫ ␳ Since there is choice in the definition of 0(p, ) and FIG. 15. Sketch of some surfaces of constant pressure and of in therefore choice in the definition of orthobaric density, ١␳ ϭ ϫ ١ situ density surrounding the location where p 0. Note the comparison between the particular orthobaric defi- the two points marked A and B that have the same pressure and in situ density but, in general, have different values of tempera- nition that one chooses and Reid’s patched isopycnals ture. Hence the conservative temperature at points A and B can- will depend on that choice. For example, if the defini- not both be described by the same function ⌰(p, ␳). tion was tuned so that the Southern Hemisphere was as

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TABLE 1. Potential densities of Reid’s (1994) 10 patched isopycnals for the Atlantic. Surface 4 is linked between hemispheres by the ␴ ϭ potential density surface 1.5 34.640 but this has been omitted from the table simply to avoid having two extra columns. Orthobaric density surfaces between ␴␷ ϭ 27.816 and ␴␷ ϭ 27.823 lie below surface 10 in the South Atlantic outcrop, while they are less dense than surface 5 at the North Atlantic outcrop. Since Reid’s surfaces 5 and 10 differ in depth by 2000 m at the equator, it is apparent that orthobaric density surfaces are far from being near neutral. Shading, fonts, and underlining are explained in the text.

␴ ␴ ␴ ␴ ␴ ␴ ␴ ␴ ␴ 0 1 2 3 4 3 2 1 0 Surface 1 26.750 31.090 26.750 2 27.300 31.938 31.938 27.440 3 27.563 32.200 32.200 27.630 4 27.675 32.355 32.376 27.777 5 27.755 32.425 36.980 41.400 41.395 36.980 32.456 27.824 6 27.770 32.445 37.013 41.440 41.440 37.017 32.485 27.846 7 27.787 32.476 37.041 41.500 45.840 41.500 37.067 32.523 27.874 8 27.800 32.487 37.057 41.538 45.880 41.539 37.099 32.548 27.892 9 27.804 32.498 37.074 41.547 45.907 41.562 37.115 32.560 27.901 10 27.815 32.502 37.080 41.564 45.920 41.572 37.126 32.569 27.908

Ϫ3 neutral as possible, then our ␳␷ ϭ 1027.816 kg m sur- biguity in defining patched potential density surfaces. face would skim just below the tenth surface in the Such ambiguity is illustrated in Fig. 14 of de Szoeke et whole Southern Hemisphere, rising to between surfaces al. (2000) where two leaves of a patched isopycnal do 9 and 10 in the pressure range 2500 dbar Ͻ p Ͻ 3500 not meet cleanly at the matching pressure of 500 dbar. dbar in the Northern Hemisphere, to between surfaces This ambiguity arises because of nonzero helicity [see 7 and 8 in the pressure range 1500 dbar Ͻ p Ͻ 2500 (33)] where the isobaric gradients of salinity and tem- ⌰ ١ ϫ ١ dbar to between layers 6 and 7 in the pressure range 500 perature are not parallel so that pS p · k is non- dbar Ͻ p Ͻ 1500 dbar before finally rising above the zero. To see this we examine the horizontal gradient of fifth layer near the sea surface in the Northern Hemi- potential density, given by sphere. The underlined values of potential densities ref- ͒ ͑ ⌰ ١͒ ͑␣ Ϫ ١͒ ͑␤ ␳ ϭ ١ erenced to the same pressure in the two hemispheres p ln ⌰ pr pS pr p . 40 help to explain how we can determine the migration of orthobaric density surfaces through Reid’s patched The horizontal gradients of two potential densities ref- isopycnals. Similarly, if one tuned the orthobaric den- erenced to different pressures will only be parallel at sity definition to suit the Northern Hemisphere, the the matching pressure (and hence no discordances will ⌰ ١ ١ ␳ ϭ Ϫ3 orthobaric density surface ␷ 1027.823 kg m would appear) when pS and p are parallel—that is, when ⌰ ١ ϫ ١ migrate across the patched isopycnals as shown by the pS p · k and the ocean’s helicity are zero. This top border of the lighter shading in Table 1. path-dependent ambiguity in patching potential density ⌫ ␳ Irrespective of the choice one makes for 0(p, ), we surfaces will not decrease as one reduces the pressure see from Table 1 that orthobaric density surfaces such interval between the matching pressures of patched po- as ␴␷ ϭ 27.816 rise progressively through six (surfaces tential density surfaces. Such helicity-caused ambiguity, 10 through 5) of Reid’s isopycnal surfaces that we know though small, (McDougall and Jackett 1988; MJ05a) is are not very different from neutral density surfaces real and is here to stay in oceanography irrespective of (Jackett and McDougall 1997). This behavior of a one’s definition of isopycnal. single orthobaric density surface crossing six of the De Szoeke et al. (2000) have interpolated the Reid (1994) patched potential density surfaces can also patched potential density definitions of Reid (1994) and be seen on Fig. 15 of de Szoeke et al. (2000); the a* of have produced discontinuities at the equator. They the fifth isopycnal at p ϭ 0 in the North Atlantic is then claimed that these discontinuities were a conse- greater than the a* of the tenth isopycnal at p ϭ 0inthe quence of the regional differentiation of Reid’s patch- South Atlantic. (Note that the caption to their Fig. 15 ing procedure. This is not the case. The interpolation has the descriptions of the full and dashed lines re- procedure adopted by de Szoeke et al. (2000) is not part versed.) of the Reid and Lynn (1971) method for forming The above discussion has considered a single neutral patched potential density surfaces. Their interpolation trajectory, such as a path along a single ocean section. procedure effectively takes Reid’s different definitions In practice there is a certain amount of real ambiguity of isopycnals that by oceanographic necessity are val- in defining neutral surfaces and this will also cause am- idly different in the two hemispheres and brings them

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together at the equator in a way that was never done by ally along orthobaric density surfaces exceeds 10Ϫ5 Reid. m2 sϪ1 in 14% of the volume of the global ocean (Fig. In contrast to what is claimed in de Szoeke et al. 5, for a lateral diffusivity of 103 m2 sϪ1). Using a cutoff (2000), the discontinuities that appear on their Fig. 17 figure of 10Ϫ4 m2 it can be shown that the false dia- do not represent non-quasi-material flow inherent to pycnal diffusivity exceeds 10Ϫ4 m2 sϪ1 as often as when ␴ Reid’s patched isopycnal technology. Rather, these dis- lateral mixing is assumed to occur along 0 surfaces continuities arise because of the nature of their own (approximately 5% of the global ocean volume). These interpolation procedure. The isopycnals selected by global statistics show that, while orthobaric density is ␴ Reid and Lynn (1971) and by Reid (1994) had pressures perhaps a little more neutral than 0, it is significantly ␴ ␴ at the equator that were close to the reference pres- less neutral than 2 (see Fig. 5a). In addition, 2 does sures (1000 dbar, 2000 dbar, etc.). In this way, there was not suffer from the non-quasi-material problem that ␴ never any switching in the vicinity of the equator be- bedevils orthobaric density as 2 is 100% quasi mate- tween the different hemispheric definitions of an isopyc- rial. The nonneutrality of orthobaric density has also nal since this switching always occurred at the matching been illustrated by showing a particular orthobaric den- pressures of 500 dbar, 1500 dbar, and so on. It is the sity surface that crosses through six of Reid’s isopycnals vertical interpolation scheme devised by de Szoeke et in the Atlantic. al. (2000) and by de Szoeke and Springer (2005) that Also we find that 0.65% of the ocean volume has causes the unphysical vertical discontinuities at the equa- orthobaric density decreasing with depth. This is to be ␴ ␴ tor such as is seen in Fig. 17 of de Szoeke et al. (2000). compared with figures of 0.25% for 1 and 2. With the De Szoeke et al. (2000) further claimed that patching sound speed anomaly from the reference sound speed potential density surfaces in the continuous limit (e.g., being say Ϫ15 m sϪ1 (as it is in the North Atlantic), a as is achieved by the neutral density concept) merely reversal in the vertical gradient of orthobaric density distributes the cross-surface mass exchange of their Fig. occurs whenever internal waves stretch the water col- 17 horizontally but does not eliminate it. The discussion umn so that N2 is less that 10Ϫ6 sϪ2. around our Figs. 9–12 shows conclusively that this is There is some freedom in the construction of an incorrect. orthobaric density variable because the reference com- ␳ pressibility, or reference sound speed c0(p, ), is chosen 7. Discussion at the discretion of the user. The above results on the A prime concern for a variable that is used as a co- neutrality (i.e., the error in the square of the density ordinate in a layered model is the amount of false di- surface slope) and vertical stability of orthobaric den- apycnal mixing that is caused by diffusing temperature sity are for the global orthobaric density variable de- and salinity along the coordinate direction. De Szoeke fined in de Szoeke et al. (2000). The fact that an et al. (2000) examined this issue and concluded that the orthobaric density surface crosses six of Reid’s (1994) angle between their orthobaric density surfaces and the isopycnals in the Atlantic is a property of any neutral tangent plane “was sufficiently small that, for orthobaric density variable and is independent of the ␳ the purposes of computing a turbulent diffusivity tensor choice of c0(p, ). (assumed to be diagonal with respect to neutral tangent We have shown that it is the existence of a pressure ١␳ ϭ ١p ϫ planes) with respect to orthobaric coordinates, it may extremum along a neutral trajectory (where be neglected except possibly near the bottom.” This 0) that ensures that no single-valued function of p and conclusion seems at odds with their Fig. 12, which ␳ can be neutral. Since neutral trajectories outcrop near shows that approximately one-half of the area of that the poles, an extremum in pressure is guaranteed. The North Atlantic section (and almost all of the bottom- corresponding limiting feature of any function of S and most 2000 m) has a fictitious dianeutral diffusivity ⌰ is the existence of an extremum in temperature along ϭ0 (McDougall⌰١ ١S ϫ greater than 10Ϫ5 m2 sϪ1 (for a lateral diffusivity of 103 a neutral trajectory where m2 sϪ1). There seems no reason to ignore the deepest and Jackett 2005b). It turns out that such extrema are 2000 m of the ocean given that modern instrumentation not as common, and we have demonstrated that a func- ␴ ⌰ can detect density inversions even in the deep ocean tion such as 2(S, ) is much closer to being neutral when averaged over 10 m, modern ocean atlases are than is orthobaric density, ␳␷ (p, ␳). vertically stable almost everywhere, and the deep ocean Orthobaric density does have one definite advantage appears as part of ocean models of both the level and over many other density definitions in that it possesses layered variety. a geostrophic streamfunction (or Montgomery poten- Our analysis of the global ocean demonstrates that tial). Before the work of Sun et al. (1999) and Hallberg the false diapycnal diffusivity caused by diffusing later- (2005) this would have been a significant advantage of

Unauthenticated | Downloaded 09/30/21 02:37 PM UTC 2074 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 35 orthobaric density, but now the methods of these pa- sion following (40)], while the different water masses pers can be used to evaluate the horizontal pressure and different compressibilities that occur in the North gradient in layered ocean models having coordinates and South Atlantic can be accommodated by finely that are functions of salinity and conservative tempera- patched potential density surfaces without the occur- ture, ƒ(S, ⌰) rence of such discordances. De Szoeke et al. (2000) have claimed that the non- We have shown that what limits the near-neutral na- quasi-material nature of orthobaric density surfaces is ture of finely patched potential density surfaces and equivalent to the flow through Reid’s patched isopyc- neutral density surfaces (where the patching is effec- nals that occurs at the junction of the leaves. De Szoeke tively done every 4° of latitude and longitude) is the et al. (2000) further stated that, in the limit as the patch- inherent path dependency of neutral trajectories (i.e., ing is done continuously, the same non-quasi-material the inherent ill-defined nature of neutral surfaces) that flow occurs through these continuously patched sur- is proportional to neutral helicity, which in turn is pro- By contrast, the nonneutral .⌰١ ١S ϫ · ١p faces as occurs through orthobaric density surfaces. We portional to have shown that these claims are both incorrect. and non-quasi-material nature of orthobaric density Rather, orthobaric density surfaces are many times less surfaces is caused not only by this helicity of oceanic quasi material than continuously patched potential den- data, but also by the different branches of compress- sity surfaces and neutral density surfaces (see Fig. 7). ibility that open up on either side of a pressure extre- In Figs. 9–12 we showed an example of a surface of mum along a neutral trajectory. This much larger con- the functional form ƒ(S, ⌰) that is 100% neutral and tribution to the non-quasi-material nature of orthobaric 100% quasi material; however, along this surface there density would occur even if the ocean’s helicity were is the unavoidable variation of orthobaric density be- everywhere identically zero so that (i) neutral surfaces cause of the different water mass relations in the two were well defined, (ii) neutral trajectories were no- hemispheres. This example stands in contrast to the where helical in nature, and (iii) there were no discor- claim by de Szoeke et al. (2000) that the non-quasi- dances between the various leaves of patched potential material flow through orthobaric density surfaces must density surfaces. While neutral density surfaces are be shared by all other types of density surfaces such as only non–quasi material to the extent that oceanic data the finely patched potential density surface we consider changes with time at a given location, orthobaric den- in Fig. 9. sity surfaces are non–quasi material to the additional It is important to realize that interhemispheric water and larger extent of the interhemispheric differences mass contrasts are not an indication of nonzero helicity. between water masses [see (30)]. The different branches of functions such as ⌰(p, ␳)do While the potential vorticity equation based on not cause problems with neutrality or quasi materiality orthobaric density seems to have an advantage over for patched potential density surfaces because the dif- several other types of potential vorticity, the non-quasi- ferent northern and southern water masses never meet material nature of orthobaric density means that the and so the helicity can remain zero even though ⌰(p, ␳) source term of potential vorticity is nonzero even when is multivalued. However, these different branches limit the flow is totally adiabatic and isohaline. the neutrality and quasi materiality of a single-valued Orthobaric density may well prove useful as a coor- function such as ␳␷(p, ␳). dinate in a descriptive or inverse study of a single ocean Figure 9 is perhaps the simplest demonstration that a basin where the reference compressibility is tuned to finely patched potential density surface can be 100% the ocean data in that basin. The existence of a Mont- neutral and 100% quasi material. De Szoeke et al. gomery potential would prove useful, and it should usu- (2000) did not make the important distinction between ally be possible to have the orthobaric density surfaces the relatively small variations in water mass that occur sufficiently neutral in a single ocean basin. The lack of at the matching pressure in a single hemisphere (which exact quasi materiality is of less importance in an in- is due to nonzero helicity) and the much larger water verse model than in a forward model that is integrated mass differences that occur between the different hemi- over many time steps, such as for climate studies. spheres and that occur even in an ocean in which he- As for a practical recommendation for interfaces for licity is everywhere zero. This distinction can be under- forward layered ocean models, it is difficult to see how stood as the thickness of the data cloud in Fig. 14 versus patched potential density surfaces, or patched ortho- the width between the southern and northern data in baric density surfaces, could be used because of the this figure. The point is that discordance at constant difficulties of the discordances that occur at the bound- pressure between differently referenced potential den- aries between the individual leaves (such discordances sity surfaces occurs because of helicity [see the discus- being due to the ocean’s helicity). A solution to this

Unauthenticated | Downloaded 09/30/21 02:37 PM UTC NOVEMBER 2005 MCDOUGALL AND JACKETT 2075 leakage between the leaves may not be impossible for Hallberg, R., 2005: A thermobaric instability of Lagrangian ver- some limited geometries and limited range of hydro- tical coordinate ocean models. Ocean Modell., 8, 279–300. graphic properties, but is not yet forthcoming. Simi- Jackett, D. R., and T. J. McDougall, 1997: A neutral density variable for the world’s oceans. J. Phys. Oceanogr., 27, 237–263. larly, while neutral density is well defined, it is not Koltermann, K. P., V. Gouretski, and K. Jancke, 2004: Atlantic 100% quasi material, it is a function of latitude and Ocean. Vol. 4, Hydrographic Atlas of the World Ocean Cir- longitude, which is a considerable drawback for for- culation Experiment (WOCE), M. Sparrow, P. Chapman, and ward modeling applications, and it is also computation- J. Gould, Eds. International WOCE Project Office. [Avail- ally expensive to evaluate. For these reasons we have able online at www.bsh.de/aktdat/mk/AIMS.] McDougall, T. J., 1984: The relative roles of diapycnal and iso- never advocated neutral density as a coordinate for for- pycnal mixing on subsurface water mass conversion. J. Phys. ward models. This paper has demonstrated that the Oceanogr., 14, 1577–1589. nonneutral and non-quasi-material nature of orthobaric ——, 1987a: Neutral surfaces. J. Phys. Oceanogr., 17, 1950–1964. density make it unsuitable for use as a coordinate in a ——, 1987b: Thermobaricity, cabbeling, and water-mass conver- forward ocean model as compared with the present best sion. J. Geophys. Res., 92, 5448–5464. ——, 1988: Neutral-surface potential vorticity. Progress in Ocean- practice, which is to use interfaces that are functions of ography, Vol. 20, Pergamon, 185–221. ␴ ⌰ salinity and conservative temperature such as 2 (S, ). ——, 1989: Streamfunctions for the lateral velocity vector in a These interfaces are totally quasi material, and the compressible ocean. J. Mar. Res., 47, 267–284. horizontal pressure gradient can now be accurately ——, 1991: Water-mass analysis with three conservative variables. evaluated in such surfaces. If the nonneutrality of such J. Geophys. Res., 96, 8687–8693. ——, 1995: The influence of ocean mixing on the absolute velocity surfaces is considered an issue for such a globally de- vector. J. Phys. Oceanogr., 25, 705–725. fined function, then rotation of the diffusion tensor ——, 2003: Potential enthalpy: A conservative oceanic variable should be considered (Griffies et al. 1998). for evaluating heat content and heat fluxes. J. Phys. Ocean- ogr., 33, 945–963. Acknowledgments. We thank Drs. Scott Springer and ——, and D. R. Jackett, 1988: On the helical nature of neutral trajectories in the ocean. Progress in Oceanography, Vol. 20, Roland de Szoeke for giving us a copy of their Pergamon, 153–183. orthobaric density software. Doctors Nathan Bindoff, ——, and W. K. Dewar, 1998: Vertical mixing and cabbeling in Scott Springer, and Roland de Szoeke provided valu- layered models. J. Phys. Oceanogr., 28, 1458–1480. able comments on a draft of this manuscript. This work ——, and ——, 2005b: The material derivative of neutral density. contributes to the CSIRO Climate Change Research J. Mar. Res., 63, 159–185. Pedlosky, J., 1979: Geophysical Fluid Dynamics. Springer-Verlag, Program. 624 pp. Reid, J. L., 1994: On the total geostrophic circulation of the North REFERENCES Atlantic Ocean: Flow patterns, tracers, and transports. Prog- ress in Oceanography, Vol. 33, Pergamon, 1–92. de Szoeke, R. A., and S. R. Springer, 2005: The all-Atlantic tem- ——, and R. J. Lynn, 1971: On the influence of the Norwegian- perature–salinity–pressure relation and patched potential Greenland and Weddell Seas upon the bottom waters of the density. J. Mar. Res., 63, 59–93. Indian and Pacific Oceans. Deep-Sea Res., 18, 1063–1088. ——, ——, and D. M. Oxilia, 2000: Orthobaric density: A ther- Schopf, P. S., and A. Loughe, 1995: A reduced gravity isopycnal modynamic variable for ocean circulation studies. J. Phys. ocean model: Hindcasts of El Niño. Mon. Wea. Rev., 123, Oceanogr., 30, 2830–2852. 2839–2863. Eden, C., and J. Willebrand, 1999: Neutral density revisited. Sun, S., R. Bleck, C. G. H. Rooth, J. Dukowicz, E. P. Chassignet, Deep-Sea Res. II, 46, 33–54. and P. Killworth, 1999: Inclusion of thermobaricity in isopy- Feistel, R., 2003: A new extended Gibbs thermodynamic potential cnic-coordinate ocean models. J. Phys. Oceanogr., 29, 2719– of seawater. Progress in Oceanography, Vol. 58, Pergamon, 2729. 43–114. Treguier, A. M., J. K. Dukowicz, and K. Bryan, 1996: Properties Gouretski, V. V., and K. P. Koltermann, 2004: WOCE global hy- of nonuniform grids used in ocean general circulation mod- drographic climatology: A technical report. Berichte des els. J. Geophys. Res., 101, 20 877–20 881. Bundesamtes für Seeschifffahrt und Hydrographie, 35/2004, Yin, F. L., and I. Y. Fung, 1991: Net diffusivity in ocean general 49 pp. circulation models with nonuniform grids. J. Geophys. Res., Griffies, S. M., A. Gnanadesikan, R. C. Pacanowski, V. Larichev, 96, 10 773–10 776. J. K. Dukowicz, and R. D. Smith, 1998: Isoneutral diffusion You, Y., and T. J. McDougall, 1990: Neutral surfaces and poten- in a z-coordinate ocean model. J. Phys. Oceanogr., 28, 805– tial vorticity in the world’s oceans. J. Geophys. Res., 95, 830. 13 235–13 261.

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