Generalized Patched Potential Density and Thermodynamic

DECEMBER 2016 T A I L L E U X 3571

Generalized Patched Potential Density and Thermodynamic Neutral Density: Two New Physically Based Quasi-Neutral Density Variables for Ocean Water Masses Analyses and Circulation Studies

RÉMI TAILLEUX University of Reading, Reading, United Kingdom

(Manuscript received 24 March 2016, in ﬁnal form 13 September 2016)

ABSTRACT

In this paper, two new quasi-neutral density variables—generalized patched potential density (GPPD) and thermodynamic neutral density gT—are introduced, which are showed to approximate Jackett and McDougall n empirical neutral density g significantly better than the quasi-material rational polynomial approximation ga previously introduced by McDougall and Jackett. In contrast to gn, gT is easily and efficiently computed for arbitrary climatologies of temperature and salinity (both realistic and idealized), has a clear physical basis rooted in the theory of available potential energy, and does not suffer from nonmaterial effects that make gn so difficult to use in water masses analysis. In addition, gT is also significantly more neutral than all known quasi-material n n T density variables, such as s2, while remaining less neutral than g . Because unlike g , g is mathematically explicit, it can be used for theoretical as well as observational studies, as well as a generalized vertical coordinate in isopycnal models of the ocean circulation. On the downside, gT exhibits inversions and degraded neutrality in the polar regions, where the Lorenz reference state is the furthest away from the actual state. Therefore, while gT represents progress over previous approaches, further work is still needed to determine whether its polar de- ficiencies can be corrected, an essential requirement for gT to be useful in Southern Ocean studies, for instance.

1. Introduction of climatological datasets for temperature and salinity devoid of spurious water masses (Lozier et al. 1994), the The problem of how best to construct a quasi-neutral construction of inverse models of the ocean circulation density variable suitably corrected for pressure is a (Wunsch 1996), the tracking and analysis of water masses longstanding fundamental issue in oceanography whose (Montgomery 1938; Walin 1982), the construction of answer is vital for many key applications ranging from isopycnal models of the ocean based on generalized co- the study of mixing to ocean climate studies. These in- ordinate systems (Griffies et al. 2000; de Szoeke 2000), clude but are not limited to the separation of mixing into the study of the residual circulation (Wolfe 2014), and the ‘‘isopycnal’’ and ‘‘diapycnal’’ components necessary for parameterization of mesoscale eddy-induced mass fluxes the construction of rotated diffusion tensors in numerical (Gent et al. 1995). ocean models (Redi 1982; Griffies 2004), the construction Physically, it is generally agreed that a suitable density variable g should possess the desirable dual thermodynamic, and dynamical attributes of defining adiabatic Denotes Open Access content. surfaces (the thermodynamic attribute) along which fluid parcels experience no net buoyancy force (the dynamical Supplemental information related to this paper is available at attribute; e.g., McDougall 1987; de Szoeke and Springer theJournalsOnlinewebsite:http://dx.doi.org/10.1175/JPO- 2000; Huang 2014). The first attribute, which is equivalent D-16-0072.s1. to material conservation (also often referred to as quasi- material conservation, meaning here conserved whenever Corresponding author address:Rémi Tailleux, Department of Meteorology, University of Reading, Earley Gate, P.O. Box 243, Reading, RG6 6BB, United Kingdom. This article is licensed under a Creative Commons E-mail: [email protected] Attribution 4.0 license.

DOI: 10.1175/JPO-D-16-0072.1

Ó 2016 American Meteorological Society Unauthenticated | Downloaded 09/29/21 09:44 AM UTC 3572 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46 u and S are also conserved), poses no difficulty as it can to improve neutrality? From a theoretical viewpoint, the always be enforced by requiring g to be a function of natural starting point for constructing a density variable are potential temperature u and salinity S only. The second the thermodynamic equations for salinity and potential attribute—usually referred to as the neutral property—is temperature (or Absolute Salinity and Conservative Tem- problematic, however, as it can only be satisfied in perature) combined into the following equation for density: special circumstances not usually encountered in the ocean. To satisfy exact neutrality, =g would need to be Dr 1 Dp Du DS 2 5 r 1 r 5 2 u S q, (3) parallel at every point to the local neutral vector Dt cs Dt Dt Dt 5 = 2 = 52 = 2 22= d g(a u b S) (g/r)( r cs p), where a and b are the thermal expansion and haline contraction where r(S, u, p) is viewed as a function of salinity, po- coefficients defined relative to u, S,andp; g is the ac- tential temperature, and pressure; q represents changes 2 celeration of gravity; cs is the squared speed of sound; in density due to molecular diffusive effects of heat and r is in situ density; and p is pressure. To understand why salt. The left-hand side of (3) defines the differential - 5 2 22 the latter property cannot be satisfied in general, it is form d dr cs dp, which in general is not perfect useful to decompose =g into components parallel and and hence not integrable because of the nonzero helicity orthogonal to d as follows: of the neutral vector; otherwise, it is well accepted that - (possibly modified via the introduction of an appro- 1 priate integration factor) would define the most natural =g 5 =r 2 = 1 5 r b= 2 a=u 1 b 2 p R b ( S ) R cs choice of quasi-neutral, pressure-corrected density var- rbd iable. Mathematically, this is equivalent to state that the 52 1 R, (1) total differential dg of any mathematically well-defined g quasi-neutral density variable g can at best be written in where b is an integrating factor, and R is a residual term the form perpendicular to d. Taking the curl of (1) and multi- 1 plying the result by d gets rid of =g and yields an g5 r 2 1 d 52r a u 2 b 1d d b d 2 dp w b( d dS) w (4) equation for the residual R, namely, |fflfflfflfflfflfflfflfflfflfflffl{zfflfflfflfflfflfflfflfflfflfflffl}cs d- rbH 2 1 d (= 3 R) 5 0, (2) g and involves a nonvanishing, residual, imperfect differential form dw, with b as an integrating factor. Equation where the term H 5 d (= 3 d) is the helicity of the (4) yields (1) upon making the following substitutions neutral vector d, which shows that exact neutrality can dg / =g, du / =u, dS / =S, and dw / R as well as by only be achieved when H 5 0, a well-known result interpreting S and u as their climatological values rather (McDougall 1987; de Szoeke and Springer 2000; Huang than their instantaneous ones. The main advantage of 2014), with Eden and Willebrand (1999) discussing some (1) is that it defines the problem in standard Euclidean of the conditions necessary for the helicity to vanish. In vector space, which makes it easy to define the ‘‘small- practice, achieving H 5 0 in the ocean would either re- ness’’ of the residual R or its orthogonality with the quire the ocean to be at rest—as r would then be a neutral vector d. Working with (4) is harder mathe- function of pressure p alone—or in absence of density- matically, as it is less easy to define a norm in the space of compensated temperature–salinity variations along differential forms that can be used to say whether dw is surfaces g 5 constant, which is equivalent to say that the small or large. ocean would then have a well-defined temperature– The aforementioned mathematical difficulty has so far salinity relationship of the form u 5 u(g) and S 5 S(g). prevented the discovery of the ‘‘right’’ way of integrating In the ocean, however, the existence of density- the density equation [(4)], with standard potential den- compensated u/S anomalies conspire with thermobari- sity, patched potential density (PPD), and orthobaric city (the pressure dependence of the thermal expansion density representing the most well-known attempts. coefficient) to make H nonzero and hence forbid None of these density variables, however, is regarded as the construction of exactly neutral density variables. fully satisfactory. In absence of any clear theoretical ar- McDougall and Jackett (1988) discuss the way the ocean gument on how best to approach (4), McDougall’s (1987) appears to conspire to keep values of helicity low. postulate that the best quasi-neutral density variable If so, what then are the physical principles determining should be one constrained to be as neutral as feasible, the degree of nonneutrality that g should have? In partic- which in practice can be constructed by means of the ular, should material conservation be retained, or sacrificed Jackett and McDougall (1997, hereinafter JMD97)

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC DECEMBER 2016 T A I L L E U X 3573 neutral density software, has been widely accepted as the since it may confound the determination of diapycnal most appealing alternative.1 mixing in inverse ocean modeling studies for instance. So Nevertheless, gn has a number of shortcomings that far, however, while it is generally agreed that a purely tend to restrict its use primarily to observational studies material function of S and u can be constructed to be quite of the present-day ocean outside such regions as the neutral over a limited region of the ocean, as showed Arctic Ocean or Mediterranean Sea, where it is currently by Eden and Willebrand (1999) for the North Atlantic not defined. Indeed, its relatively high computational cost Ocean, McDougall and Jackett (2005b) have speculated makes its use prohibitive in numerical ocean modeling that this is fundamentally impossible to achieve for the studies; its lack of mathematically explicit form forbids its global ocean, after failing to construct a quasi-material n use in theoretical studies of the ocean circulation, and its rational polynomial approximation ga of g , whose neu- nonmaterialitymakesitsuseininversestudiesofocean trality appeared to be no better than that of s2, while being mixing or in the analysis of water masses using Walin’s neither a good approximation of gn, nor of its gradient. (1982) approach conceptually problematic, owing to the The main novelty of the present paper is to show that difficulty of evaluating thermobaric dianeutral dispersion JMD97 empirical neutral density gn, despite being pri- rigorously. Moreover, the physical basis for gn remains marily based on heuristic considerations, actually contains arguably quite unclear. Indeed, density variables—such useful information about how best to integrate (4).Thisis as potential density or orthobaric density, for instance— shown here by showing that gn is very close to a physically tend to be defined as purely thermodynamic concepts based quasi-neutral density variable that outperforms all having (or not) desirable dynamical properties when used known density variables in terms of neutrality. This vari- for recasting the equations of motion in thermodynamic able is also materially conserved and naturally approxi- n coordinates, for example, de Szoeke (2000). Thus, de mates g significantly better than ga. Such a variable is Szoeke and Springer’s (2000) orthobaric density is de- called thermodynamic neutral density and is a function fined as purely thermodynamic variable function of in situ of the Lorenz neutral density that enters the theory of density and pressure only, which dynamically defines an available potential energy, whose construction for a re- exact geostrophic streamfunction. In contrast, JMD97’s alistic ocean with a fully nonlinear equation of state was construction of gn tends to emphasize (approximate) recently discussed by Saenz et al. (2015).Tothatend,our dynamics over thermodynamics by defining it primarily in paper proceeds in two steps: The first step, detailed in terms of the equation b 52d dx ’ 0, where b is meant section 2, provides a new look at the concept of patched to represent the buoyancy of a single parcel experiencing potential density, which is the concept that historically adiabatic and isohaline lateral displacements dx.The prompted the construction of neutral density and ortho- thermodynamic properties of gn remain unclear, how- baric density. This section argues that the classical ex- ever, and a controversial topic [e.g., McDougall and pression for patched potential density is not a useful one Jackett (2005b) vs de Szoeke and Springer (2009)]. One for lacking any information about the actual patching of several difficulties is that neutral trajectories obtained process whereby density surfaces in different depth ranges as solutions of d dx 5 0 implicitly require nonmaterial are joined up at discontinuity points. An improved PPD, sources of heat and salt, thus violating the assumptions called generalized patched potential density (GPPD), underlying the definition of buoyancy b 52d dx, since, which is significantly less discontinuous than the original as showed by McDougall (1987), they generally end up PPD and which explicitly accounts for the patching pro- at a different vertical position than they originate from if cess, is constructed. The advantage of GPPD is to make integrated over a closed loop (around the main gyre of immediately clear what its continuous limit should be. the North Atlantic ocean for instance). Thermodynamic neutral density is one particular example Although JMD97 chose not to enforce materiality as a of continuously differentiable analog of GPPD, whose way to maximize the neutrality of their variable gn,both construction, comparison with other density variables, McDougall and Jackett (2005b) and de Szoeke and and neutral properties are discussed in section 3. Section 4 Springer (2009) seem to agree that nonmateriality is an discusses the results and their implications. undesirable feature of a quasi-neutral density variable,

2. A generalized patched potential density explicitly accounting for the patching process 1 One can distinguish between the NDFK, which is associated with the reference dataset used by JMD97’s software, and NDSK, a. Statement of the problem which is obtained by constructing neutral paths to parcels already labeled in the reference dataset. The distinction is not important Let us assume, like JMD97, de Szoeke and Springer for the present purposes. (2000), and many others before, that the concept of

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC 3574 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46 patched potential density holds the key for un- as feasible through the successive removal of discontinu- derstanding how best to construct a quasi-neutral den- ities, ultimately leading to the following posterior form of sity variable corrected for pressure. To that end, let us PPD: first recall that the standard definition of PPD is po- GPPD 5 5 2 tential density referenced to a piecewise constant pres- g PPDposterior s(S, u, pijk) sijk , (6) sure field, namely, where pijk and sijk are possibly three-dimensional 5 # # PPDprior s[S, u, pr(zi)], zi z zi11 , (5) piecewise constant fields. We call (6) GPPD and regard it as the postpatching process (posterior) or true where i 5 1, ..., N 2 1, where N is the number of discrete form of PPD. As we show below, a careful choice of the depth levels entering the construction of PPD. To be piecewise constant pijk and sijk can make GPPD signif- 5 specific, let us assume z1 0andzN is a depth typical of the icantly less discontinuous than PPDprior, thus enabling abyssal ocean. Here, we use the suffix prior to draw the GPPD to appear reasonably continuous when plotted attention of the reader to the fact that such a definition of if enough discrete elements are chosen. PPD is merely a front window for what PPD is really b. Description of the patching process as the about, given that (5) does not actually tell us anything successive removal of discontinuities about the patching process whereby density surfaces in different depth ranges are joined up at points of disconti- Our approach to the patching process aims to make nuity. Although the patching process itself can be de- PPDprior as continuous as feasible without altering sig- scribed in mathematical terms, it would be complicated nificantly =PPDprior, which is what makes PPDprior quasi and tedious to do so. For the present purposes, we adopt a neutral. To that end, we initiate the patching process by much simpler approach to the patching process, which we first subtracting a piecewise constant density offset in believe captures its essence while being mathematically each depth range as follows: clearer. Our approach stems from the realization that the (1) 5 2 patching process is fundamentally about dealing with the PPD s[S, u, pr(zk)] soffset[pr(zk)], (7) discontinuous character of PPDprior, in the sense that if ‘ that PPDprior were continuous, one would just trace water where pr(z)andsoffset(p) are assumed to be C functions (1) mass properties along surfaces of constant PPDprior.As of depth and pressure, respectively. The jump in PPD (1) a result, we redefine the patching process as a process across a discontinuity, denoted [PPD (zk, zk11)], fundamentally aimed at making PPDprior as continuous becomes

[PPD(1)](z , z ) 5 s[S , u , p (z )] 2 s[S , u , p (z )] 2 s [p (z )] 1 s [p (z )] k k11 " b b r k b b # r k11 offset r k offset r k11 1 ds ’ 2 offset 2 1 d 2 2 (p) [pr(zk) pr(zk11)] O( p ), (8) cs (Sb, ub, p) dp

where Sb and ub are salinity and potential temperature with xi, i 5 1, ... , Ni, yj, j 5 1, ... , Nj, a series of values along the surface of discontinuity. The natural discrete points in the horizontal directions aimed at choice to reduce the discontinuity is to choose capturing the spatial variations in the u/S relation-

ship. The problem, however, is that making soffset ds 1 offset (p) 5 , (9) vary with horizontal position must in turn introduce 2 (2) dp cs (Sb, ub, p) horizontal discontinuities in PPD , which can only be corrected by making pr vary with horizontal po- but unless Sb and ub obey a well-defined u/S relation- sition as well, leading to the following (and last) 5 5 ship of the form Sb Sb(p)andub ub(p), the ap- modification: proach will only succeed at removing the discontinuity locally, not globally. To cure the problem, it is nec- PPD(3) 5 sfS, u, p [x , y , p (z )]g essary to make the density offset vary with horizontal r i j r k 2 position as well, suggesting the following second soffset[xi, yj, pr(zk)], (11) modification: which is consistent with the form of generalized patched PPD(2) 5 s[S, u, p (z )] 2 s [x , y , p (z )], (10) r k offset i j r k potential density [(6)] given above. Obviously, this is as

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC DECEMBER 2016 T A I L L E U X 3575 far as the patching process can go, as we have run out of pressure field close to the reference pressure field pr(zk) options for further curing discontinuities. used in PPDprior and a density offset a function of both horizontal position and p , the optimization procedure c. Validation r naturally chooses a pressure field that can depart occa-

The above description of the patching process is ar- sionally strongly from pr(zk), with a density offset guably only qualitative. In practice, a full implementa- function that is simply a one-dimensional function of the tion of the method would require writing down explicit latter. The associated plot for gGPPD is given in the top- equations for the horizontal and vertical density jumps left panel of Fig. 1 for the 308W latitude–depth section in as well as providing an explicit procedure for con- the Atlantic Ocean, which can be compared with the straining the number of discrete elements and the values corresponding section for gn in the top-right panel. The of the piecewise constant pijk and sijk in (6). This is not strong similarity between the two figures is striking, further pursued here, however, as our primary aim is to given that the ability of gGPPD to reproduce the main use the concept of GPPD as a stepping stone for clari- features of gn is achieved with only 7 3 11 5 77 discrete fying the continuous limit of PPD and introducing the reference pressures pjk; the visual agreement is further concept of thermodynamic neutral density discussed in confirmed by the scatterplot of gn against gGPPD de- the next section. Before we do that, however, we first picted in the bottom panel of Fig. 4 (see below), which seek to validate the concept of GPPD. Specifically, if our shows a near-perfect correlation between the two hypothesis that (6) represents the true or revealed form quantities [the outliers seemingly originating from of PPD, it should be possible to construct the piecewise somewhat strange values of WOCE gn in enclosed seas, constant fields pijk and sijk to obtain an accurate ap- for which the use of JMD97 neutral density software is a proximation of JMD97 empirical neutral density gn; priori not valid]. A histogram of the differences gGPPD 2 gn indeed, as shown by the latter authors, gn is known to (blue bars in top panel of Fig. 4) shows that gGPPD ap- 2 behave as PPD; if so, gn should also behave like GPPD. proximates gn to better than 0.01 kg m 3 in most of the The following aims to show that this is indeed the case ocean, which is remarkable, as this is much better than and that a good agreement can, in fact, be achieved what is achieved by McDougall and Jackett’s (2005b) by choosing pijk and sijk to vary with latitude and variable ga (red bars in top panel), which was specifically depth only. constructed to best approximate gn. n To that end, using the g field supplied as part of the Intriguingly, the structure of the pjk’s is in fact much Gouretski and Koltermann (2004) WOCE dataset, we more reminiscent of that of the reference pressures that computed the 2D fields pijk and sijk, minimizing the misfit fluid parcels would have in their reference state of between gn and gGPPD using all possible data points for minimum potential energy that have been recently de- n which g is defined for an a priori–given partition Vjk. scribed in some recent advances in APE theory by Through trial and error, we settled on the particular two- Tailleux (2013b) and Saenz et al. (2015). The possibility dimensional partition of the ocean volume depicted in the to use APE theory to provide a physical basis for pr is top-left panel of Fig. 2 (shown below), with Dz 5 500 m confirmed in the next section and suggests that part of and Dy ’ 208, using a least squares approach to find the the structure of neutral density can be explained in optimal values of pjk and sjk in each subdomain. The terms of Lorenz reference density. main intent here is only to illustrate the feasibility of GPPD, not to explore systematically the sensitivity of the results to the different possible choices of volume 3. Continuous limit of PPD and connection with Lorenz theory of available potential energy partition or constraints on pijk and sijk. The results are illustrated in Fig. 2. Interestingly, the a. Implications for the continuous limit of PPD top-left and bottom-left panels strongly suggest that sjk A classical result of analysis is that all continuous is primarily controlled by pjk at leading order, which is confirmed by the regression analysis depicted in the functions can be viewed as the limit of piecewise con- bottom-right panel. Although it would be in principle stant functions. It follows that by making the number of possible to modify our optimization procedure to con- discrete elements forming the volume partition Vijk arbitrarily large, the piecewise constant pressure and strain the piecewise pressure values pijk to be close to actual pressures, this was not done here, in order to see density offsets can be assumed to converge toward / / whether the procedure would do it on its own or not. continuous fields pijk pr(x) and sijk sr(x), respectively. This in turn implies Rather, pijk was just imposed to lie within the range of pressures encountered in the ocean. It is therefore in- / 5 s u 2 s teresting to see that rather than choosing a piecewise GPPD GPPDcontinuous [S, , pr(x)] r(x), (12)

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC 3576 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46

FIG. 1. (top left) GPPD based on the GPPD reference pressure and density offsets depicted in the top left and bottom left of Fig. 2, respectively, at 308W in the Atlantic Ocean. (top right) Neutral density gn at the same longitude. (bottom left) Thermodynamic neutral density based on the reference pressure depicted in the top-right panel of Fig. 2. (bottom right) McDougall and Jackett (2005b) materially conserved approximation to gn. which is hence also continuous. In the following, we Motivated by the results of the previous section further constrain the form of the density offset by as- drawing a link between the structure of pijk and that of suming it to be a function of pr alone, that is, sr(x) 5 the reference pressure entering Lorenz APE theory, we sr1d[pr(x)], as in the previous section. Mathematically, introduce a new quasi-neutral density variable, called the continuous limit of =(GPPD) is not so well deﬁned, thermodynamic neutral density gT, deﬁned as since in general all what can be ascertained for the limit T 5 LZ 2 LZ of a piecewise constant function is that it be continuous, g (S, u) s[S, u, pr (S, u)] sr1d[pr (S, u)], (14) not continuously differentiable (differentiability is as- LZ sociated with piecewise linear functions, not just piece- where pr (S, u) is the reference pressure that a parcel wise constant). In other words, it is usually not true that would have if brought in a notional reference state of rest

=(GPPD) converges uniformly toward =(GPPDcontinuous), obtained by means of an adiabatic and isohaline re- where arrangement of the actual state. As shown recently by Tailleux (2013b) and Saenz et al. (2015), the reference = 5 = 1 = LZ GPPDcontinuous rS S ru u pressure pr that fluid parcels would have in the Lorenz " # reference state of minimum potential energy rLZ(z)isthe 1 ds r 1 2 r1d = solution of the level of neutral buoyancy (LNB) equation 2 pr . (13) cs (S, u, pr) dp LZ 5 LZ r[S, u, pr (zr)] rr (zr), (15) As shown below, this mathematical difficulty is reflected in the fact that the function gT introduced below does a where the possible time dependence of the reference better job at approximating gn than its gradient. state, for example, Tailleux (2013a), is neglected for the

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC DECEMBER 2016 T A I L L E U X 3577

FIG. 2. (top left) The latitude–depth dependent reference pressure seen by GPPD depicted in the top panel of Fig. 1. (top right) The reference pressure associated with Lorenz reference state underlying thermodynamic neutral density depicted in the bottom left of Fig. 1. (bottom left) Density offsets entering the construction of GPPD. (bottom right) Scatterplot of the GPPD reference pressure against GPPD density offset, showing the linear dependence of one on the other. moment. Future work, however, should aim to establish between the two variables, in order to make gT traceable when and where the temporal variations of Lorenz ref- to gn as defined in Huber et al. (2015). Traceability is erence state matter. For a time-independent reference important in order to interpret the remaining differ- state, the LNB equation [(15)] implies that the reference ences between the two variables as due to differences in depth of fluid parcels zr 5 zr(S, u) is a materially con- the physics rather than due to some of the arbitrary served quantity; solving (15) at all points in the ocean choices that enter the construction of density variables. provides the following explicit construction for the This was done here by means of a joint pdf analysis of LZ n continuous reference pressure field pr(x), namely, the respective distributions of r(S, u, pr ) and g , with sr1d(pr) constructed so as to minimize the misfit between 5 LZ T n pr(x) pr fzr[S(x), u(x)]g. (16) g and g . At leading order, sr1d(pr) is found to behave primarily as a linear function of zr by working in LZ The reference density profile rr (z) was estimated for the context of the Boussinesq approximation, using the WOCE dataset following the methodology detailed 23 pr 52r0gzr,withr0 5 1035.407 15 kg m and g 5 2 in Saenz et al. (2015), with an example of the resulting 9.81 m s 2,thevalueofr being obtained by regressing 8 0 pr(x)fieldat30W in the Atlantic Ocean being illus- the pressure and depth fields provided as part of the trated in the top-right panel of Fig. 2. WOCE dataset. After some trial and error, we finally T n b. Comparison between g and g settled on the following fit for sr1d(pr): To compare gT with gn, we calibrated the unknown s (p ) 5 a 1 bz 2 P (z ), (17) density offset function sr1d(pr) to minimize the differences r1d r r oly r

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC 3578 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46

FIG. 3. The piecewise polynomial function of zR entering the construction of sr1d(zr) (see text for details; thin solid line). Lo- n 2 1 FIG. 4. (top) Histogram of the decimal logarithm of the absolute cation of the maximum of the joint pdf between g s[S, u, pr(zr)] n GPPD n 1 value of the differences between g and g (blue), between g ajzrj b and jzRj (blue and orange dots). T n and g (green), and between g and ga (brown). (bottom) Scat- GPPD T n terplots of g , g , and ga against WOCE g . The straight line is n the 1:1 line of equation g 5 gproxy, where the latter is either one of with a 520.023 364 812 862 605 and b 5 T GPPD g , g or ga. 0.004 527 584 358 902, where the leading-order linear behavior of sr1d(pr) was further corrected by the T piecewise polynomial function Poly depicted in Fig. 3. variables, which reveal that g is, in general, better than This piecewise polynomial is made up of two distinct gGPPD at approximating gn, although it also reveals a few polynomials, whose slopes at the intersection point instances of rather large differences between gT and gn very discontinuously. Attempts at removing this dis- that do not exist for gGPPD. continuity degraded the agreement between gn and gT Another way to compare gT and gn is directly in (u, S) in the deep ocean, suggesting that gn might be func- space. Although gn is not materially conserved, it is tionally different in the deep ocean relative to the rest nevertheless possible to write it as a sum of a materially n of the ocean, perhaps as the result of difficulties with conserved part gmaterial(S, u) plus some residual dg.For n the method of characteristics underlying its construc- the present purposes, we estimated gmaterial(S, u)asthe tion in presence of bottom topographic features. The bin average of gn in (u, S)space,usingDS 5 0.1 psu and distribution for gT obtained from such a procedure is Du 5 0.18C for the binning, which is equivalent to defining n depicted in the bottom left of Fig. 1 for the same gmaterial as the materially conserved function of u and S Atlantic Ocean section at 308W previously used. An that best approximates gn in a least squares sense. atlas providing plots of gT every 108 of longitude, along Figure 5, top-left, top-right, and bottom-left panels show n n T with the corresponding plots for g and McDougall and gmaterial, g , and their residual, respectively. For compar- n Jackett (2005b) variable ga is also made available as ison, the residual between gmaterial and ga is also provided supplemental material. Clearly, gT does much better on the bottom-right panel, which shows significantly n T n than ga at capturing the main features and details of g . higher values. Remarkably, g and gmaterial appear to The main differences primarily occur in the upper re- exhibit the same functional dependence on S and u for gion of the ACC where gT displays some inversions not most of the ocean water masses, suggesting that the seen in gn. At the same time, it seems important to nonmateriality of gn might be the primary cause for the point out that as explained in JMD97, the values of gn observed differences between gT and gn, even though in the Southern Ocean are not obtained from the actual the residual gT 2 gn appears to have a rather complex Levitus data but from modified ones, which are found structure. Since the estimation of the nonmateriality necessary for correctly labeling neutral densities of the of gn has proven so far technically complex and con- second kind in such a region and which may explain some troversial [see de Szoeke and Springer (2009) vs of differences between gT and gn seen there. Apart from McDougall and Jackett (2005b)], the present results this issue, Fig. 4 (bottom panel) shows that gT and gn are are interesting as they might point to a potentially otherwise extremely well correlated. The upper panel much simpler way to quantify the nonmateriality of gn, shows a histogram of the differences between the two whichweplanoninvestigatinginfuturestudies.

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC DECEMBER 2016 T A I L L E U X 3579

n n FIG. 5. (top left) Materially conserved part of g obtained by bin-averaging g in (u, S) space. (top right) The quasi-material Lorenz neutral density bin averaged in the same way as gn. (bottom left) Difference between (top left) and (top right). (bottom right) As in bottom left, but for ga.

To conclude this section, it is important to point out This approach, however, is beyond the scope of the that the structure of the differences between gT and gn is present paper and will be discussed in a subsequent somewhat sensitive to the way—by no means unique— study, along with the importance of retaining or not the that the function sr1d(pr) is constructed and hence that temporal variations of Lorenz reference state. Note that these differences should only be regarded as indicative an approach such as (18) becomes necessary when rather than definitive. Indeed, it is important to note that constructing thermodynamic neutral density for water the WOCE gn is a neutral density of the second kind mass distributions differing from present-day ones, since n n (NDSK) rather than of the first kind and not necessarily calibrating sr1d(pr) to mimic g works only when g is as neutral as could be. Moreover, there might be alter- available (recall that the neutral density software is only native ways to construct sr1d(pr) that would result in an valid for present-day climatologies). even better agreement between gn and gT. On the other c. How neutral is gT relative to other density hand, it is also important to recognize that rather than variables? constructing sr1d(pr) to minimize the differences between gT and gn, one might prefer to define it based on Since JMD97’s construction of neutral density focuses physical arguments. The most natural approach would on =gn and its closeness to neutrality rather than on gn be in terms of a globally defined u/S relationship pa- itself, it is of interest to examine to what extent the good T n rameterized in terms of pr, that is, of the form Sr(pr) and agreement between the values of g and g demon- ur(pr), which would yield strated in the previous section also extends to the gradients. That this should be so is not mathematically ds 1 r1d ( p ) 5 . (18) guaranteed because it is easy to find counterexamples of r 2 u dp cs [Sr( pr), r( pr), pr] two continuously differentiable functions, f(x) and g(x)

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC 3580 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46

T FIG. 6. Probability distribution functions for the sine of the angle between =g and the neutral vector compared n with that of various other density variables: (a) s0, (b) s2, (c) ga, and (d) g . being approximately equal to each other without this k=g 3 dk jsin(=g, d)j 5 . (19) being true of their derivatives.2 Moreover, as mentioned k=gkkdk before, the process of defining the continuous limit of n =(GPPD) is not as well defined as it is for GPPD itself. As expected (from Fig. 6d), the WOCE g is the more As a result, scatterplots of jj=gTjj versus jj=gnjj or of neutral of all density variables, but the main new finding T angle(=gT, d) versus angle(=gn, d) are expected to ex- here is that g is a surprisingly close second, clearly T n hibit significantly more scatter than plots of g versus g , outperforming s0, s2, and ga, while achieving a degree n which was indeed verified. These are not shown as they of neutrality comparable in many ways to that of g . T are deemed to not be very informative, even though they Importantly, g thus appears as the first globally de- T tend to suggest that g performs better than ga at fined materially conserved density variable to pass n approximating =g . McDougall and Jackett’s (2005b) ‘‘kiss of death’’ s2 A more telling diagnostic is illustrated in Fig. 6, whose neutrality test, which they used to fail the de Szoeke and aim is to illustrate the relative degree of neutrality of gT Springer (2000) orthobaric density. Perhaps more as- n T relative to that of s0, s2, ga, and g , which is based on tonishing, however, is that g naturally outperforms computing the data-based frequency distribution of the ga, a materially conserved density variable that was sine of the angle made between the gradient of a given specifically constructed by McDougall and Jackett n density variable and the neutral vector: (2005b) to best approximate g and its gradient. The result is important because it demonstrates that McDougall and Jackett (2005a,b) significantly under-

2 estimated the ability of a materially conserved variable For instance, the L2 norm of the difference between f(x) 5 1 and g(x) 5 1 1 « sin(x/«) is bounded by the arbitrarily small pa- to approximate neutral density. The question that the 0 0 rameter «, whereas the L2 norm of f (x) 2 g (x) 52cos(x/«) is only present ﬁndings raises, and which future studies should bounded at best by 1 regardless of «. aim to clarify, is whether gT is the optimal way to

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC DECEMBER 2016 T A I L L E U X 3581

reference state and the actual state. Because of thermobaricity, this vertical ordering of fluid parcels may occasionally undergo significant modifications in regions where the actual state becomes too far away from Lorenz reference state, causing gT to develop inversions, as is the case in the polar regions. Our hypothesized link between gn and gT, therefore, is likely to exist only in those regions of the ocean where the vertical gradients of the two quantities have both the same sign and both correctly 2 FIG. 7. Schematics of the argument establishing that neutral predict ocean stability as measured by N . density needs to be a function of Lorenz reference density when constrained to be materially conserved. 4. Discussion n construct a materially conserved approximation of g or In this paper, we have reexamined several issues whether an even better way exists? pertaining to the construction of quasi-neutral density d. A posteriori rationalization of the relevance variables, starting with a fresh look at how best to define of Lorenz reference state to the theory of the continuous limit of patched potential density, a point quasi-neutral density variables of contention between JMD97 and de Szoeke and Springer (2000). To that end, it was found essential to n T The strong agreement found between g and g con- redefine patched potential density because even though firms our hypothesis that the empirical neutral density the latter is commonly defined as potential density ref- procedure designed by McDougall (1987) contains im- erenced to a piecewise constant reference pressure field portant information about the physics of quasi-neutral s[S, u, pr(zi)], zi # z # zi11, this formula obscures the density variables, which the present results suggest point whole machinery of the actual patching process whereby to Lorenz APE theory. Can this be rationalized a pos- different potential density surfaces are joined up at teriori? To see this, let us imagine that we have been points of discontinuity. As a result, we introduced the able to find a solution g(S, u) to the neutral density concept of generalized patched potential density equation constrained to be materially conserved, and let (GPPD) us show that it must necessarily be a function of Lorenz GPPD 5 2 reference density. To that end, let us consider the set of g s(S, u, pijk) sijk (20) all possible surfaces g(S, u) 5 constant; each of these surfaces can be plotted individually as depicted in the as the true or posterior form of PPD by reinterpreting left panel of Fig. 7 and will in general have a complicated the patching process as being primarily about the suc- shape in the actual state of the ocean. Because all the cessive removal of the discontinuities of the prior form surfaces g(S, u) 5 constant are materially conserved, of PPD. Mathematically, GPPD relies on a three- each of the parcels making up such surfaces will remain dimensional discrete partition of the total ocean vol- on such surfaces in any adiabatic and isohaline re- ume and the specification of the three-dimensional arrangements of the actual state, including the notional piecewise constant pressure fields pijk and sijk. GPPD state of rest entering the Lorenz theory of available is significantly less discontinuous than PPD, so that it potential energy. We know, however, that in a state of will look reasonably continuous when plotted. As a re- rest, the isosurfaces of any solution to the neutral density sult, one may in principle trace water mass properties equation must coincide with constant geopotential sur- simply along surfaces of constant gGPPD, as one would faces, as otherwise they would not be neutral. This im- naturally do with any standard continuous density var- plies therefore that g(S, u) must be a function of Lorenz iables such as s2. While practitioners of PPD may have a reference density and hence that Lorenz APE theory is hard time reconciling their approach to the patching the natural way to think about quasi-neutral density process with ours, or to accept that gGPPD really rep- variables if constrained to be materially conserved, thus resents the true form of PPD, our approach is vindi- confirming that the differences between in gn and gT cated by showing that it is possible to construct gGPPD must represent a measure of the nonmaterial conser- so that it approximates the WOCE gn dataset to better 2 2 vation of gn. than 10 2 kg m 3 over most of the ocean, which is sig- An important caveat, however, is that the above proof nificantly better than what can be achieved by McDougall is probably only valid as far as the natural vertical or- and Jackett’s (2005b) quasi-material rational polynomial n dering of fluid parcels remains the same in the Lorenz approximation ga.Ifg behaves both like PPD and

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC 3582 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46

GPPD, GPPD must represent a valid alternative way to time, although not on space. Moreover, because of look at PPD. thermobaricity and the existence of multiple levels of If one accepts our proposal to regard GPPD as the neutral buoyancy in some parts of u/S space, it is in true form of PPD, then the problem of how best to de- principle possible for the Lorenz reference state to fine the continuous limit of PPD becomes trivial, the change with time purely as the result of adiabatic GPPD 5 latter being then necessarily given by gcontinuous s[S, u, changes, which could potentially cause adiabatic ver- 2 GPPD 5 2 pr(x)] soffset(x)orgcontinuous s[S, u, pr(x)] soffset[x, y, tical dispersion with no signature in microstructure n pr(x)]. Again, this conclusion is vindicated by the possi- measurements, as is believed to be the case for g (but bility of constructing a very good (continuously differ- for physically quite different reasons). entiable) approximation of the WOCE gn dataset by T using as pr the reference position of a fluid parcel in The fact that the physically based g is naturally ca- Lorenz reference state, thus motivating the introduction pable of approximating gn significantly more accurately of a new materially conserved quasi-neutral density var- than McDougall and Jackett’s (2005b) rational poly- n iable, called thermodynamic neutral density, defined as nomial approximation ga, while also sharing most of g ’s key attributes, strongly suggests that Lorenz reference T 5 LZ 2 LZ g (S, u) s[S, u, pr (S, u)] sr1d[pr (S, u)], (21) state should play a more important role in the theory of neutral density than previously realized, at least in those where the function sr1d(pr) was defined as a piecewise regions of the ocean where the vertical gradients of the polynomial function of pr calibrated to minimize the two quantities both have the same sign and both cor- differences with WOCE gn. The main finding here is that rectly predict ocean stability. This view appears to be gT approximates gn and its gradient considerably better supported, at least partly, by the fact that whereas than ga(S, u), which was specifically designed to best neutral density has so far represented the main basis for approximate gn. But the ability of gT to mimic gn does thinking about how to define isopycnal and diapycnal not stop there. In addition to approximating gn very directions in the ocean, it is the theory of available po- well, gT possesses shares additional features that until tential energy that has formed the main basis for the now were thought to be the sole prerogative of gn, rigorous study of diapycnal mixing in the stratified namely, as follows: turbulent mixing community, following the pioneering work of Winters et al. (1995). Moreover, while meso- T d The variable g is the first physically based globally scale eddy parameterizations generally rely on isopycnal defined density variable that significantly outperforms directions based on the local neutral tangent plane, Gent all other density variables, apart from gn, in terms of et al.’s (1995) view that mesoscale eddies should act as a neutrality; in particular, it passes the McDougall and sink of APE suggests that such parameterizations might T Jackett (2005b) s2 neutrality kiss of death test. be more naturally formulated based on g . On the other T d The ACC region poses similar problems to both g hand, it is important to point out that Winters et al.’s and gn. Thus, the ACC region is associated with the (1995) APE framework has been so far validated only possibility of multiple neutral paths for gn (JMD97), for a linear equation of state, for which the concept of whereas it is associated with multiple levels of neutral density is unambiguously and uniquely defined; in con- buoyancy for gT (Saenz et al. 2015). Moreover, it is trast, the number of quasi-neutral pressure-corrected where gT displays inversions not seen in gn, whereas it material density variables of the form g(S, u) in a ther- is where the Levitus dataset used by JMD97 needs to mobaric ocean in the presence of density-compensated be modified in order for their neutral density software u/S anomalies is potentially infinite. Moreover, it can to function correctly. also be argued that it is the probability density function n T d Like g , g possesses interhemispheric differences in (PDF) attribute of the Lorenz reference state that is water mass properties, so that it is not affected by what really the property that matters for diagnosing mixing McDougall and Jackett (2005a) consider to be a rigorously rather than its connection to available po- challenge for quasi-neutral density variables. tential energy. Thus, one could argue in the oceanic case n T d Like g , g is affected by both cabbeling and thermo- that diagnosing mixing rigorously could equally well be baricity, in contrast to standard potential density, as achieved by analyzing the temporal behavior of the PDF

discussed in Iudicone et al. (2008). of s0, s2,ors4 [although how to relate the effective n d As is well known, g is a function of thermodynamic diffusivity in PDF space, e.g., Winters et al. (1995),to variables u, S,andp as well as horizontal position, observed values of diapycnal mixing in physical space whereas if the time dependence of Lorenz reference is a priori not straightforward]. Moreover, while the state is retained, gT(S, u, t) is also dependent on vertical gradient of Lorenz reference density is in

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC DECEMBER 2016 T A I L L E U X 3583 general an exact predictor of the stability of a stratified Acknowledgments. This work was supported by fluid (as measured by the sign of N2) both for a linear NERC Grant NE/K016083/1: ‘‘Improving simple cli- equation of state and nonlinear equations of state mate models through a traceable and process-based function of temperature and pressure alone, this is not analysis of ocean heat uptake (INSPECT).’’ Extensive the case in the ocean, as evidenced by the presence of discussions with Trevor McDougall significantly con- inversions exhibited by gT in the ACC region. On this tributed to clarify many of the issues discussed here. basis, it would appear that even though gT appears to Comments by Magnus Hieronymus, David Marshall, represent an important development in the theory of Geoff Stanley, Juan Saenz, Stephan Riha, and Paul neutral density, it is unlikely to be the last word on the Barker significantly improved clarity and presentation. matter. Tailleux (2016) recently developed a new ther- Data and software needed to reproduce the results modynamic approach to quasi-neutral density variables presented here are made freely available by the author and conjectured that a purely quasi-neutral density upon request. variable potentially significantly more neutral than gT might exist. Future work should therefore be aimed at REFERENCES testing Tailleux’s (2016) conjecture, which, if valid, de Szoeke, R. A., 2000: Equations of motion using thermodynamic would have important implications for the whole field. coordinates. J. Phys. Oceanogr., 30, 2184–2829, doi:10.1175/ T From a practical viewpoint, g has the advantage over 1520-0485(2001)031,2814:.2.0.CO;2. gn of being significantly easier to compute, following ——, and S. R. Springer, 2000: Orthobaric density: A thermodynamic recent progress in our understanding of how to construct variable for ocean circulation studies. J. Phys. Oceanogr., 30, , . the Lorenz reference state in a cheap and computa- 2830–2852, doi:10.1175/1520-0485(2001)031 2830: 2.0.CO;2. ——, and ——, 2009: The materiality and neutrality of neutral tionally efficient way as discussed by Saenz et al. (2015), density and orthobaric density. J. Phys. Oceanogr., 39, 1779– which physically amounts to mapping water masses’ 1799, doi:10.1175/2009JPO4042.1. volume in thermohaline (u/S) space onto physical Eden, C., and J. Willebrand, 1999: Neutral density revisited. Deep- space, a much simpler and cleaner approach than that Sea Res. II, 46, 33–54, doi:10.1016/S0967-0645(98)00113-1. based on sorting fluid parcels previously proposed by Gent, P. R., J. Willebrand, T. J. McDougall, and J. C. McWilliams, 1995: Parameterizing eddy-induced tracer transports in ocean Huang (2005). Importantly, the construction of the circulation models. J. Phys. Oceanogr., 25, 463–474, doi:10.1175/ Lorenz reference state does not rely on any integration 1520-0485(1995)025,0463:PEITTI.2.0.CO;2. along characteristics that still form the basis for con- Gouretski, V. V., and K. P. Koltermann, 2004: WOCE global hy- structing quasi-neutral surfaces, for example, Klocker drographic climatology. Berichte des Bundesamtes für See- et al. (2009), which is arguably not very well suited to shifffahrt und Hydrographie Tech. Rep. 35/2004, 49 pp. Griffies, S. M., 2004: Fundamentals of Ocean Models. Princeton the construction of a density variable owing to the University Press, 424 pp. complicated geometry of the ocean. In contrast, the ——, and Coauthors, 2000: Developments in ocean climate modeling. computation of neutral density still relies on using Ocean Modell., 2, 123–192, doi:10.1016/S1463-5003(00)00014-7. the pre–International Thermodynamic Equation Of Huang, R. X., 2005: Available potential energy in the world’s oceans. Seawater—2010 (TEOS-10) JMD97 gn software, which J. Mar. Res., 63, 141–158, doi:10.1357/0022240053693770. ——, 2014: Adiabatic density surface, neutral density surface, po- is only capable of computing neutral density of the tential density surface, and mixing path. J. Trop. Oceanogr., second kind for present-day climatologies, whereas the 33, 1–19, doi:10.11978/j.issn.1009-5470.2014.04.001. holy grail for gn software would be being able to compute Huber, M., R. Tailleux, D. Ferrerira, T. Kuhlbrodt, and J. M. neutral density of the first kind (NDFK) for arbitrary Gregory, 2015: A traceable physical calibration of the verti- climatologies of temperature and salinity by adhering to cal advection-diffusion equation for modeling ocean heat uptake. Geophys. Res. Lett., 42, 2333–2342, doi:10.1002/ the most recent TEOS-10 standards. We conjecture that 2015GL063383. the present results should be useful to devise new ways to Iudicone, D., G. Madec, and T. J. McDougall, 2008: Water-mass make progress toward that holy grail. transformations in a neutral density framework and the key The present results suggest that it would be advan- role of light penetration. J. Phys. Oceanogr., 38, 1357–1376, tageous to use gT for the kind of isentropic analyses doi:10.1175/2007JPO3464.1. Jackett, D. R., and T. J. McDougall, 1997: A neutral density vari- pioneered by Montgomery (1938) or for water mass able for the world’s oceans. J. Phys. Oceanogr., 27, 237–263, analyses following Walin (1982), as recently extended doi:10.1175/1520-0485(1997)027,0237:ANDVFT.2.0.CO;2. by Iudicone et al. (2008),aswellasthenaturalvertical Klocker, A., T. J. McDougall, and D. R. Jackett, 2009: A new density coordinate for use in Young’s (2012) thickness- method for forming approximately neutral surfaces. Ocean weighted average formalism or for studying the At- Sci., 5, 155–172, doi:10.5194/os-5-155-2009. Lozier, M. S., M. S. McCartney, and W. B. Owens, 1994: Anoma- lantic meridional overturning circulation in density lous anomalies in averaged hydrographic data. J. Phys. Oce- coordinates, which we hope to demonstrate in future anogr., 24, 2624–2638, doi:10.1175/1520-0485(1994)024,2624: studies. AAIAHD.2.0.CO;2.

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC 3584 JOURNAL OF PHYSICAL OCEANOGRAPHY VOLUME 46

McDougall, T. J., 1987: Neutral surfaces. J. Phys. Oceanogr., 17, 1950– Tailleux, R., 2013a: Available potential energy and exergy in 1964, doi:10.1175/1520-0485(1987)017,1950:NS.2.0.CO;2. stratified fluids. Annu. Rev. Fluid Mech., 45,35–58,doi:10.1146/ ——, and D. R. Jackett, 1988: On the helical nature of neutral annurev-fluid-011212-140620. trajectories in the ocean. Prog. Oceanogr., 20, 153–183, ——, 2013b: Available potential energy density for a multicom- doi:10.1016/0079-6611(88)90001-8. ponent Boussinesq fluid with a nonlinear equation of state. ——, and ——, 2005a: An assessment of orthobaric density in the J. Fluid Mech., 735, 499–518, doi:10.1017/jfm.2013.509. global ocean. J. Phys. Oceanogr., 35, 2054–2075, doi:10.1175/ ——, 2016: Neutrality versus materiality: A thermodynamic theory JPO2796.1. of neutral surfaces. Fluids, 1, 32, doi:10.3390/fluids1040032. ——, and ——, 2005b: The material derivative of neutral density. Walin, G., 1982: On the relation between sea-surface heat flow and J. Mar. Res., 63, 159–185, doi:10.1357/0022240053693734. thermal circulation in the ocean. Tellus, 34, 187–195, Montgomery, R. B., 1938: Circulation in the upper layers of the doi:10.1111/j.2153-3490.1982.tb01806.x. southern North Atlantic deduced with use of isentropic analy- Winters, K. B., P. N. Lombard, J. J. Riley, and E. A. d’Asaro, 1995: sis. Papers in Physical Oceanography and Meteorology, Vol. 6, Available potential energy and mixing in density stratified fluids. Massachusetts Institute of Technology and Woods Hole J. Fluid Mech., 289, 115–128, doi:10.1017/S002211209500125X. Oceanographic Institute, 55 pp., doi:10.1575/1912/1093. Wolfe, C. L., 2014: Approximations to the ocean’s residual circu- Redi, M. H., 1982: Oceanic isopycnal mixing by coordinate lation in arbitrary tracer coordinates. Ocean Modell., 75, 20– rotation. J. Phys. Oceanogr., 12, 1154–1158, doi:10.1175/ 35, doi:10.1016/j.ocemod.2013.12.004. 1520-0485(1982)012,1154:OIMBCR.2.0.CO;2. Wunsch, C., 1996: The Ocean Circulation Inverse Problem. Cam- Saenz, J. A., R. Tailleux, E. D. Butler, G. O. Hughes, and K. I. C. bridge University Press, 442 pp. Oliver, 2015: Estimating Lorenz’s reference state in an ocean Young, W. R., 2012: An exact thickness-weighted average formu- with a nonlinear equation of state for seawater. J. Phys. Oce- lation of the Boussinesq equations. J. Phys. Oceanogr., 42, anogr., 45, 1242–1257, doi:10.1175/JPO-D-14-0105.1. 692–707, doi:10.1175/JPO-D-11-0102.1.

Unauthenticated | Downloaded 09/29/21 09:44 AM UTC