## 1. Introduction

The density gradients associated with the global thermohaline circulation are created at the ocean surface and, to a minor extent, at the bottom boundaries by buoyancy fluxes that force large diapycnal fluxes. Dianeutral mixing due to external mechanical power consumption alters these gradients, thus supplying the potential energy needed to maintain the circulation. The nonlinear nature of the equation of state for seawater also produces dianeutral fluxes that need to be taken into consideration. Despite its importance, a complete picture of dianeutral fluxes in the ocean is not yet available (Wunsch and Ferrari 2004).

The lack of oceanic measurements and of an accurate theoretical framework is not the only reason for this incomplete picture of how water masses are formed and transformed. Diagnostic methods for water-mass formation based on buoyancy balance have been developed only recently, even for numerical models. The classical method for estimating the water-mass formation rate is generally the thermodynamic method, which is based on divergence of the air–sea flux transformation (e.g., Walin 1982; Tziperman 1986; Speer and Tziperman 1992), extended to take mixing into account (Garrett et al. 1995; Marshall et al. 1999; Nurser et al. 1999). The local subduction rate (e.g., Stommel 1979; Marshall et al. 1993; Blanke et al. 2001, 2002), calculated kinematically and integrated between isopycnals from one boundary to the other, should equal the formation rate in the absence of diffusive processes (Marshall et al. 1999; Large and Nurser 2001).

This paper is an attempt to generalize the classical approaches, in particular the one in Marshall et al. (1999), by the use of neutral densities and the inclusion of internal sources/sinks of buoyancy that were not previously considered. There are, in fact, two main drawbacks in the current methods. The first is the use of surface-referenced potential density (e.g., Large and Nurser 2001; Marshall et al. 1999), which is clearly not suitable for the analysis of domains that include deep layers (Fig. 1). Also, the transports due to purely internal nonlinear processes such as cabbeling or thermobaricity (as described by McDougall 1987b) cannot be properly evaluated on potential density surfaces. This problem can be overcome using a more general density variable, namely, neutral density (McDougall 1987a; Jackett and McDougall 1997), at present widely used for the analysis of oceanic data. We will show in this study that the use of the standard approach (e.g., Tziperman 1986) but using neutral density surfaces (e.g., Sloyan and Rintoul 2001) is not strictly correct. A more accurate formulation is developed here that is fortunately quite easy to implement.

The second improvement on current practice is the treatment of the solar shortwave contribution to near-surface heat fluxes. In contrast to the other components of the air–sea interaction, the solar shortwave radiation penetrates into the subsurface layers, with an exponential decay having an *e*-folding depth varying from less than 1 to ∼20 m (e.g., Mobley 1994), depending on the wavelength and the transparency or water type. This depth-dependent aspect of the solar radiation has not been considered in previous analyses (e.g., Large and Nurser 2001); the implicit assumption previously has been that the heating occurs only within the mixed layer. This is not generally the case (Fig. 2; see also Ohlmann et al. 1996) and, especially at midlatitudes in summer or in the equatorial and tropical upwelling systems, its impact, which depends also on water turbidity (Mobley 1994), is not negligible (e.g., Miller et al. 2003; Marzeion et al. 2005). Therefore, we also include that effect in the diagnostics of dianeutral fluxes presented here.

In this paper, besides the methodology, we present its application to the fields produced by an ice–ocean coupled model. The aim is first to validate the methodology and then to illustrate the impact of the new theory on the estimate of the water-mass transformations in some prominent cases. The paper is organized as follows: section 2 is devoted to the presentation of the method; in section 3, the impact of the inclusion of the solar penetrative shortwave flux is considered, with applications to the tropics; section 4 discusses the implications of using a neutral density framework, with applications to the Southern Ocean; and the last section contains the conclusions. Appendix A describes the assumptions in the computation of the neutral density evolution in terms of the temperature and salinity tendencies; appendix B presents the implementation and validation of the diagnostics in a global ice–ocean coupled model; and appendix C summarizes the model configuration.

## 2. Theoretical developments

Walin (1982) derived elegant relations between water-mass formation and diffusive and radiative (nonadvective) heat fluxes, combining heat and volume budgets for an isothermal layer. This derivation has lately been repeated in terms of potential density (Tziperman 1986; Speer and Tziperman 1992; Marshall et al. 1999; Nurser et al. 1999; Viúdez 2000).

Here, we first establish the conservation equation of (neutral) density. Then we rederive the Walin formulation directly from the equation of conservation of (neutral) density (i.e., without using the volume budget).

### a. Density evolution equation in terms of potential temperature and salinity

*τ*=

*θ*or

*S*, can be expressed aswhere

*d*

_{τ}is the tendency due to mixing processes and

*f*

_{τ}is the tendency due to external forcing.

*Q*is the net heat flux across the sea surface,

*I*

_{0}is the total surface solar radiation (i.e., the portion of the surface heat flux that penetrates inside the ocean),

*I*=

*I*(

**x**) is the solar irradiance flux that reaches the interior, SST is the sea surface temperature,

*e*is the net surface freshwater flux (evaporation minus precipitation and river runoffs),

*T*

_{e}is the temperature of the surface freshwater budget (i.e., the temperature of the different freshwater sources),

*G*is the geothermal heat flux across the ocean sea floor (

*z*= −

*H*), and

*δ*is the Dirac operator (homogeneous to the inverse of a length).

**k**is the diffusive tensor and

*FS*

_{θ}and

*FS*

_{S}represent sources and sinks due to double-diffusive convection of both the “diffusive” and salt fingering varieties.

*ρ*, defined at a reference pressure chosen as the local pressure

*p*

_{r}, iswhere

*α*is the thermal expansion coefficient and

*β*is the saline contraction coefficient for

*potential*temperature, whose relationships with the standard coefficients

*α*′ and

*β*′ for in situ temperature are (Gill 1982; McDougall 1987a)Using (1), Eq. (3) can be expressed in terms of nonadvective processes and external forcing acting on

*θ*and

*S:*or more conciselyTo relate the material derivative of locally referenced potential density in (5) to the corresponding changes of neutral density

*γ*, we draw on the work of McDougall and Jackett (2005) who derive a comprehensive expression for the material derivative of neutral density, repeated here as (A1) in appendix A. This expression shows that, even in the absence of irreversible mixing processes, neutral density is not materially conserved. However, as discussed in appendix A, this lack of exact material conservation is usually not numerically significant compared with the effects of irreversible mixing (which cause the material derivatives of potential temperature and salinity,

*Dθ*/

*Dt*and

*DS*/

*Dt*). From (A4) of appendix A, we then have

*b*is the ratio of the spatial gradients of neutral density and locally referenced potential density, |

**∇**

*γ*|/|

**∇**

*ρ*|, and is a nontrivial function of space: the

*b*depends on the thermodynamic properties of seawater and the water-mass distribution in the World Ocean, and often

*b*can be different from unity, as discussed in section 4. (In the following, the ratio

*γ*/

*ρ*will be assumed to be equal to 1.) By multiplying (5) by

*b*, we obtain the evolution equation for neutral density; that is, we obtain the evolution of a

*global*variable instead of that of a

*local*one:

### b. Water-mass formation and transformation

Let us consider a limited area of the ocean with an open boundary (Fig. 3). We denote *S*_{γ} as the surface with neutral density *γ* over the limited domain; *V*_{γ} the volume sandwiched between two surfaces *S*_{γ}; the ocean bottom *H*; the ocean surface OS_{γ}; *ψ*_{γ} and *E*_{γ} the volume fluxes of fluid entering/exiting the domain *V*_{γ} across the open boundary and surface, respectively; and Ω_{γ} the total volume flux across the *S*_{γ} (i.e., the dianeutral volume flux). The sign convention is that all the fluxes are positive if entering *V*_{γ}.

*ω*

_{γ}and

**∇**

_{⊥}

*γ*, defined as the velocity across the moving

*S*

_{γ}and the gradient of

*γ*in the direction normal to

*S*

_{γ}. In this framework, the material derivative of

*γ*,reduces to the product

*ω*

_{γ}

**∇**

_{⊥}

*γ*so that, from (7)

*ω*

_{γ}, can be expressed as

The total volume flux Ω_{γ} across *S*_{γ} is thus the summation of the flux *ω*_{γ}*ds* over *S*_{γ}: Ω_{γ} = ∫∫*S*_{γ} *γ*(*d*_{γ} + *f*_{γ}) (**∇**_{⊥}*γ*)^{−1} *ds*.

*D*and

*F*, the volume-integrated effect of mixing and forcing over

*V*

_{γ}. Here Ω

_{γ}is usually referred to as the water-mass transformation from water masses lighter than

*γ*into water masses denser than

*γ*, while

*M*

_{γ}= ∂Ω

_{γ}/∂

*γ*, the convergence of Ω

_{γ}, is referred to as the water-mass formation at a given density

*γ*(Walin 1982; Speer and Tziperman 1992). Therefore, ∂

*D*/∂

*γ*and ∂

*F*/∂

*γ*are the transformations driven by ocean physics and external forcing, respectively. Finally, we note that, setting

*b*= 1 and substituting

*γ*by

*ρ*in (7) to (10), we obtain a potential density formulation of water-mass formation/transformation.

Equation (10) expresses the fact that a cross-dianeutral volume flux (transformation) directed from light to dense, Ω_{γ} > 0, requires a density supply either by ocean physics or by the forcing. This expression abridges the works of Nurser et al. (1999) and Marshall et al. (1999) with the addition that here we are in the neutral density framework and that (10) does not represent the diapycnal density flux across the extended isopycnal made up of the isopycnal and its surface extension, but rather it is the volume-integrated effect of the right-hand side of (3). This has several important consequences that will be discussed in sections 3 and 4. The implementation of (10) for application to oceanographic databases (in situ data, model outputs) is presented in appendix B.

*V*

_{γ}, we have the equation for the water-mass evolution in a neutral density framework:

## 3. Surface fluxes

*F*/∂

*γ*in Eq. (10) can be written fromThe key point here is that the forcing term does not reduce to a surface integral as in the classical theory but the last term takes into account the three-dimensional nature of the forcing.

### a. Relation with previous theories

*F*/∂

*ρ*, whereIn practice, it corresponds to integrating the surface buoyancy flux at the outcrop.

There are several differences between (12) and (13). The first is that (12) assumes a neutral density framework, while the diagnostic (12) is usually applied in a potential density framework, as discussed in section 4. Here, setting *b* = 1 and substituting *γ* by *ρ* in (7) to (10) and using *α* = *α*(*θ*, *S*, 0), *β* = *β*(*θ*, *S*, 0), a potential density formulation for external forcings of water-mass formation/transformation can be derived.

Second, in (12) the surface volume flux plays a role only as driver of a heat flux, via the temperature difference *T*_{e} − SST. Suppose an ocean without salt and forced only by precipitation, the temperature of which is equal to the SST. In this case, (12) gives *F* = 0; that is, there will be no transformation of water masses. On the other hand, *F* ≠ 0 in (13) because of the last term in the parentheses. This term only adds extra volume in each density class, just as Ψ_{γ} does, and for this reason we moved the corresponding term in the neutral density framework to the left-hand side of (11) (see also Large and Nurser 2001).

Third, (12) takes into account the geothermal heating [third term in (12)] that represents a significant forcing for the lower limb of the thermohaline circulation (e.g., Scott et al. 2001; Dutay et al. 2003; Adkins et al. 2005).

Finally, the last term in (12), associated with the penetrative solar irradiance *I*, is omitted in (13), where all the heat exchanged with the atmosphere is supposed to be captured at the surface, that is, in the ocean mixed layer. Thus, there is in (13) the implicit assumption that internal sources of buoyancy are neglected (e.g., Marshall et al. 1999; Donners et al. 2005).

*E*(

**x**,

*λ*) is given by the Beer–Lambert relation:in which

*K*

_{w}(

**x**,

*λ*) and

*K*

_{chl}(

**x**,

*λ*) are the spectral diffuse attenuation coefficients due to pure water and chlorophyll-related materials, respectively. The inverse of the attenuation coefficients, or penetration depth, varies with wavelength from a few microns to a few tens of meters. The net irradiance flux

*I*(

**x**) = ∫

*λ*

_{sol}

*E*(

**x**,

*λ*)

*dλ*, integrated over the solar spectrum, thus heats roughly the first 100 m of the ocean and in several cases reaches the (highly) stratified layer below the mixed layer (e.g., Siegel et al. 1995; Ohlmann et al. 1996). As a consequence, part of the solar flux entering at a given location warms waters at much higher densities than the surface density. Finally, the total solar irradiance is often parameterized as a sum of exponentials (e.g., Ohlmann and Siegel 2000).

Next, consider the case of the summer upper stratification of an ideal ocean basin (a vertical slice of an infinite ocean; Fig. 4) in which a well-developed seasonal pycnocline is present; that is, subsurface densities are much higher than the surface density. The oceanic region is here partitioned into finite volume isopycnal or neutral layers of thickness Δ*ρ* (here we do not distinguish between locally referenced potential density and neutral density) and we will focus on the isopycnal layer *ρ* of volume *V* (*V* = *V*_{1} + *V*_{2}), outcropping in the region *x*_{1} < *x* ≤ *x*_{2}.

*x*

_{1}<

*x*≤

*x*

_{2}issince the irradiance at the bottom is attenuated to zero in open ocean waters, where

*H*≫ 100 m. Taking the limit,the standard treatment for solar flux forcing in (13) is recovered (e.g., Speer and Tziperman 1992; Nurser et al. 1999), and (13) thus presumes that 1) the whole solar energy entering at the outcrop of the layer

*ρ*is associated with the volume entire

*V*of the same layer

*ρ*and 2) the warming of the layer is limited to its subsurface region (i.e., the mixed layer).

*ρ*with volume

*V*is

_{A}The terms above can be interpreted as (i) the total irradiance entering at the outcrop; (ii) the irradiance exiting the isopycnal layer and warming the deeper layers (*V*_{3}) in the outcrop region of *ρ*; (iii) the irradiance entering the layer *ρ* in the stratified region outside the outcrop; and finally (iv) the irradiance leaving the layer in the latter region. The contribution to the water-mass transformation by shortwave radiation in (12) is obtained taking the limit *FR*_{ρ} = lim_{Δρ→0}(1/Δ*ρ*)*R*_{ρ}.

In the equation for *R*_{ρ} above the term (i) is, in fact, equal to *F _{R}* derived from (14) above. The classical formulation thus assumes that the other three terms, related to the solar energy penetrating the thermocline, are negligible. In other words, in a highly stratified ocean the solar warming in the outcrop region could be overestimated using the classical treatment by an amount given by (ii), while the warming of adjacent deeper (denser) isopycnals could be, in principle, underestimated by an amount given by (iii) minus (iv). As a result, in warm seasons (13) leads to overestimation of the warming at the lowest outcropping densities and an underestimation of the warming of intermediate (subsurface) layers.

Moreover, below the thin mixed layers, which are typical of upwelling regions of the tropics or of some highly stratified marginal seas, such as the Mediterranean Sea (e.g., de Boyer Montegut et al. 2004; D’Ortenzio et al. 2005), part of the solar energy can reach the permanent pycnocline. The oceanic region directly affected by the air–sea interaction can thus be thicker than the maximum mixed layer depth over the year, usually assumed as the boundary with the interior (e.g., Marshall et al. 1999).

Finally, when the assumption of steady state applies, mixing can be estimated using volume conservation, that is, as the residual between the water mass formation via surface forcing and the net flux exiting the domain, as can be the case in inversion of in situ data. A likely consequence of the use of (13) is thus an overestimation of the diapycnal mixing acting on the isopycnal layer, as discussed in the following.

### b. The impact of the penetrative character of the solar irradiance

The impact of the inclusion of the penetrative character of the surface forcing on the estimate of surface forcing and mixing has been estimated via the analysis of the outputs of a global ice–ocean model (see appendix C). The amount of solar irradiance reaching the base of the mixed layer is indeed significant almost everywhere (Fig. 5). It has its maximum in the equatorial upwelling regions. The subtropical gyres are also affected as well as the upwelling regions at the eastern boundaries. As expected, the deep heating is less important in the Southern Ocean and, in particular, along the path of the Antarctic Circumpolar Current, where the winds are strong and thick mixed layers are always present. The “Antarctic Divergence” also receives a significant amount of heat.

More quantitatively, the global transformations due to surface forcings have been computed with both the classical and the new approach and are presented in Fig. 6. (Note that in this global estimate, Northern Hemisphere and Southern Hemisphere formations sometimes cancel out.) To ease the comparison with previous studies, here we use a straightforward variation of (12) that makes use of potential density for the computation of water-mass formation and of *σ*_{0} for the water-mass classification, as discussed in section 2b. Using the classical approach [Eq. (13), Fig. 6a], we obtain a general picture of the transformations, which presents a maximal transformation of 90 Sv (Sv ≡ 10^{6} m^{3} s^{−1}) at 21.5 *σ*_{0} (tropical waters), due to warming and freshening, while cooling promotes the formation of about 40 Sv subpolar mode and intermediate water (IW) and of about 15 Sv of deep, bottom, and marginal seas waters. Freshening participates in forming the Subantarctic Mode Waters (SAMW) and Antarctic Intermediate Waters (AAIW) in the Southern Ocean. The global pattern is in general agreement with a previous estimate by Speer et al. (1995) that used climatologies of in situ data and of surface forcings (their Fig. 3a: They did not include data from the region south of 30°S and thus missed the water-mass formation in the Southern Ocean). The global transformations evaluated using (12) (Fig. 6b) demonstrate that, using the previous approach, the formation of both light and mode waters (MWs) is overestimated by several tens of Sverdrups (warming is overestimated for light waters while it is underestimated for the densities of the layers underneath, as discussed in section 3a) and the mismatch amounts to a factor of ∼2. The disagreement remains significant for intermediate water formation, while it has no impact on the densest water masses. Interestingly, we found that the formation of the lightest water masses is basically due to freshening. In fact, it is associated with the formation of barrier layers in the tropics (e.g., de Boyer Montegut et al. 2004).

Most of the transformation on a global scale occurs in the tropics, which is characterized by large positive heat fluxes via shortwave radiation and by the presence of large regions of upwelling (e.g., Schott et al. 2004) in which a large upward vertical advection creates relatively shallow mixed layers (e.g., de Boyer Montegut et al. 2004). A first glance of the impact of the penetration of surface heating below and at the base of the mixed layer can be obtained from Fig. 7 in which the diapycnal fluxes across the 25.5 isopycnal (in the range of the subtropical cells) are presented. The vertical mixing (which is the main mixing mechanism) lightens the water masses almost everywhere in the tropical areas; it is especially large in the eastern equatorial upwellings and significant in the whole tropical region. A net solar-induced upwelling is also noticeable in the eastern regions of the tropics, a flux that is clearly smaller than the diffusive flux but absolutely not negligible.

We focus here on the radiative heat flux in the tropical Pacific Ocean, which represents the region of the largest warm water formation on a global scale, and more specifically on the region bounded by 40°S–40°N, 170°W and the west coast of the Americas (see Fig. 7). This region includes most of the Pacific Subtropical Mode Water (STMW) formation regions (Hanawa and Talley 2001), waters that are implicated in the subtropical cell circulation (Schott et al. 2004). First, we note that the pattern of the diffusive fluxes across 25.5 kg m^{−3} roughly describes the regions of upwelling (negative values) and downwelling (positive values) (see Plate 2 in Schott et al. 2004). They present a maximum at the base of the mixed layer in the cold tongue at the equator. The upward diffusive transport is significant also in the regions of the equatorial countercurrents, specifically along 4°S and 4°N in the central and western part of the basin, and in the region immediately southeastward of Central America. The solar irradiance warms the subsurface waters in the cold tongue and in the adjacent regions.

In general and more quantitatively, the region is characterized by a large warming for almost all of the density range (Fig. 8a), while freshening forms the lightest tropical waters and evaporation opposes the warming in the range between 22.5 and 25.5 *σ*_{0}. The net result is the production of about 40 Sv of water lighter than 23.0 *σ*_{0}. Considering the ensemble of the transformation processes, we observe that this large positive buoyancy flux for the tropical waters is partly opposed by mixing (Fig. 8b). At larger densities (>24.5 *σ*_{0}), less exposed to surface fluxes, mixing instead promotes the transformations of dense STMW and deeper waters (mostly SAMW) into dense STMW that lighten further because of the surface warming. The net result is the formation of about 15 Sv of light water (<23 *σ*_{0}) at the expense of water denser than 25.5 *σ*_{0} (STMW and SAMW, essentially). The impact of the solar penetrative heating on the transformations can be appreciated in Fig. 8c, in which the total transformation due to surface forcings is compared with the same estimate computed using (13). The former formulation produces an overestimation of the warming for the water lighter than 24.5 *σ*_{0} and an apparent cooling at greater densities, a cooling due to the missing solar heating input below the mixed layer that is somehow expected because the region includes the extratropics. Thus, by including the solar heating we obtain a considerably different view of the net annual budget of the forcing on the Pacific water-mass transformations.

The role of the equatorial upwelling, that is, the transformations occurring in the equatorial box B in Fig. 8d, is also examined. This box has fixed meridional boundaries at 10°S and 10°N so as to avoid the tropical cells and to seize the midpoint of the subtropical cells (Hazeleger et al. 2001; Schott et al. 2004). The forcing (Fig. 8d) presents a large warming centered at about 23.0 *σ*_{0} and extending to 26.0 *σ*_{0}. The freshwater input forms the lightest waters, as in the global analysis above. By comparing Figs. 8b and 8d, it can easily be spotted that most of the upwelling of dense waters in the region discussed here occurs in the equatorial area (see Fig. 8d). The estimate of the net STMW upwelling is coherent but somehow smaller than in experimental estimates [Schott et al. (2004); e.g., Sloyan et al. (2003), using an inverse model, gives 24 ± 4 Sv across the 23.0 *σ*_{0} between 2°S and 2°N in the central Pacific], while here we found an upwelling of ∼5 Sv of water denser than 26.3–26.5 *σ*_{0}, which is the densest upwelling layer in the experimental estimates (Sloyan et al. 2003; Schott et al. 2004). Interestingly, the net water-mass formation in the region (∼25 Sv across 23.0 *σ*_{0}) is larger than in the case of the entire domain (Fig. 8b). The residual is mostly due to the thermodynamic processes associated with the subtropical cells (the formation in A and C and destruction in B of the STMW) that adds ∼7 Sv across 23.0 *σ*_{0} at north and ∼6 Sv across 24.0 *σ*_{0} at south (not shown).

The classical approach (Fig. 8e) gives for the equatorial region a reduced warming at STMW densities and, again, an overestimation of the warming for the lightest water masses (barrier layers). Interestingly, this latter estimate quite closely resembles the only available previous estimate, reported in Large and Nurser (2001), of transformations occurring in the Pacific between 10°S and 10°N (including also the warm pool west of 170°W: here not included). Using in situ data and forcing climatologies, they show that a large warming characterizes the lightest tropical waters and that there is no warming for densities larger than 25.0 *σ*_{0}. By considering Figs. 8e and 6 (which included the warm pool) we conclude that the rationale in discussing estimates of Speer et al. (1995) also applies here.

Finally, estimating the mixing as the residual between the surface forcing and the net formation rates measured at boundary sections [as in inverse model estimates, e.g., Sloyan et al. (2003)], we observe that the classical approach overestimates the diffusive upwelling of STMW across the 24.5 *σ*_{0} by ∼6 Sv (see Fig. 8e).

## 4. The neutral density framework

### a. Isoneutral versus isopycnal approaches

*D*/∂

*γ*instead includes all the effects of ocean physics that appear in (3). In fact,

*d*can be written asthat is, the tendency due to mixing processes can be expressed here in terms of the divergence of isoneutral and dianeutral fluxes (subscript

_{γ}*I*and ⊥, respectively) plus the sources and sinks due to double-diffusive processes such as salt fingering and diffusive convection. The first term on the right-hand side, in particular, includes the thermobaricity and cabbeling transport through

*S*

_{γ}(McDougall 1987b). Indeed, even if

*d*

_{θ}and

*d*

_{S}are expressed in flux form,

*D*cannot be transformed into fluxes across the boundary of

*V*

_{γ}in the potential density framework. In other words, the nonlinear terms in the equation of state (expressed here through

*α*and

*β*being functions of

*θ*,

*S*, and pressure) are able to change the density of water masses in

*V*

_{γ}and thus Ω

_{γ}, the water-mass transformation.

*f*

_{ρ}= 0) and where we can assume

**∇**

_{⊥}∼ ∂/∂

*z*. Consequently,

*Dγ*/

*Dt*≡

*ω*

_{γ}

**∇**

_{⊥}

*γ*≈

*ω*

_{γ}∂

*γ*/∂

*z*and

*b*= |

**∇**

*γ*|/|

**∇**

*ρ*| ∼ |∂

*γ*/∂

*z*|/|∂

*ρ*/∂

*z*|, and thus

*ω*

_{γ}corresponds to the dianeutral velocity

*e*in McDougall (1987b) and thus 2) Ω

_{γ}, as defined in (10), contains all the known physics of the ocean interior. In particular, in this case

*d*

_{ρ}is given bywhere the relation

*α*

**∇**

_{I}

*θ*=

*β*

**∇**

_{I}

*S*has been used to simplify the result. The first term on the right, related to the nonlinearity of the equation of state, can be rewritten as (McDougall 1984)

Cabbeling and thermobaricity are important in several oceanic regions, as discussed in McDougall (1987b) (see also Marsh 2000; You and McDougall 1990; Adkins et al. 2005). From the equation above it is evident that the use of a potential density framework implies an imprecise evaluation of the effect of cabbeling and the impossibility of including the effect of thermobaricity in the transformations.

### b. The b factor

*b*is a complicated function of

*θ, S, p*, latitude and longitude, and the local gradients of

*θ*and

*S*. From McDougall [(1988); see Eq. (47) and the discussion below Eq. (31) on p. 199 thereof], we find that

*b*varies along neutral trajectories according towhere

*N*is the Brunt–Väisälä frequency and

*k*is simply proportional to the thermobaric coefficient

*T*

_{b}according toTo a good approximation we may replace the surface in which the gradients are taken in (17) from being in the neutral tangent plane to being taken in neutral density surfaces so that it becomesThis equation says that the natural logarithm of

*b*varies along a neutral density surface in proportion to the thermobaric nonlinearity of the equation of state and in response to variations of temperature and to variations of the height of the neutral density surface. The particular combination of the epineutral gradients of temperature and height,

**∇**

_{γ}

*θ*−

*θ*

_{z}

**∇**

_{γ}

*z*, is equal to the horizontal temperature gradient

**∇**

_{H}

*θ*.

An explanation of the need for *b* to vary spatially can be gleaned by considering the case in which there are no compensating gradients of salinity and temperature along neutral density surfaces so that **∇**_{γ}*θ* = 0. In this case, the neutral density surfaces coincide with potential density surfaces. When considering the lateral motion of fluid between pairs of such surfaces, one finds [as proven in section 7.1 of McDougall (1988)] that the vertical integral of *N* ^{2} between the two density surfaces varies in space: an aspect first realized by Veronis (1972). The spatial variation of *b* as given by (19) is required so that the vertical integral of the product *bN* ^{2} between the two density surfaces does not vary in space; that is, the spatial variation in *b* has the role of compensating for the less than desirable nature of *N* ^{2}, and it is interesting that such compensation is needed even if there are no water-mass variations along the density surfaces.

The variation of the natural logarithm of *b* is most pronounced in the Southern Ocean. This region is characterized by 1) a significant rise and large variations of the thickness of deep neutral surfaces, which are uplifted and squeezed by the upwelling in the Antarctic divergence (see Fig. 9), and 2) very dense water–mass formation along Antarctica and in the Weddell Sea, which is largely influenced by thermobaricity. Therefore, *b* is significantly different from unity in the regions of highest surface density and, in general, in regions characterized by the highest density values (Fig. 10). Small deviations from unity also occur along the Subantarctic Front, especially in the Pacific sector of the Southern Ocean.

### c. Diagnosis of the transformations occurring in the Southern Ocean

Using neutral densities instead of potential density implies several differences in the formulation of the water-mass transformations, from taking into account the variation of *α* and *β* with pressure to the addition of the *b* factor. To show the impact of the use of neutral densities with respect to potential density in water-mass formation/transformation and of the inclusion of *b* in the neutral framework, we commence by evaluating the total dianeutral fluxes in the Southern Ocean (here the region south of 30°S) using a long simulation performed with a global ice–ocean model (see appendixes B and C). See Table 1 for a definition of the water masses used in the following; a detailed analysis is presented in Iudicone et al. (2008)]. The choice of this basin is dictated by the occurrence of the largest range of density values that makes the basin the most difficult to be dealt with in standard potential density approaches (e.g., Sloyan and Rintoul 2001).

An overall picture of the transformations in the model Southern Ocean computed via the use of (10) is shown in Fig. (11a), where the mixing and the surface fluxes are both included as well as transports through the basin boundaries (see appendix B). The overall distribution of the transport is consistent with observational estimates (e.g., Talley et al. 2003), with discrepancies in the absolute values, the main discrepancy being a weak Upper Circumpolar Deep Water (UCDW)/Antarctic Bottom Water (AABW) bottom cell [about 10–15 Sv, two to three times smaller than in situ inversions depending on the method; e.g., Sloyan and Rintoul (2001)]. As for the surface fluxes, thermocline and mode water are transformed into denser waters (more than 40 Sv are formed in the class 26.0 < *γ*′ < 27.2), while a negative buoyancy input, due to a freshwater excess (not shown), is largely dominating the transformations at higher classes (27.0 < *γ*′ < 27.8). The effect of brine rejection is also noticeable at the highest densities. The overall pattern of surface forcing transformations is consistent with climatologies (e.g., Large and Nurser 2001) and other studies (e.g., Sloyan and Rintoul 2001), apart from the values larger than 28.0 due to the presence of an ice model in our model configuration and not included in the data analysis. Dianeutral and isoneutral mixing tend to densify the water masses (the sign is always positive except for the densest SAMW and densest water masses). A large diapycnal transport (net production is 20 Sv) is observed for the IW/UCDW, whose large diapycnal fluxes at the surface are almost completely compensated by internal mixing, finally giving a net budget at 30°S of only 3–4 Sv for the IW. Sloyan and Rintoul (2001) found a similar large transformation of SAMW/IW into UCDW due to mixing (∼31 Sv). At higher densities, the mixing of a large amount of UCDW/Lower Circumpolar Deep Water (LCDW), the peak in the curve at about 28.0 *γ*′ that corresponds to a destruction of about 11 Sv of dense UCDW and light LCDW and the formation of dense LCDW/AABW, and extremely dense shelf water (>3 Sv) finally produce AABW (the deepest cell).

The evaluation of the water-mass transformations in the simulated Southern Ocean in a potential density framework is shown in Fig. 11b (see also Marsh et al. 2000). The impact of ocean processes at densities larger than 27 *γ*′ is clearly different. Dense water masses are no longer clearly discernible and the associated transformations are apparently inverted, with a net loss of buoyancy for all water masses denser than 27 *γ*′ (net upwelling). The mismatch between the two approaches is due to the inability of the potential density to discriminate water mass at depth (which are compressed into a shorter density range) and the use of the surface-referenced *α* and *β* in the case of the potential density framework (see discussion in section 4a). To evaluate the impact of the latter assumption, implicit in using potential densities, we repeated the assessment of the dianeutral fluxes using (10), as in Fig. 11a, except that we introduced surface-referenced *α* and *β*. As shown in Fig. 11c, the same change of sign in the estimation of mixing in the CDW class and the absence of AABW formation as in Fig. 11b is observed. The decomposition of the water mass transformations due to mixing in Fig. (11a) into vertical and isoneutral mixing (Fig. 11d) shows that, as expected, the nonlinearity of the equation of state affects the isoneutral mixing related to cabbeling and thermobaricity (section 4a).

We finally consider the impact of the *b* factor alone on our estimates to illustrate how the use of neutral densities indeed requires the estimate of *b*. To evaluate the role of *b*, the estimate of the water mass transformation using (12) has been repeated setting *b* = 1 everywhere (Fig. 11e; see also appendix B). In the latter case a mismatch between the two estimates occurs for water with the highest densities (*γ*′ > 27.6), which actually occupies most of the basin volume.

## 5. Conclusions

In this paper we have presented a generalization of the current method for evaluating water-mass transformation due to boundary forcing and interior mixing. The use of a global density variable, neutral density, allows the classical method of estimating water-mass transformation using surface fluxes (Tziperman 1986) to be extended accurately to the whole oceanic domain. A further development was the inclusion of internal sources of mixing and buoyancy via penetrative fluxes. The method, relatively easy to implement, was tested with encouraging results (noise is relatively small and the estimate of the transports compare positively) for the estimation of water-mass formations/transformations reproduced by an ice–ocean model.

We focused first on the significance of a correct evaluation of the effect of solar radiance in the water-mass transformations. The impact of solar shortwave radiation on the water masses below the mixed layer is indeed important, as can be deduced from Fig. 2; this has not previously been considered in quantitative estimates. In fact, the application to the tropics showed that owing to the shallow mixed layer in the regions of upwelling, solar irradiance into the interior can be comparable to vertical diffusion in promoting the transformation of thermocline and mode waters into subsurface waters. It is noteworthy that the term related to solar radiation in (11) can be included in the inversion of in situ data.

The application to the OGCM also showed that the use of a neutral density framework is necessary when dealing with the analysis of thermodynamics of the global thermohaline circulation. Finally, the value of *b* has to be carefully estimated, and we propose the use of horizontal gradients as an efficient and relatively simple approach to evaluate *b*, an approach that is suitable also for the mixed layer.

## Acknowledgments

The authors thank Dr. Lynne Talley and the two anonymous reviewers for their most constructive and knowledgeable comments.

This work is funded in part by the Programe National d’Etude du Climate (PNEDC) and CANOPO PEA. This work contributes to the CSIRO Climate Change Research Program.

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## APPENDIX A

### The Material Derivative of Neutral Density

*γ*is given bywhere

*F*is defined byand the third line and the

**V**·

**S**term in (A1) is due to the scalar product of the horizontal velocity

**V**and the difference in slope between the neutral tangent plane and the neutral density surface in the labeled reference dataset,

**s**≡

**∇**

_{n}

*z*−

^{r}**∇**

_{γ}

*z*. In McDougall and Jackett (1988), it was shown that this contribution to (A1) is particularly small. Here

^{r}*T*

_{b}is the thermobaric parameter,

*T*

_{b}=

*β*∂(

*α*/

*β*)/∂

*p*≈ 2.7 × 10

^{−12}K

^{−1}Pa

^{−1}, defined in terms of the variation of the thermal expansion coefficient

*α*and saline contraction coefficient

*β*with pressure.

In (A1) (*θ* − *θ ^{r}*) and (

*p*−

*p*) are the differences in potential temperature and pressure between a parcel’s potential temperature and pressure and those of the labeled dataset [of Jackett and McDougall’s (1997) neutral density algorithm] that are neutrally related to the parcel; that is, the parcel (

^{r}*S*,

*θ*,

*p*) and the datum (

*S*,

^{r}*θ*,

^{r}*p*) in the labeled dataset have the same density when moved adiabatically and isentropically to the average pressure

^{r}*= 0.5 (*p

*p*+

*p*). For reasonable values of (

^{r}*θ*−

*θ*) and (

^{r}*p*−

*p*) (such as, say, 2°C and 150 dbar) the factor

^{r}*F*is close to unity and the first bracket pair in the last line of (A1) can be approximated by

*g*

^{−1}

*N*

^{2}.

*a*(

*)*p

*θ*

^{r}_{z}−

*β*(

*)*p

*S*

^{r}_{z}is similar to

*g*

^{−1}

*N*

^{2}and

**V**·

**∇**

_{γ}

*θ*≈

^{r}*O*(

*θ̇*). The last part of (A3) uses typical values of the thermal expansion coefficient and of the pressure difference of 2 × 10

^{−4}K

^{−1}and 150 dbar (1.5 MPa), respectively. Hence we conclude that for many purposes we may approximate (A1) by the simpler expression [see also Eq. (5) of McDougall (1995)]where

*α*and

*β*are evaluated at the local pressure and

*ρ*is the locally referenced potential density.

## APPENDIX B

### Discretization and Implementation

Here we describe the practical method to compute the total dianeutral transport across neutral density surfaces due to both interior mixing processes and surface buoyancy fluxes. Briefly, we will develop an expression for net transformations [see (10)]. As discussed in section 3, the method examines the evolution equation of neutral density via the tendencies of potential temperature and salinity; thus, the key point is to evaluate these latter terms.

*i*,

*j*,

*k*) and integrating

*F*+

*D*over small volumes

*V*

_{i,j,k}, we have

*γ*, the integral on the right in (10) can then be rewritten aswhere Π

_{γ−γ′}is the boxcar function Π

_{γ−γ′}= 1 for

In practice, we evaluate the irreversible production of “density” in the above equations by examining the corresponding production terms of potential temperature and salinity via (4). Each term in (B1) is thus easily evaluated from the tendencies of *θ* and *S* (by substituting the terms in brackets, as discussed above), via the locally referenced expansion coefficients [see Eq. (A4) in appendix A]. We remind one here that (B1) does not imply the projection of the forcing and mixing terms on the neutral surface and that the standard Cartesian projection can be used.

Finally, for the diffusive flux terms, the formula derived here uses the same mathematical formalism of Marshall et al. (1999; see their appendix for a thorough discussion), with the additional feature that we include the factor *b* = |**∇***γ*|/|**∇***ρ*|, which needs a further effort in the computation (see below). Entrainment/detrainment processes are naturally included if a prognostic treatment is used for static instabilities (e.g., higher diffusivity). A separate diagnostic has to be used instead when dealing with convective adjustment algorithms.

#### The implementation of the method for the ocean model outputs

Once the tendency terms for temperature and salinity are stored, the implementation of (12) via (B1) is straightforward, with the exception of the *b* factor that has no state equation and has to be estimated empirically, as discussed in section 4. Its estimate is, in fact, quite tricky because of the noise in the direct computation. After several tests, we have chosen to estimate *b* by taking the local average of the *horizontal* ratio Δ*γ*/Δ*ρ* in every grid point and for every model output. This avoids problems in the mixed layer. In the end, an annual mean *b* field was derived. The time-averaging procedure considerably reduced the noise in the estimate at the expenses of neglecting the seasonal variability.

To validate the application of the new formulation to the ocean model, we computed the budget of transformations for the Southern Ocean, here bounded to the north by 30°S, where *b* can be greater than unity. As a first check for the correct implementation of the methodology and of the steadiness of the ocean simulation, the residuals of all the terms of the neutral density equation [i.e., the terms in (10) minus the advection terms] have been computed (not shown). The overall sum is zero (there is no net residual), but some noise exists at intermediate densities when considering the residual decomposed into neutral density classes. The source of the noise is in the discretization of density values due to the use of time mean values for the tendencies and to the finite number of grid points; that is, the bin size is too small in some cases. These issues are significant, essentially for the computation of the surface heat fluxes.

_{γ}+

*E*

_{γ}= −Ω

_{γ}, whereand

*ϕ*(

*γ*′) and

*e*(

*γ*′) are the volume transports (as function of neutral density) at 30°S and at the surface, respectively.

In Fig. B1 the values of −(Ψ_{γ} + *E*_{γ}), decomposed into the same density classes as Ω_{γ}, are compared to Ω_{γ}. The superposition of the two curves is rather good. The main discrepancies occur at about 27.3–27.6 *γ*′ and 28.0 *γ*′, due to both the binning on finite-size bins and the noise in the estimate of the *b* factor.

We finally consider the impact of the *b* factor alone on our estimates to illustrate how the use of neutral densities indeed requires the estimate of *b*. In the latter case a mismatch between the two estimates occurs for water with the highest densities (*γ*′ > 27.6), water that actually occupies most of the basin volume (Fig. B1). The estimate of the surface forcing for densities close to 27.3 *γ*′ is also sensitive to the use of *b*, and it is actually improved if *b* = 1 is used for these water masses. As discussed in section 4b, the *b* term is larger than unity along the subpolar front, with values particularly large (1.2–1.4) in the South Pacific sector of the Southern Ocean. This latter point is rather puzzling and remains unexplained.

## APPENDIX C

### The Ice–Ocean Coupled Model

The method has been applied to the final year of a 1500-yr simulation of a state-of-the-art global ice–ocean coupled system (ORCA2-LIM; see Madec 1999; Timmermann et al. 2005; Iudicone et al. 2008); the model includes a parameterization of eddy transport and isoneutral diffusion and bottom boundary layer diffusion of tracers. Surface forcings are included via the use of the bulk formulae, which include a penetrative formulation for the solar shortwave radiation (the same as in Manizza et al. 2005). A geothermal flux is also imposed at the bottom. The model has virtually reached a steady state and has realistic tracer and transport values (Iudicone et al. 2008). Mean model tendencies for *θ* and *S* have been stored every 14.3 days, and neutral density computed via the Jackett and McDougall (1997) routines.

Water mass definitions for the Southern Ocean analysis. Included are STW, MW, IW, UCDW, LCDW, and AABW. Neutral density ranges are generally as in Sloyan and Rintoul (2001). Only the upper limit for IW has been extended to 27.8 *γ*′ after inspection of the characteristics of the simulated water masses.