It is well-known that in situ temperature is not a satisfactory measure of the “heat content” of a water parcel because the in situ temperature of a water parcel changes as the ambient pressure changes (i.e. if a water parcel is transported to a different depth, i.e. pressure, in the ocean). This change is of the order of 0.1 ∘C as pressure changes 1000 dbar and is large relative to the precision of 0.01 ∘C required to understand deep-ocean circulation patterns. The utility of in situ temperature lies in the fact that it is easily measured with a thermometer and that air–sea boundary heat fluxes are to some degree proportional to in situ temperature differences.

Traditionally, potential temperature has been used as an improved measure of ocean heat content. Potential temperature is defined as the temperature that a parcel would have if moved isentropically and without exchange of mass to a fixed reference pressure (usually taken to be surface atmospheric pressure), and it can be calculated from measured ocean in situ temperatures using empirical correlation equations based on laboratory measurements. However, the enthalpy of seawater varies nonlinearly with temperature and salinity (Fig. 1), and this variation results in non-conservative behaviour under mixing (McDougall, 2003; Sect. A.17 of IOC et al., 2010). The ocean's potential temperature is subject to internal sources and sinks – it is not conservative.

With the development of a Gibbs function for seawater, based on empirical fits to measurements of known thermodynamic properties (Feistel, 2008; IOC et al., 2010), it became possible to apply a more rigorous theory for quasi-equilibrium thermodynamics to study heat content problems in the ocean. As a practical matter, calculations can now be made that allow for an estimate of the magnitude of non-conservative terms in the ocean circulation. By integrating over water depth these production rates can be expressed as an equivalent heat flux per unit area.

Non-conservation of potential temperature was found to be equivalent to a root mean square surface heat flux of about 60 mW m-2 (Graham and McDougall, 2013) and an average value of 16 mW m-2 (see below). These numbers can be compared to a present-day estimated global-warming surface heat flux imbalance of between 300 and 470 mW m-2 (Zanna et al., 2019; von Schuckmann et al., 2020). By comparison, the globally averaged rate of increase in temperature due to the dissipation of kinetic energy is equivalent to a surface flux of approximately 10 mW m-2. These equivalent heat fluxes and subsequent similar values are gathered into Table 1 for reference. In the context of a conceptual ocean model being driven by known heat fluxes, the presence of the non-conservation of potential temperature causes sea surface temperature (SST) errors seasonally in the equatorial region of about 0.5 K (0.5 ∘C), while the error (in all seasons) at the outflow of the Amazon is 1.8 K (see Sect. 9 of McDougall, 2003). With different boundary conditions (such as restoring boundary conditions) the error in assuming that potential temperature is conservative is split in different proportions between (a) the potential temperature values and (b) the potential temperature fluxes.

No single alternative thermodynamic variable is available that is both independent of pressure and conservative under mixing. For example, specific entropy is produced in the ocean interior when mixing occurs, with the depth-integrated production being equivalent to an imbalance in the air–sea heat flux of a root mean square value of about 500 mW m-2 (Graham and McDougall, 2013), while, apart from the dissipation of kinetic energy, enthalpy is conservative under mixing at constant pressure, but enthalpy is intrinsically pressure-dependent.

However, it was found that a constructed variable, potential enthalpy (McDougall, 2003), has a mean non-conservation error in the global ocean of only about 0.3 mW m-2 (this is the mean value of an equivalent surface heat flux, equal to the depth-integrated interior production of potential enthalpy that is additional to the production due to the dissipation of kinetic energy; Graham and McDougall, 2013). The potential enthalpy, h̃0, is the enthalpy of a water parcel after being moved adiabatically and at constant salinity to the reference pressure 0 dbar at which the temperature is equal to the potential temperature, θ, of the water parcel: 1h̃0SA,θ=hSA,θ,0dbar. In Eq. (1) the function h is the specific enthalpy of TEOS-10 (defined as a function of Absolute Salinity, in situ temperature, and sea pressure), whereas h̃0 is the potential enthalpy function, and the tilde implies that the temperature input to this function is potential temperature, θ. By way of comparison, the area-averaged geothermal input of heat into the ocean bottom is about 86 mW m-2, and the interior heating of the ocean due to viscous dissipation, is equivalent to a mean surface heat flux of about 3 mW m-2 (Graham and McDougall, 2013). Remi Tailleux (personal communication, 2021) has suggested that the dissipation of kinetic energy in the ocean may be as much as 3 times as large as this value at approximately 10 mW m-2. Thus, we conclude that potential enthalpy, although not a theoretically ideal conservative parameter, can be treated as such for many present purposes in oceanography. If at some stage in the future a source term were to be added to the evolution equation for Conservative Temperature, the most important contribution would be that due to the dissipation of kinetic energy, being a factor of ∼10–30 larger than the non-conservation of Conservative Temperature due to other diffusive contributions (namely the terms in the last two lines of Eq. 38 of Graham and McDougall, 2013).

Since potential enthalpy was not a widely understood property, a decision was made in the development of TEOS-10 to adopt Conservative Temperature, Θ, which has units of temperature and is proportional to potential enthalpy: 2Θ=Θ̃SA,θ=h̃0SA,θ/cp0, where the proportionality constant cp0≡3991.86795711963 J kg-1 K-1 was chosen so that the average value of Conservative Temperature at the ocean surface matched that of potential temperature. Although in hindsight other choices (e.g. with fewer significant digits) might have been more useful, this value of cp0 is now built into the TEOS-10 standard.

Note that at specific locations in the ocean, in particular at low salinities and high temperatures, Θ and θ can differ by more than 1 ∘C (Fig. 2); the difference is a strongly nonlinear function of temperature and salinity. Θ is, by definition, independent of adiabatic and isohaline changes in pressure.

This question is answered in Sects. A17 and A18 of the TEOS-10 manual (IOC et al., 2010) as well as McDougall (2003) and Graham and McDougall (2013). The answer is that potential enthalpy referenced to the sea surface pressure, h0, which is a variable that is (almost totally) conservative in the real ocean, is not simply a linear function of potential temperature, θ, and Absolute Salinity, SA (and note that both enthalpy and entropy are unknown and unknowable up to separate linear functions of Absolute Salinity). If potential enthalpy were a linear function of potential temperature and Absolute Salinity then the “heat content” per unit mass of seawater could be accurately taken to be proportional to potential temperature, and the isobaric specific heat capacity at zero sea pressure would be a constant. As an example of the nonlinearity of h̃0(SA,θ), the isobaric specific heat at the sea surface pressure cp(SA,θ,0dbar)≡hθ0 varies by 6 % across the full range of temperatures and salinities found in the world ocean (Fig. 1). By way of contrast, the potential enthalpy of an ideal gas is proportional to its potential temperature.

Another way of treating heat in an ocean model is to continue carrying potential temperature as its temperature variable but to (i) use the variable isobaric heat capacity at the sea surface to relate the air–sea heat flux to an air–sea flux of potential temperature and (ii) to evaluate the non-conservative source terms of potential temperature and add these source terms to the potential temperature evolution equation during the ocean model simulation (Tailleux, 2015).

However, it is not possible to accurately choose the value of the isobaric heat capacity at the sea surface that is needed when θ is the model's temperature variable. This issue arises because of the unresolved variations in the sea surface salinity (SSS) and SST (for example, unresolved rain events that temporarily lower the SSS), together with the nonlinear dependence of the isobaric specific heat on salinity and temperature. Because of such unresolved correlations, the air–sea heat flux would be systematically misestimated. Nor is it possible to accurately estimate the non-conservative source terms of θ in the ocean interior. This problem arises because the source terms are the product of a turbulent flux and a mean gradient. In a mesoscale eddy-resolved ocean model (or even finer scale) it is not clear how to find the eddy flux of θ, as this depends on how the averaging is done in space and time. Furthermore, when analysing the output of such an ocean model, one would need to find a way of dealing with the contributions from source terms that are not expressible in the form of flux convergences when, for example, estimating the meridional heat transport.

We conclude that the idea that ocean models could retain potential temperature θ as the model's temperature variable, rather than adopt the TEOS-10 recommendation of using Conservative Temperature Θ, is unworkable because (1) the air–sea heat flux cannot be accurately evaluated, (2) the non-conservative source terms that appear in the θ evolution equation cannot be estimated accurately, and (3) the ocean section-integrated heat fluxes cannot be accurately calculated. When contemplating upgrading ocean model physics, rather than retaining the EOS-80 and treating the temperature variable as being potential temperature and having to add estimates of the non-conservative production terms to the temperature evolution equation, it is clearly much simpler and more accurate to instead adopt the TEOS-10 equation of state and to treat the model's temperature variable as Conservative Temperature, as recommended by IOC et al. (2010).

This question is addressed in McDougall (2003), in Sect. A18 of the TEOS-10 manual (IOC et al., 2010), and in Graham and McDougall (2013). The first step in addressing the non-conservation of Θ is to find a thermophysical variable that is conserved when fluid parcels mix. McDougall (2003) and Graham and McDougall (2013) showed that when fluid parcels are brought together adiabatically and isentropically to mix at pressure pm, it is the potential enthalpy hm referenced to the pressure pm of a mixing event that is conserved, apart from the dissipation of kinetic energy, ε. From this knowledge they constructed the evolution equation for Conservative Temperature (as well as for potential temperature and for entropy).

By contrast, Tailleux (2010, 2015) assumed that it was the total energy, being the sum of internal energy, kinetic energy, and geopotential, that is conserved when fluid parcels mix in the ocean. However, as shown by McDougall et al. (2003), the -∇⋅(Pu) term on the right-hand side of the evolution equation for total energy is non-zero when integrated over the mixing region so that total energy is not a conservative variable. For a variable to possess the “conservative property”, it is not sufficient that the material derivative of that property is given by the divergence of a flux. Rather, what is needed is the material derivative of a conservative variable to be equal to the divergence of a flux that is zero in the absence of mixing at that location. That is, the flux whose divergence appears on the right-hand side of the evolution equation of a conservative variable must be a diffusive flux (whether a molecular or a turbulent type of diffusive flux). This feature allows one to integrate over a region in which a mixing event is occurring and be confident that there is no flux through the bounding area that lies outside the fluid that is being mixed. This is not possible for Total Energy because even when integrating out to a quiescent surface that encloses an isolated patch of turbulent mixing, the flux divergence term -∇⋅(Pu) can still be non-zero there. Note that both contraction-on-mixing and wave processes contribute to -∇⋅(Pu).

Tailleux (2010, 2015) treated this non-conservative term, -∇⋅(Pu), as though it were a conservative term in all their evolution equations, so these papers actually arrived at the correct evolution equations for Θ, θ, and η (for example, Eq. B7 of Tailleux, 2010, and Eq. B10 of Graham and McDougall, 2013, are identical). However, these equations were written in terms of the molecular fluxes of heat and salt, and the Tailleux (2010, 2015) papers did not find a way to use these expressions to evaluate the non-conservation of Θ, θ, and η in a turbulently mixed ocean. This was done in Sect. 3 of Graham and McDougall (2013).

While enthalpy is conserved when mixing occurs at constant pressure, it does not possess the “potential” property; rather, an adiabatic and isohaline change in pressure causes a change in enthalpy according to h^P=v, where v is the specific volume. This property is illustrated in Fig. 3 where it is seen that for an adiabatic and isohaline increase in pressure of 1000 dbar, the increase in enthalpy is the same as that caused by an increase in Conservative Temperature of more than 2.4 ∘C. If enthalpy variations at constant pressure were a linear function of Absolute Salinity and Conservative Temperature, the contours in Fig. 3 would be parallel equidistant straight lines, and Conservative Temperature would be totally conservative. Since this is not the case, this figure illustrates the (small) non-conservation of Conservative Temperature. Further discussion and evaluation of the non-conservation of Conservative Temperature can be found in McDougall (2003) and Graham and McDougall (2013).

To a degree of approximation which is useful for many purposes, the dissolved matter in seawater (“sea salt”) can be treated as a material of uniform composition, whose globally integrated absolute salinity (i.e. the grams of solute per kilogram of seawater) changes only due to the addition and removal of fresh water through rain, evaporation, and river inflow. This property is because the processes that govern the addition and removal of the constituents of sea salt have extremely long timescales relative to those that affect the pure water component of seawater. We can thus treat the total ocean salt content as approximately constant but subject to spatially and temporally varying boundary fluxes of fresh water that give rise to salinity gradients.

The utility of this definition of uniform composition of sea salt lies in its conceptual simplicity, which is well-suited to theoretical and numerical ocean modelling at timescales of up to hundreds of years. However, to the demanding degree required for observing and understanding deep-ocean pressure gradients, sea salt is neither uniform in composition nor a conserved variable, and its absolute amount cannot be measured precisely in practice. The repeatable precision of various technologies used to estimate salinity can be as small as 0.002 g kg-1, but the non-ideal nature of seawater means that these estimates can be different by as much as 0.025 g kg-1 relative to the true Absolute Salinity in the open ocean, and as much 0.1 g kg-1 in coastal areas (Pawlowicz, 2015).

The most important interior source and sink factors governing changes in the composition of sea salt are biogeochemical processes that govern the biological uptake of dissolved nutrients, calcium, and carbon in the upper ocean, as well as the remineralization of these substances from sinking particles at depth. At present it is thought that changes resulting from hydrothermal vent activity, fractionation from sea ice formation, and multi-component molecular diffusion processes are of local importance only, but little work has been done to quantify this.

To address this problem, TEOS-10 defines a Reference Composition of seawater and several slightly different salinity variables that are necessary for different purposes to account for the variable composition of sea salt. The TEOS-10 Absolute Salinity, SA, is the absolute salinity of Reference Composition Seawater of a measured density (note that capitalization of variable names denotes a precise definition in TEOS-10). It is the salinity variable that is designed to be used to accurately calculate density using the TEOS-10 Gibbs function.

Preformed Salinity, S*, is the salinity of a seawater parcel with the effects of biogeochemical processes removed, which is somewhat analogous to a chlorinity-based salinity estimate. It is thus a conservative tracer of seawater suitable for modelling purposes, but it neglects the spatially variable small portion of sea salt involved in biogeochemical processes that is required for the most accurate density estimates. Since the original measurements of specific volume to which both EOS-80 and TEOS-10 were fitted were made on samples of Standard Seawater with a composition close to the Reference Composition, the Reference Salinity of these samples is also the same as Preformed Salinity.

Ocean observational databases contain a completely different variable: Practical Salinity. This variable, which predates TEOS-10, is essentially based on a measure of the electrical conductance of seawater normalized to conditions of fixed temperature and pressure by empirical correlation equations between the ranges of 2 and 42 PSS-78 and scaled so that ocean salinity measurements that have been made through a variety of technologies over the past 120 years are numerically comparable. Practical Salinity measurement technologies involve a certified reference material called IAPSO Standard Seawater, which for our purposes can be considered the best available artefact representing seawater of Reference Composition.

Practical Salinity was not designed for numerical modelling purposes and does not accurately represent the mass fraction of dissolved matter. We can link Practical Salinity, SP, to the Absolute Salinity of Reference Composition seawater (so-called Reference Salinity, SR) using a fixed scale factor, uPS, so that 3SR=uPSSPwhereuPS≡(35.16504/35)gkg-1. Conversions to and between the other “salinity” definitions, however, involve knowledge about spatial and temporal variations in seawater composition. Fortunately, the largest component of these changes occurs in a set of constituents involved in biogeochemical processes, whose co-variation is known to be strongly correlated. Thus, the Absolute Salinity of real seawater can be determined globally to useful accuracy from the Reference Salinity by the addition of a single parameter: the so-called Absolute Salinity Anomaly, δSA, 4SA=SR+δSA, which has been tabulated in a global atlas for the current ocean (McDougall et al., 2012) and is estimated in coastal areas by considering the effects of river salts (Pawlowicz, 2015). It can also be determined from measurements of either density or of carbon and nutrients (IOC et al., 2010; Ji et al., 2021). For the purposes of numerical ocean modelling, the Absolute Salinity Anomaly could in theory be obtained by separately tracking the carbon cycle and nutrients and applying known correction factors, but we are not aware of any attempts to do so.

Chemical modelling (Pawlowicz, 2010; Pawlowicz et al., 2011; Wright et al., 2011; Pawlowicz et al., 2012) suggests the approximate relation 5SA-S*≈1.35δSA≡1.35SA-SR, and these relationships are schematically illustrated in Fig. 4. The magnitude of the Absolute Salinity Anomaly is around -0.005 to +0.025 g kg-1 in the open ocean relative to a mean Absolute Salinity of about 35 g kg-1. The correction it implies may be important when initializing models or comparing them with observations, but its major effect is likely in producing biases in calculated isobaric density gradients.

The density of seawater is the most important thermodynamic property affecting oceanic motions, since its spatial changes (along with changes to the sea surface height) give rise to pressure gradients which are the primary driving force for currents within the ocean interior through the hydrostatic relation. The “traditional” equation of state is known as EOS-80 (UNESCO, 1981) and is standardized as a function of Practical Salinity and in situ temperature, ρ=ρ(SP,t,p), which has 41 numerical terms. An additional equation (the adiabatic lapse rate) is required for conversion of temperature to potential temperature. However, for ocean models, the EOS-80 is usually taken to be the 41-term expression written in terms of potential temperature, ρ=ρ̃(SP,θ,p), of Jackett and McDougall (1995), where the tilde indicates that this rational function fit was made with Practical Salinity SP and potential temperature θ as the input salinity and temperature variables.

The current standard for describing the thermodynamic properties of seawater, known as TEOS-10, provides an equation of state, v=1/ρ=v(SA,t,p), in the form of a function which involves 72 coefficients (IOC et al., 2010) and is an analytical pressure derivative of the TEOS-10 Gibbs function. However, for ocean models using TEOS-10 the equation of state used is one of those in Roquet et al. (2015): the 55-term equation of state, ρ=ρ^(SA,Θ,z), used by Boussinesq models and the 75-term polynomial for specific volume, v=v^(SA,Θ,p), used by non-Boussinesq ocean models.

In this paper we will not concentrate on the distinction between Boussinesq and non-Boussinesq ocean models, and henceforth we will take the third input to the equation of state to be pressure, even though for a Boussinesq model it is in fact a scaled version of depth as per the energetic arguments of Young (2010). By the same token, we will cast the discussion in terms of the in situ density, even though the non-Boussinesq models have as their equation of state a polynomial for the specific volume, v=1/ρ.

For seawater of Reference Composition, both the TEOS-10 and EOS-80 fits ρ=ρ^(SA,Θ,p) and ρ=ρ̃(SP,θ,p) are almost equally accurate (see Sect. A5 of IOC et al., 2010, and note the comparison between Figs. A5.1 and A5.2 therein). That is, if we set δSA=0 and use Eq. (3) to relate Practical and Reference Salinities (which in this case are the same as Preformed Salinities), the numerical density values of in situ density calculated using EOS-80 are not significantly different to those using TEOS-10 in the open ocean (the differences are significant for brackish waters).

This being the case, we can see from Sects. A5 and A20 of the TEOS-10 manual (IOC et al., 2010) that 58 % of the data deeper than 1000 dbar in the world ocean would have the thermal wind misestimated by ∼2.7 % due to ignoring the difference between Absolute and Reference Salinities. No ocean model has addressed this deficiency to date, but McCarthy et al. (2015) studied the influence of using Absolute Salinity versus Reference Salinity in calculating the overturning circulation in the North Atlantic. They found that the overturning stream function changed by 0.7 Sv at a depth of 2700 m relative to a mean value at this depth of about 7 Sv, i.e. a 10 % effect. Because we argue that the salinity variable in ocean models is best interpreted as being Preformed Salinity, S*, the neglect of the distinction between Preformed and Absolute Salinities in ocean models means that they misestimate the overturning stream function by 1.35 (see Fig. 4) times 0.7 Sv, namely ∼1 Sv, i.e. a 13.5 % effect.

Sensible, latent, and longwave radiative fluxes are affected by near-surface turbulence and are usually calculated using bulk formulae involving air and sea surface water temperatures (the air and sea in situ temperatures), as well as other parameters (e.g. the latent heat involves the isobaric evaporation enthalpy, commonly called the latent heat of evaporation, which is actually a weak function of temperature and salinity; see Eq. 6.28 of Feistel et al., 2010, and Eq. 3.39.7 of IOC et al., 2010). The total air–sea heat flux, Q, is then translated into a water temperature change by dividing by a heat capacity cp0, which has always been taken to be constant in numerical models (Griffies et al., 2016). Although this method is appropriate for Conservative Temperature, CT (assuming that the TEOS-10 value is used for cp0), it is not appropriate when potential temperature is being considered. The flux of potential temperature into the surface of the ocean should be Q divided by cp(S*,θ,0). The use of a constant specific heat capacity, in association with the interpretation of the ocean's temperature variable as being potential temperature, means that the ocean has received a different amount of heat than the atmosphere actually delivers to the ocean, and this issue will be explored in Sect. 3.

When precipitation (P) occurs at the sea surface, this addition of fresh water brings with it the associated potential enthalpy h(SA=0,t,0dbar) per unit mass of fresh water, where t is the in situ temperature of the raindrops as they arrive at the sea surface. The temperature at which rain enters the ocean is not yet treated consistently in coupled models, and Sect. K1.6 of Griffies et al. (2016) suggests that this effect could be equivalent to an area-averaged extra air–sea heat flux of between -150 and -300 mW m-2, representing a heat loss for the ocean.

In deciding how to numerically model the ocean, an explicit choice must be made about the equation of state, and one would think that this choice would have implications about the precise meaning of the temperature and salinity variables in the model, which we will call Tmodel and Smodel, respectively. We can divide ocean models into two general classes: EOS-80 models and TEOS-10 models.

Under this option the model's temperature variable Tmodel is treated as being potential temperature θ; this is the prevailing interpretation to date. With this interpretation of Tmodel one wonders whether Conservative Temperature Θ should be calculated from the model's (assumed) potential temperature before calculating (i) the global Ocean Heat Content as the volume integral of ρcp0Θ and (ii) the advective meridional heat transport as the area integral of ρcp0Θv at constant latitude, where v is the northward velocity. This question was not clearly addressed in Griffies et al. (2016), and here we emphasize one of the main conclusions of the present paper, namely that ocean heat content and meridional heat transports should be calculated using the model's prognostic temperature variable. Any subsequent conversion from one temperature variable to another (such as potential to conservative) in order to calculate heat content and heat transport is incorrect and confusing and should not be attempted.

Under this option the ocean model's temperature variable is taken to be Conservative Temperature Θ. The air–sea flux of potential enthalpy is then correctly ingested into the ocean model using the fixed specific heat cp0, and the mixing processes in the model correctly conserve Conservative Temperature. Hence, the second, fourth, and fifth items listed in Sect. 2 are handled correctly, except for the following caveat. In the coupled model, the bulk formulae that set the air–sea heat flux at each time step use the uppermost model temperature as the sea surface temperature as input. So with the option 2 interpretation of the model's temperature variable as being Conservative Temperature, these bulk formulae are not being fed the SST (which at the sea surface is equal to the potential temperature θ). The difference between these temperatures is Θ-θ, which is the negative of what we plot in Fig. 2. This is a caveat with this option 2 interpretation, namely that the bulk formula that the model uses to determine the air–sea flux at each time step is a little different to what was intended when the parameters of the bulk formulae were chosen. This is a caveat regarding what was intended by the coupled modeller rather than what the coupled model experienced. That is, with this option 2 interpretation, the air–sea heat flux, while being a little bit different than what might have been intended, does arrive in the ocean properly; there is no non-conservative production or destruction of heat at the air–sea boundary as there is in option 1.

Regarding the remaining two items involving temperature listed in Sect. 2, we can dismiss the fifth item, since any small difference in the initial values, set at the beginning of the lengthy spin-up period, between potential temperature and Conservative Temperature will be irrelevant after the long spin-up integration.

This then leaves the first point, namely that the model used the equation of state that expects potential temperature as its temperature input, ρ̃(S*/uPS,θ,p), but under this option 2 we are interpreting the model's temperature variable as being Conservative Temperature. In the remainder of this section we address the magnitude of this effect, namely the use of ρ̃(S*/uPS,Θ,p) versus the correct density ρ̃(S*/uPS,θ,p), which is almost the same as ρ^(S*,Θ,p). Note that, as discussed in Sect. 3 above, the salinity argument of the TEOS-10 equation of state is taken to be S*, while that of the EOS-80 equation of state is taken to be S*/uPS. These salinity variables are simply proportional to each other, and they have the same influence in both equations of state.

Under this option 2 we are interpreting the model's temperature variable as being Conservative Temperature, so the density value that the model calculates from its equation of state is deemed to be ρ̃(S*/uPS,Θ,p), whereas the density should be evaluated as ρ^(S*,Θ,p); we remind ourselves that the hat over the in situ density function indicates that this is the TEOS-10 equation of state written with Conservative Temperature as its temperature input. To be clear, under EOS-80 and under TEOS-10 the in situ density of seawater of Reference Composition has been expressed by two different expressions, 7ρ=ρ̃S*/uPS,θ,p=ρ^S*,Θ,p, both of which are very good fits to the in situ density (hence the equals signs); the increased accuracy of the TEOS-10 equation for density was mostly due to the refinement of the salinity variable, and the increase in the accuracy of TEOS-10 versus EOS-80 for Standard Seawater (Millero et al., 2008) was minor by comparison except for brackish seawater.

We need to ask what error will arise from calculating in situ density in the model as ρ̃(S*/uPS,Θ,p) instead of as the correct TEOS-10 version of in situ density, ρ^(S*,Θ,p). The effect of this difference on calculations of the buoyancy frequency and even the neutral tangent plane is likely small, so we concentrate on the effect of this difference on the isobaric gradient of in situ density (the thermal wind).

Given that under this option 2 the model's temperature variable is being interpreted as Conservative Temperature, Θ, the model-calculated isobaric gradient of in situ density is 8ρ̃S*∇PS*+ρ̃θ∇PΘ, whereas the correct isobaric gradient of in situ density is actually 9ρ^S*∇PS*+ρ^Θ∇PΘ. Notice that here and henceforth we drop the scaling factor uPS from the gradient expressions such as Eq. (8). In any case, this scaling factor cancels from the expression, but we simply drop it for ease of looking at the equations; we can imagine that the EOS-80 equation of state is written in terms of S* (which would simply require that a first line is added to the computer code that divides the salinity variable by uPS).

The model's error in evaluating the isobaric gradient of in situ density is then the difference between the two equations above, namely 10errorin∇Pρ=ρ̃S*-ρ^S*∇PS*+ρ̃θ-ρ^Θ∇PΘ. The relative error here in the temperature derivative of the equations of state can be written approximately as 11ρ̃θ-ρ^Θ/ρ^Θ=α̃θ/α^Θ-1, which is the difference from unity of the ratio of the thermal expansion coefficient with respect to potential temperature to that with respect to Conservative Temperature. This ratio, α̃θ/α^Θ, can be shown to be equal to cp(S*,θ,0)/cp0, and we know (from Fig. 1) that this varies by 6 % in the ocean. This ratio is plotted in Fig. 7a. In regions of the ocean that are very fresh, a relative error in the contribution of the isobaric temperature gradient to the thermal wind will be as large as 6 %, while in most of the ocean this relative error will be less than 0.5 %.

Now we turn our attention to the relative error in the salinity derivative of the equation of state, which, from Eq. (10), can be written approximately as 12ρ̃S*-ρ^S*/ρ^S*=β̃θ/β^Θ-1, and the ratio, β̃θ/β^Θ, has been plotted (at p=0 dbar) in Fig. 7b. This figure shows that the relative error in the salinity derivative, (ρ̃S*-ρ^S*)/ρ^S*, is an increasing (approximately quadratic) function of temperature, being approximately zero at 0 ∘C, 1 % error at 20 ∘C, and 2 % error at 30 ∘C. An alternative derivation of these implications of Eq. (10) is given in Appendix B.

We conclude that under option 2, wherein the temperature variable of an EOS-80-based model (whose polynomial equation of state expects to have potential temperature as its input temperature) is interpreted as being Conservative Temperature, there are persistent errors in the contribution of the isobaric salinity gradient to the isobaric density gradient that are approximately proportional to temperature squared, with the error being approximately 1 % at a temperature of 20 ∘C (mostly due to the salinity derivative of in situ density at constant potential temperature being 1 % different to the corresponding salinity derivative at constant Conservative Temperature). Larger fractional errors in the contribution of the isobaric temperature gradient to the thermal wind equation do occur (of up to 6 %), but these are restricted to the rather small volume of the ocean that is quite fresh.

In Fig. 8 we have evaluated how much the meridional isobaric density gradient changes in the upper 1000 dbar of the world ocean when the temperature argument in the expression for density is switched from θ to Θ. As explained above, this switch is almost equivalent to the density difference between calling the EOS-80 and the TEOS-10 equations of state using the same numeric inputs for each. We find that 19 % of these data has the isobaric density gradient changed by more than 1 % when switching from θ to Θ. The median value of the percentage error is 0.22 %; that is, 50 % of the data shallower than 1000 dbar has the isobaric density gradient changed by more than 0.22 % when switching from using EOS-80 to TEOS-10, with the same numerical temperature input, which we are interpreting as being Θ.

Figure 8 should not be interpreted as being the extra error involved with taking Tmodel to be Conservative Temperature in EOS-80 ocean models because, due to the lack of interior non-conservative source terms, the interpretation of Tmodel as being potential temperature is already incorrect by an amount that scales as Θ minus θ. Rather, Fig. 8 illustrates the error in an EOS-80 model due to the use of an equation of state that is not appropriate to the way that its temperature variable is treated in the model.

Under option 1 wherein Tmodel is interpreted as potential temperature, there is a non-conservation of heat at the sea surface, with the ocean seeing one heat flux and the atmosphere immediately above it seeing another, with 5 % of the differences in these heat fluxes being larger than approximately ±100 mW m-2 and a net imbalance of 16 mW m-2.

Under option 2 wherein Tmodel is interpreted as Conservative Temperature, the air–sea flux imbalance does not arise, but two other inaccuracies arise. First, under option 2 the bulk formulae that determine part of the air–sea flux are based on the surface values of Θ rather than of θ (for which the bulk formulae are designed). Second, the isobaric density gradient in the upper ocean is typically different by ∼1 % to the isobaric density gradient that would be found if the TEOS-10 equation of state had been adopted in these models. These two aspects of option 2 are considered less serious than not conserving heat at the sea surface by up to ±100 mW m-2. Neither of the two inaccuracies that arise under option 2 are fundamental thermodynamic errors. Rather, they are equivalent to the ocean modeller choosing (i) slightly different bulk formulae and (ii) a slightly different equation of state. The constants in the bulk formulae are very poorly known so that the switching from θ to Θ in their use will be well within their uncertainty (Cronin et al., 2019,) while the ∼1 % change to the isobaric density gradient due to using the different equations of state is at the level of our knowledge of the equation of state of seawater (see the discussion section below).

We conclude that option 2 wherein the Tmodel in EOS-80 models is interpreted as Conservative Temperature is much preferred as it treats the air–sea heat flux in a manner consistent with the first law of thermodynamics, and the treatment of Tmodel as being a conservative variable in the ocean interior is more consistent with it being Conservative Temperature than being potential temperature. These same two features of ocean models mean that Tmodel cannot be accurately interpreted as potential temperature, since both the surface flux boundary condition and the lack of the non-conservative source terms in the ocean interior mean that these ocean models continually force Tmodel away from being potential temperature, even if it was initialized as such.

How does this paper differ from the recommendations in Griffies et al. (2016)? That paper recommended that the ocean heat content and meridional transport of heat should be calculated using the model's temperature variable and the model's value of cp0, and we strenuously agree. However, in the present paper we argue that the temperature variable carried by an EOS-80-based ocean model should be interpreted as being Conservative Temperature and not be interpreted as being potential temperature. This idea was raised as a possibility in Griffies et al. (2016), but the issue was left unclear in that paper. For example, Sect. D2 of Griffies et al. (2016) recommends that TEOS-10-based models archive potential temperature (as well as their model variable, Conservative Temperature) “in order to allow meaningful comparisons” with the output of the EOS-80-based models. We now disagree with this suggestion; the thesis of the present paper is that the temperature variables of both EOS-80- and TEOS-10-based models are already directly comparable – they should both be interpreted as being Conservative Temperature, and they should both be compared with Conservative Temperature from observations. The fact that the model's temperature variable is labelled “thetao” in EOS-80 models and “bigthetao” in TEOS-10-based models we now see as very likely to cause confusion, since we are recommending that the temperature outputs of both types of ocean models should be interpreted as Conservative Temperature.

The present paper also diverges from Griffies et al. (2016) in the way that the salinity variables in CMIP ocean models should be interpreted and thus compared to observations. Griffies et al. (2016) interpret the salinity variable in TEOS-10-based ocean models as being Reference Salinity SR, whereas we show that these models actually carry Preformed Salinity S* but have errors in their calculation of densities. Similarly, Griffies et al. (2016) interpret the salinity variable in EOS-80-based ocean models as being Practical Salinity SP, whereas we show that these models actually carry S*/uPS: that is, Preformed Salinity divided by the constant, uPS. This distinction between the present paper and Griffies et al. (2016) is negligible in the upper ocean where Preformed Salinity is almost identical to Reference Salinity (because the composition of seawater in the upper ocean is close to Reference Composition), but in the deeper parts of the ocean, the distinction is not negligible; for example, based on the work of McCarthy et al. (2015) we have shown that the use of Absolute Salinity versus Preformed Salinity leads to ∼1 Sv difference in the meridional overturning stream function in the North Atlantic at a depth of 2700 m. However, in this deeper part of the ocean, even though the difference between Absolute Salinity and Preformed Salinity is not negligible, the difference between Preformed Salinity and Reference Salinity (which the TEOS-10-based ocean models have to date assumed their salinity variable to be) is smaller in the ratio 0.35/1.35=0.26 (see Fig. 4). That is, if the salinity output of a TEOS-10-based ocean model was taken to be Reference Salinity, the error would be only a quarter of the difference between Absolute Salinity and Preformed Salinity, a difference which limits the accuracy of the isobaric density gradients in the deeper parts of ocean models (see Fig. 4). A similar remark applies to EOS-80-based ocean models if their salinity output is regarded as being Practical Salinity instead of being (as we propose) S*/uPS.

In Table 1 we summarize the effects of uncertainties in physical or numerical processes in estimating ocean heat content or its changes. The first two rows are the rate of warming (expressed in m Wm-2 averaged over the sea surface) due to anthropogenic global warming and due to geothermal heating. The third row is an estimate of the surface heat flux equivalent of the depth-integrated rate of dissipation of turbulent kinetic energy, and the fourth is an estimate of the neglected net flux of potential enthalpy at the sea surface due to the evaporation and precipitation of water occurring at different temperatures.

The next (fifth) row is the consequence of considering the scenario in which all the radiant heat is absorbed into the ocean at a pressure of 25 dbar rather than at the sea surface. The derivative of specific enthalpy with respect to Conservative Temperature at 25 dbar, h^Θ, is cp0 times the ratio of the absolute in situ temperature at 25 dbar, (T0+t), to the absolute potential temperature, (T0+θ), at this pressure (see Eq. A11.15 of IOC et al., 2010). The ratio of h^Θ to cp0 at 25 dbar is typically different to unity by 6×10-6, and taking a typical rate of radiative heating of 100 W m-2 over the ocean's surface leads to 0.6 mW m-2 as the area-averaged rate of misestimation of the surface flux of Conservative Temperature for this assumed pressure of penetrative radiation. Since this is so small, the use of cp0 (rather than h^Θ) to convert the divergence of the radiative heat flux into a flux of Conservative Temperature is well-supported, providing the correct diagnostics are used for the calculation (such diagnostic issues may be responsible for the heat budget closure issues identified by Irving et al., 2021).

The next six rows of Table 1 list the mean and twice the standard deviation of the volume-integrated non-conservative production of Conservative Temperature, potential temperature, and specific entropy (all in mW m-2) at the sea surface. The following two rows are the results we have found in this paper for the air–sea heat flux error that is made if the EOS-80 temperature is taken to be potential temperature.

The final three rows show that ocean models, being cast in flux divergence form with heat fluxes being passed between one grid box and the next, do not have appreciable numerical errors in deducing air–sea fluxes from changes in the volume-integrated heat content.

The estimate from Graham and McDougall (2013) of -10 mW m-1 is for the net interior production of θ, so this is a net destruction. A steady state requires this amount of extra flux of θ at the sea surface (so it can be consumed in the interior). Our estimate of this extra flux of θ at the sea surface is 16 mW m-2, which is only a little larger than the estimate of Graham and McDougall (2013).

In summary, this paper has argued for the following guidelines for analysing the CMIP model runs. We should interpret the prognostic temperature variable of all CMIP models (whether they are based on the EOS-80 or the TEOS-10 equation of state) as being Conservative Temperature, compare the model's prognostic temperature with the Conservative Temperature, Θ, of observational data, calculate the ocean heat content as the volume integral of the product of (i) in situ density (for non-Boussinesq models or reference density for Boussinesq), (ii) the model's prognostic temperature, Θ, and (iii) the model's value of cp0, interpret the salinity variable of the model output as being Preformed Salinity S* for TEOS-10-based ocean models and S*/uPS for EOS-80-based ocean models (so it is advisable to post-multiply the salinity output of EOS-80 models by uPS in order to have the salinity outputs of all types of CMIP models as Preformed Salinity S*), and compare the model's salinity variable with Preformed Salinity, S*, calculated from ocean observations.

Sea surface temperature should be taken as the model's prognostic temperature in the case of EOS-80 models (since this is the temperature that was used in the bulk formulae), and as the calculated and stored values of potential temperature in the case of TEOS-10 models.

Ensure that all required fixed variables, such as cp0, (Boussinesq) reference density, seawater volume, and the freezing equation are saved to the CMIP archives alongside the prognostic temperature and salinity variables so that analysts have all components required to accurately interpret the model fields. In addition, providing the full-depth OHC time series for each simulation would provide a quantified target for analysts to compare and contrast changes across models and simulations.

Note that this sixth recommendation for EOS-80-based models exposes an unavoidable inconsistency in that the surface values of the model's prognostic temperature are best regarded internally in the ocean model as being Conservative Temperature, but we cannot avoid the fact that this same temperature was used as the sea surface (in situ) temperature in the bulk formulae during the running of such ocean models. Issues such as these will not arise when all ocean models have been converted to the TEOS-10 equation of state.

How then should the model's salinity and temperature outputs, S* and Θ, be used to evaluate dynamical concepts such as stream functions and dynamic height? The answer most consistent with the running of a numerical model is to use the equation of state that the model used together with the model's temperature and salinity outputs on the native grid of the model. This method is important when studying detailed dynamical balances in ocean model output. But since we now have the output salinity and temperature of both EOS-80 and TEOS-10 models being the same (namely S* and Θ), there is an efficiency and simplicity argument to analyse the output of all these models in the same manner using algorithms from the Gibbs SeaWater (GSW) Oceanographic Toolbox of TEOS-10 (McDougall and Barker, 2011). Doing these model intercomparisons often involves interpolating the model outputs to depths (or pressures) different than those used in the original ocean model, therefore incurring some interpolation errors. While the use of the GSW software means that the in situ density will be calculated slightly differently than in some of the forward models, thus affecting the thermal wind and sea level rise, these differences are small, as can be seen by comparing Figs. A5.1 and A5.2 of the TEOS-10 manual (IOC et al., 2010). Hence, we think that it is viable for most purposes to evaluate density and dynamic height using the GSW Oceanographic Toolbox, with the input salinity to this GSW code being the model's Preformed Salinity and the temperature input being the Conservative Temperature, which as we have argued, are the model's prognostic salinity and temperature variables.

Another issue that may arise is when a TEOS-10-based model has been run with Conservative Temperature, but the monthly mean Conservative Temperature output has been converted into potential temperature before sending the model output to the CMIP archive. What is the damage done if this inaccurately averaged value of potential temperature is converted back to Conservative Temperature using only the monthly mean potential temperature and salinity? While such an issue is perhaps an operational detail that takes us some distance from our intention of writing an academic paper about these issues, nevertheless we show Fig. 9, which indicates that transforming between these monthly averaged values is not a serious issue for relatively coarse-resolution ocean models.