The international Thermodynamic Equation of Seawater 2010 (TEOS-10)
defined the enthalpy and entropy of seawater, thus enabling the global ocean heat content to be calculated as the volume integral of the product of in situ density,

However, many ocean models in the Coupled Model Intercomparison Project Phase 6 (CMIP6) as well as all models that contributed to earlier phases, such as CMIP5, CMIP3, CMIP2, and CMIP1, used EOS-80 (Equation of State – 1980) rather than the updated TEOS-10, so the question arises of how the salinity and temperature variables in these models should be physically interpreted, with a particular focus on comparison to TEOS-10-compliant observations. In this article we address how heat content, surface heat fluxes, and the meridional heat transport are best calculated using output from these models and how these quantities should be compared with those calculated from corresponding observations. We conclude that even though a model uses the EOS-80, which expects potential temperature as its input temperature, the most appropriate interpretation of the model's temperature variable is actually Conservative Temperature. This perhaps unexpected interpretation is needed to ensure that the air–sea heat flux that leaves and arrives in atmosphere and sea ice models is the same as that which arrives in and leaves the ocean model.

We also show that the salinity variable carried by present TEOS-10-based models is Preformed Salinity, while the salinity variable of EOS-80-based models is also proportional to Preformed Salinity. These interpretations of the salinity and temperature variables in ocean models are an update on the comprehensive Griffies et al. (2016) paper that discusses the interpretation of many aspects of coupled Earth system models.

Numerical ocean models simulate the ocean by calculating the acceleration of fluid parcels in response to various forces, some of which are related to spatially varying density fields that affect pressure, as well as solving transport equations for the two tracers on which density depends, namely temperature (the CMIP6 variables identified as thetao or bigthetao) and dissolved matter (“salinity”, CMIP6 variable so). For computational reasons it is useful for the numerical schemes involved to be conservative, meaning that the amount of heat and salt in the ocean changes only due to the area-integrated fluxes of heat and salt that cross the ocean's boundaries; in the case of salt, this is zero. This conservative property is guaranteed for ocean models to within computational truncation error since these numerical models are designed using finite-volume integrated tracer conservation (e.g. see Appendix F in Griffies et al., 2016). It is only by ensuring such conservation properties that scientists can reliably make use of numerical ocean models for the long (centuries and longer) simulations required for climate and Earth system studies.

However, this apparent numerical success ignores some difficult theoretical issues with the equation set being numerically solved. Here, we are concerned with issues related to the properties of seawater that have only recently been widely recognized because of research resulting in the Thermodynamic Equation of Seawater 2010 (TEOS-10). These issues mean that the intercomparison of different models, and comparison with ocean observations, needs to be undertaken with care.

In particular, it is widely recognized that the traditional measure of heat content per unit mass in the ocean (with respect to an arbitrary reference state), the so-called potential temperature, is not a conservative variable (McDougall, 2003). Hence, the time change in potential temperature at a point in space is not determined solely by the convergence of the potential temperature flux at that point. Furthermore, the non-conservative nature of potential temperature means that the potential temperature of a mixture of water masses is not the mass average of the initial potential temperatures since potential temperature is “produced” or “destroyed” by mixing within the ocean's interior. This empirical fact is an inherent property of seawater (e.g. McDougall, 2003; Graham and McDougall, 2013), so treating potential temperature as a conservative tracer (as well as making certain other assumptions related to the modelling of heat and salt) results in contradictions, which have been built into most numerical ocean models to varying degrees.

These contradictions have existed since the beginning of numerical ocean modelling but have generally been ignored or overlooked because many other oceanographic and numerical factors were of greater concern. However, as global heat budgets and their imbalances are now a critical factor in understanding climate changes, it is important to examine the consequences of these assumptions and perhaps correct them even at the cost of introducing problems elsewhere. These concerns are particularly important when heat budgets are being compared between different models and with similar calculations made with observed conditions in the real ocean.

The purpose of this paper is to describe these theoretical difficulties, to estimate the magnitude of errors that result, and to make recommendations about resolving them both in current and future modelling efforts. For example, the insistence that a model's temperature variable is potential temperature involves errors in the air–sea heat flux in some areas that are as large as the mean rate of current global warming. A simple re-interpretation of the model's temperature variable overcomes this inconsistency and allows coupled climate models to conserve heat.

The reader who wants to skip straight to the recommendations on how the salinity and temperature outputs of CMIP models should be interpreted can go straight to Sect. 6.

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

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

Summary of the impact of various processes and modelling errors on
the global ocean heat budget and its imbalance (units: mW m

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

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

Since potential enthalpy was not a widely understood property, a decision was made in the development of TEOS-10 to adopt Conservative Temperature,

Note that at specific locations in the ocean, in particular at low salinities and high temperatures,

Contours (in

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,

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

We conclude that the idea that ocean models could retain potential temperature

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

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

Tailleux (2010, 2015) treated this non-conservative term,

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

Contours of

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

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,

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,

Chemical modelling (Pawlowicz, 2010; Pawlowicz et al., 2011; Wright et al., 2011; Pawlowicz et al., 2012) suggests the approximate relation

Number line of salinity, illustrating the differences between Preformed Salinity

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,

The current standard for describing the thermodynamic properties of
seawater, known as TEOS-10, provides an equation of state,

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,

For seawater of Reference Composition, both the TEOS-10 and EOS-80 fits

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

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,

When precipitation (

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

One class of CMIP ocean model is based around EOS-80, and these models have
the following characteristics.

The model's equation of state,

The air–sea heat flux is delivered to and from the ocean using a constant isobaric specific heat,

Other ocean models have begun to implement TEOS-10 features. These models
generally have the following characteristics.

The model's equation of state,

At each time step of the model, the value of potential temperature at the sea surface (i.e. SST) is calculated from the

The air–sea heat flux is delivered to and from the ocean using the TEOS-10 constant isobaric specific heat,

There is one CMIP6 ocean model that we are aware of, ACCESS-CM2 (Australian Community Climate and Earth System Simulator; Bi et al., 2020), whose equation of state is written in terms of Conservative Temperature, but the salinity argument in the equation of state is Practical Salinity. The salinity in this model is initialized with atlas values of Practical Salinity.

From the above it is clear that there are small but significant theoretical incompatibilities between different models and between models and the observed ocean. These issues become apparent when dealing with the technicalities of intercomparisons, and various choices must be made. We now consider the implications of these different choices and provide recommendations for best practices.

Note that the samples whose measured specific volumes were incorporated into
both the EOS-80 and TEOS-10 equations of state were of Standard Seawater
whose composition is close to Reference Composition. Consequently, the
EOS-80 and TEOS-10 equations of state were constructed with Preformed
Salinity,

For an ocean model that has no non-conservative interior source terms
affecting the evolution of its salinity variable and that is initialized at
the sea surface with Preformed Salinity, the only interpretation for the
model's salinity variable is Preformed Salinity, and the use of the TEOS-10
equation of state will then yield the correct specific volume. Furthermore,
whether the model is initialized with values of Absolute Salinity, Reference
Salinity, or Preformed Salinity, these initial salinity values are nearly
identical in the upper ocean, and any differences between the three initial
conditions in the deeper ocean would be largely diffused away within the
long spin-up period. That is, in the absence of the non-conservative
biogeochemical source terms that would be needed to model Absolute Salinity
and to force it away from being conservative (or the smaller source terms
that would be needed to maintain Reference Salinity), the model's salinity
variable will drift towards being Preformed Salinity. Hence, we conclude
that, after the long spin-up phase, the salinity variable of a TEOS-10-based
ocean model is accurately interpreted as being preformed salinity

Likewise, the prognostic salinity variable after a long spin-up period of an
EOS-80-based model is most accurately interpreted as being Preformed
Salinity divided by

We clearly need more estimates of the magnitude of the dynamic effects of the variable seawater composition, but for now we might take a change in 1 Sv in the meridional transport of deepwater masses in each ocean basin (based on the Atlantic work of McCarthy et al., 2015) as an indication of the magnitude of the effect of neglecting the effects of biogeochemistry on salinity. At this stage of model development, since all models are equally deficient in their thermophysical treatment of salinity, at least this aspect does not present a problem as far as making comparisons between CMIP models.

From the details described above, both types of numerical ocean models suffer from some internal contradictions with thermodynamical best practice. For example, for the EOS-80-based models, if

One use for these models is to calculate heat budgets and heat fluxes –
both at the surface and between latitudinal bands, and inherent to CMIP is
the idea that these different models should be intercompared. The question
of how this intercomparison should be done, however, was not clearly addressed in Griffies et al. (2016). Here we begin the discussion by
considering two different options for interpreting

Under this option the model's temperature variable

There are several thermodynamic inconsistencies that arise from option 1. First, the ocean model has assumed in its spin-up phase (for perhaps a
millennium) that

The second inconsistent aspect of option 1 is that the air–sea flux of heat
is ingested into the ocean model during both the spin-up stage and
the subsequent transient response phase, as though the model's temperature
variable is proportional to potential enthalpy. For example, consider some
time during the year at a particular location where the sea surface is fresh
(a river outflow or melted ice). During this time, any heat that the
atmosphere loses or gains should have affected the potential temperature of the upper layers of the ocean using a specific heat that is 6 % larger
than

This second inconsistent aspect of option 1 can be restated as follows. The adoption of potential temperature as the model's temperature variable means that there is a discontinuity in the heat flux of the coupled air–sea system right at the sea surface; for every Joule of heat (i.e. potential enthalpy) that the atmosphere gives to the ocean, under this option 1 interpretation, up to 6 % too much heat arrives in the ocean over relatively fresh waters. In this way, the adoption of potential temperature as the model temperature variable ensures that the coupled ocean–atmosphere system will not conserve heat. Rather, there appear to be non-conservative sources and sinks of heat right at the sea surface where heat is unphysically manufactured or destroyed.

The third inconsistent aspect is a direct consequence of the second; namely,
if one is tempted to post-calculate Conservative Temperature

Here we quantify the air–sea flux errors involved with assuming that

The ACCESS-CM2 zonally integrated

Figure 5e shows that, with

Figure 5d and f show that much of this spread is due to the variation of the
isobaric specific heat capacity in salinity, with the remainder due to the
variation of this heat capacity with temperature. We note that if this
analysis were performed with a model that resolved individual rain showers
and the associated freshwater lenses on the ocean surface, then these
episodes of very fresh water at the sea surface would be expected to increase the calculated values of

While a heat flux error of

In Appendix A we enquire whether the way that EOS-80 models treat their fluid might be made to be thermodynamically correct for a fluid other than seawater. We find that it is possible to construct such a thermodynamic definition of a fluid with the aim that its treatment in EOS-80 models is
consistent with the laws of thermodynamics. This fluid has the same specific
volume as seawater for given values of salinity, potential temperature, and
pressure, but it has different expressions for both enthalpy and entropy.
This fluid also has a different adiabatic lapse rate and therefore a different relationship between in situ and potential temperatures. However, this exercise in thermodynamic abstraction does not alter the fact that, as a model of the real ocean and with the temperature variable being interpreted as being potential temperature, the EOS-80 models have

Since CMIP6 is centrally concerned with how the planet warms, it is advisable to adopt a framework wherein heat fluxes and their consequences are respected. That is, we regard it as imperative to avoid non-conservative sources of heat at the sea surface. It is the insistence that the temperature variable in EOS-80-based models is potential temperature that implies that the ocean receives a heat flux from the atmosphere that is larger by

Under this option the ocean model's temperature variable is taken to be
Conservative Temperature

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,

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

We need to ask what error will arise from calculating in situ density in the model as

Given that under this option 2 the model's temperature variable is being
interpreted as Conservative Temperature,

The model's error in evaluating the isobaric gradient of in situ density is then the difference between the two equations above, namely

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

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

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

The northward density gradient at constant pressure (the horizontal axis) for data in the global ocean atlas of Gouretski and Koltermann (2004) for

Figure 8 should not be interpreted as being the extra error involved with
taking

Under option 1 wherein

Under option 2 wherein

We conclude that option 2 wherein the

Now that we have argued that

We have made the case that the salinity variable in CMIP ocean models that
have been spun up for several centuries is Preformed Salinity

We have made the case that it is advisable to avoid non-conservative sources
of heat at the sea surface. It is the prior interpretation of the temperature variable in EOS-80-based models as being potential temperature that implies that the ocean receives a heat flux that is larger by

A consequence of this new interpretation of the prognostic temperature
variable of all CMIP ocean models as being Conservative Temperature means
that the EOS-80-based models suffer from a relative error of

We must also acknowledge that all models have ignored the difference between Preformed Salinity, Reference Salinity, and Absolute Salinity (which is the salinity variable from which density is accurately calculated). As discussed in IOC et al. (2010), Wright et al. (2011), and McDougall and Barker (2011), glossing over these issues of the spatially variable composition of sea salt, which is the same as glossing over the effects of biogeochemistry on salinity and density, means that all our ocean and climate models have errors in their thermal wind (vertical shear of horizontal velocity) that globally exceed 2.7 % for half the ocean volume deeper than 1000 m. In the deep North Pacific Ocean, the misestimation of thermal wind is many times this 2.7 % value. The recommended way of incorporating the spatially varying composition of seawater into ocean models appears as Sect. A20 in the TEOS-10 manual (IOC et al., 2010) and as Sect. 9 in McDougall and Barker (2011), with ocean models needing to carry a second salinity type variable. While it is true that this procedure has the effect of relaxing the model towards the non-standard seawater composition of today's ocean, it is clearly advantageous to make a start with this issue by incorporating the non-conservative source terms that apply to the present ocean rather than to continue to ignore the issue altogether. As explained in these references, once the modelling of ocean biogeochemistry matures, the difference between the various types of salinity can be calculated in real time in an ocean model without the need for referring to historical data.

Nevertheless, we acknowledge that no published ocean model to date has attempted to include the influence of biogeochemistry on salinity and density, and therefore we recommend that the salinity from both observations and model output be treated as Preformed Salinity

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

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

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

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,

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

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

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,

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

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,

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.

The RMSE (K) in evaluating Conservative Temperature from the
CMIP6 archived monthly averaged values of potential temperature and salinity compared with averaging the instantaneous values of Conservative Temperature for a month at the

Ocean models have always assumed a constant isobaric heat capacity and have traditionally assumed that the model's temperature variable is whatever temperature the equation of state was designed to accept. Here we enquire whether there is a way of justifying option 1 thermodynamically in the sense that option 1 would be totally consistent with thermodynamic principles for a fluid that is different to real seawater.

That is, we pursue the idea that these EOS-80-based ocean models are not actually models of seawater but are models of a slightly different fluid. We require a fluid that is identical to seawater in some respects, such as having the same dissolved material (Millero et al., 2008) and the same issues around Absolute Salinity, Preformed Salinity, and Practical Salinity, as well as the same in situ density as real seawater (at given values of Absolute Salinity, potential temperature, and pressure). But we require the expression for the enthalpy of this new fluid to be different to that of real seawater.

The difference that we envisage between real seawater and this new fluid is
that, at zero pressure, the enthalpy of the new fluid is given exactly by
the constant value

The enthalpy of this new fluid is then given by (since

With these definitions (Eqs. A1 and A3) of enthalpy and entropy of our
new fluid, we have completely defined all the thermophysical properties of
the fluid (see Appendix P of IOC et al., 2010, for a discussion). Many aspects of the fluid are different to seawater, including the adiabatic lapse rate (and hence the relationship between in situ and potential temperatures), since the adiabatic lapse rate is given by

We conclude that this is indeed a conceptual way of forcing the EOS-80-based models to be consistent with thermodynamic principles. That is, we have shown that these EOS-80 models are not models of seawater, but they do accurately model a different fluid whose thermodynamic definition we have given in Eqs. (A1) and (A3). This new fluid interacts with the atmosphere in the way that EOS-80 models have assumed to date; the potential temperature of this new fluid is correctly mixed in the ocean in a conservative fashion, and the equation of state is written in terms of the model's temperature variable, namely potential temperature.

Hence, we have constructed a fluid which is thermodynamically different to seawater, but it does behave exactly as these EOS-80 models treat their model seawater. That is, we have constructed a new fluid for which, if seawater had these thermodynamic characteristics, the EOS-80 ocean models would have correct thermodynamics while being able to interpret the model's temperature variable as potential temperature.

But this does not change the fact that in order to make these EOS-80 models thermodynamically consistent in this way we have ignored the real variation at the sea surface of the isobaric specific heat capacity – a variation that we know can be as large as 6 %.

Hence, we do not propose this non-seawater explanation as a useful rationalization of the behaviour of EOS-80-based ocean models. Rather, it
seems less dramatic and more climatically relevant to adopt the simpler
interpretation of option 2. Under this option we accept that the model is
modelling actual seawater, that the model's temperature variable is in fact
Conservative Temperature, and that there are some errors in the equation of
state of these EOS-80 models that amount to errors of the order of 1 % in
the thermal wind relation throughout much of the upper (warm) ocean. That is, so long as we interpret the temperature variable of these EOS-80-based models as Conservative Temperature, they are fine except that they have used an incorrect equation of state; they have used

Equation (10) is an expression for the error in the isobaric density gradient
when Conservative Temperature is used as the input temperature variable to
the EOS-80 equation of state (which expects its input temperature to be potential temperature). An alternative accurate expression to Eq. (9) for the isobaric density gradient is

This paper has not run any ocean or climate models, so it has not produced any such computer code. Processed data and code to produce the CMIP6 ACCESS-CM2 PI Control results in Figs. 5, 6, and 9 are published online (Zenodo,

This paper has not produced any model data. Processed data and code to produce the CMIP6 ACCESS-CM2 PI Control results in Figs. 5, 6, and 9 are published online (Zenodo,

TJM devised this new way of interpreting CMIP ocean model variables. PMB and RMH provided figures for the paper, and all authors contributed to the concepts and the writing of the paper.

Publisher's note: Copernicus Publications remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

We have benefitted from comments and suggestions from Baylor Fox-Kemper, Sjoerd Groeskamp, Casimir de Lavergne, John Krasting, Fabien Roquet, Geoff Stanley, Jan-Erik Tesdal, Rainer Feistel and Remi Tailleux. This paper contributes to the tasks of the Joint SCOR/IAPSO/IAPWS Committee on the Thermophysical Properties of Seawater. Trevor J. McDougall, Paul M. Barker, and Ryan M. Holmes gratefully acknowledge Australian Research Council support through grant FL150100090.

This research has been supported by the Australian Research Council (grant no. FL150100090). The work of Paul J. Durack was prepared by Lawrence Livermore National Laboratory (LLNL) under contract no. DE-AC52-07NA27344 and is a contribution to the US Department of Energy, Office of Science, Earth and Environmental System Sciences Division, Regional and Global Modeling and Analysis Program (LLNL release no. LLNL-JRNL-823462). We acknowledge the World Climate Research Programme, which, through its Working Group on Coupled Modeling, coordinated and promoted CMIP6. We thank the climate modelling groups for producing and making available their model output, the Earth System Grid Federation (ESGF) for archiving the data and providing access, and the multiple funding agencies who support CMIP6 and ESGF. Data analysis was undertaken using facilities at the National Computational Infrastructure (NCI), which is supported by the Australian Government.

This paper was edited by Robert Marsh and reviewed by Baylor Fox-Kemper and Remi Tailleux.