## 1. Introduction

The problem of how best to model advection and diffusion of “heat” in the ocean is not an easy one to solve. First, it is difficult to define what “heat” actually is in the ocean. Second, it is not a trivial task to derive a variable whose advection and diffusion can be regarded as the advection and diffusion of “heat.” It is common practice in oceanography to regard the first law of thermodynamics as the difference between the conservation equation of total energy and the evolution equation for kinetic energy plus gravitational potential energy. What is left is an expression that at leading order can be construed as an evolution equation for “heat” [Landau and Lifshitz 1959; McDougall 2003; Griffies 2004; Intergovernmental Oceanographic Commision (IOC) et al. 2010].

*ρ*is density,

*C*is the thermodynamic variable in question,

**∇**· is the divergence operator,

**u**is the three-dimensional velocity vector,

*D*/

*Dt*= ∂/∂

*t*+

**u**·

**∇**is the material derivative operator following the fluid motion, and

**F**

^{C}is the flux of

*C*caused by molecular diffusive processes. The subscript

*t*denotes a partial derivative with respect to time.

Since its introduction to the field of oceanography by Helland-Hansen (1911), potential temperature has been treated as the variable whose advection and diffusion can be interpreted as the advection and diffusion of heat. Ocean models have to date treated potential temperature as if it were a conservative property; however, when fluid parcels mix at fixed pressure, nonconservative source terms of potential temperature arise, mostly because the isobaric heat capacity of seawater, which appears in the evolution equation of potential temperature, varies by up to 5% with salinity, and also because enthalpy depends nonlinearly not only on potential temperature but also on salinity (McDougall 2003). If the nonconservative source terms of potential temperature were not ignored, then the errors in ocean models carrying potential temperature as the default temperature variable would be eliminated (although, in this instance, ocean models would need to carry the isobaric heat capacity in the form *c _{p}*(

*S*

_{A},

*θ*,

*p*= 0 dbar), which is used at the sea surface to relate the air–sea heat flux to the flux of potential temperature and which varies by 5% at the sea surface because it is a function of

_{r}*S*

_{A}and

*θ*, rather than the constant heat capacity

*h*

^{0}, referenced to the pressure

*p*= 0 as

*h*is specific enthalpy,

*S*

_{A}is Absolute Salinity, Θ is Conservative Temperature (see below),

*p*is the excess of the absolute pressure over the fixed pressure

*p*

_{0}= 0.101 325 MPa, ∫ is the integral operator, and

*υ*is specific volume. The overhat notation is used to denote a variable that is a function of Conservative Temperature. Values for the arbitrary constants that arise in the Gibbs function definition of the variables in Eq. (2) are listed in IOC et al. (2010). We choose potential enthalpy referenced to the pressure

*p*= 0 so that the error in equating the air-sea heat flux with the flux of

*h*

^{0}is minimized. Referencing potential enthalpy to

*p*= 0 is also appropriate because enthalpy is unknown up to a linear function of salinity, and since there is no flux of salt at the sea surface, the choice of that linear function of salinity will not affect the air–sea heat flux of

*h*

^{0}. A temperature variable called Conservative Temperature, denoted Θ, was defined to be proportional to potential enthalpy such that

^{−1}K

^{−1}(IOC et al. 2010). Although it is possible to reference potential enthalpy to pressures other than the sea surface, Conservative Temperature is always defined in terms of potential enthalpy referenced at the sea surface. The evolution equation for potential enthalpy almost exactly satisfies the first law of thermodynamics, and since the air–sea heat flux is equal to the air–sea flux of potential enthalpy, potential enthalpy can be accurately treated as the “heat content” of seawater (IOC et al. 2010).

By treating potential temperature as if it were conservative, ocean models have in some sense inadvertently interpreted their prognostic temperature variable as Conservative Temperature. McDougall (2003) and Tailleux (2010) have done early work in quantifying the errors introduced into ocean models by ignoring the nonconservative source terms of potential temperature. In particular, McDougall (2003) found that typical errors in ocean models in ignoring the nonconservative source terms of *θ* are ±0.1°C, although the errors can be as large as 1.4°C (with Θ > *θ*) in areas that are warm and fresh, such as where the fresh Amazon River water enters the ocean. This behavior can be seen from Fig. 1, which shows contours of the difference between *θ* and Θ in degrees Celsius at the sea surface. IOC et al. (2010) estimated that Conservative Temperature is about 100 times more conservative than potential temperature. This was deduced through careful examination of their Figs. A.17.1 and A.18.1. More recent work by Tailleux (2010) found that Conservative Temperature is about 20 times more conservative than potential temperature. However, in the literature to date, namely in Tailleux (2010) and McDougall (2003), the relative errors in ignoring the nonconservative source terms of potential temperature and Conservative Temperature have been estimated either from generalizing the laminar evolution equations of each variable to the turbulent ocean, or by using parcel mixing arguments in a turbulent ocean following a method by Fofonoff (1985). No work to date has explicitly derived complete evolution equations for potential temperature and Conservative Temperature in a turbulent ocean, including their respective nonconservative source terms, nor have the errors incurred by ocean models in assuming the variables are conservative been quantified.

In section 2, the forms of the first law of thermodynamics in terms of specific entropy, potential temperature, and Conservative Temperature are derived. The turbulent forms of the evolution equations for specific entropy, potential temperature, and Conservative Temperature are derived in section 3. The source terms of these evolution equations are quantified using the output of a coarse-resolution oceanic general circulation model in section 4. Since from the second law of thermodynamics it is known that entropy is a nonconservative variable, entropy, along with the dissipation of kinetic energy, provide valuable benchmarks of nonconservation with which to compare the nonconservation of potential temperature and Conservative Temperature.

## 2. The First Law of Thermodynamics in terms of *θ*, Θ, and *η*

*ρ*is density,

*D*/

*Dt*= ∂/∂

*t*+

**u**·

**∇**is the material derivative operator following the fluid motion,

*h*is specific enthalpy,

*p*is the excess of the absolute pressure over the fixed pressure,

*p*

_{0}= 0.101 325 MPa,

*u*is specific internal energy,

*T*

_{0}= 273.15 K,

*t*is in-situ temperature,

*η*is specific entropy,

*μ*is the relative chemical potential,

*S*

_{A}is Absolute Salinity,

**F**

^{R}is the flux of heat at the ocean boundaries and by radiation,

**F**

^{Q}is the molecular diffusion of heat,

*ɛ*is the rate of dissipation of kinetic energy, and

*η*,

*θ*, and Θ (IOC et al. 2010). The first law of thermodynamics can be regarded as the difference between the conservation equation of total energy and the evolution equation for the sum of kinetic energy and potential energy, leaving what can be approximately interpreted as an evolution equation for heat in the ocean (Landau and Lifshitz 1959; McDougall 2003; Griffies 2004; IOC et al. 2010). The physical interpretation of the first law of thermodynamics is that the specific internal energy

*u*of a fluid parcel can change due to (McDougall 2003) (i) the work in changing the fluid parcel’s volume at some pressure (

*p*

_{0}+

*p*)/

*ρ*

^{2}

*Dρ*/

*Dt*; (ii) the divergence of the flux of heat via

*ɛ*; and (iv) nonconservative biogeochemical processes.

*υ*is specific volume. Hence, Eqs. (7)–(10) become

*S*

_{A}and

*η*are

*η*by substituting the relations from Eq. (17) into Eq. (13), obtaining

*θ*, holding

*S*

_{A}and

*p*constant, we obtain

*p*= 0, we see that

*c*is the specific isobaric heat capacity, and using the expression for

_{p}*S*

_{A}, holding

*θ*and

*p*constant, we obtain

*θ*can be expressed as

*S*

_{A}and

*p*constant, we obtain

*p*= 0, we see that

*S*

_{A}, holding Θ and

*p*constant, and taking into account the relations from Eq. (17) and that from the fundamental thermodynamic relation

By inspecting the above forms of the first law of thermodynamics, an approximate ranking of the nonconservative natures of *η*, *θ*, and Θ can be made. The ocean circulation can be viewed as a series of adiabatic and isohaline movements of seawater parcels followed by isolated turbulent mixing events, which occur at fixed pressure. Note that we assume that mixing between parcels occurs when the parcels have the same spatial location, that is, at some fixed pressure. The final irreversible part of a turbulent mixing event involves molecular diffusion which destroys tracer variance. We are ignoring pressure perturbations, for example due to sound waves, during this irreversible molecular diffusion. As each of *η*, *θ*, and Θ are conserved during the adiabatic and isohaline movements, the nonconservation of *η*, *θ*, and Θ will depend upon the extent to which the coefficients that multiply the material derivatives of *η*, *θ*, Θ, and *S*_{A} in Eqs. (18), (23), and (28) vary at fixed pressure (McDougall 2003). These coefficients, namely *η*, *θ*, and Θ respectively, at each pressure in the cast, we can infer that the source terms of *η* and *θ* vary considerably more than the source terms of Θ and that the greatest variation occurs at the sea surface (in fact, it will be shown that the source terms of Θ are exactly equal to 0 at the sea surface, whereas the source terms of *θ* and *η* achieve their maximums at the sea surface). Hence we conclude that Θ is the most conservative variable, followed by *θ* and *η*.

## 3. Turbulent evolution equations

The preceding discussion shows that nonconservative source terms of Θ, *θ*, and *η* exist during mixing events. It follows that to derive the full evolution equations for Θ, *θ*, and *η*, including their nonconservative source terms, it is first necessary to obtain the turbulently averaged forms of the evolution equations for Absolute Salinity and enthalpy at fixed pressure. These are derived using a similar approach to the one undertaken in IOC et al. (2010).

### a. The evolution equation for S_{A}

*ρ*

_{0}is constant density, usually assigned a value of 1035 kg m

^{−3}(Griffies 2004). The overbar notation indicates temporally averaged variables, and the double-primed notation indicates deviations of the instantaneous quantity from its density-weighted averaged value. Absolute Salinity has been density-weighted averaged, so that

*e*is the dianeutral velocity, and

*e*on the neutral tangent plane (that is,

*K*and the dianeutral diffusivity

*D*. The height at which Eq. (31) is evaluated is the average height of the density value

*θ*, and

*η*in a turbulent ocean. The left-hand side is the derivative following the motion of the thickness-weighted

*S*

_{A}with respect to the thickness-weighted horizontal velocity

*S*

_{A}, where

*γ*is the vertical gradient of a suitable density variable (such as neutral density, or locally referenced potential density). The overhat notation is not used in what follows to avoid confusion with notation denoting whether a property is a function of Θ. However, it should be kept in mind that the quantities in the subsequent expressions are turbulently-averaged quantities.

_{z}### b. The evolution equation for potential enthalpy h^{m} referenced to the general pressure p^{m}, at p^{m}

*h*referenced to the pressure

^{m}*p*is defined by

^{m}*p*is chosen,

^{m}*h*is a function of only

^{m}*S*

_{A}and Θ [actually

*h*, from Eq. (33) we find

^{m}*p*=

*p*the first law of thermodynamics, Eq. (10), becomes

^{m}*h*is that at

^{m}*p*=

*p*, the evolution equation for

^{m}*h*is of the conservative form except for the term

^{m}*ρɛ*in the dissipation of kinetic energy. Because

*h*is a “potential” variable in that it is unchanged under adiabatic and isohaline changes in pressure, it is possible to undertake the same two-stage-averaging process as used in the previous subsection and to take the turbulent fluxes of

^{m}*h*to be down the respective mean gradients of

^{m}*h*. Performing this two-stage averaging process, we find that the Boussinesq form of the mesoscale averaged equation is

^{m}*h*, where

^{m}*γ*is the vertical gradient of a suitable density variable. Having derived the evolution equations for

_{z}*S*

_{A}and

*h*, the full evolution equations for Θ,

^{m}*θ*, and

*η*in a turbulently mixed ocean are now derived.

### c. The evolution equation for *Θ*

*p*,

^{m}*h*, is expressed in the form

^{m}*p*=

*p*[recall that Conservative Temperature Θ is defined with respect to 0 dbar, see Eqs. (2) and (3)], it follows that the spatial and temporal gradients in Eq. (36) can be written in terms of the spatial and temporal gradients of

^{m}*S*

_{A}and Θ, namely

*m*is no longer needed on the partial derivatives of enthalpy here because this equation applies at all pressures. The epineutral fluxes in Eq. (39) can be written in terms of the turbulent epineutral mesoscale fluxes rather than their parameterized versions as follows:

**F**

^{Θ}and

**F**

^{S}are the epineutral fluxes of Θ and

*S*

_{A}respectively.

Note that in this subsection we have taken the averaged enthalpy to be a function of the averaged Absolute Salinity and the averaged Conservative Temperature in the form

### d. The evolution equation for θ

*θ*in the functional form

*θ*following the appropriately averaged mean velocity is the sum of the conservation equation for Θ following the appropriately averaged mean velocity, Eq. (39), multiplied by

*S*

_{A}following the appropriately averaged mean velocity, Eq. (32), multiplied by

*θ*in a turbulent ocean, and in particular, the form of the nonconservative source terms, is one of the key results of this paper.

*θ*is derived:

### e. The evolution equation for *η*

*η*in the form

*S*

_{A}following the appropriately averaged mean velocity, Eq. (32); namely,

*η*in a turbulent ocean becomes

*η*is positive definite. This is further discussed in appendix C.

## 4. Quantifying the nonconservative source terms

### a. Dataset

For our calculations, we require a dataset with salinity, temperature, and pressure as well as credible values of the turbulent diffusivities *K* and *D*. We use the same model output from a standard configuration of version 4 of the Modular Ocean Model (MOM4; Griffies et al. 2004) as used in Klocker and McDougall (2010a,b), with a resolution of 1° × 1°. This model carries Conservative Temperature rather than potential temperature, practical salinity, an isopycnal diffusivity *K* of 1000 m^{2} s^{−1}, and uses the vertical mixing scheme by Bryan and Lewis (1979). The evolution equations for Θ, *θ*, and *η* in both the laminar and turbulent cases are developed here in terms of Absolute Salinity. This is the salinity argument of the modern Gibbs function, and although the MOM4 dataset used in this section carries practical salinity rather than Absolute Salinity, for consistency with the notation used throughout the paper so far and for laying the foundation for future work in this area, the model’s salinity variable will be interpreted as Absolute Salinity, denoted *S*_{A} which we have simply calculated as practical salinity times 35.165 04 g kg^{−1}/35. The model uses the Jackett et al. (2006) equation of state. This equation of state has been superseded by the more accurate and comprehensive Thermodynamic Equation of Seawater 2010 (TEOS-10), described in IOC et al. (2010); however, for consistency with the model data, all calculations are performed using the earlier definition for the equation of state.

*ɛ*is calculated using the Osborn (1980) relation between the diapycnal diffusivity

*D*and the square of the buoyancy frequency

*N*

^{2}; namely,

### b. Quantifying the nonconservative source terms of Θ, θ, and η

^{−2})

*I*denotes the variable whose source terms are being integrated, and

*z*denotes the fixed reference depth to which the integrals are calculated.

_{r}*P*

_{Θ},

*P*, and

_{θ}*P*represent the source terms of Θ,

_{η}*θ*, and

*η*respectively in Eqs. (39), (44), and (48). Unless otherwise indicated, the integrals are evaluated from the sea floor (where

*z*≈ −5000 m). Also,

_{b}*ρ*

_{0}is the constant density 1035 kg m

^{−3}, and

^{−1}K

^{−2}.

We consider the integrals of the source terms of Θ, *θ*, and *η*, and *ɛ*, as per Eqs. (50)–(53), from the sea floor (*z _{b}* ≈ −5000 m) to the sea surface (

*z*≈ 0 m). In the cases of Θ,

_{r}*θ*, and

*η*, the resulting expressions are

*I*

_{Θ}(0),

*I*(0), and

_{θ}*I*(0), which can be interpreted as the errors in the air–sea heat flux (mW m

_{η}^{−2}) in assuming that the variables Θ,

*θ*, and

*η*are conservative. In the case of

*ɛ*, the resulting expression is

*I*(0), which represents the air–sea heat flux that warms the water column at the same rate as does the interior turbulent rate of dissipation of kinetic energy. For this study, an arbitrary depth of 300 m is applied globally such that above 300 m the horizontal gradients

_{ɛ}**∇**

_{z}of temperature and salinity are used in place of the epineutral gradients

**∇**

_{n}, while below 300 m, epineutral gradients of temperature and salinity are used. Given the model data used is of coarse resolution, it is sufficient for our purposes to choose a fixed value for the mixed layer depth and a depth of 300 m is chosen to encapsulate the dynamics of the mixed layer. Deep convection regions where the slope might usually be infinite are not encountered in this study, also due to the coarse resolution of the model data.

#### 1) Map of *I*_{Θ}(0)

The magnitude of the depth-integrated source terms, namely *I*_{Θ}(0), is shown in Fig. 4. The mean of *I*_{Θ}(0) is 0.318 mW m^{−2}, and the maximum value is 13.8 mW m^{−2}. In general, the largest values of *I*_{Θ}(0) are found in areas where strong currents are present [for example, the Gulf Stream, the Antarctic Circumpolar Current (ACC) region, and the Kuroshio Current], whereas smaller values of *I*_{Θ}(0) occur at high latitudes in the Southern Hemisphere and in the Bering Sea. The dominant terms contributing to *I*_{Θ}(0) are the *K*_{1} and *D*_{1} terms in Eq. (39), taking into account that horizontal gradients rather than epineutral gradients are calculated in the mixed layer (the upper 300 m). It is clear that *K*_{1} and *D*_{1}, integrated with respect to depth from the sea floor to the sea surface, represent the error in the air–sea heat flux due to ignoring the effect of cabbeling since *K*_{1} and *D*_{1} terms are here and henceforth denoted *K*_{1}(0) and *D*_{1}(0) and are illustrated in Figs. 4b and 4c. The mean of *K*_{1}(0) is 0.176 mW m^{−2}, the mean of *D*_{1}(0) is 0.127 mW m^{−2}, and both *K*_{1}(0) and *D*_{1}(0) are strictly positive. The second terms in the expressions that are multiplied by *K* and *D* in Eq. (39) are integrated with respect to depth from the sea floor to the sea surface and are denoted *K*_{2}(0) and *D*_{2}(0). These terms can be negative since the horizontal gradients of *S*_{A} and Θ can be of opposite sign in the upper ocean. However, the magnitude of *K*_{2}(0) + *D*_{2}(0) is up to 20 times smaller than *K*_{1}(0) + *D*_{1}(0), which means that the overall sign of *I*_{Θ}(0) is positive. The third terms in the expressions that are multiplied by *K* and *D* in Eq. (39), integrated with respect to depth and denoted *K*_{3}(0) and *D*_{3}(0) (these terms are sometimes called the “heat of mixing” terms), are on average 278 and 224 times smaller than *K*_{1}(0) and *D*_{1}(0) respectively, although the spatial patterns are similar.

Figures 4d and 4e show the integral of the nonconservative source terms of Θ from −5000 to −1800 m, denoted *I*_{Θ}(−1800), and from −1800 to −500 m, denoted *I*_{Θ}(−500) − *I*_{Θ}(−1800), respectively. The relative magnitude of the values of each of these plots is of the same order as in *I*_{Θ}(0); however, the mean value of *I*_{Θ}(−1800) is 3.20 × 10^{−2} mW m^{−2} and the mean value of *I*_{Θ}(−500) − *I*_{Θ}(−1800) is 7.73 × 10^{−2} mW m^{−2}, so it can be seen that the production of Θ in the upper 500 m of the ocean dominates *I*_{Θ}(0). Both *I*_{Θ}(−1800) and *I*_{Θ}(−500) − *I*_{Θ}(−1800) are dominated by the vertical flux terms in Eq. (39).

#### 2) Map of *I*_{θ}(0)

_{θ}

Figure 5 illustrates the error in the air–sea heat flux in assuming that *θ* is conservative. The mean value of *I _{θ}*(0) is −9.62 mW m

^{−2}, the magnitude of which is approximately 30 times greater than the mean value of

*I*

_{Θ}(0), the maximum value is 678 mW m

^{−2}, and the minimum value is −1465 mW m

^{−2}. So, the maximum absolute error in assuming

*θ*is conservative is approximately 106 times greater than the maximum absolute error in assuming Θ is conservative. In general, negative values occur at high latitudes and in the equatorial regions, whereas larger, positive values dominate the mid-latitudes.

Similarly to the previous section, we denote the depth-integrated terms from the expression for the source terms of *θ* in Eq. (44) as *K*_{1}(0), *K*_{2}(0), *K*_{3}(0), *D*_{1}(0), *D*_{2}(0), and *D*_{3}(0), where horizontal rather than epineutral gradients are calculated in the top 300 m of the ocean. It can be seen that the *K*_{1}(0) and *D*_{1}(0) terms are positive definite and the *K*_{3}(0) and *D*_{3}(0) terms are negative definite, while the *K*_{2}(0) and *D*_{2}(0) terms can take both positive and negative values due to the sign of the temperature and salinity gradients in each term and hence *θ* can be alternatively produced and consumed in turbulent mixing. The “heat of mixing” terms, *K*_{3}(0) and *D*_{3}(0), are smaller than the remaining terms in the expression for the evolution of *θ* by up to 16 times. Unlike in the Θ case, cabbeling is not the dominant term contributing to *K*_{1}(0) and *D*_{1}(0); rather, the magnitude of the *K*_{1}(0) and *D*_{1}(0) terms and the *K*_{2}(0) and *D*_{2}(0) terms to *I _{θ}*(0).

#### 3) Map of *I*_{η}(0)

_{η}

The error in assuming that *η* is conservative *I _{η}*(0) is illustrated in Fig. 6. An approximate conversion factor of 14 J kg

^{−1}K

^{−2}has been used to express entropy in the same units as Θ and

*θ*. We have done this so that the nonconservative source terms of entropy can be expressed in units of W m

^{−2}. The mean value of

*I*(0) is 379 mW m

_{η}^{−2}and the maximum value is 7290 mW m

^{−2}, which are approximately 1191 times the mean and 528 times the maximum values in

*I*

_{Θ}(0) respectively. The spatial patterns in

*I*(0) are similar to those in the production of Θ, including that the majority of the contribution to

_{η}*I*(0) occurs in the upper 500 m of the ocean, along the intensified boundary currents and in the ACC. The

_{η}*K*

_{1}(0) and

*D*

_{1}(0) terms are the depth-integrated

*K*

_{1}and

*D*

_{1}terms as defined in Eq. (48) and are the dominant terms contributing to

*I*(0). Furthermore, the

_{η}*K*

_{1}(0) and

*D*

_{1}(0). The contribution of

*K*

_{3}(0) and

*D*

_{3}(0) to the overall production of

*η*is small.

*K*

_{2}(0) and

*D*

_{2}(0) alternatively attain positive and negative values due to the fact that

**∇**

_{z}

*S*

_{A}·

**∇**

_{z}Θ can be negative in the upper ocean, as

*η*

_{Θ}and

#### 4) Map of *I*_{ɛ}(0)

_{ɛ}

Figure 7 shows the magnitude of *I _{ɛ}*(0), the dissipation of kinetic energy into heat, integrated to the sea surface. The mean value of

*I*(0) is 2.99 mW m

_{ɛ}^{−2}, and the maximum is 18.5 mW m

^{−2}, which occurs in the Arctic Ocean.

*I*(0) shows a strong dependence on latitude, with low values between 50° latitude and 70° latitude and below approximately −50° latitude, and high values around 30° latitude and −30° latitude. These patterns are predominantly due to the square of the buoyancy frequency

_{ɛ}*N*

^{2}, although the diapycnal diffusivity

*D*dominates the spatial patterns in the equatorial region.

#### 5) Comparison of the maps of *I*_{Θ}(0), *I*_{θ}(0), *I*_{η}(0), and *I*_{ɛ}(0)

_{θ}

_{η}

_{ɛ}

From these maps it can be concluded that *I*_{Θ}(0) is the smallest of the variables Θ, *θ*, and *η*. The mean values of *I*_{Θ}(0), *I _{θ}*(0),

*I*(0), and

_{η}*I*(0) can be compared to the mean geothermal heat flux, which is approximately 86.4 mW

_{ɛ}^{−2}(Emile-Geay and Madec 2009). So, the error in assuming that Θ is conservative is of the order of 270 times smaller than the mean geothermal heat flux. By contrast, the nonconservative production of

*θ*is approximately 9 times smaller than the geothermal heat flux, and the nonconservative production of

*η*is approximately 4 times larger than the mean geothermal heat flux. Ignoring the turbulent kinetic energy dissipation rate in ocean models introduces an error approximately 29 times smaller than the error in ignoring the geothermal heat flux.

#### 6) Frequency distributions

Another way of expressing the nonconservation of Θ, *θ*, and *η* and the magnitude of *ɛ* is to plot the frequency distributions of *I*_{Θ}(0), *I _{θ}*(0),

*I*(0), and

_{η}*I*(0). For convenience of comparison, the log

_{ɛ}_{10}of each of these variables has been calculated and its frequency plotted in Fig. 8. Here, 95% of the values of

*I*

_{Θ}(0) occur within the range 0 to 0.892 mW m

^{−2}. By comparison, 95% of the values of

*I*(0) occur within the range 0 to 5.37 mW m

_{ɛ}^{−2}and 95% of the values of

*I*(0) occur within the range 0 to 1070 mW m

_{η}^{−2}.

*I*(0) is the only variable that is not positive definite, with a mean of −9.62 mW m

_{θ}^{−2}and a standard deviation of 59.8 mW m

^{−2}so that approximately 95% of

*I*(0) occurs within the range −130 to 110 mW m

_{θ}^{−2}.

The approach in Tailleux (2010) used to estimate the magnitude of the nonconservative source terms of Θ and *θ* involved replacing the laminar fluxes of the evolution equations for Θ and *θ* (derived in appendix B along with the laminar evolution equation for *η*) with turbulent fluxes so that an extra term approximately ⅕ the magnitude of *ɛ* was subtracted from the respective nonconservative source terms on the right-hand side of Eqs. (39) and (44). Hence, this approach would change the mean value of the nonconservative source terms of Θ and *θ* by approximately 0.598 mW m^{−2}. This term is larger than the mean value of *I*_{Θ}(0), which is 0.318 mW m^{−2}. We conclude that the substitution of turbulent fluxes into the laminar evolution equation for Θ is inappropriate at leading order.

In summarizing the 95th percentiles for the depth integrals to the sea surface in Eqs. (50)–(53), it can be seen that Conservative Temperature is approximately 120 times more conservative than potential temperature and 1200 times more conservative than entropy. Of particular note is the fact that treating Conservative Temperature as being conservative in a model is of less concern (by a factor of 6) than neglecting the turbulent rate of dissipation of kinetic energy.

#### 7) Meridional heat fluxes

The meridional heat fluxes for Θ, *θ*, *η*, and *ɛ*, shown in Fig. 9, are calculated by taking the area integral of *θ*, and *η* are conservative variables. For the three sign definite variables *I*_{Θ}(0), *I _{η}*(0), and

*I*(0), the gradient of increase in the heat fluxes is always positive. The plots show a strong dependence on latitude, with distinct changes in the rate of increase at about −50° latitude and 50° latitude, corresponding to sharp changes in the magnitude of

_{ɛ}*I*

_{Θ}(0),

*I*(0),

_{θ}*I*(0), and

_{η}*I*(0). The meridional heat flux for

_{ɛ}*I*(0) is also a strong function of latitude. It is informative to compare the ratios of the maximum values of the meridional heat fluxes for

_{θ}*I*

_{Θ}(0),

*I*(0),

_{η}*I*(0), and

_{θ}*I*(0). Then it can be seen that the magnitude of

_{ɛ}*I*

_{Θ}(0) is approximately 9.8 times smaller than for

*I*(0), 17 times smaller than for

_{ɛ}*I*(0) and over 1200 times smaller than for

_{θ}*I*(0).

_{η}### c. The sign definiteness of the production of *Θ*, η, and ɛ

In this subsection, the sign definiteness of each of the variables is considered. Based on our choice of fixed reference pressure (0 dbar), the nonconservative source terms of *θ* can be positive or negative throughout the ocean so that ocean models can locally overestimate or underestimate *θ*. The nonconservative production of entropy is positive definite, which follows from the second law of thermodynamics, and the turbulent dissipation rate of kinetic energy is also positive definite. The fact that turbulent mixing in the ocean always produces entropy is proven in section A.16 of IOC et al. (2010).

*S*

_{A}–Θ–

*p*space in which TEOS-10 is defined. Note the following exact expressions for

*p*.

^{m}The sign definite nature of the nonconservative production terms has also been found for specific volume; in this case the production terms are negative definite. It is interesting to note that the nonconservative source terms of density, related to specific volume by *ρ* = 1/*υ*, are not necessarily sign definite. This can easily be seen from inspection of the second-order partial derivatives of *ρ* derived in terms of *υ*, as there is no requirement for *ρ* or its second-order partial derivatives to be sign definite, and in fact, the nonconservative source terms of *ρ* are not sign definite using the TEOS-10 equation of state. The conditions on the sign definiteness of specific volume are derived in appendix C.

## 5. Discussion and conclusions

A key finding of this paper is that the error in the air–sea heat flux incurred by ocean models in assuming that Conservative Temperature Θ is a conservative thermodynamic property is approximately 120 times smaller than the corresponding error for potential temperature. This factor of 120 applies to the rms of the depth-integrated source terms. Furthermore, the error in the air–sea heat flux incurred by ocean models in assuming Θ is conservative is approximately 6 times smaller than the error incurred by ocean models in ignoring the rate of dissipation of turbulent kinetic energy in the conservation equation for “heat.” Entropy is the least conservative of the three variables Θ, *θ*, and *η*, and the error in the air–sea heat flux in ignoring the source terms of entropy is approximately 1200 times greater than the corresponding error for Conservative Temperature. Figure 10 summarizes the ratios of the upper bound of the 95th percentiles of the depth-integrated source terms for *ɛ*, *θ*, *η* compared with that of Θ. Even at great depth in the ocean (for example 4000 dbar), Fig. 2 implies that Θ is more locally conservative than *θ* by approximately an order of magnitude. Of course, given that at depth the salinity and temperature gradients are small, the magnitude of the source terms is considerably smaller at depth than closer to the sea surface. This issue is discussed in further detail in IOC et al. (2010) in their appendix A.18. It is well known, from the second law of thermodynamics, that entropy is produced during irreversible mixing processes, hence the error in assuming that entropy is conservative is positive definite. By definition, the dissipation of kinetic energy is also positive definite. Conversely, the error in assuming that potential temperature is a conservative thermodynamic property is not sign definite; potential temperature is alternatively produced and consumed during turbulent mixing. From examination of the three inequalities in Eq. (54), we find that the source terms of Conservative Temperature are positive definite for our choice of its fixed reference pressure, 0 dbar. This is true for the full TEOS-10 Gibbs function as well as for the 48-term approximation of IOC et al. (2010) and is true over the full range of applicability of each equation of state. However, this does not imply that the nonconservative source terms of potential enthalpy are always positive definite; rather, for different choices of fixed reference pressure *p ^{m}*, potential enthalpy may be alternatively produced and consumed during turbulent mixing. Since Conservative Temperature is defined as being proportional to potential enthalpy referenced to

*p*= 0 dbar, then we can say that the nonconservative source terms of Conservative Temperature are positive definite.

Conservative temperature can be regarded as a conservative variable with a high degree of accuracy. Furthermore, because potential enthalpy possesses the “potential” property and because the air–sea heat flux is equal to the air–sea flux of potential enthalpy, then potential enthalpy is the most accurate variable to represent the “heat content” per unit mass of seawater. Warren (1999, 2006) argues that because enthalpy is unknown up to a linear function of salinity, it is only possible to talk of a flux of “heat” through an ocean section if the fluxes of mass and salt through that section are both zero. This argument is incorrect; it is perfectly valid to regard the flux of potential enthalpy across an ocean section as “heat flux” irrespective of whether there are nonzero fluxes of mass and salt across the section. This issue is discussed at greater length in McDougall (2003) and IOC et al. (2010). Some model components of coupled models, such as the atmosphere and rivers, carry enthalpy rather than potential enthalpy as heat content. In such cases, when heat is transferred from the atmosphere or from rivers to the ocean, if the ocean is not carrying potential enthalpy as heat content, then a bias in the air–sea heat flux is introduced. This bias is removed if potential enthalpy is regarded as “heat content” by all model components in coupled models.

If Eq. (C12) for the nonconservative production of *δθ* of appendix C were to be used to quantify the errors in oceanographic practice incurred by assuming that *θ* is a conservative variable, one might select property contrasts that were typical of a prominent oceanic front and decide that because *δθ* is small at this one front, then the issue can be ignored. But the observed properties in the ocean result from a large and indeterminate number of such prior mixing events and the nonconservative production of *θ* accumulates during each of these mixing events, often in a sign definite fashion. How can we possibly estimate the error that is made by treating potential temperature as a conservative variable during all of these unknowably many past individual mixing events? As explained in IOC et al. (2010), the difference between *θ* and Θ (as seen in Fig. 1), is the error caused by ignoring the nonconservative source terms in *θ* in all these past mixing events.

*θ*. The ratio of the mean gradients of Θ and

*θ*can be expressed as (see appendix A.14 IOC et al. 2010)

*α*

^{Θ}/

*β*

^{Θ}is a function of pressure. It can also be shown that

*θ*can be 1.2% less and 1.6% greater than that of Θ for typical

*S*

_{A}and Θ ranges in the oceans.

Tailleux (2010) estimated the nonconservative production of Θ and *θ* by replacing the laminar fluxes of the instantaneous evolution equations for Θ and *θ* with turbulent fluxes. Because of the pressure dependence of enthalpy in the laminar forms, this approach contributes an extra term to the turbulent evolution equations for Θ and *θ* approximately ⅕ of the size of *ɛ* as shown in appendix B. The extra term is relatively small in the *θ* equation but dominates the true nonconservation of Θ in the Θ evolution equation. The substitution of a turbulent diffusivity in place of a laminar one is a potentially dangerous thing to do, as we demonstrate in appendix B where we find that doing this does not guarantee that entropy is always produced under turbulent mixing processes. McDougall and Garrett (1992) found a corresponding difference between the behavior of molecular and turbulent fluxes, namely that the molecular flux divergences of heat and salt contribute to the divergence of the three-dimensional velocity vector, while the corresponding turbulent heat and salt flux divergences do not. We are thus comfortable with finding a different expression for the irreversible nonconservative production of Conservative Temperature than that of Tailleux (2010). The results of this paper rely heavily on the physical idea that for turbulent mixing to occur, the fluid parcels must be in contact with each other, sharing the same location in latitude, longitude, and pressure. This seems a robust concept and we are encouraged that our results show a striking similarity to the more standard results of mixing between two water parcels at fixed pressure [compare Eqs. (39) and (C9)]. However, this idea of mixing taking place at a series of constant pressures is indeed an assumption that we have no independent way of checking. For example, we have ignored the small variations of pressure that occur as the turbulent motion occurs all the way down to the molecular scale. However, it must be realized that these same small pressure variations occur in the standard thought experiment when two parcels are mixed in a laboratory situation, leading to Eq. (C9). So, while the idea of turbulent mixing occurring at a fixed pressure is indeed an assumption, it seems a convincing one, and we prefer it to simply replacing a molecular flux with a turbulent one; an approach that does not guarantee that the second law of thermodynamics is obeyed (see appendix B).

R. Tailleux (2012, personal communication) and Tailleux (2012) have made the point that, in a steady-state ocean, the volume integrated nonconservative production of Conservative Temperature is very nearly equal to minus the volume integrated thermodynamic work by expansion/contraction. Conservative Temperature has this special property because the flux of heat through the sea surface is exactly proportional to the flux of Conservative Temperature (whereas this proportionality is not true of potential temperature or of specific entropy). Here, we generalize the derivation of Tailleux to the case where there is a geothermal heat flux into the ocean (as well as there being heat fluxes both into and out of the ocean at the sea surface).

*u*has been written in the divergence form. Integrating this equation over the whole ocean volume, while assuming a steady state, and taking the mass flux of water across the air–sea interface to be zero, we find that

*F*

^{geo}is the geothermal heat flux into the ocean at the sea floor and

*F*

^{surf}is the air–sea heat flux exchanged with the atmosphere (defined positive when the heat flux is upwards, leaving the ocean, and hence cooling the ocean). This air–sea heat flux

*F*

^{surf}includes radiation as well as sensible and latent heat. Note that the material derivative

*Dυ*/

*Dt*in the integrand of Eq. (58) is the instantaneous material derivative; it is not the coarse-grained version that we usually use in oceanographic practice.

*F*

^{geo}at an individual point on the sea floor is

*t*−

*θ*)/(

*T*

_{0}+

*t*) is very small in the ocean (never much more than 10

^{−3}) so that the last term in Eq. (60) is unlikely to be important. The term in the geothermal heat flux cannot be so easily dismissed because the geothermal heat flux occurs at the sea floor where (

*t*−

*θ*)/(

*T*

_{0}+

*t*) is the largest (whereas the largest values of

*ε*occur shallower in the water column where (

*t*−

*θ*)/(

*T*

_{0}+

*t*) is very small indeed). We have shown in this paper that the irreversible production

*p*

_{0}+

*p*)

*Dυ*/

*Dt*, is always negative.

Since the nonconservative production of Conservative Temperature

The results of this paper highlight some of the deficiencies in present ocean modeling practice regarding the use of potential temperature, namely, that potential temperature *θ* is not a conservative property and the evolution equation for potential temperature that is currently carried in ocean models mistakenly neglects the nonconservative source terms. Nonconservative source terms also arise in the evolution equation of Conservative Temperature; however, they are significantly smaller in magnitude than the corresponding source terms of potential temperature. To minimize the errors incurred by ocean models in neglecting the nonconservative source terms, it is therefore recommended that the default temperature variable becomes Conservative Temperature Θ rather than potential temperature *θ* and the output temperature of the model be compared to observed Conservative Temperature data rather than to potential temperature data.

## Acknowledgments

Paul Barker is thanked for sharing several computer codes related to the TEOS-10 equation of state. We would like to thank Remi Tailleux and Steve Griffies for their insightful comments on the draft manuscript.

## APPENDIX A

### Alternative Derivation of Evolution Equations

In this paper we have assumed relations such as *S*_{A} and Θ are the appropriately averaged Absolute Salinity and Conservative Temperature (thickness-weighted mean values, averaged in density coordinates). Recall that *h ^{m}* is potential enthalpy referenced to pressure

*p*. However, evaluating enthalpy with the instantaneous values of

^{m}*S*

_{A}and Θ and then averaging enthalpy will give a different average enthalpy than evaluating enthalpy with the pre-averaged values of

*S*

_{A}and Θ. Here we explore whether this distinction is important in the context of this paper. As a preliminary observation, we note the encouraging similarity between the nonconservative source terms in the parcel-mixing approach of appendix C to those in the evolution equation approach in the bulk of the paper [i.e. compare Eqs. (C9) and (39)]. This observation suggests that any differences that arise from the different forms of averaging discussed in this appendix will be small.

*p*=

*p*, making the Boussinesq approximation and ignoring the molecular and radiative heat fluxes, the instantaneous first law of thermodynamics is [from Eq. (35)]

^{m}*D*/

*Dt*= ∂/∂

*t*+

**u**·

**∇**. Now, we expand the partial derivatives

The averaging in this appendix has been Eulerian averaging at fixed pressure. The same additional source terms, Eq. (A5), are obtained by thickness-weighted averaging in density coordinates. We have chosen to illustrate this effect using Eulerian averages simply because it requires less algebra in the Eulerian averaging case.

## APPENDIX B

### The Laminar Evolution Equations

Tailleux (2010) estimated the magnitudes of the nonconservative source terms of Conservative Temperature and potential temperature from the respective laminar evolution equations by taking down-gradient turbulent flux formulations for **F**^{Θ}, **F**^{θ}, and **F**^{S}. This appendix compares the laminar evolution equations for Θ and *θ* with their turbulent evolution equations, Eqs. (39) and (45), to check the validity of the approach in Tailleux (2010).

*θ*, and

*η*, namely, Eqs. (28), (23), and (18) can be rearranged to be expressed as

*ρDS*

_{A}/

*Dt*= −

**∇**·

**F**

^{S}, ignoring the nonconservative source of Absolute Salinity due to biogeochemistry

*θ*, and

*η*respectively. The first law of thermodynamics in the form Eq. (35) at

*p*=

*p*shows that the molecular diffusive flux of

^{m}**F**

^{R}+

**F**

^{Q}(the molecular viscosity also affects the evolution of

*h*through the

^{m}*ɛ*term). The functional form of

*h*is also given by

^{m}**F**

^{R}+

**F**

^{Q}gives

*p*and expresses how the molecular diffusion of Conservative Temperature is related to the molecular (and radiative) fluxes of heat and salt. Using the functional forms

*h*with respect to

*η*and

*S*

_{A}; namely,

*θ*and

*η*respectively are given by

These are the forms of the nonconservative source terms of Θ, *θ*, and *η*, corresponding to those derived recently by Tailleux (2010). Note that they are derived by rearranging the first law of thermodynamics using the fundamental thermodynamic relation; the second law of thermodynamics has not been invoked. The second law of thermodynamics states that entropy is always produced during mixing processes. This means that the sum of the last three terms in Eq. (B12) must always be positive. For molecular diffusion this is the case, but we will show at the end of this appendix that this is not the case if the fluxes in Eq. (B12) are simply taken to be turbulent fluxes down their respective gradients.

We now compare the laminar equations (B10) and (B11) with their turbulent equivalents, Eqs. (39) and (45) to check whether replacing the molecular fluxes of **F**^{Θ}, **F**^{θ}, and **F**^{S} with their corresponding turbulent fluxes is valid.

*θ*, or

*η*and

*S*

_{A}have no separate dependence on pressure. For the case of Θ, these extra terms are found by evaluating the pressure dependence of

*K*

**∇**

_{n}Θ has the exactly horizontal component −

*K*

**∇**

_{n}Θ and a vertical component because the neutral tangent plane has a slope

**∇**

_{n}

*z*with respect to the horizontal. In Cartesian coordinates then, the lateral turbulent flux of Θ along the neutral tangent plane, −

*K*

**∇**

_{n}Θ, is

**k**is the unit vector in the vertical direction, and the total flux of Θ due to both epineutral and vertical diffusion is

**∇**

_{n}·

**b**=

**∇**

_{z}·

**b**+

**b**

_{z}·

**∇**

_{n}

*z*, for any two-dimensional vector

**b**.

*S*

_{A}and

*p*, such that

*α*

^{Θ}is the thermal expansion coefficient and

*β*

^{Θ}is the saline contraction coefficient, so the extra terms in Eq. (B13) contribute to

*D*Θ/

*Dt*in Eq. (B10) the amount

*α*

^{Θ}

**∇**

_{n}Θ =

*β*

^{Θ}

**∇**

_{n}

*S*

_{A}, the epineutral terms in the above expression cancel. Invoking the hydrostatic relation, the vertical terms become

*N*

^{2}=

*g*(

*α*

^{Θ}Θ

_{z}−

*β*

^{Θ}

*S*

_{Az}), so in the turbulent case for Θ, the pressure dependence of

*D*Θ/

*Dt*equation (B10) the term

*θ*, it can be seen that the pressure dependence of

*Dθ*/

*Dt*equation (B11) the term

*ɛ*=

*DN*

^{2}/Γ. Substituting this into equations (B28) and (B29) contributes the following terms to Eqs. (B10) and (B11) for Θ and

*θ*, respectively,

*ɛ*,

*θ*a negative value about ⅕ of the size of the last term in equations (B10) and (B11). These terms in Eq. (B30), while properly being in the laminar equations (B10) and (B11), should not appear in the turbulent forms of those equations (when the laminar diffusive fluxes

**F**

^{Θ},

**F**

^{θ}, and

**F**

^{S}are (arbitrarily) replaced by turbulent fluxes of Θ,

*θ*, and

*S*

_{A}). We conclude that the substitution of turbulent fluxes into the laminar diffusive fluxes is inappropriate at leading order.

We return to a discussion of Eq. (B12) and ask whether the sum of the second and third terms on the right-hand side is always guaranteed to be positive if these fluxes are simply changed form being the molecular fluxes (which contain the Dufour, Soret, and baro-diffusion terms) to being turbulent fluxes directed down their respective gradients of the mean properties. Consider the case where the turbulent flux of entropy is zero [see Eq. (B9)], but there is a turbulent salt flux. Part of the third term on the right of Eq. (B12) is −*μ _{p}*

**F**

^{S}·

**∇**

*p*and this changes sign with the sign of the salt flux, so that the second law of thermodynamics is not guaranteed to be obeyed (note that the Gibbs relations show that

*υ*times the saline contraction coefficient). By contrast, it has been shown in section A.16 of IOC et al. (2010) that the turbulent mixing process as advocated in the present paper (namely the conservation of enthalpy when turbulent mixing occurs at constant pressure) does always produce entropy and hence does obey the second law of thermodynamics.

## APPENDIX C

### The Mixing of Two Seawater Parcels at Constant Pressure

*Dp*/

*Dt*= 0. Although there will be dissipation of kinetic energy due to viscosity, the contribution is small and so the term

*ρɛ*will be ignored in this discussion. The rate of increase of enthalpy due to the interior source term of Absolute Salinity caused by remineralization is also small, so it too will be ignored. The system is assumed to be closed, that is, there are no radiative or boundary heat fluxes (

**F**

^{R}=

**F**

^{Q}). Then, the volume integral of the right hand side of Eq. (C1) is equal to zero, implying that the volume integrated amount of

*ρh*is constant during mixing at fixed pressure. This is a very important result in oceanography. The fact that enthalpy is conserved during mixing at fixed pressure forms the basis for understanding the flux of heat and “heat content” in the oceans. However, it does not follow that enthalpy can adequately represent heat content because it is only conserved during mixing at fixed pressure and it does not have the “potential” property.

*δC*denotes the nonconservative production of scalar

*C*during mixing at fixed pressure.

#### a. The nonconservative production of *Θ*

*δ*Θ under mixing at fixed pressure is derived by considering enthalpy in the functional form

*p*=

*p*dbar denotes the fixed pressure at which the mixing occurs. Then

^{m}*S*

_{A}and Θ of the mixed parcel,

*S*

_{A}and Θ, respectively, noting that the enthalpy of the mixed parcel is given by

*p*=

*p*. It is clear from the above equation and from the definition of potential enthalpy, namely

^{m}*δ*Θ is positive definite if the following three inequalities are satisfied:

#### b. The nonconservative production of θ

*θ*is derived. First,

*S*

_{A}and

*θ*of the mixed fluid at fixed pressure

*p*, retaining terms to second order. Then, equations (C2)–(C4) and (C6) are invoked, yielding

^{m}*θ*and

*S*

_{A}in the ocean,

*δθ*is not sign definite.

#### c. The nonconservative production of η

*S*

_{A},

*h*, and

*p*, namely,

*S*

_{A}and

*h*of the mixed fluid at fixed pressure

*p*, retaining terms to second order. Equations (C2)–(C4) and (C7) are used to find

^{m}*η*be strictly produced during mixing. In their appendix A.16, IOC et al. (2010) derived three constraints on the functional form of entropy

*g*(

*S*

_{A},

*t*,

*p*), namely,

#### d. The nonconservative production of υ

*S*

_{A},

*h*, and

*p*, namely,

*S*

_{A}and

*h*of the mixed fluid at fixed pressure

*p*, retaining terms to second order. Equations (C2)–(C4) and (C8) are used to find

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