Conservative temperature

Conservative temperature ${\displaystyle (\Theta )}$ is a thermodynamic property of seawater. It is derived from the potential enthalpy and is recommended under the TEOS-10 standard (Thermodynamic Equation of Seawater - 2010) as a replacement for potential temperature as it more accurately represents the heat content in the ocean.

Conservative temperature was initially proposed by Trevor McDougall in 2003. The motivation was to find an oceanic variable representing the heat content that is conserved during both pressure changes and turbulent mixing. In-situ temperature ${\displaystyle T}$ is not sufficient for this purpose, as the compression of a water parcel with depth causes an increase of the temperature despite the absence of any external heating. Potential temperature ${\displaystyle \theta }$ can be used to combat this issue, as it is referenced to a specific pressure and so ignores these compressive effects. In fact, potential temperature is a conservative variable in the atmosphere for air parcels in dry adiabatic conditions, and has been used in ocean models for many years. However, turbulent mixing processes in the ocean destroy potential temperature, sometimes leading to large errors when it is assumed to be conservative.

By contrast, the enthalpy of the parcel is conserved during turbulent mixing. However, it suffers from a similar problem to the in-situ temperature in that it also has a strong pressure dependence. Instead, potential enthalpy is proposed to remove this pressure dependence. Conservative temperature is then proportional to the potential enthalpy.

The fundamental thermodynamic relation is given by:

$${\displaystyle dh-{\frac {1}{\rho }}dp=T\,d\sigma +\mu \,dS}$$

where ${\displaystyle h}$ is the specific enthalpy, ${\displaystyle p}$ is the pressure, ${\displaystyle \rho }$ is the density, ${\displaystyle T}$ is the temperature, ${\displaystyle \sigma }$ is the specific entropy, ${\displaystyle S}$ is the salinity and ${\displaystyle \mu }$ is the relative chemical potential of salt in seawater.

During a process that does not lead to the exchange of heat or salt, entropy and salinity can be assumed constant. Therefore, taking the partial derivative of this relation with respect to pressure yields:

$${\displaystyle \left({\partial h \over \partial p}\right)_{S,\,\sigma }={\frac {1}{\rho }}}$$