Conservative temperature
is a thermodynamic property of
seawater
Seawater, or salt water, is water from a sea or ocean. On average, seawater in the world's oceans has a salinity of about 3.5% (35 g/L, 35 ppt, 600 mM). This means that every kilogram (roughly one liter by volume) of seawater has appro ...
. It is derived from the potential
enthalpy
Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
and is recommended under the
TEOS-10
TEOS-10 (Thermodynamic Equation of Seawater - 2010) is the international standard for the use and calculation of the thermodynamic properties of seawater, humid air and ice. It supersedes the former standard EOS-80 (Equation of State of Seawater 1 ...
standard (Thermodynamic Equation of Seawater - 2010) as a replacement for
potential temperature
The potential temperature of a parcel of fluid at pressure P is the temperature that the parcel would attain if adiabatically brought to a standard reference pressure P_, usually . The potential temperature is denoted \theta and, for a gas well-ap ...
as it more accurately represents the
heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
content in the ocean.
Motivation
Conservative temperature was initially proposed by
Trevor McDougall
__NOTOC__
Trevor John McDougall FAGU is a physical oceanographer specialising in ocean mixing and the thermodynamics of seawater. He is Scientia Professor of Ocean Physics in the School of Mathematics and Statistics at the University of New S ...
in 2003. The motivation was to find an oceanic variable representing the
heat
In thermodynamics, heat is defined as the form of energy crossing the boundary of a thermodynamic system by virtue of a temperature difference across the boundary. A thermodynamic system does not ''contain'' heat. Nevertheless, the term is ...
content that is conserved during both pressure changes and turbulent mixing.
In-situ temperature
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
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
Enthalpy , a property of a thermodynamic system, is the sum of the system's internal energy and the product of its pressure and volume. It is a state function used in many measurements in chemical, biological, and physical systems at a constant ...
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.
Derivation
Potential enthalpy
The
fundamental thermodynamic relation
In thermodynamics, the fundamental thermodynamic relation are four fundamental equations which demonstrate how four important thermodynamic quantities depend on variables that can be controlled and measured experimentally. Thus, they are essentiall ...
is given by:
where
is the
specific enthalpy,
is the
pressure
Pressure (symbol: ''p'' or ''P'') is the force applied perpendicular to the surface of an object per unit area over which that force is distributed. Gauge pressure (also spelled ''gage'' pressure)The preferred spelling varies by country a ...
,
is the
density
Density (volumetric mass density or specific mass) is the substance's mass per unit of volume. The symbol most often used for density is ''ρ'' (the lower case Greek letter rho), although the Latin letter ''D'' can also be used. Mathematicall ...
,
is the
temperature
Temperature is a physical quantity that expresses quantitatively the perceptions of hotness and coldness. Temperature is measured with a thermometer.
Thermometers are calibrated in various temperature scales that historically have relied o ...
,
is the specific
entropy
Entropy is a scientific concept, as well as a measurable physical property, that is most commonly associated with a state of disorder, randomness, or uncertainty. The term and the concept are used in diverse fields, from classical thermodyna ...
,
is the
salinity
Salinity () is the saltiness or amount of salt (chemistry), salt dissolved in a body of water, called saline water (see also soil salinity). It is usually measured in g/L or g/kg (grams of salt per liter/kilogram of water; the latter is dimensio ...
and
is the relative
chemical potential
In thermodynamics, the chemical potential of a species is the energy that can be absorbed or released due to a change of the particle number of the given species, e.g. in a chemical reaction or phase transition. The chemical potential of a speci ...
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:
By integrating this equation, the potential enthalpy
is defined as the enthalpy at a reference pressure
:
Here the enthalpy and density are defined in terms of the three state variables: salinity, potential temperature and pressure.
Conversion to conservative temperature
Conservative temperature
is defined to be directly proportional to potential enthalpy. It is rescaled to have the same units (
Kelvin
The kelvin, symbol K, is the primary unit of temperature in the International System of Units (SI), used alongside its prefixed forms and the degree Celsius. It is named after the Belfast-born and University of Glasgow-based engineer and ...
) as the in-situ temperature:
where
= 3989.24495292815 J kg
−1K
−1 is a reference value of the
specific heat capacity
In thermodynamics, the specific heat capacity (symbol ) of a substance is the heat capacity of a sample of the substance divided by the mass of the sample, also sometimes referred to as massic heat capacity. Informally, it is the amount of heat t ...
, chosen to be as close as possible to the spatial average of the heat capacity over the entire ocean surface.
Conservative properties of potential enthalpy
Conservation form
The
first law of thermodynamics
The first law of thermodynamics is a formulation of the law of conservation of energy, adapted for thermodynamic processes. It distinguishes in principle two forms of energy transfer, heat and thermodynamic work for a system of a constant am ...
can be written in the form:
or equivalently:
where
denotes the
internal energy
The internal energy of a thermodynamic system is the total energy contained within it. It is the energy necessary to create or prepare the system in its given internal state, and includes the contributions of potential energy and internal kinet ...
,
represents the flux of heat and
is the rate of dissipation, which is small compared to the other terms and can therefore be neglected. The operator
is the
material derivative
In continuum mechanics, the material derivative describes the time rate of change of some physical quantity (like heat or momentum) of a material element that is subjected to a space-and-time-dependent macroscopic velocity field. The material der ...
with respect to the fluid flow
, and
is the
nabla Nabla may refer to any of the following:
* the nabla symbol ∇
** the vector differential operator, also called del, denoted by the nabla
* Nabla, tradename of a type of rail fastening system (of roughly triangular shape)
* ''Nabla'' (moth), a ge ...
operator.
In order to show that potential enthalpy is conservative in the ocean, it must be shown that the first law of thermodynamics can be rewritten in
conservation form Conservation form or ''Eulerian form'' refers to an arrangement of an equation or system of equations, usually representing a hyperbolic system, that emphasizes that a property represented is conserved, i.e. a type of continuity equation. The term i ...
. Taking the material derivative of the equation of potential enthalpy yields:
where
and
. It can be shown that the final two terms on the right-hand side of this equation are as small or even smaller than the dissipation rate discarded earlier
and the equation can therefore be approximated as:
Combining this with the first law of thermodynamics yields the equation:
which is in the desired conservation form.
Comparison to potential temperature
Given that conservative temperature was initially introduced to correct errors in the oceanic heat content, it is important to compare the relative errors made by assuming that conservative temperature is conserved to those originally made by assuming that potential temperature is conserved. These errors occur from non-conservation effects that are due to entirely different processes; for conservative temperature heat is lost due to
work done
In physics, work is the energy transferred to or from an object via the application of force along a displacement. In its simplest form, for a constant force aligned with the direction of motion, the work equals the product of the force stren ...
by compression, whereas for potential temperature this is due to surface fluxes of heat and freshwater.
It can be shown that these errors are approximately 120 times smaller for conservative temperature than for potential temperature, making it far more accurate as a representation of the conservation of heat in the ocean.
Usage
TEOS-10 framework
Conservative temperature is recommended under the TEOS-10 framework as the replacement for potential temperature in ocean models.
Other developments in TEOS-10 include:
* Replacement of practical salinity with the absolute salinity
as the primary salinity variable,
*Introduction of preformed salinity as a conservative variable under
biogeochemical
Biogeochemistry is the scientific discipline that involves the study of the chemical, physical, geological, and biological processes and reactions that govern the composition of the natural environment (including the biosphere, the cryosphere, th ...
processes,
* Defining all oceanic variables with respect to the
Gibbs function
In thermodynamics, the Gibbs free energy (or Gibbs energy; symbol G) is a thermodynamic potential that can be used to calculate the maximum amount of work that may be performed by a thermodynamically closed system at constant temperature and pre ...
.
Models
Conservative temperature has been implemented in several
ocean general circulation models such as those involved in the
Coupled Model Intercomparison Project
In climatology, the Coupled Model Intercomparison Project (CMIP) is a collaborative framework designed to improve knowledge of climate change. It was organized in 1995 by the Working Group on Coupled Modelling (WGCM) of the World Climate Resear ...
Phase 6 (CMIP6).
However, as these models have predominantly used potential temperature in previous generations, not all models have decided to switch to conservative temperature.
References
{{Reflist
Physical oceanography