Potential temperature
Encyclopedia
The potential temperature of a parcel
of fluid at pressure
is the temperature that the parcel would acquire if adiabatically
brought to a standard reference pressure
, usually 1000 millibars. The potential temperature is denoted 
and, for air, is often given by
where
is the current absolute temperature
(in K) of the parcel,
is the gas constant
of air, and
is the specific heat capacity at a constant pressure. This equation is often known as Poisson's equation.
and oceanography
. The reason that it is used
in both fluids is that changes in pressure result in warmer fluid residing under colder fluid- examples being the fact that air temperature drops as one climbs a mountain and water temperature can increase with depth in very deep ocean trenches and
within the ocean mixed layer
. When potential temperature is used instead, these apparently unstable conditions vanish.
Under almost all circumstances, potential temperature increases upwards in the atmosphere, unlike actual temperature which may increase or decrease. Potential temperature is conserved for all dry adiabatic processes, and as such is an important quantity in the planetary boundary layer
(which is often very close to being dry adiabatic).
Potential temperature is a useful measure of the static stability of the unsaturated atmosphere. Under normal, stably stratified conditions, the potential temperature increases with height,
and vertical motions are suppressed. If the potential temperature decreases with height,
the atmosphere is unstable to vertical motions, and convection
is likely. Since convection acts to quickly mix the atmosphere and return to a stably stratified state, observations of decreasing potential temperature with height are uncommon, except while vigorous convection is underway or during periods of strong insolation
. Situations in which the equivalent potential temperature
decreases with height, indicating instability in saturated air, are much more common.
Since potential temperature is conserved under adiabatic or isentropic air motions, in steady, adiabatic flow lines or surfaces of constant potential temperature act as streamlines or flow surfaces, respectively. This fact is used in isentropic analysis, a form of synoptic analysis which allows visualization of air motions and in particular analysis of large-scale vertical motion. http://www.comet.ucar.edu/class/faculty/Jun05_2000/docs/schneider/htm/index.html http://weather.cod.edu/analysis/analysis.isentropic.html
form of the first law of thermodynamics
can be written as:
where
denotes the enthalpy
change,
the temperature, 
the change in entropy
,
the specific volume, and 
the pressure.
For adiabatic processes, the change in entropy is 0 and the 1st law simplifies to:
For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state
,
can be substituted into the 1st law
yielding, after some rearrangement:
where the
was used and both terms were divided by the product 
Integrating
yields:
and solving for
, the temperature a parcel would acquire if moved adiabatically to the pressure level 
, you get:
Air parcel
In fluid dynamics, within the framework of continuum mechanics, a fluid parcel is a very small amount of fluid, identifiable throughout its dynamic history while moving with the fluid flow. As it moves, the mass of a fluid parcel remains constant, while—in a compressible flow—its volume may...
of fluid at pressure
is the temperature that the parcel would acquire if adiabaticallyAdiabatic process
In thermodynamics, an adiabatic process or an isocaloric process is a thermodynamic process in which the net heat transfer to or from the working fluid is zero. Such a process can occur if the container of the system has thermally-insulated walls or the process happens in an extremely short time,...
brought to a standard reference pressure
, usually 1000 millibars. The potential temperature is denoted and, for air, is often given by
where
is the current absolute temperatureTemperature
Temperature is a physical property of matter that quantitatively expresses the common notions of hot and cold. Objects of low temperature are cold, while various degrees of higher temperatures are referred to as warm or hot...
(in K) of the parcel,
is the gas constantGas constant
The gas constant is a physical constant which is featured in many fundamental equations in the physical sciences, such as the ideal gas law and the Nernst equation. It is equivalent to the Boltzmann constant, but expressed in units of energy The gas constant (also known as the molar, universal,...
of air, and
is the specific heat capacity at a constant pressure. This equation is often known as Poisson's equation.Contexts
The concept of potential temperature applies to any stratified fluid. It is most frequently used in the atmospheric sciencesAtmospheric sciences
Atmospheric sciences is an umbrella term for the study of the atmosphere, its processes, the effects other systems have on the atmosphere, and the effects of the atmosphere on these other systems. Meteorology includes atmospheric chemistry and atmospheric physics with a major focus on weather...
and oceanography
Oceanography
Oceanography , also called oceanology or marine science, is the branch of Earth science that studies the ocean...
. The reason that it is used
in both fluids is that changes in pressure result in warmer fluid residing under colder fluid- examples being the fact that air temperature drops as one climbs a mountain and water temperature can increase with depth in very deep ocean trenches and
within the ocean mixed layer
Mixed layer
The oceanic or limnological mixed layer is a layer in which active turbulence has homogenized some range of depths. The surface mixed layer is a layer where this turbulence is generated by winds, cooling, or processes such as evaporation or sea ice formation which result in an increase in salinity...
. When potential temperature is used instead, these apparently unstable conditions vanish.
Comments
Potential temperature is a more dynamically important quantity than the actual temperature. This is because it is not affected by the physical lifting or sinking associated with flow over obstacles or large-scale atmospheric turbulence. A parcel of air moving over a small mountain will expand and cool as it ascends the slope, then compress and warm as it descends on the other side- but the potential temperature will not change in the absence of heating, cooling, evaporation, or condensation (processes that exclude these effects are referred to as dry adiabatic). Since parcels with the same potential temperature can be exchanged without work or heating being required, lines of constant potential temperature are natural flow pathways.Under almost all circumstances, potential temperature increases upwards in the atmosphere, unlike actual temperature which may increase or decrease. Potential temperature is conserved for all dry adiabatic processes, and as such is an important quantity in the planetary boundary layer
Planetary boundary layer
The planetary boundary layer , also known as the atmospheric boundary layer , is the lowest part of the atmosphere and its behavior is directly influenced by its contact with a planetary surface. On Earth it usually responds to changes in surface forcing in an hour or less...
(which is often very close to being dry adiabatic).
Potential temperature is a useful measure of the static stability of the unsaturated atmosphere. Under normal, stably stratified conditions, the potential temperature increases with height,
and vertical motions are suppressed. If the potential temperature decreases with height,
the atmosphere is unstable to vertical motions, and convection
Atmospheric convection
Atmospheric convection is the result of a parcel-environment instability, or temperature difference, layer in the atmosphere. Different lapse rates within dry and moist air lead to instability. Mixing of air during the day which expands the height of the planetary boundary layer leads to...
is likely. Since convection acts to quickly mix the atmosphere and return to a stably stratified state, observations of decreasing potential temperature with height are uncommon, except while vigorous convection is underway or during periods of strong insolation
Insolation
Insolation is a measure of solar radiation energy received on a given surface area in a given time. It is commonly expressed as average irradiance in watts per square meter or kilowatt-hours per square meter per day...
. Situations in which the equivalent potential temperature
Equivalent potential temperature
Equivalent potential temperature, commonly referred to as theta-e \left, is a quantity related to the stability of a column of air in the atmosphere....
decreases with height, indicating instability in saturated air, are much more common.
Since potential temperature is conserved under adiabatic or isentropic air motions, in steady, adiabatic flow lines or surfaces of constant potential temperature act as streamlines or flow surfaces, respectively. This fact is used in isentropic analysis, a form of synoptic analysis which allows visualization of air motions and in particular analysis of large-scale vertical motion. http://www.comet.ucar.edu/class/faculty/Jun05_2000/docs/schneider/htm/index.html http://weather.cod.edu/analysis/analysis.isentropic.html
Potential temperature perturbations
The atmospheric boundary layer (ABL) potential temperature perturbation is defined as the difference between the potential temperature of the ABL and the potential temperature of the free atmosphere above the ABL. This value is called the potential temperature deficit in the case of a katabatic flow, because the surface will always be colder than the free atmosphere and the PT perturbation will be negative.Derivation
The enthalpyEnthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
form of the first law of thermodynamics
Thermodynamics
Thermodynamics is a physical science that studies the effects on material bodies, and on radiation in regions of space, of transfer of heat and of work done on or by the bodies or radiation...
can be written as:
where
denotes the enthalpyEnthalpy
Enthalpy is a measure of the total energy of a thermodynamic system. It includes the internal energy, which is the energy required to create a system, and the amount of energy required to make room for it by displacing its environment and establishing its volume and pressure.Enthalpy is a...
change,
the temperature, the change in entropyEntropy
Entropy is a thermodynamic property that can be used to determine the energy available for useful work in a thermodynamic process, such as in energy conversion devices, engines, or machines. Such devices can only be driven by convertible energy, and have a theoretical maximum efficiency when...
,
the specific volume, and the pressure.For adiabatic processes, the change in entropy is 0 and the 1st law simplifies to:
For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state
Equation of state
In physics and thermodynamics, an equation of state is a relation between state variables. More specifically, an equation of state is a thermodynamic equation describing the state of matter under a given set of physical conditions...
,
can be substituted into the 1st lawyielding, after some rearrangement:
where the
was used and both terms were divided by the product Integrating
Integral
Integration is an important concept in mathematics and, together with its inverse, differentiation, is one of the two main operations in calculus...
yields:
and solving for
, the temperature a parcel would acquire if moved adiabatically to the pressure level , you get:
Related quantities
The Brunt–Väisälä frequency is a closely related quantity that uses potential temperature and is used extensively in investigations of atmospheric stability.External links
- Eric Weisstein's World of Physics at Wolfram Research
See also
- Wet-bulb potential temperatureWet-bulb potential temperatureWet-bulb potential temperature, sometimes referred to as pseudo wet-bulb potential temperature, is the temperature attained by mass of air brought adiabatically to saturation and then carried along moist-adiabat to 1000 mb ....
- Atmospheric thermodynamicsAtmospheric thermodynamicsAtmospheric thermodynamics is the study of heat to work transformations in the earth’s atmospheric system in relation to weather or climate...