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{{Short description|Temperature that a fluid would attain if adiabatically brought to a standard reference pressure}}
The '''potential temperature''' of a [[Air parcel|parcel]] of fluid at pressure <math>P</math> is the temperature that the parcel would attain if [[Adiabatic process|adiabatically]] brought to a standard reference pressure <math>P_{0}</math>, usually {{convert|1000|hPa|mb|abbr=on}}. The potential temperature is denoted <math>\theta</math> and, for a gas well-approximated as [[ideal gas|ideal]], is given by

: <math> \theta = T \left(\frac{P_0}{P}\right)^{R/c_p}, </math>

where <math>T</math> is the current absolute [[temperature]] (in K) of the parcel, <math>R</math> is the [[specific gas constant]] of air, and <math>c_p</math> is the [[specific heat capacity]] at a constant pressure.
<math>R/c_p = 0.286</math> for air (meteorology). The reference point for potential temperature in the ocean is usually at the ocean's surface which has a water pressure of 0 dbar.<ref name="Talley">{{cite book |last1=Talley |first1=Lynne D. |title=Descriptive Physical Oceanography |date=2011 |publisher=Elsevier |location=Boston |pages=29–65 |isbn=9780750645522 |edition=Sixth |url=https://www.sciencedirect.com/book/9780750645522/descriptive-physical-oceanography |language=en}}</ref> The potential temperature in the ocean doesn't account for the varying heat capacities of seawater, therefore it is not a conservative measure of heat content.<ref name="Talley">{{cite book |last1=Talley |first1=Lynne D. |title=Descriptive Physical Oceanography |date=2011 |publisher=Elsevier |location=Boston |pages=29–65 |isbn=9780750645522 |edition=Sixth |url=https://www.sciencedirect.com/book/9780750645522/descriptive-physical-oceanography |language=en}}</ref> Graphical representation of potential temperature will always be less than the actual temperature line in a temperature vs depth graph.<ref name="Talley">{{cite book |last1=Talley |first1=Lynne D. |title=Descriptive Physical Oceanography |date=2011 |publisher=Elsevier |location=Boston |pages=29–65 |isbn=9780750645522 |edition=Sixth |url=https://www.sciencedirect.com/book/9780750645522/descriptive-physical-oceanography |language=en}}</ref>

== Contexts ==
The concept of potential temperature applies to any stratified fluid. It is most frequently used in the [[atmospheric sciences]] and [[oceanography]].<ref>{{cite book|url=https://www.academia.edu/download/35363446/book.pdf|title=Introduction To Physical Oceanography |first=Robert H.|last=Stewart|date=September 2008|publisher=Academia|chapter=6.5: Density, Potential Temperature, and Neutral Density|pages=83–88|access-date=March 8, 2017}}{{dead link|date=July 2022|bot=medic}}{{cbignore|bot=medic}}</ref> The reason that it is used
in both fields is that changes in pressure can result in warmer fluid residing under colder fluid – examples being dropping air temperature with altitude and increasing water temperature with depth in very deep ocean trenches and
within the ocean [[mixed layer]]. When the potential temperature is used instead, these apparently unstable conditions vanish as a parcel of fluid is invariant along its isolines. In the oceans, the potential temperature referenced to the surface will be slightly less than the in-situ temperature (the temperature that a water volume has at the specific depth that the instrument measured it in) since the expansion due to reduction in pressure leads to cooling.<ref name="Talley">{{cite book |last1=Talley |first1=Lynne D. |title=Descriptive Physical Oceanography |date=2011 |publisher=Elsevier |location=Boston |pages=29–65 |isbn=9780750645522 |edition=Sixth |url=https://www.sciencedirect.com/book/9780750645522/descriptive-physical-oceanography |language=en}}</ref> The numeric difference between the in situ and potential temperature is almost always less than 1.5 degrees Celsius. However, it's important to use potential temperature when comparing temperatures of water from very different depths.<ref name="Talley">{{cite book |last1=Talley |first1=Lynne D. |title=Descriptive Physical Oceanography |date=2011 |publisher=Elsevier |location=Boston |pages=29–65 |isbn=9780750645522 |edition=Sixth |url=https://www.sciencedirect.com/book/9780750645522/descriptive-physical-oceanography |language=en}}</ref>

== 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]] (which is often very close to being dry adiabatic).

[[File:Potential temperature and hydrostatic stability.jpg|thumb|Potential temperature and hydrostatic stability]]
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,<ref name=Moore/>

: <math>\frac{\partial \theta}{\partial z} > 0</math>

and vertical motions are suppressed. If the potential temperature decreases with height,<ref name=Moore/>

:<math>\frac{\partial \theta}{\partial z} < 0</math>

the atmosphere is unstable to vertical motions, and [[Atmospheric convection|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, streaklines, and pathlines|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.<ref name=Moore>{{cite web|url=http://rams.atmos.colostate.edu/at540/fall03/isen.pdf|title=Isentropic Analysis Techniques: Basic Concepts|author=Dr. James T. Moore (Saint Louis University Dept. of Earth & Atmospheric Sciences)|publisher= COMET COMAP |date=August 5, 1999|access-date=March 8, 2017}}</ref>

==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 [[enthalpy]] form of the first law of [[thermodynamics]] can be written as:

: <math> dh = T \, ds + v \, dp, </math>

where <math>dh</math> denotes the [[enthalpy]] change, <math>T</math> the temperature, <math>ds</math> the change in [[entropy]], <math>v</math> the specific volume, and <math>p</math> the pressure.

For adiabatic processes, the change in entropy is 0 and the 1st law simplifies to:

: <math> dh = v \, dp. </math>

For approximately ideal gases, such as the dry air in the Earth's atmosphere, the [[equation of state]], <math> pv = RT </math> can be substituted into the 1st law
yielding, after some rearrangement:

: <math> \frac{dp}{p} = {\frac{c_p}{R}\frac{dT}{T}}, </math>

where the <math> dh = c_{p}dT </math> was used and both terms were divided by the product <math> pv </math>

[[Integral|Integrating]] yields:

: <math> \left(\frac{p_1}{p_0}\right)^{R/c_p} = \frac{T_1}{T_0}, </math>

and solving for <math>T_{0}</math>, the temperature a parcel would acquire if moved adiabatically to the pressure level <math>p_{0}</math>, you get:

: <math> T_0 = T_1 \left(\frac{p_0}{p_1}\right)^{R/c_p} \equiv \theta. </math>

== Potential virtual temperature ==

The potential [[virtual temperature]] <math>\theta_{v}</math>, defined by

: <math> \theta_v = \theta \left( 1 + 0.61 r - r_L \right), </math>

is the theoretical potential temperature of the dry air which would have the same density as the humid air at a standard pressure P<sub>0</sub>. It is used as a practical substitute for density in buoyancy calculations. In this definition <math>\theta</math> is the potential temperature, <math>r</math> is the mixing ratio of water vapor, and <math>r_L</math> is the mixing ratio of liquid water in the air.

== Related quantities ==
The [[Brunt–Väisälä frequency]] is a closely related quantity that uses potential temperature and is used extensively in investigations of atmospheric stability.

== See also ==
* [[Wet-bulb potential temperature]]
* [[Atmospheric thermodynamics]]
* [[Conservative temperature]]
* [[Equivalent potential temperature]]

== References ==
{{reflist}}

== Bibliography ==
* M K Yau and R.R. Rogers, ''Short Course in Cloud Physics, Third Edition'', published by Butterworth-Heinemann, January 1, 1989, 304 pages. {{ISBN|9780750632157}} {{ISBN|0-7506-3215-1}}

== External links ==
* [http://scienceworld.wolfram.com/physics/PotentialTemperature.html Eric Weisstein's World of Physics] at Wolfram Research

{{Meteorological variables}}

[[Category:Atmospheric thermodynamics]]
[[Category:Meteorological quantities]]
[[Category:Physical oceanography]]

Potential temperature - Revision history

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