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The potential temperature of a parcel of fluid at pressure

P

displaystyle P

is the temperature that the parcel would attain if adiabatically brought to a standard reference pressure

P

0

displaystyle P_ 0

, usually 1000 millibars. The potential temperature is denoted

θ

displaystyle theta

and, for a gas well-approximated as ideal, is given by

θ = T

(

P

0

P

)

R

/

c

p

,

displaystyle theta =Tleft( frac P_ 0 P right)^ R/c_ p ,

where

T

displaystyle T

is the current absolute temperature (in K) of the parcel,

R

displaystyle R

is the gas constant of air, and

c

p

displaystyle c_ p

is the specific heat capacity at a constant pressure.

R

/

c

p

= 0.286

displaystyle R/c_ p =0.286

for air (meteorology).

Contents

1 Contexts
2 Comments
3
**Potential temperature** perturbations
4 Derivation
5 Potential virtual temperature
6 Related quantities
7 See also
8 References
9 Bibliography
10 External links

Contexts[edit]
The concept of potential temperature applies to any stratified fluid.
It is most frequently used in the atmospheric sciences and
oceanography.[1] 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.
Comments[edit]
**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).
**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,[2]

∂ θ

∂ z

> 0

displaystyle frac partial theta partial z >0

and vertical motions are suppressed. If the potential temperature decreases with height,[2]

∂ θ

∂ z

< 0

displaystyle frac partial theta partial z <0

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.[2]
**Potential temperature** perturbations[edit]
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[edit]
The enthalpy form of the first law of thermodynamics can be written
as:

d h = T

d s + v

d p ,

displaystyle dh=T,ds+v,dp,

where

d h

displaystyle dh

denotes the enthalpy change,

T

displaystyle T

the temperature,

d s

displaystyle ds

the change in entropy,

v

displaystyle v

the specific volume, and

p

displaystyle p

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

d h = v

d p .

displaystyle dh=v,dp.

For approximately ideal gases, such as the dry air in the Earth's atmosphere, the equation of state,

p v = R T

displaystyle pv=RT

can be substituted into the 1st law yielding, after some rearrangement:

d p

p

=

c

p

R

d T

T

,

displaystyle frac dp p = frac c_ p R frac dT T ,

where the

d h =

c

p

d T

displaystyle dh=c_ p dT

was used and both terms were divided by the product

p v

displaystyle pv

Integrating yields:

(

p

1

p

0

)

R

/

c

p

=

T

1

T

0

,

displaystyle left( frac p_ 1 p_ 0 right)^ R/c_ p = frac T_ 1 T_ 0 ,

and solving for

T

0

displaystyle T_ 0

, the temperature a parcel would acquire if moved adiabatically to the pressure level

p

0

displaystyle p_ 0

, you get:

T

0

=

T

1

(

p

0

p

1

)

R

/

c

p

≡ θ .

displaystyle T_ 0 =T_ 1 left( frac p_ 0 p_ 1 right)^ R/c_ p equiv theta .

Potential virtual temperature[edit] The potential virtual temperature

θ

v

displaystyle theta _ v

, defined by

θ

v

= θ

(

1 + 0.61 r −

r

L

)

,

displaystyle theta _ v =theta left(1+0.61r-r_ L right),

is the theoretical potential temperature of the dry air which would have the same density as the humid air at a standard pression P0. It is used as a practical substitute for density in buoyancy calculations. In this definition

θ

displaystyle theta

is the potential temperature,

r

displaystyle r

is the mixing ratio of water vapor, and

r

L

displaystyle r_ L

is the mixing ratio of liquid water in the air.
Related quantities[edit]
The
**Brunt–Väisälä frequency** is a closely related quantity that
uses potential temperature and is used extensively in investigations
of atmospheric stability.
See also[edit]

Wet-bulb potential temperature Atmospheric thermodynamics Conservative temperature

References[edit]

^ Stewart, Robert H. (September 2008). "6.5: Density, Potential
Temperature, and Neutral Density". Introduction To Physical
**Oceanography**

Oceanography (pdf). Academia. pp. 83–88. Retrieved March 8,
2017.
^ a b c Dr. James T. Moore (Saint Louis University Dept. of Earth
& Atmospheric Sciences) (August 5, 1999). "
**Isentropic**

Isentropic Analysis
Techniques: Basic Concepts" (pdf). COMET COMAP. Retrieved March 8,
2017.

Bibliography[edit]

M K Yau and R.R. Rogers, Short Course in
**Cloud**

Cloud Physics, Third Edition,
published by Butterworth-Heinemann, January 1, 1989, 304 pages. EAN
9780750632157 ISBN 0-7506-3215-1

External links[edit]

Eric Weisstein's World of Physics at Wolfram Research

v t e

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