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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

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Meteorological data and variables

General

Adiabatic processes Advection Buoyancy Lapse rate Lightning Surface solar radiation Surface weather analysis Visibility Vorticity Wind Wind
Wind
shear

Condensation

Cloud Cloud
Cloud
condensation nuclei (CCN) Fog Convective condensation level (CCL) Lifted condensation level
Lifted condensation level
(LCL) Precipitation Water vapor

Convection

Convective available potential energy
Convective available potential energy
(CAPE) Convective inhibition
Convective inhibition
(CIN) Convective instability Convective momentum transport Convective temperature (Tc) Equilibrium level
Equilibrium level
(EL) Free convective layer
Free convective layer
(FCL) Helicity K Index Level of free convection
Level of free convection
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Lifted index
(LI) Maximum parcel level (MPL) Bulk Richardson number (BRN)

Temperature

Dew point
Dew point
(Td) Dew point
Dew point
depression Dry-bulb temperature Equivalent temperature (Te) Forest fire weather index Haines Index Heat index Humidex Humidity Relative humidity
Relative humidity
(RH) Mixing ratio Potential temperature (θ) Equivalent potential temperature
Equivalent potential temperature
(θe) Sea surface temperature
Sea surface temperature
(SST) Thermodynamic temperature Vapor pressure Virtual temperature Wet-bulb temperature Wet-bulb potential temperature Wind
Wind
chill

Pressure

Atmospheric pressure Baroclinity Barotropicity Pressure
Pressure
gradient Pressure-gr