Potential temperature

The potential temperature is the temperature of the gas , adiabatically reduced to standard pressure , usually 10 ⁵ Pa . Denoted by the Greek letter $\theta$ ${\ displaystyle \ theta}$ $\ theta$ and is expressed through the following equation, also called the Poisson equation :

\theta =T\left({\frac {p_{0}}{p}}\right)^{R/C_{p}},

{\ displaystyle \ theta = T \ left ({\ frac {p_ {0}} {p}} \ right) ^ {R / C_ {p}},}

{\ displaystyle \ theta = T \ left ({\ frac {p_ {0}} {p}} \ right) ^ {R / C_ {p}},}

Where $T$ ${\ displaystyle T}$ $T$ - temperature in Kelvin , $R$ ${\ displaystyle R}$ $R$ - gas constant and $C_{p}$ ${\ displaystyle C_ {p}}$ $C_p$ - specific heat in the isobaric process .

Derivation of the equation

We use the equation of the first law of thermodynamics :

dh=T\,ds+v\,dp,

{\ displaystyle dh = T \, ds + v \, dp,}

Where $d h$ ${\ displaystyle dh}$ denotes the change in enthalpy , $T$ ${\ displaystyle T}$ - absolute temperature $d s$ ${\ displaystyle ds}$ - entropy change, $v$ ${\ displaystyle v}$ - specific volume and $p$ ${\ displaystyle p}$ - pressure. For adiabatic processes, the change in entropy is zero, and the equation takes the form:

dh=v\,dp.

{\ displaystyle dh = v \, dp.}

Substitute in the above expression the equation of state of an ideal gas

pv=RT,

{\ displaystyle pv = RT,}

and also considering that

dh=C_{p}dT,

{\ displaystyle dh = C_ {p} dT,}

we get

{\frac {dp}{p}}={{\frac {C_{p}}{R}}{\frac {dT}{T}}}.

{\ displaystyle {\ frac {dp} {p}} = {{\ frac {C_ {p}} {R}} {\ frac {dT} {T}}}.}

After integration we have:

\left({\frac {p_{1}}{p_{0}}}\right)^{R/C_{p}}={\frac {T_{1}}{T_{0}}},

{\ displaystyle \ left ({\ frac {p_ {1}} {p_ {0}}} \ right) ^ {R / C_ {p}} = {\ frac {T_ {1}} {T_ {0}} },}

Expressing from here $T_{0}$ ${\ displaystyle T_ {0}}$ we get:

T_{0}=T_{1}\left({\frac {p_{0}}{p_{1}}}\right)^{R/C_{p}}\equiv \theta .

{\ displaystyle T_ {0} = T_ {1} \ left ({\ frac {p_ {0}} {p_ {1}}} \ right) ^ {R / C_ {p}} \ equiv \ theta.}

It is useful to take into account that

{R/C_{p}}={\frac {\kappa -1}{\kappa }}

{\ displaystyle {R / C_ {p}} = {\ frac {\ kappa -1} {\ kappa}}}

where

\kappa \equiv {\frac {C_{p}}{C_{v}}}

{\ displaystyle \ kappa \ equiv {\ frac {C_ {p}} {C_ {v}}}}

- adiabatic index .

The concept of potential temperature is used in meteorology . A similar concept, taking into account the equation of state of sea water, is in oceanology .

Literature

Potential temperature in the Meteorological Dictionary

Potential temperature

Derivation of the equation

See also

Literature

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