Potential temperature

The state variable potential temperature  θ is a fictitious temperature measure. In meteorology and oceanography, it is used to make the temperature of air or water of different heights or depths comparable with one another, i.e. H. different pressures to be considered. The potential temperature is a measure of the sum of internal energy  (local temperature) and potential energy  (height / depth) and was introduced in 1888 by Wilhelm von Bezold .

context

If one climbs up in the earth's atmosphere , one observes a drop in temperature. A pressure drop can also be observed. This phenomenon initially appears to be in contradiction with the experience that warm air rises and cold air sinks, but can be explained with the help of the potential temperature.

calculation

The vertical movements of gas or liquid parcels represent adiabatic changes of state with a good approximation . If these closed parcels are moved adiabatically to a normal pressure p ₀  (1000  mbar ), the air or liquid takes the potential due to its compressibility and the work associated with it Temperature. In general, air particles in the atmosphere move in a first approximation on surfaces of the same potential temperature. These areas are called isentropes .

Dry potential temperature

Free of condensation and evaporation , the dry potential temperature does not change during adiabatic processes.

From the adiabatic equation

${\displaystyle {\frac {\mathrm {d} T}{T}}={\frac {R_{L}}{c_{p}}}\cdot {\frac {\mathrm {d} p}{p}}}$

followed by integration of  p ₀ to  p (wherein T (p ₀ ) = θ ) and solving for  θ the dry potential temperature : ${\displaystyle \theta }$

${\displaystyle \theta =T\cdot \left({\frac {p_{0}}{p}}\right)^{R_{L} \over c_{p}}}$

The individual symbols stand for the following quantities :

• c _p : specific heat capacity of air = 1005 J / (kg K) at constant pressure,
• R _L : specific gas constant for dry air = 287 J / (kgK),
• T - absolute temperature ,
• p - pressure .

Moisture potential temperature

If condensation and evaporation occur, the moisture potential temperature is introduced analogously, i.e. the temperature that an air parcel would assume at saturation if it were brought to a normal pressure p ₀ in a humid adiabatic manner : ${\displaystyle \theta _{SW}}$

${\displaystyle \theta _{SW}=T\cdot \left({\frac {p_{0}}{p}}\right)^{\beta \cdot R_{L} \over c_{p}}}$

With

• β - variable factor, which must be less than 1.

The static stability of the stratification can be seen from the gradient of  θ .

The same applies to the atmosphere in the ocean

${\displaystyle \theta \left(S,T,p,p_{0}\right)=T_{0}-\int _{p_{0}}^{p}\Gamma _{ad}\left(S,\Theta ,p\right)\,\mathrm {d} p}$

The individual symbols stand for the following quantities:

• S - salinity ,
• T ₀ - surface temperature at normal pressure,
• ${\displaystyle \Gamma _{ad}}$- adiabatic temperature gradient .

Relationships

The potential temperature can be directly related to the entropy : ${\displaystyle S}$

${\displaystyle C_{p}\cdot {\frac {\mathrm {d} \theta }{\theta }}=\mathrm {d} S}$

As a result, isentropes are not only isolines with the same entropy, but also with the same potential temperature.

Another relationship arises with the dry adiabatic temperature gradient Γ and the geometric temperature gradient ${\displaystyle \gamma ={\frac {\mathrm {d} T}{\mathrm {d} z}}:}$

${\displaystyle {\frac {\mathrm {d} \theta }{\mathrm {d} z}}={\frac {\theta }{T}}\cdot \left(\Gamma -\gamma \right)}$

With

• the vertical coordinate  z .

Apart from the case of a dry adiabatic atmosphere stratification (Γ = γ) the difference Γ - γ is always positive:

${\displaystyle \Gamma -\gamma >0}$,

the dry potential temperature increases with altitude:

${\displaystyle \Rightarrow {\frac {\mathrm {d} \theta }{\mathrm {d} z}}>0}$.

This is true even if  T remains constant , as is the case above the tropopause . ${\displaystyle \left(\gamma ={\frac {\mathrm {d} T}{\mathrm {d} z}}=0\right)}$

See also

• Pseudopotential temperature
• Virtual temperature
• Potential density

literature

• Wilhelm von Bezold: On the thermodynamics of the atmosphere. Second message. Potential temperature. Vertical temperature gradient. Compound convection. In: Session reports of the Royal Prussian Academy of Sciences in Berlin. Born in 1888, 1189–1206.
• Guide for training in the German weather service.