Inatmospheric thermodynamics, thevirtual temperature (${\displaystyle T_v}$) of a moistair parcel is thetemperature at which a theoretical dryair parcel would have a totalpressure anddensity equal to the moist parcel of air.^[1]The virtual temperature of unsaturated moist air is always greater than the absolute air temperature, however, as the existence of suspended cloud droplets reduces the virtual temperature.

The virtual temperature effect is also known as the vapor buoyancy effect.^[2] It has been described to increase Earth's thermal emission by warming the tropical atmosphere.^[3][4]

In atmosphericthermodynamic processes, it is often useful to assume air parcels behave approximatelyadiabatically, and approximatelyideally. Thespecific gas constant for the standardized mass of one kilogram of a particular gas is variable, and described mathematically as

${\displaystyle R_x=\frac{R^{\ast }}{M_x},}$

where${\displaystyle R^{\ast }}$ is the molar gas constant, and${\displaystyle M_x}$ is the apparentmolar mass of gas${\displaystyle x}$ in kilograms per mole. The apparent molar mass of a theoretical moist parcel inEarth's atmosphere can be defined in components ofwater vapor and dry air as

${\displaystyle M_{\text{air}}=\frac{e}{p}{M}_v+\frac{p_d}{p}{M}_d,}$

with${\displaystyle e}$ beingpartial pressure of water,${\displaystyle p_d}$ dryair pressure, and${\displaystyle M_v}$ and${\displaystyle M_d}$ representing the molar masses of water vapor and dry air respectively. The total pressure${\displaystyle p}$ is described byDalton's law of partial pressures:

${\displaystyle p=p_d+e.}$

Rather than carry out these calculations, it is convenient to scale another quantity within the ideal gas law to equate the pressure and density of a dry parcel to a moist parcel. The only variable quantity of the ideal gas law independent of density and pressure is temperature. This scaled quantity is known as virtual temperature, and it allows for the use of the dry-airequation of state for moist air.^[5] Temperature has an inverse proportionality to density. Thus, analytically, a highervapor pressure would yield a lower density, which should yield a higher virtual temperature in turn.^{[citation needed]}

Consider a moist air parcel containing masses${\displaystyle m_d}$ and${\displaystyle m_v}$ of dry air and water vapor in a given volume${\displaystyle V}$. The density is given by

${\displaystyle \rho =\frac{m_d+m_v}{V}={\rho }_d+{\rho }_v,}$

where${\displaystyle {\rho }_d}$ and${\displaystyle {\rho }_v}$ are the densities the dry air and water vapor would respectively have when occupying the volume of the air parcel. Rearranging the standard ideal gas equation with these variables gives

${\displaystyle e={\rho }_v{R}_{v}T}$ and${\displaystyle p_d={\rho }_d{R}_{d}T.}$

Solving for the densities in each equation and combining with the law of partial pressures yields

${\displaystyle \rho =\frac{p-e}{R_{d}T}+\frac{e}{R_{v}T}.}$

Then, solving for${\displaystyle p}$ and using${\displaystyle ϵ=\frac{R_d}{R_v}=\frac{M_v}{M_d}}$ is approximately 0.622 in Earth's atmosphere:

${\displaystyle p=\rho R_d{T}_v,}$

where the virtual temperature${\displaystyle T_v}$ is

${\displaystyle T_v=\frac{T}{1-\frac{e}{p}(1-ϵ)}.}$

We now have a non-linearscalar for temperature dependent purely on theunitless value${\displaystyle e/p}$, allowing for varying amounts of water vapor in an air parcel. This virtual temperature${\displaystyle T_v}$ in units ofkelvin can be used seamlessly in any thermodynamic equation necessitating it.

Often the more easily accessible atmospheric parameter is themixing ratio${\displaystyle w}$. Through expansion upon the definition of vapor pressure in the law of partial pressures as presented above and the definition of mixing ratio:

${\displaystyle \frac{e}{p}=\frac{w}{w+ϵ},}$

which allows

${\displaystyle T_v=T\frac{w+ϵ}{ϵ(1+w)}.}$

Algebraic expansion of that equation, ignoring higher orders of${\displaystyle w}$ due to its typical order in Earth's atmosphere of${\displaystyle {10}^{-3}}$, and substituting${\displaystyle ϵ}$ with its constant value yields the linear approximation

${\displaystyle T_v\approx T(1+0.608w).}$

With the mixing ratio${\displaystyle w}$ expressed in g/g.^[6]

An approximate conversion using${\displaystyle T}$ in degreesCelsius and mixing ratio${\displaystyle w}$ in g/kg is^[7]

${\displaystyle T_v\approx T+\frac{w}{6}.}$

Knowing that specific humidity${\displaystyle q}$ is given in terms of mixing ratio${\displaystyle w}$ as${\displaystyle q=\frac{w}{1+w}}$, then we can write mixing ratio in terms of the specific humidity as${\displaystyle w=\frac{q}{1-q}}$.We can now write the virtual temperature${\displaystyle T_v}$ in terms of specific humidity as${\displaystyle T_v=T\frac{\frac{q}{1-q}+ϵ}{ϵ(1+\frac{q}{1-q})}}$

Simplifying the above will reduce to

${\displaystyle T_v=T[\frac{q}{ϵ}+(1-q)]}$

and using the value of${\displaystyle ϵ=0.622}$, then we can write

${\displaystyle T_v=T(0.608q+1)}$

Virtual potential temperature is similar topotential temperature in that it removes the temperature variation caused by changes in pressure. Virtual potential temperature is useful as a surrogate for density in buoyancy calculations and in turbulence transport which includes vertical air movement.

A moist air parcel may also contain liquid droplets and ice crystals in addition to water vapor. A net mixing ratio${\displaystyle w_T}$ can be defined as the sum of the mixing ratios of water vapor${\displaystyle w}$, liquid${\displaystyle w_i}$, and ice${\displaystyle w_l}$ present in the parcel. Assuming that${\displaystyle w_i}$ and${\displaystyle w_l}$ are typically much smaller than${\displaystyle w}$, adensity temperature of a parcel${\displaystyle T_{\rho }}$ can be defined, representing the temperature at which a theoretical dry air parcel would have the a pressure and density equal to a moist parcel of air while accounting for condensates:^[8]: 113

${\displaystyle T_{\rho }=T\frac{1+w/ϵ}{1+w_T}}$

Virtual temperature is used in adjustingCAPE soundings for assessing available convectivepotential energy fromskew-T log-P diagrams. The errors associated with ignoring virtual temperature correction for smaller CAPE values can be quite significant.^[9] Thus, in the early stages of convective storm formation, a virtual temperature correction is significant in identifying thepotential intensity intropical cyclogenesis.^[10]

• Wallace, John M.; Hobbs, Peter V. (2006).Atmospheric Science.ISBN 0-12-732951-X.

• Atmospheric thermodynamics
• Meteorological quantities
• Atmospheric temperature
• Atmospheric pressure
• Humidity and hygrometry

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