In atmospheric thermodynamics, the virtual temperature ( T v {\displaystyle T_{v}} ) of a moist air parcel is the temperature at which a theoretical dry air parcel would have a total pressure and density equal to the moist parcel of air. 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. It has been described to increase Earth's thermal emission by warming the tropical atmosphere.
Introduction
Description In atmospheric thermodynamic processes, it is often useful to assume air parcels behave approximately adiabatically, and approximately ideally. The specific gas constant for the standardized mass of one kilogram of a particular gas is variable, and described mathematically as
R x = R ∗ M x , {\displaystyle R_{x}={\frac {R^{*}}{M_{x}}},}
where R ∗ {\displaystyle R^{*}} is the molar gas constant, and M x {\displaystyle M_{x}} is the apparent molar mass of gas x {\displaystyle x} in kilograms per mole. The apparent molar mass of a theoretical moist parcel in Earth's atmosphere can be defined in components of water vapor and dry air as
M air = e p M v + p d p M d , {\displaystyle M_{\text{air}}={\frac {e}{p}}M_{v}+{\frac {p_{d}}{p}}M_{d},}
with e {\displaystyle e} being partial pressure of water, p d {\displaystyle p_{d}} dry air pressure, and M v {\displaystyle M_{v}} and M d {\displaystyle M_{d}} representing the molar masses of water vapor and dry air respectively. The total pressure p {\displaystyle p} is described by Dalton's law of partial pressures:
p = p d + e . {\displaystyle p=p_{d}+e.}
Purpose 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-air equation of state for moist air. Temperature has an inverse proportionality to density. Thus, analytically, a higher vapor pressure would yield a lower density, which should yield a higher virtual temperature in turn.
Derivation Consider a moist air parcel containing masses m d {\displaystyle m_{d}} and m v {\displaystyle m_{v}} of dry air and water vapor in a given volume V {\displaystyle V} . The density is given by
ρ = m d + m v V = ρ d + ρ v , {\displaystyle \rho ={\frac {m_{d}+m_{v}}{V}}=\rho _{d}+\rho _{v},}
where ρ d {\displaystyle \rho _{d}} and ρ v {\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
e = ρ v R v T {\displaystyle e=\rho _{v}R_{v}T} and p d = ρ d R d T . {\displaystyle p_{d}=\rho _{d}R_{d}T.}
Solving for the densities in each equation and combining with the law of partial pressures yields
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