Amalgamation property

In atmospheric thermodynamics, the virtual temperature $T_{v}$ 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.

Introduction

Description

In atmospheric thermodynamic processes, it is often useful to assume air parcels behave approximately adiabatic, and thus approximately ideally. The gas constant for the standardized mass of one kilogram of a particular gas is dynamic, and described mathematically as:

${R_{x}}=1000{\frac {R^{*}}{M_{x}}}\,,$

where $R^{*}$ is the universal gas constant and $M_{x}$ is the apparent molecular weight of gas $x$. The apparent molecular weight of a theoretical moist parcel in Earth's atmosphere can be defined in components of dry and moist air as:

${M_{air}}={\frac {e}{p}}M_{v}+{\frac {p_{d}}{p}}M_{d}\,,$

with $e$ water vapor pressure, $p_{d}$ dry air pressure, and $M_{v}$ and $M_{d}$ representing the molecular weight of water and dry air respectively. The total pressure $p$ is described by Dalton's Law of Partial Pressures:

${p}={p_{d}}+{e}\,.$

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 an air parcel containing masses $m_{d}$ and $m_{v}$ of water vapor in a given volume $V$. The density is given by:

${\rho }={\frac {m_{d}+m_{v}}{V}}=\rho _{d}+\rho _{v}\,,$

where $\rho _{d}$ and $\rho _{v}$ are the densities of 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:

${e}=\rho _{v}R_{v}T\,$ and ${p_{d}}=\rho _{d}R_{d}T\,.$

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

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

Then, solving for $p$ and using $\textstyle \epsilon ={\frac {R_{d}}{R_{v}}}={\frac {M_{v}}{M_{d}}}$ is approximately 0.622 in Earth's atmosphere:

${p}={\rho }R_{d}T_{v}\,,$

where the virtual temperature $T_{v}$ is:

${T_{v}}={\frac {T}{1-{\frac {e}{p}}(1-{\epsilon })}}\,.$

We now have a non-linear scalar for temperature dependent purely on the unitless value $\scriptstyle {\frac {e}{p}}\,,$ allowing for varying amounts of water vapor in an air parcel. This virtual temperature $T_{v}$ in units of Kelvin can be used seamlessly in any thermodynamic equation necessitating it.

Variations

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

${\frac {e}{p}}={\frac {w}{w+{\epsilon }}}\,,$

which allows:

${T_{v}}=T{\frac {w+\epsilon }{\epsilon (1+w)}}\,.$

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

${T_{v}}\approx T(1+0.61w)\,.$

An approximate conversion using $T$ in degrees Celsius and mixing ratio $w$ in g/kg is:

${T_{v}}\approx T+{\frac {w}{6}}\,.$

Uses

Virtual temperature is used in adjusting CAPE soundings for assessing available convective potential energy from Skew-T log-P diagrams. The errors associated with ignoring virtual temperature correction for smaller CAPE values can be quite significant. Thus, in the early stages of convective storm formation, a virtual temperature correction is significant in identifying the potential intensity in tropical cyclogenesis.

Further reading

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