Virtual Temperature of Moist Air

up:: 034 MOC Atmospheric Physics

Given a parcel of moist air, its virtual temperature is the temperature that “would be” required for it to behave as an ideal gas, with dry air’s specific constant . Given a(n effective) temperature (and consequent pressure ), it’s given as

Proof

Given the Ideal Gas Law for dry air¹

and for water vapor:²

as well as defining

We have that the density of moist air is composed of its water vapor density and its “dry”/non-water-vapor density:

Substituting for the respective ideal gas laws, for a given temperature :

Due to Dalton’s partial pressure law, we have that

and, thus, isolating variables with dry air’s constant, we have

is called the virtual temperature of air, i.e. the temperature that air “should have” to behave itself as an ideal gas³.

Sanity check

If the air is too saturated with water vapor, we have that , and thus

and thus the water vapor’s gas law is recovered.

If air is too dry, we have that and⁴

i.e. dry air’s gas law is recovered.

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References

• WALLACE, John Michael; HOBBS, Peter Victor. Atmospheric science: an introductory survey. 2nd ed ed. Amsterdam Paris: Academic press, 2006.

Footnotes

1. the specific air constant for dry air. ↩

2. Usually is denoted as . I’m writing it as to make it more legible for me ( as in “water vapor”). I also avoided naming it (as in “water vapor”) to not mistake this subscript with virtual temperature’s . ↩

3. With dry air’s specific air constant . ↩

4. Dalton’s partial pressures law. ↩