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 $R_d$. Given a(n effective) temperature $T$ (and consequent pressure $p$), it’s given as

$$T_v:=\frac{1}{T}\left(1-(1-ϵ)\frac{p_w}{p}\right)$$

Given the Ideal Gas Law for dry air¹

$$p_d={\rho }_d{R}_{d}T$$

and for water vapor:²

$$p_w={\rho }_w{R}_{w}T$$

as well as defining

$$ϵ:=\frac{R_d}{R_w}$$

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

$$\rho =\frac{m_w+m_d}{V}={\rho }_w+{\rho }_d$$

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

$$\rho =\frac{1}{T}\left(\frac{p_d}{R_d}+\underset{=\frac{p_w}{R_w}}{\underbrace{p_w\frac{ϵ}{R_d}}}\right)$$

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

$$p=p_d+p_w$$

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

$$\rho =\frac{p}{R_d}\underset{:=T_v}{\underbrace{\frac{1}{T}\left(1-(1-ϵ)\frac{p_w}{p}\right)}}$$

$T_v$ 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 $p\to p_w$, and thus

$$\begin{array}{rl}\rho & =\frac{p_w}{R_d}\frac{1}{T}(1-(1-ϵ))\\ & =\frac{p_w}{T}\frac{ϵ}{R_d}\\ & =\frac{p_w}{T}\frac{1}{R_w}\end{array}$$

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

If air is too dry, we have that $\frac{p_w}{p}\to 0$ and⁴ $p=p_w+p_d\to p_d$

$$\rho =\frac{p_d}{R_d}\frac{1}{T}$$

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. $R_d$ the specific air constant for dry air. ↩

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

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

4. Dalton’s partial pressures law. ↩