Virtual Temperature (Density Temperature of Moist Air)

Moist air is lighter than dry air, because water vapour is lighter than the dry air it displaces. At the same temperature and pressure, a humid parcel therefore weighs less than a dry one. The virtual temperature turns that fact into a temperature: it is the temperature dry air would need to have the same pressure and density as the moist sample. The dry-air equation of state, $p = \rho R_d T_v$, then works for moist air once $T_v$ replaces $T$. Because the vapour makes the air lighter, $T_v$ sits a little above $T$, and with no vapour at all the two are equal.

The terms are few. $T$ is the air temperature and $r$ is the mixing ratio of the vapour, the mass of vapour per mass of dry air. $\varepsilon = 0.622$ is the ratio of the gas constants of dry air and of water vapour. The full definition from the American Meteorological Society's Glossary of Meteorology is $T_v=T\,(1+r/\varepsilon)/(1+r)$, and this page uses it. A common shortcut, which the Glossary offers for typical observed values, is $T_v\approx T\,(1+0.61\,r)$. Because the formula multiplies $T$, the temperature must be absolute: 35 C is worked as 308.15 K. The ratio is a plain mass ratio, so 30 g/kg is 0.03.

The worked example is air at 35 C with a mixing ratio of 30 g/kg, the case in Roland Stull's Practical Meteorology. Stull gets 40.6 C from the 0.61 shortcut: 308.15 K times 1.0183 is 313.789 K, or 40.64 C. The full definition gives 40.45441 C, about 40.5, which is 0.18 K lower. That gap is the size of the approximation, and it is why the page's check against Stull allows a tolerance of 0.4 percent. Run backwards, the formula is self-consistent: a mixing ratio of 30 g/kg makes 308.15 K look like 313.60441 K, and a virtual temperature of 40.45441 C at 30 g/kg means an air temperature of 35 C.

The layer thickness from the hypsometric equation, and the density of the air from a pressure and a temperature, both need the virtual temperature and not the air temperature. One limit: liquid water and ice carried in the air make it heavier and lower the virtual temperature (the Glossary's loaded version divides by $1 + r_v + r_l$ instead). This page counts the vapour only.