Virtual Temperature Calculator

Virtual temperature is a bookkeeping device that lets meteorologists keep using the dry-air ideal gas law, p = rho * Rd * T with Rd = 287.05 J/kg/K, on air that actually contains water vapor. A water molecule has a molar mass of about 18.02 g/mol against roughly 28.96 g/mol for the dry-air mixture, so replacing nitrogen and oxygen with vapor makes a parcel lighter at fixed temperature and pressure. Rather than carry a moisture-dependent gas constant, the moisture effect is folded into a slightly inflated temperature: Tv = T * (1 + 0.61 * w), where T is in Kelvin and w is the mixing ratio in kg of vapor per kg of dry air. The coefficient 0.61 is the approximation of (Md/Mw) - 1, since 28.96/18.02 - 1 equals about 0.608. The correction is small but never negligible where it matters. A saturated tropical boundary layer at 30 degrees C carries w near 27 g/kg and picks up roughly 5 K of virtual correction; a mid-latitude winter airmass at -5 degrees C with 2 g/kg gains only about 0.3 K. By 500 hPa the correction is usually under 1 K, and in the stratosphere it vanishes. The main caveat is that the linear 0.61 form is a first-order expansion that degrades above about 40 g/kg, and it accounts only for vapor. Inside cloud, suspended liquid and ice load the parcel and push density the other way, which requires the density temperature Tv * (1 - wL) instead. Sensor error also propagates: radiosonde humidity uncertainty of a few percent RH is usually the dominant term in Tv error, not the temperature bead.

The virtual temperature concept grew out of nineteenth-century work on the thermodynamics of moist air. Once Rudolf Clausius and others had put the gas laws on firm footing and John Dalton had established that a mixture of gases exerts the sum of its partial pressures, it became clear that humid air could not simply be treated with the dry-air gas constant. Guldberg and Mohn in Norway and later Wilhelm von Bezold and Julius von Hann in the German-language meteorological tradition of the 1880s and 1890s worked through the consequences of vapor content for density and buoyancy, and the practice of substituting an equivalent dry-air temperature emerged as the simplest way to avoid recomputing the gas constant for every parcel. The device became indispensable once upper-air sounding networks appeared. Radiosondes, developed in the early 1930s by Pavel Molchanov in the Soviet Union and refined for operational use in the United States and Europe over that decade, delivered routine profiles of temperature, humidity, and pressure, and the hypsometric equation used to convert those profiles into geopotential heights requires mean virtual temperature rather than mean temperature. When numerical weather prediction became operational after the 1950 Charney, Fjortoft, and von Neumann ENIAC integration, virtual temperature was built into the thermodynamic cores of the models. It remains standard in convective diagnostics: the virtual temperature correction to parcel buoyancy, emphasized in work by Doswell and Rasmussen in the mid-1990s, measurably changes computed CAPE in warm moist environments.