Virtual Temperature (Tv)

In the last section we defined the equation of state for a dry atmosphere. In the real atmosphere, the water content is highly variable, from 0-4% by volume. The proportionality constant (R) for dry air is 287 J kg-1K-1, while R for pure water vapor is 461 J kg-1K-1. But, the atmosphere is rarely ever dry and is never pure water vapor. Thus, the value for R in the real atmosphere is always somewhere between 287 and 461. As the moisture content of a sample of air changes, so does the proportionality constant for that sample. The value for R, though not diffucult to calculate, is not a joy to calculate either. For moist air, using the equation of state could become a bit cumbersome since you would need to compute a new R value each time the moisture content changed. Necessity is the mother of invention and for this reason virtual temperature (Tv) was invented. We have stated many times that less dense or lighter air rises, and we have generally referred to the less dense air as warmer air. To make matters more interesting, we will now inform you that a sample of moist air is less dense than a sample of dry air at the same volume, temperature, and pressure. The reason for this is water vapor has a smaller apparent molecular weight than dry air. In other words, one mole (6.02 x 1023 molecules) of water vapor weighs less than one mole of dry air. The definition of virtual temperature (Tv) is the temperature to which a sample of dry air must be heated in order to have the same density as a sample of moist air at the same pressure. The virtual temperature can be calculated using the equation

Tv = (1+0.61w)T

Virtual Temperature = (1 + (0.61 x Mixing Ratio)) x Temperature


where w is the mixing ratio in units of kg water vapor/kg dry air and T is in degrees Kelvin. (NOTE: the mixing ratio is usually recorded in g/kg and has to be converted to kg/kg when used in computations.) The mixing ratio can easily be obtained from meteorological charts using only the dew point temperature and pressure. Calculating Tv is much simpler than calculating R. The virtual temperature allows us to use the equation of state for dry air by substituting Tv for T. The equation of state for moist air then becomes

P = Rd Tv

Pressure x Specific Volume = Dry Gas Constant x Virtual Temperature

where Rd is the proportionality constant for dry air. The virtual temperature simplifies the complications that the presence of water vapor in the atmosphere creates. It allows us to convert a moist sample of air into a theoretical dry one. While we may be able to get by without using Tv, it makes life easier. Virtual temperature is a term that is used extensively in meteorology. We generally always use Tv instead of T when speaking of a moist atmospheric environment.

Calculating
Virtual
Temperature (Tv)

 		This link will provide you with a "pop-up" Javascript calculator that will enable you to compare the effects varying amounts of moisture in the air have on the virtual temperature. Because this calculator is written in Javascript and uses frames, it requires Netscape 2.02 (or a more recent version).


Once you have calculated the virtual temperature a few times using this on-line calculator, you might want to refer back to Computing With the Equation of State and substitute the temperature with your virtual temperature calculations.

On to Potential Temperature

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