Virtual temperature

The virtual temperature (symbol: ) is a temperature measurement, which finds application in theoretical meteorology and numerical weather models. It is the temperature which would have dry air in order to have the same density as wet air at a lower real temperature and pressure. The virtual temperature of an air parcel is no real observable size and always higher than its actual measurable temperature.

Motivation and Background

Preview

Since water vapor has a molar mass of only about 18.01528 g / mol, dry air contrast, about 28.9644 g / mol, water vapor is about 0,622 times as heavy as dry air of the same number of particles. However, this also means that moist air is lighter than dry air at the same particle as composed the different gases independently after Dalton's law to a gas mixture. The water vapor capacity of the air is here in warm air than in cold air. The behavior of ideal gases is described by the equation of state by the dependent variables of pressure, volume, amount of substance and temperature. In the lowest region of the atmosphere, called the troposphere, the temperature decreases with latitude -dependent atmospheric temperature from up to a height of 8 to 15 km. This gradient is the water vapor content of the air dependent, because the more water vapor in the air becomes, the more is the condensation energy of water (the energy that is released during condensation of a gas ) offset the heat radiation of the earth. Would not water vapor exists in the atmosphere, the temperature would drop to the comparatively high trockenadiabatischen temperature gradient with height and this would also be the same as the virtual temperature. In reality, however, is almost always water vapor present, which is why we also speak of a slightly lower feuchtadiabatischen temperature gradient. The molar proportion of water vapor varies, depending on weather and climate, between close to 0 and up to about 4 percent ( not to be confused with the relative humidity). The lower the water vapor content of the air is, the smaller the difference between the two gradients and the closer real and virtual temperature together.

Virtual temperature

The prerequisite for the virtual temperature condition is the same density of the real humid air and the notional dry air without any water vapor. The dry air can in this case only have the same density as moist air lighter when heated this or lowers along the temperature gradient, which corresponds to a reduction in height equals. When assembling as a thought experiment, a dry air parcel before and this lowers slowly, so there is a height or a temperature at which the density of dry air would be equal to the density of moist air. This height or above the temperature gradient converted this temperature is called the virtual temperature. It also follows that moist air behaves exactly like dry air of the virtual temperature and you can use via the detour of their calculation furnished to dry air standard formulas so, without having to consider these the real humidity. One can in this way thus reduce the equations of meteorology to a state variable and thus simplify it considerably.

Generalized virtual temperature

In the troposphere and the lower stratosphere is the composition of the air, except for the water vapor content, almost constant. In more than 80 kilometers above the gas mixture begins to segregate, and by photodissociation due to high-energy solar radiation diatomic gases such as oxygen and nitrogen are partially offset in the atomic state. In addition, the increased ozone content comes in the middle and upper stratosphere. This leads to a change in the ( averaged over all of the components ) Molecular weight of air. This can, occasionally referred to in the same way as the change in water vapor content by a well with TM, generalized virtual temperature are presented.

Calculation

Cloudless conditions ( water vapor):

Cloudy conditions ( in addition to water vapor, liquid water and ice are taken into account ):

To be used are the following sizes:

  • : Temperature in K
  • : Ratio of the specific gas constant of water vapor and dry air
  • : Specific humidity in kg / kg
  • : Water vapor mixing ratio
  • : Liquid water mixing ratio
  • : Vapor pressure in Pa
  • : Air pressure in Pa

Derivation

First, the equations for the density of the steam, the dry and humid air are prepared:

The virtual temperature is defined as the temperature which would have dry air, so that it has the same density as the hot air:

This can be with equation 2.3. be equated and solved for the virtual temperature:

If one defines and uses the relationship, follows with:

And from equation 1.2.; further follows equation 1.1. about the relationship.

Equation 1.3. is given by Equation 2.5. using for.

Will take into account in the density of the moist air and the mass of liquid water and ice, it is possible, for example, Equation 2.1. modified as follows:

Analogous to the above derivation is thus obtained as a modified equation 2.6.

And further equation 1.4.

In addition to the above the following variables are used:

  • R = 8.314472 J / (mol · K): Universal gas constant
  • MWasserdampf = 18.01528 g / mol: Molar mass of pure water
  • Mtrockene air = 28.9644 g / mol: Molar mass of dry air ( value of the standard atmosphere)

As a result.

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