Virtual temperature

The virtual temperature (symbol:) is a temperature measure that is used in theoretical meteorology and in numerical weather models . It is the temperature that dry air would have to have in order to have the same density as moist air at a lower real temperature and the same pressure . The virtual temperature of an air parcel is not a real observable variable and is always higher than its actually measurable temperature. ${\displaystyle T_{v}}$

Motivation and Background

Preview

Since water vapor has a molar mass of only around 18.01528 g / mol , while dry air has around 28.9644 g / mol, water vapor is around 0.622 times as heavy as dry air with the same number of particles . However, this also means that moist air is lighter than dry air with the same number of particles, since the individual gases are composed independently of one another to form a gas mixture according to Dalton's law . The water vapor capacity of the air is greater in warm air than in cold air. The behavior of ideal gases is described with their equation of state by the dependent variables pressure , volume , amount of substance and temperature . In the lowest part of the earth's atmosphere , the so-called troposphere , the temperature decreases with a latitude-dependent atmospheric temperature gradient up to an altitude of 8 to 15 km. This gradient depends on the water vapor content of the air, because the more water vapor there is in the air, the more the heat radiation from earth is compensated by the condensation energy of the water (the energy that is released when a gas condenses ). If no water vapor existed in the atmosphere, the temperature with the comparatively high dry adiabatic temperature gradient would decrease with altitude and would also be identical to the virtual temperature. In reality, on the other hand, water vapor is almost always present, which is why one speaks of a slightly lower moisture adiabatic temperature gradient . The molar proportion of water vapor varies, depending on the weather and climate, between almost 0 and up to around 4 percent (this must not be confused with the relative humidity ). The lower the amount of water vapor in the air, the smaller the difference between the two gradients and the closer the real and virtual temperatures are to one another.

Virtual temperature

The condition assumed for the virtual temperature is the same density of the real humid air and the fictitious dry air without any water vapor . The dry air can only have the same density as lighter moist air if it is heated or lowered along the temperature gradient, which is equivalent to a decrease in altitude. If one imagines a dry air package as a thought experiment and slowly lowers it, there is a height at which the density of the dry air would be equal to the density of the moist air. The temperature calculated using the temperature gradient from this altitude is called the virtual temperature. From this it also follows that moist air behaves in the same way as dry air of the virtual temperature and one can use the standard formulas set up for dry air via the detour of their calculation without having to take the real air humidity into account. In this way you can reduce the equations of meteorology by one state variable and thereby simplify them noticeably.

Generalized virtual temperature

In the troposphere and the lower stratosphere, the composition of the air, apart from the water vapor content, is almost constant. At an altitude of more than 80 kilometers, the gas mixture begins to separate, and through photodissociation as a result of high- energy solar radiation, diatomic gases such as oxygen and nitrogen are partially transformed into an atomic state. In addition, there is the increased ozone content in the middle and upper stratosphere. This leads to a change in the molecular mass of the air (averaged over all components). This can be represented in the same way as the change in the water vapor content by means of a generalized virtual temperature , sometimes also referred to as T _M.

calculation

Cloudless conditions (water vapor only):

${\displaystyle T_{\mathrm {v} }=T\cdot \left(1+\left(\textstyle {\frac {1}{\varepsilon }}-1\right)\cdot s\right)=T\cdot \left(1+0{,}6078\cdot s\right)\qquad \left(1.1.\right)}$

${\displaystyle T_{\mathrm {v} }=T\cdot {\frac {1+{\frac {r_{\mathrm {v} }}{\varepsilon }}}{1+r_{\mathrm {v} }}}\approx T\cdot \left(1+0{,}6078\cdot r_{\mathrm {v} }\right)\qquad \left(1.2.\right)}$

${\displaystyle T_{\mathrm {v} }=T\cdot {\frac {1}{1-\left(1-\varepsilon \right)\cdot {\frac {e}{p}}}}\approx T\cdot \left(1+0{,}3780\cdot \textstyle {\frac {e}{p}}\right)\qquad \left(1.3.\right)}$

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

${\displaystyle T_{\mathrm {v} }=T\cdot {\frac {1+{\frac {r_{\mathrm {v} }}{\varepsilon }}}{1+r_{\mathrm {v} }+r_{\ell }}}\approx T\cdot \left(1+0{,}6078\cdot r_{\mathrm {v} }-r_{\ell }\right)\qquad \left(1.4.\right)}$

The following sizes are to be used:

• ${\displaystyle T}$: Temperature in K
• ${\displaystyle \varepsilon }$: Ratio of the specific gas constants of water vapor and dry air ${\displaystyle \left(M_{\mathrm {Wasserdampf} }\,/\,M_{\mathrm {trockene\;Luft} }\approx 0{,}6220\right)}$
• ${\displaystyle s}$: specific humidity in kg / kg ${\displaystyle \left(m_{\mathrm {Wasserdampf} }\,/\,m_{\mathrm {feuchte\;Luft} }\right)}$
• ${\displaystyle r_{\mathrm {v} }}$: Water vapor mixing ratio ${\displaystyle \left(m_{\mathrm {Wasserdampf} }\,/\,m_{\mathrm {trockene\;Luft} }\right)}$
• ${\displaystyle r_{\ell }}$: Liquid water mixing ratio ${\displaystyle \left(m_{\mathrm {Fl{\ddot {u}}ssigwasser} }\,/\,m_{\mathrm {trockene\;Luft} }\right)}$
• ${\displaystyle e}$: Vapor pressure in Pa
• ${\displaystyle p}$: Air pressure in Pa

Derivation

First, the equations for the densities of water vapor, dry and humid air are set up:

${\displaystyle \rho _{\mathrm {Wasserdampf} }={\frac {M_{\mathrm {Wasserdampf} }\cdot e}{R\cdot T}}\qquad \left(2.1.\right)}$

${\displaystyle \rho _{\mathrm {trockene\;Luft} }={\frac {M_{\mathrm {trockene\;Luft} }\cdot \left(p-e\right)}{R\cdot T}}\qquad \left(2.2.\right)}$

${\displaystyle \rho _{\mathrm {feuchte\;Luft} }=\rho _{\mathrm {Wasserdampf} }+\rho _{\mathrm {trockene\;Luft} }={\frac {M_{\mathrm {trockene\;Luft} }}{R\cdot T}}\left(p-\left(1-{\frac {M_{\mathrm {Wasserdampf} }}{M_{\mathrm {trockene\;Luft} }}}\right)e\right)\qquad \left(2.3.\right)}$

The virtual temperature is defined as the temperature that dry air would have to have in order for it to have the same density as humid air:

${\displaystyle \rho _{\mathrm {feuchte\;Luft} }={\frac {M_{\mathrm {trockene\;Luft} }\cdot p}{R\cdot T_{\mathrm {v} }}}\qquad \left(2.4.\right)}$

This can be done with equation 2.3. equated and resolved according to the virtual temperature : ${\displaystyle T_{\mathrm {v} }}$

${\displaystyle T_{\mathrm {v} }={\frac {T}{1-{\frac {e}{p}}\cdot \left(1-{\frac {M_{\mathrm {Wasserdampf} }}{M_{\mathrm {trockene\;Luft} }}}\right)}}\qquad \left(2.5.\right)}$

If one defines the relation and used with the following : ${\displaystyle \varepsilon =\textstyle {\frac {M_{\mathrm {Wasserdampf} }}{M_{\mathrm {trockene\;Luft} }}}}$ ${\displaystyle r_{\mathrm {v} }=\varepsilon \cdot \textstyle {\frac {e}{p-e}}}$ ${\displaystyle \textstyle {\frac {p}{e}}=1+\textstyle {\frac {\varepsilon }{r_{\mathrm {v} }}}}$

${\displaystyle T_{\mathrm {v} }=T\cdot {\frac {1+{\frac {\varepsilon }{r_{\mathrm {v} }}}}{\left(1+{\frac {\varepsilon }{r_{\mathrm {v} }}}\right)-\left(1-\varepsilon \right)}}\qquad \left(2.6.\right)}$

and from this equation 1.2 .; further follows equation 1.1. about the relationship . ${\displaystyle \textstyle {\frac {1}{r_{\mathrm {v} }}}=\textstyle {\frac {1}{s}}-1}$

Equation 1.3. results from equation 2.5. using for . ${\displaystyle 1/(1-q)\approx 1+q}$ ${\displaystyle q=e/P\ll 1}$

If you want to take into account the mass of liquid water and ice in the density of the moist air, you can z. B. Equation 2.1. modify as follows:

${\displaystyle \rho _{\mathrm {Wasser(-dampf)} }={\frac {m_{\mathrm {Wasserdampf} }+m_{\mathrm {Wasser} }}{V_{\mathrm {Wasserdampf} }}}={\frac {m_{\mathrm {Wasserdampf} }+m_{\mathrm {Wasser} }}{m_{\mathrm {Wasserdampf} }}}\cdot {\frac {M_{\mathrm {Wasserdampf} }\cdot e}{R\cdot T}}}$

${\displaystyle =\left(1+{\frac {r_{\ell }}{r_{\mathrm {v} }}}\right)\cdot {\frac {M_{\mathrm {Wasserdampf} }\cdot e}{R\cdot T}}}$

Analogous to the above derivation, the modified equation 2.6 results.

${\displaystyle T_{\mathrm {v} }=T\cdot {\frac {1+{\frac {\varepsilon }{r_{\mathrm {v} }}}}{\left(1+{\frac {\varepsilon }{r_{\mathrm {v} }}}\right)-\left(1-\varepsilon \cdot \left(1+{\frac {r_{\ell }}{r_{\mathrm {v} }}}\right)\right)}}}$

and further equation 1.4.

In addition to the above, the following sizes are used:

• R = 8.314472 J / ( mol · K ): universal gas constant
• M _{water vapor} = 18.01528 g / mol: Molar mass of pure water
• M _{dry air} = 28.9644 g / mol: Molar mass of dry air (value of the standard atmosphere )

From this it follows . ${\displaystyle \varepsilon =\textstyle {\frac {M_{\mathrm {Wasserdampf} }}{M_{\mathrm {trockeneLuft} }}}=0{,}62198}$

literature

• Michael Hantel: Introduction to Theoretical Meteorology . Springer, Berlin / Heidelberg 2013.

Web links

• http://www.top-wetter.de/calculator.htm : Web form for calculating the most important meteorological parameters