number density derivations

1. number density for air component from mass density:

   symbol	description	unit	variable name
   \(M_{x}\)	molar mass for air component x	\(\frac{g}{mol}\)
   \(n_{x}\)	number density for air component x (e.g. \(n_{O_{3}}\))	\(\frac{molec}{m^3}\)	<species>_number_density {:}
   \(N_A\)	Avogadro constant	\(\frac{1}{mol}\)
   \(\rho_{x}\)	mass density for air component x (e.g. \(\rho_{O_{3}}\))	\(\frac{kg}{m^3}\)	<species>_density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n_{x} = \frac{\rho_{x}N_{A}}{10^{-3}M_{x}}\]

2. number density for total air from mass density:

   symbol	description	unit	variable name
   \(M_{air}\)	molar mass for total air	\(\frac{g}{mol}\)	molar_mass {:}
   \(n\)	number density for total air	\(\frac{molec}{m^3}\)	number_density {:}
   \(N_A\)	Avogadro constant	\(\frac{1}{mol}\)
   \(\rho\)	mass density for total air	\(\frac{kg}{m^3}\)	density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n = \frac{\rho N_{A}}{10^{-3}M_{air}}\]

3. number density for total air from pressure and temperature

   symbol	description	unit	variable name
   \(k\)	Boltzmann constant	\(\frac{kg m^2}{K s^2}\)
   \(n\)	number density	\(\frac{molec}{m^3}\)	number_density {:}
   \(p\)	pressure	\(Pa\)	pressure {:}
   \(T\)	temperature	\(K\)	temperature {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n = \frac{p}{kT}\]

4. number density from volume mixing ratio

   symbol	description	unit	variable name
   \(n\)	number density of total air	\(\frac{molec}{m^3}\)	number_density {:}
   \(n_{x}\)	number density for air component x (e.g. \(n_{O_{3}}\))	\(\frac{molec}{m^3}\)	<species>_number_density {:}
   \(\nu_{x}\)	volume mixing ratio for air component x (e.g. \(n_{O_{3}}\))	\(ppv\)	<species>_volume_mixing_ratio {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n_{x} = \nu_{x}n\]

5. number density from volume mixing ratio dry air

   symbol	description	unit	variable name
   \(n_{dry\_air}\)	number density of dry air	\(\frac{molec}{m^3}\)	dry_air_number_density {:}
   \(n_{x}\)	number density for air component x (e.g. \(n_{O_{3}}\))	\(\frac{molec}{m^3}\)	<species>_number_density {:}
   \(\bar{\nu}_{x}\)	volume mixing ratio for air component x (e.g. \(n_{O_{3}}\))	\(ppv\)	<species>_volume_mixing_ratio_dry_air {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n_{x} = \bar{\nu}_{x}n_{dry\_air}\]

6. number density for air component from column number density

   symbol	description	unit	variable name
   \(c_{x}\)	column number density for air component x (e.g. \(c_{O_{3}}\))	\(\frac{molec}{m^2}\)	<species>_column_number_density {:}
   \(n_{x}\)	number density for air component x (e.g. \(n_{O_{3}}\))	\(\frac{molec}{m^3}\)	<species>_number_density {:}
   \(z^{B}(l)\)	altitude boundaries (\(l \in \{1,2\}\))	\(m\)	altitude_bounds {:,2}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n_{x} = \frac{c_{x}}{\lvert z^{B}(2) - z^{B}(1) \rvert}\]

7. number density for total air from dry air number density and H2O number density

   symbol	description	unit	variable name
   \(n\)	number density	\(\frac{molec}{m^3}\)	number_density {:}
   \(n_{dry\_air}\)	number density of dry air	\(\frac{molec}{m^3}\)	dry_air_number_density {:}
   \(n_{H_{2}O}\)	number density for H2O	\(\frac{molec}{m^3}\)	H2O_number_density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n = n_{dry\_air} + n_{H_{2}O}\]

8. number density for dry air from total air number density and H2O number density

   symbol	description	unit	variable name
   \(n\)	number density	\(\frac{molec}{m^3}\)	number_density {:}
   \(n_{dry\_air}\)	number density of dry air	\(\frac{molec}{m^3}\)	dry_air_number_density {:}
   \(n_{H_{2}O}\)	number density for H2O	\(\frac{molec}{m^3}\)	H2O_number_density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n_{dry\_air} = n - n_{H_{2}O}\]

9. number density for H2O from total air number density and dry air number density

   symbol	description	unit	variable name
   \(n\)	number density	\(\frac{molec}{m^3}\)	number_density {:}
   \(n_{dry\_air}\)	number density of dry air	\(\frac{molec}{m^3}\)	dry_air_number_density {:}
   \(n_{H_{2}O}\)	number density for H2O	\(\frac{molec}{m^3}\)	H2O_number_density {:}

   The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

   \[n_{H_{2}O} = n - n_{dry\_air}\]

10. number density for total air from column number density

    symbol	description	unit	variable name
    \(c\)	column number density	\(\frac{molec}{m^2}\)	column_number_density {:}
    \(n\)	number density	\(\frac{molec}{m^3}\)	number_density {:}
    \(z^{B}(l)\)	altitude boundaries (\(l \in \{1,2\}\))	\(m\)	altitude_bounds {:,2}

    The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

    \[n = \frac{c}{\lvert z^{B}(2) - z^{B}(1) \rvert}\]

11. number density for air component from partial pressure and temperature

    symbol	description	unit	variable name
    \(k\)	Boltzmann constant	\(\frac{kg m^2}{K s^2}\)
    \(n_{x}\)	number density for air component x (e.g. \(n_{O_{3}}\))	\(\frac{molec}{m^3}\)	<species>_number_density {:}
    \(p_{x}\)	partial pressure for air component x (e.g. \(p_{O_{3}}\))	\(Pa\)	<species>_partial_pressure {:}
    \(T\)	temperature	\(K\)	temperature {:}

    The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

    \[n_{x} = \frac{p_{x}}{kT}\]

12. surface number density for total air from surface pressure and surface temperature

    symbol	description	unit	variable name
    \(k\)	Boltzmann constant	\(\frac{kg m^2}{K s^2}\)
    \(n_{surf}\)	surface number density	\(\frac{molec}{m^3}\)	surface_number_density {:}
    \(p_{surf}\)	surface pressure	\(Pa\)	surface_pressure {:}
    \(T_{surf}\)	surface temperature	\(K\)	surface_temperature {:}

    The pattern : for the dimensions can represent {vertical}, {latitude,longitude}, {latitude,longitude,vertical}, {time}, {time,vertical}, {time,latitude,longitude}, {time,latitude,longitude,vertical}, or no dimensions at all.

    \[n_{surf} = \frac{p_{surf}}{kT_{surf}}\]