mass density derivations
mass density for air component from number 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.
\[\rho_{x} = \frac{10^{-3}n_{x}M_{x}}{N_{A}}\]mass density for total air from number 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.
\[\rho = \frac{10^{-3}n M_{air}}{N_{A}}\]mass density for air component from column mass density:
symbol description unit variable name \(z^{B}(l)\) altitude boundaries (\(l \in \{1,2\}\)) \(m\) altitude_bounds {:,2} \(\rho_{x}\) mass density for air component x (e.g. \(\rho_{O_{3}}\)) \(\frac{kg}{m^3}\) <species>_density {:} \(\sigma_{x}\) column mass density for air component x (e.g. \(c_{O_{3}}\)) \(\frac{kg}{m^2}\) <species>_column_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.
\[\rho_{x} = \frac{\sigma_{x}}{\lvert z^{B}(2) - z^{B}(1) \rvert}\]mass density for total air from dry air mass density and H2O mass density
symbol description unit variable name \(\rho\) mass density \(\frac{kg}{m^3}\) density {:} \(\rho_{dry\_air}\) mass density of dry air \(\frac{kg}{m^3}\) dry_air_density {:} \(\rho_{H_{2}O}\) mass density for H2O \(\frac{kg}{m^3}\) H2O_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.
\[\rho = \rho_{dry\_air} + \rho_{H_{2}O}\]mass density for dry air from total air mass density and H2O mass density
symbol description unit variable name \(\rho\) mass density \(\frac{kg}{m^3}\) density {:} \(\rho_{dry\_air}\) mass density of dry air \(\frac{kg}{m^3}\) dry_air_density {:} \(\rho_{H_{2}O}\) mass density for H2O \(\frac{kg}{m^3}\) H2O_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.
\[\rho_{dry\_air} = \rho - \rho_{H_{2}O}\]mass density for H2O from total air mass density and dry air mass density
symbol description unit variable name \(\rho\) mass density \(\frac{kg}{m^3}\) density {:} \(\rho_{dry\_air}\) mass density of dry air \(\frac{kg}{m^3}\) dry_air_density {:} \(\rho_{H_{2}O}\) mass density for H2O \(\frac{kg}{m^3}\) H2O_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.
\[\rho_{H_{2}O} = \rho - \rho_{dry\_air}\]mass density for total air from column mass density:
symbol description unit variable name \(z^{B}(l)\) altitude boundaries (\(l \in \{1,2\}\)) \(m\) altitude_bounds {:,2} \(\rho\) mass density for total air \(\frac{kg}{m^3}\) density {:} \(\sigma\) column mass density for total air \(\frac{kg}{m^2}\) column_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.
\[\rho = \frac{\sigma}{\lvert z^{B}(2) - z^{B}(1) \rvert}\]