mass mixing ratio derivations

1. mass mixing ratio from mass density

   symbol	description	unit	variable name
   \(\rho\)	mass density of total air	\(\frac{kg}{m^3}\)	density {:}
   \(\rho_{x}\)	mass density for air component x (e.g. \(\rho_{O_{3}}\))	\(\frac{kg}{m^3}\)	<species>_density {:}
   \(q_{x}\)	mass mixing ratio for air component x with regard to total air	\(\frac{kg}{kg}\)	<species>_mass_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.

   \[q_{x} = \frac{\rho_{x}}{\rho}\]

2. mass mixing ratio from volume mixing ratio

   symbol	description	unit	variable name
   \(M_{air}\)	molar mass for total air	\(\frac{g}{mol}\)	molar_mass {:}
   \(M_{x}\)	molar mass for air component x	\(\frac{g}{mol}\)
   \(q_{x}\)	mass mixing ratio of quantity x with regard to total air	\(\frac{kg}{kg}\)	<species>_mass_mixing_ratio {:}
   \(\nu_{x}\)	volume mixing ratio of quantity x with regard to total air	\(ppv\)	<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.

   \[q_{x} = \nu_{x}\frac{M_{x}}{M_{air}}\]

3. mass mixing ratio dry air from mass density

   symbol	description	unit	variable name
   \(\rho_{dry\_air}\)	mass density of dry air	\(\frac{kg}{m^3}\)	dry_air_density {:}
   \(\rho_{x}\)	mass density for air component x (e.g. \(\rho_{O_{3}}\))	\(\frac{kg}{m^3}\)	<species>_density {:}
   \(\bar{q}_{x}\)	mass mixing ratio for air component x with regard to dry air	\(\frac{kg}{kg}\)	<species>_mass_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.

   \[q_{x} = \frac{\rho_{x}}{\rho_{dry\_air}}\]

4. mass mixing ratio dry air from volume mixing ratio dry air

   symbol	description	unit	variable name
   \(M_{dry\_air}\)	molar mass for dry air	\(\frac{g}{mol}\)
   \(M_{x}\)	molar mass for air component x	\(\frac{g}{mol}\)
   \(\bar{q}_{x}\)	mass mixing ratio of quantity x with regard to dry air	\(\frac{kg}{kg}\)	<species>_mass_mixing_ratio_dry_air {:}
   \(\bar{\nu}_{x}\)	volume mixing ratio of quantity x with regard to dry air	\(ppv\)	<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.

   \[\bar{q}_{x} = \bar{\nu}_{x}\frac{M_{x}}{M_{dry\_air}}\]