volume mixing ratio derivations

1. volume mixing ratio from number density

   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 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.

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

2. volume mixing ratio from mass 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.

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

3. volume mixing ratio from partial pressure

   symbol	description	unit	variable name
   \(p\)	pressure	\(Pa\)	pressure {:}
   \(p_{x}\)	partial pressure for air component x (e.g. \(p_{O_{3}}\))	\(Pa\)	<species>_partial_pressure {:}
   \(\nu_{x}\)	volume mixing ratio for air component 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.

   \[\nu_{x} = \frac{p_{x}}{p}\]

4. volume mixing ratio dry air from number density

   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 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.

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

5. volume mixing ratio dry air from mass 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.

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

6. volume mixing ratio dry air from partial pressure

   symbol	description	unit	variable name
   \(p_{dry\_air}\)	partial pressure of dry air	\(Pa\)	dry_air_partial_pressure {:}
   \(p_{x}\)	partial pressure for air component x (e.g. \(p_{O_{3}}\))	\(Pa\)	<species>_partial_pressure {:}
   \(\bar{\nu}_{x}\)	volume mixing ratio for air component 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.

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