# 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 of air component x (e.g. \(n_{O_{3}}\)) | \(\frac{molec}{m^3}\) | <species>_number_density {:}
   \(\nu_{x}\) | volume mixing ratio if 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 of total air | \(\frac{g}{mol}\) | molar_mass {:}
   \(M_{x}\) | molar mass of 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 of air component x (e.g. \(p_{O_{3}}\)) | \(Pa\) | <species>_partial_pressure {:}
   \(\nu_{x}\) | volume mixing ratio of 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 from volume mixing ratio dry air

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\nu_{x}\) | volume mixing ratio of air component x with regard to total air | \(ppv\) | <species>_volume_mixing_ratio {:}
   \(\nu_{dry\_air}\) | volume mixing ratio of dry air with regard to total air | \(ppv\) | dry_air_volume_mixing_ratio {:}
   \(\bar{\nu}_{x}\) | volume mixing ratio of 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.

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

5. 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 of air component x (e.g. \(n_{O_{3}}\)) | \(\frac{molec}{m^3}\) | <species>_number_density {:}
   \(\bar{\nu}_{x}\) | volume mixing ratio of 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.

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

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

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(M_{dry\_air}\) | molar mass of dry air | \(\frac{g}{mol}\) |
   \(M_{x}\) | molar mass of 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.

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

7. 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 of air component x (e.g. \(p_{O_{3}}\)) | \(Pa\) | <species>_partial_pressure {:}
   \(\bar{\nu}_{x}\) | volume mixing ratio of 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.

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

8. volume mixing ratio dry air from volume mixing ratio

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\nu_{x}\) | volume mixing ratio of air component x with regard to total air | \(ppv\) | <species>_volume_mixing_ratio {:}
   \(\nu_{dry\_air}\) | volume mixing ratio of dry air with regard to total air | \(ppv\) | dry_air_volume_mixing_ratio {:}
   \(\bar{\nu}_{x}\) | volume mixing ratio of 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.

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

9. dry air volume mixing ratio from H2O volume mixing ratio

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\nu_{H_{2}O}\) | volume mixing ratio of H2O with regard to total air | \(ppv\) | H2O_volume_mixing_ratio {:}
   \(\nu_{dry\_air}\) | volume mixing ratio of dry air with regard to total air | \(ppv\) | dry_air_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.

   \[\nu_{dry\_air} = 1 - \nu_{H_{2}O}\]

10. dry air volume mixing ratio from H2O volume mixing ratio dry air

    symbol | description | unit | variable name
    --- | --- | --- | ---
    \(\bar{\nu}_{H_{2}O}\) | volume mixing ratio of H2O with regard to dry air | \(ppv\) | H2O_volume_mixing_ratio_dry_air {:}
    \(\nu_{dry\_air}\) | volume mixing ratio of dry air with regard to total air | \(ppv\) | dry_air_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.

    \[\nu_{dry\_air} = \frac{1}{1 + \bar{\nu}_{H_{2}O}}\]

11. H2O volume mixing ratio from dry air volume mixing ratio

    symbol | description | unit | variable name
    --- | --- | --- | ---
    \(\nu_{H_{2}O}\) | volume mixing ratio of H2O with regard to total air | \(ppv\) | H2O_volume_mixing_ratio {:}
    \(\nu_{dry\_air}\) | volume mixing ratio of dry air with regard to total air | \(ppv\) | dry_air_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.

    \[\nu_{H_{2}O} = 1 - \nu_{dry\_air}\]