# 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 of air component x (e.g. \(\rho_{O_{3}}\)) | \(\frac{kg}{m^3}\) | <species>_density {:}
   \(q_{x}\) | mass mixing ratio of 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 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.

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

3. mass mixing ratio from mass mixing ratio dry air

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(q_{x}\) | mass mixing ratio of air component x with regard to total air | \(\frac{kg}{kg}\) | <species>_mass_mixing_ratio {:}
   \(q_{dry\_air}\) | mass mixing ratio of dry air with regard to total air | \(\frac{kg}{kg}\) | dry_air_mass_mixing_ratio {:}
   \(\bar{q}_{x}\) | mass mixing ratio of 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.

   \[q_{x} = \bar{q}_{x}q_{dry\_air}\]

4. 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 of air component x (e.g. \(\rho_{O_{3}}\)) | \(\frac{kg}{m^3}\) | <species>_density {:}
   \(\bar{q}_{x}\) | mass mixing ratio if 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.

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

5. mass mixing ratio dry air from volume 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.

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

6. mass mixing ratio dry air from mass mixing ratio

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(q_{x}\) | mass mixing ratio of air component x with regard to total air | \(\frac{kg}{kg}\) | <species>_mass_mixing_ratio {:}
   \(q_{dry\_air}\) | mass mixing ratio of dry air with regard to total air | \(\frac{kg}{kg}\) | dry_air_mass_mixing_ratio {:}
   \(\bar{q}_{x}\) | mass mixing ratio of 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.

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

7. dry air mass mixing ratio from H2O mass mixing ratio

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(q_{H_{2}O}\) | mass mixing ratio of H2O with regard to total air | \(\frac{kg}{kg}\) | H2O_mass_mixing_ratio {:}
   \(q_{dry\_air}\) | mass mixing ratio of dry air with regard to total air | \(\frac{kg}{kg}\) | dry_air_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.

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

8. dry air mass mixing ratio from H2O mass mixing ratio

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\bar{q}_{H_{2}O}\) | mass mixing ratio of H2O with regard to dry air | \(\frac{kg}{kg}\) | H2O_mass_mixing_ratio_dry_air {:}
   \(q_{dry\_air}\) | mass mixing ratio of dry air with regard to total air | \(\frac{kg}{kg}\) | dry_air_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.

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

1. H2O mass mixing ratio from dry air mass mixing ratio

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(q_{H_{2}O}\) | mass mixing ratio of H2O with regard to total air | \(\frac{kg}{kg}\) | H2O_mass_mixing_ratio {:}
   \(q_{dry\_air}\) | mass mixing ratio of dry air with regard to total air | \(\frac{kg}{kg}\) | dry_air_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.

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