# column mass mixing ratio derivations

1. column mass mixing ratio from column mass density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\sigma\) | column mass density of total air | \(\frac{kg}{m^3}\) | column_density {:}
   \(\sigma_{x}\) | column mass density of air component x (e.g. \(\sigma{O_{3}}\)) | \(\frac{kg}{m^3}\) | <species>_column_density {:}
   \(q_{x}\) | column mass mixing ratio of quantity x with regard to total air | \(\frac{kg}{kg}\) | <species>_column_mass_mixing_ratio {:}

   The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

   \[q_{x} = \frac{\sigma_{x}}{\sigma}\]

2. column mass mixing ratio dry air from column mass density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\sigma_{dry\_air}\) | column mass density of dry air | \(\frac{kg}{m^3}\) | dry_air_column_density {:}
   \(\sigma_{x}\) | column mass density of air component x (e.g. \(\sigma{O_{3}}\)) | \(\frac{kg}{m^3}\) | <species>_column_density {:}
   \(\bar{q}_{x}\) | column mass mixing ratio of quantity x with regard to dry air | \(\frac{kg}{kg}\) | <species>_column_mass_mixing_ratio_dry_air {:}

   The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

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

3. column mass mixing ratio from column 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 | \(\frac{g}{mol}\) |
   \(q_{x}\) | column mass mixing ratio of quantity x with regard to total air | \(\frac{kg}{kg}\) | <species>_column_mass_mixing_ratio {:}
   \(\nu_{x}\) | column volume mixing ratio of quantity x with regard to total air | \(ppv\) | <species>_column_volume_mixing_ratio {:}

   The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

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

4. column mass mixing ratio dry air from column 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 | \(\frac{g}{mol}\) |
   \(\bar{q}_{x}\) | column mass mixing ratio of quantity x with regard to dry air | \(\frac{kg}{kg}\) | <species>_column_mass_mixing_ratio_dry_air {:}
   \(\bar{\nu}_{x}\) | column volume mixing ratio of quantity x with regard to dry air | \(ppv\) | <species>_column_volume_mixing_ratio_dry_air {:}

   The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

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

5. stratospheric column mass mixing ratio dry air from stratospheric column 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 | \(\frac{g}{mol}\) |
   \(\bar{q}_{x}\) | stratospheric column mass mixing ratio of quantity x with regard to dry air | \(\frac{kg}{kg}\) | stratospheric_<species>_column_mass_mixing_ratio_dry_air {:}
   \(\bar{\nu}_{x}\) | stratospheric column volume mixing ratio of quantity x with regard to dry air | \(ppv\) | stratospheric_<species>_column_volume_mixing_ratio_dry_air {:}

   The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

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

6. tropospheric column mass mixing ratio dry air from tropospheric column 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 | \(\frac{g}{mol}\) |
   \(\bar{q}_{x}\) | tropospheric column mass mixing ratio of quantity x with regard to dry air | \(\frac{kg}{kg}\) | tropospheric_<species>_column_mass_mixing_ratio_dry_air {:}
   \(\bar{\nu}_{x}\) | tropospheric column volume mixing ratio of quantity x with regard to dry air | \(ppv\) | tropospheric_<species>_column_volume_mixing_ratio_dry_air {:}

   The pattern : for the dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

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