column mass mixing ratio derivations
====================================

   .. _derivation_column_mass_mixing_ratio_from_column_mass_density:

#. column mass mixing ratio from column mass density

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

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

   .. math::

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


   .. _derivation_column_mass_mixing_ratio_dry_air_from_column_mass_density:

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

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

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

   .. math::

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


   .. _derivation_column_mass_mixing_ratio_from_column_volume_mixing_ratio:

#. column mass mixing ratio from column volume mixing ratio

   =============== ======================================== ===================== ==========================================
   symbol          description                              unit                  variable name
   =============== ======================================== ===================== ==========================================
   :math:`M_{air}` molar mass of total air                  :math:`\frac{g}{mol}` `molar_mass {:}`
   :math:`M_{x}`   molar mass of air component x            :math:`\frac{g}{mol}`
   :math:`q_{x}`   column mass mixing ratio of quantity x   :math:`\frac{kg}{kg}` `<species>_column_mass_mixing_ratio {:}`
                   with regard to total air
   :math:`\nu_{x}` column volume mixing ratio of quantity x :math:`ppv`           `<species>_column_volume_mixing_ratio {:}`
                   with regard to total air
   =============== ======================================== ===================== ==========================================

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

   .. math::

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


   .. _derivation_column_mass_mixing_ratio_dry_air_from_column_volume_mixing_ratio_dry_air:

#. column mass mixing ratio dry air from column volume mixing ratio dry air

   ===================== ======================================== ===================== ==================================================
   symbol                description                              unit                  variable name
   ===================== ======================================== ===================== ==================================================
   :math:`M_{dry\_air}`  molar mass of dry air                    :math:`\frac{g}{mol}`
   :math:`M_{x}`         molar mass of air component x            :math:`\frac{g}{mol}`
   :math:`\bar{q}_{x}`   column mass mixing ratio of quantity x   :math:`\frac{kg}{kg}` `<species>_column_mass_mixing_ratio_dry_air {:}`
                         with regard to dry air
   :math:`\bar{\nu}_{x}` column volume mixing ratio of quantity x :math:`ppv`           `<species>_column_volume_mixing_ratio_dry_air {:}`
                         with regard to dry air
   ===================== ======================================== ===================== ==================================================

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

   .. math::

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


   .. _derivation_stratospheric_column_mass_mixing_ratio_dry_air_from_stratospheric_column_volume_mixing_ratio_dry_air:

#. stratospheric column mass mixing ratio dry air from stratospheric column volume mixing ratio dry air

   ===================== ======================================== ===================== ================================================================
   symbol                description                              unit                  variable name
   ===================== ======================================== ===================== ================================================================
   :math:`M_{dry\_air}`  molar mass of dry air                    :math:`\frac{g}{mol}`
   :math:`M_{x}`         molar mass of air component x            :math:`\frac{g}{mol}`
   :math:`\bar{q}_{x}`   stratospheric column mass mixing ratio   :math:`\frac{kg}{kg}` `stratospheric_<species>_column_mass_mixing_ratio_dry_air {:}`
                         of quantity x with regard to dry air
   :math:`\bar{\nu}_{x}` stratospheric column volume mixing ratio :math:`ppv`           `stratospheric_<species>_column_volume_mixing_ratio_dry_air {:}`
                         of quantity x with regard to dry air
   ===================== ======================================== ===================== ================================================================

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

   .. math::

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


   .. _derivation_tropospheric_column_mass_mixing_ratio_dry_air_from_tropospheric_column_volume_mixing_ratio_dry_air:

#. tropospheric column mass mixing ratio dry air from tropospheric column volume mixing ratio dry air

   ===================== ======================================= ===================== ===============================================================
   symbol                description                             unit                  variable name
   ===================== ======================================= ===================== ===============================================================
   :math:`M_{dry\_air}`  molar mass of dry air                   :math:`\frac{g}{mol}`
   :math:`M_{x}`         molar mass of air component x           :math:`\frac{g}{mol}`
   :math:`\bar{q}_{x}`   tropospheric column mass mixing ratio   :math:`\frac{kg}{kg}` `tropospheric_<species>_column_mass_mixing_ratio_dry_air {:}`
                         of quantity x with regard to dry air
   :math:`\bar{\nu}_{x}` tropospheric column volume mixing ratio :math:`ppv`           `tropospheric_<species>_column_volume_mixing_ratio_dry_air {:}`
                         of quantity x with regard to dry air
   ===================== ======================================= ===================== ===============================================================

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

   .. math::

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

