mass density derivations

  1. mass density of air component from number density

    symbol

    description

    unit

    variable name

    \(M_{x}\)

    molar mass of air component x

    \(\frac{g}{mol}\)

    \(n_{x}\)

    number density of air component x (e.g. \(n_{O_{3}}\))

    \(\frac{molec}{m^3}\)

    <species>_number_density {:}

    \(N_A\)

    Avogadro constant

    \(\frac{1}{mol}\)

    \(\rho_{x}\)

    mass density of air component x (e.g. \(\rho_{O_{3}}\))

    \(\frac{kg}{m^3}\)

    <species>_density {:}

    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.

    \[\rho_{x} = \frac{10^{-3}n_{x}M_{x}}{N_{A}}\]
  2. mass density of total air from number density

    symbol

    description

    unit

    variable name

    \(M_{air}\)

    molar mass of total air

    \(\frac{g}{mol}\)

    molar_mass {:}

    \(n\)

    number density of total air

    \(\frac{molec}{m^3}\)

    number_density {:}

    \(N_A\)

    Avogadro constant

    \(\frac{1}{mol}\)

    \(\rho\)

    mass density of total air

    \(\frac{kg}{m^3}\)

    density {:}

    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.

    \[\rho = \frac{10^{-3}n M_{air}}{N_{A}}\]
  3. mass density of air component from column mass density

    symbol

    description

    unit

    variable name

    \(z^{B}(l)\)

    altitude boundaries (\(l \in \{1,2\}\))

    \(m\)

    altitude_bounds {:,2}

    \(\rho_{x}\)

    mass density of air component x (e.g. \(\rho_{O_{3}}\))

    \(\frac{kg}{m^3}\)

    <species>_density {:}

    \(\sigma_{x}\)

    column mass density of air component x (e.g. \(c_{O_{3}}\))

    \(\frac{kg}{m^2}\)

    <species>_column_density {:}

    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.

    \[\rho_{x} = \frac{\sigma_{x}}{\lvert z^{B}(2) - z^{B}(1) \rvert}\]
  4. mass density of total air from dry air mass density and H2O mass density

    symbol

    description

    unit

    variable name

    \(\rho\)

    mass density

    \(\frac{kg}{m^3}\)

    density {:}

    \(\rho_{dry\_air}\)

    mass density of dry air

    \(\frac{kg}{m^3}\)

    dry_air_density {:}

    \(\rho_{H_{2}O}\)

    mass density of H2O

    \(\frac{kg}{m^3}\)

    H2O_density {:}

    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.

    \[\rho = \rho_{dry\_air} + \rho_{H_{2}O}\]
  5. mass density of dry air from total air mass density and H2O mass density

    symbol

    description

    unit

    variable name

    \(\rho\)

    mass density

    \(\frac{kg}{m^3}\)

    density {:}

    \(\rho_{dry\_air}\)

    mass density of dry air

    \(\frac{kg}{m^3}\)

    dry_air_density {:}

    \(\rho_{H_{2}O}\)

    mass density of H2O

    \(\frac{kg}{m^3}\)

    H2O_density {:}

    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.

    \[\rho_{dry\_air} = \rho - \rho_{H_{2}O}\]
  6. mass density of H2O from total air mass density and dry air mass density

    symbol

    description

    unit

    variable name

    \(\rho\)

    mass density

    \(\frac{kg}{m^3}\)

    density {:}

    \(\rho_{dry\_air}\)

    mass density of dry air

    \(\frac{kg}{m^3}\)

    dry_air_density {:}

    \(\rho_{H_{2}O}\)

    mass density of H2O

    \(\frac{kg}{m^3}\)

    H2O_density {:}

    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.

    \[\rho_{H_{2}O} = \rho - \rho_{dry\_air}\]
  7. mass density of total air from column mass density

    symbol

    description

    unit

    variable name

    \(z^{B}(l)\)

    altitude boundaries (\(l \in \{1,2\}\))

    \(m\)

    altitude_bounds {:,2}

    \(\rho\)

    mass density of total air

    \(\frac{kg}{m^3}\)

    density {:}

    \(\sigma\)

    column mass density of total air

    \(\frac{kg}{m^2}\)

    column_density {:}

    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.

    \[\rho = \frac{\sigma}{\lvert z^{B}(2) - z^{B}(1) \rvert}\]