# column mass density derivations

1. column mass density of total air from dry air column mass density and H2O column mass density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\sigma\) | column mass density | \(\frac{kg}{m^2}\) | column_density {:}
   \(\sigma_{dry\_air}\) | column mass density of dry air | \(\frac{kg}{m^2}\) | dry_air_column_density {:}
   \(\sigma_{H_{2}O}\) | column mass density of H2O | \(\frac{kg}{m^2}\) | H2O_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.

   \[\sigma = \sigma_{dry\_air} + \sigma_{H_{2}O}\]

2. column mass density of dry air from total air column mass density and H2O column mass density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\sigma\) | column mass density | \(\frac{kg}{m^2}\) | column_density {:}
   \(\sigma_{dry\_air}\) | column mass density of dry air | \(\frac{kg}{m^2}\) | dry_air_column_density {:}
   \(\sigma_{H_{2}O}\) | column mass density of H2O | \(\frac{kg}{m^2}\) | H2O_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.

   \[\sigma_{dry\_air} = \sigma - \sigma_{H_{2}O}\]

3. column mass density of H2O from total air column mass density and dry air column mass density

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\sigma\) | column mass density | \(\frac{kg}{m^2}\) | column_density {:}
   \(\sigma_{dry\_air}\) | column mass density of dry air | \(\frac{kg}{m^2}\) | dry_air_column_density {:}
   \(\sigma_{H_{2}O}\) | column mass density of H2O | \(\frac{kg}{m^2}\) | H2O_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.

   \[\sigma_{H_{2}O} = \sigma - \sigma_{dry\_air}\]

4. column mass density of air component from 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.

   \[\sigma_{x} = \rho_{x} \lvert z^{B}(2) - z^{B}(1) \rvert\]

5. column mass density of total air from 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.

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

6. column mass density of air component from column number density

   This conversion applies to both total columns as well as partial column profiles.

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(c_{x}\) | column number density of air component x (e.g. \(c_{O_{3}}\)) | \(\frac{molec}{m^2}\) | <species>_column_number_density {:}
   \(M_{x}\) | molar mass of air component x | \(\frac{g}{mol}\) |
   \(N_A\) | Avogadro constant | \(\frac{1}{mol}\) |
   \(\sigma_{x}\) | column mass density of air component x (e.g. \(\sigma_{O_{3}}\)) | \(\frac{kg}{m^2}\) | <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.

   \[\sigma_{x} = \frac{10^{-3}c_{x}M_{x}}{N_{A}}\]

7. column mass density of total air from column number density

   This conversion applies to both total columns as well as partial column profiles.

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(c\) | column number density of total air | \(\frac{molec}{m^2}\) | column_number_density {:}
   \(M_{air}\) | molar mass of total air | \(\frac{g}{mol}\) | molar_mass {:}
   \(N_A\) | Avogadro constant | \(\frac{1}{mol}\) |
   \(\sigma\) | 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.

   \[\sigma = \frac{10^{-3}c M_{air}}{N_{A}}\]

8. column mass density of total air from pressure profile and surface pressure

   symbol | description | unit | variable name
   --- | --- | --- | ---
   \(\bar{g}\) | mean gravity of profile | \(\frac{m}{s^2}\) |
   \(g\) | nominal gravity at sea evel | \(\frac{m}{s^2}\) |
   \(g_{h}\) | gravity at specific heiht | \(\frac{m}{s^2}\) |
   \(p^{B}(i,l)\) | pressure boundaries (\(l \in \{1,2\}\)) | \(Pa\) | pressure_bounds {:,vertical,2}
   \(p_{surf}\) | surface pressure | \(Pa\) | surface_pressure {:}
   \(R\) | local earth curvature radius | \(m\) |
   \(z(i)\) | altitude | \(m\) | altitude {:,vertical}
   \(\phi\) | latitude | \(degN\) | latitude {:}
   \(\sigma\) | column mass density of total air | \(\frac{kg}{m^2}\) | column_density {:}

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

   \begin{eqnarray} g & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)} {\sqrt{1 - 0.00669437999013{\sin}^2(\frac{\pi}{180}\phi)}} \\ g_{h}(i) & = & g\left(\frac{R}{R + z(i)}\right)^2 \\ \bar{g} & = & \frac{\sum_{i}{p^{B}(i,0)-p^{B}(i,1)}}{\sum_{i}{\frac{p^{B}(i,0)-p^{B}(i,1)}{g_{h}(i)}}} \\ \sigma & = & \frac{p_{surf}}{\bar{g}} \end{eqnarray}