column mass density derivations
- 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. \[\sigma = \sigma_{dry\_air} + \sigma_{H_{2}O}\]
- 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. \[\sigma_{dry\_air} = \sigma - \sigma_{H_{2}O}\]
- 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. \[\sigma_{H_{2}O} = \sigma - \sigma_{dry\_air}\]
- 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. \[\sigma_{x} = \rho_{x} \lvert z^{B}(2) - z^{B}(1) \rvert\]
- 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. \[\sigma = \rho \lvert z^{B}(2) - z^{B}(1) \rvert\]
- 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. \[\sigma_{x} = \frac{10^{-3}c_{x}M_{x}}{N_{A}}\]
- 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. \[\sigma = \frac{10^{-3}c M_{air}}{N_{A}}\]
- 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}