column number density derivations
total column number density of air component from partial column number density profile:
symbol description unit variable name \(c_{x}\) total column number density of air component x (e.g. \(c_{O_{3}}\)) \(\frac{molec}{m^2}\) <species>_column_number_density {:} \(c_{x}(i)\) column number density profile of air component x (e.g. \(c_{O_{3}}(i)\)) \(\frac{molec}{m^2}\) <species>_column_number_density {:,vertical} The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.
\[c_{x} = \sum_{i}{c_{x}(i)}\]total column number density of total air from partial column number density profile:
symbol description unit variable name \(c\) total column number density of total air \(\frac{molec}{m^2}\) column_number_density {:} \(c(i)\) column number density profile of total air \(\frac{molec}{m^2}\) column_number_density {:,vertical} The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.
\[c_{x} = \sum_{i}{c_{x}(i)}\]tropospheric column number density of air component from partial column number density profile and altitude:
symbol description unit variable name \(c_{x}\) tropospheric column number density of air component x (e.g. \(c_{O_{3}}\)) \(\frac{molec}{m^2}\) tropospheric_<species>_column_number_density {:} \(c_{x}(i)\) column number density profile of air component x (e.g. \(c_{O_{3}}(i)\)) \(\frac{molec}{m^2}\) <species>_column_number_density {:,vertical} \(z_{TP}\) tropopause altitude \(m\) tropopause_altitude {:} \(z^{B}(l)\) altitude boundaries (\(l \in \{1,2\}\)) \(m\) altitude_bounds {:,2} The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.
\[\begin{split}c_{x} = \sum_{i}{\begin{cases} z^{B}(2) \leq z_{TP}, & c_{x}(i) \\ z^{B}(1) < z_{TP} < z^{B}(2), & c_{x}(i) \frac{z_{TP} - z^{B}(1)}{z^{B}(2) - z^{B}(1)} \\ z_{TP} \leq z^{B}(1), & 0 \end{cases}}\end{split}\]stratospheric column number density of air component from partial column number density profile and altitude:
symbol description unit variable name \(c_{x}\) stratospheric column number density of air component x (e.g. \(c_{O_{3}}\)) \(\frac{molec}{m^2}\) stratospheric_<species>_column_number_density {:} \(c_{x}(i)\) column number density profile of air component x (e.g. \(c_{O_{3}}(i)\)) \(\frac{molec}{m^2}\) <species>_column_number_density {:,vertical} \(z_{TP}\) tropopause altitude \(m\) tropopause_altitude {:} \(z^{B}(l)\) altitude boundaries (\(l \in \{1,2\}\)) \(m\) altitude_bounds {:,2} The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.
\[\begin{split}c_{x} = \sum_{i}{\begin{cases} z^{B}(2) \leq z_{TP}, & 0 \\ z^{B}(1) < z_{TP} < z^{B}(2), & c_{x}(i) \frac{z^{B}(2) - z_{TP}}{z^{B}(2) - z^{B}(1)} \\ z_{TP} \leq z^{B}(1), & c_{x}(i) \end{cases}}\end{split}\]tropospheric column number density of air component from partial column number density profile and pressure:
symbol description unit variable name \(c_{x}\) tropospheric column number density of air component x (e.g. \(c_{O_{3}}\)) \(\frac{molec}{m^2}\) tropospheric_<species>_column_number_density {:} \(c_{x}(i)\) column number density profile of air component x (e.g. \(c_{O_{3}}(i)\)) \(\frac{molec}{m^2}\) <species>_column_number_density {:,vertical} \(p_{TP}\) tropopause pressure \(Pa\) tropopause_pressure {:} \(p^{B}(l)\) pressure boundaries (\(l \in \{1,2\}\)) \(Pa\) pressure_bounds {:,2} The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.
\[\begin{split}c_{x} = \sum_{i}{\begin{cases} p^{B}(2) \geq p_{TP}, & c_{x}(i) \\ p^{B}(1) > p_{TP} > p^{B}(2), & c_{x}(i) \frac{\ln(p^{B}(1)) - \ln(p_{TP})}{\ln(p^{B}(1)) - \ln(p^{B}(2))} \\ p_{TP} \geq p^{B}(1), & 0 \end{cases}}\end{split}\]stratospheric column number density of air component from partial column number density profile and pressure:
symbol description unit variable name \(c_{x}\) stratospheric column number density of air component x (e.g. \(c_{O_{3}}\)) \(\frac{molec}{m^2}\) stratospheric_<species>_column_number_density {:} \(c_{x}(i)\) column number density profile of air component x (e.g. \(c_{O_{3}}(i)\)) \(\frac{molec}{m^2}\) <species>_column_number_density {:,vertical} \(p_{TP}\) tropopause pressure \(Pa\) tropopause_pressure {:} \(p^{B}(l)\) pressure boundaries (\(l \in \{1,2\}\)) \(Pa\) pressure_bounds {:,2} The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.
\[\begin{split}c_{x} = \sum_{i}{\begin{cases} p^{B}(2) \geq p_{TP}, & 0 \\ p^{B}(1) > p_{TP} > p^{B}(2), & c_{x}(i) \frac{\ln(p_{TP}) - \ln(p^{B}(2))}{\ln(p^{B}(1)) - \ln(p^{B}(2))} \\ p_{TP} \geq p^{B}(1), & c_{x}(i) \end{cases}}\end{split}\]column number density of total air from dry air column number density and H2O column number density
symbol description unit variable name \(c\) column number density \(\frac{molec}{m^2}\) column_number_density {:} \(c_{dry\_air}\) column number density of dry air \(\frac{molec}{m^2}\) dry_air_column_number_density {:} \(c_{H_{2}O}\) column number density of H2O \(\frac{molec}{m^2}\) H2O_column_number_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.
\[c = c_{dry\_air} + c_{H_{2}O}\]column number density of dry air from total air column number density and H2O column number density
symbol description unit variable name \(c\) column number density \(\frac{molec}{m^2}\) column_number_density {:} \(c_{dry\_air}\) column number density of dry air \(\frac{molec}{m^2}\) dry_air_column_number_density {:} \(c_{H_{2}O}\) column number density of H2O \(\frac{molec}{m^2}\) H2O_column_number_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.
\[c_{dry\_air} = c - c_{H_{2}O}\]column number density of H2O from total air column number density and dry air column number density
symbol description unit variable name \(c\) column number density \(\frac{molec}{m^2}\) column_number_density {:} \(c_{dry\_air}\) column number density of dry air \(\frac{molec}{m^2}\) dry_air_column_number_density {:} \(c_{H_{2}O}\) column number density of H2O \(\frac{molec}{m^2}\) H2O_column_number_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.
\[c_{H_{2}O} = c - c_{dry\_air}\]column number density of air component from number density:
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 {:} \(n_{x}\) number density of air component x (e.g. \(n_{O_{3}}\)) \(\frac{molec}{m^3}\) <species>_number_density {:} \(z^{B}(l)\) altitude boundaries (\(l \in \{1,2\}\)) \(m\) altitude_bounds {:,2} 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.
\[c_{x} = n_{x} \lvert z^{B}(2) - z^{B}(1) \rvert\]column number density of total air from number density:
symbol description unit variable name \(c\) column number density of total air \(\frac{molec}{m^2}\) column_number_density {:} \(n\) number density of total air \(\frac{molec}{m^3}\) number_density {:} \(z^{B}(l)\) altitude boundaries (\(l \in \{1,2\}\)) \(m\) altitude_bounds {:,2} 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.
\[c = n \lvert z^{B}(2) - z^{B}(1) \rvert\]column number density of air component from column mass 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. \(n_{O_{3}}\)) \(\frac{molec}{m^2}\) <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.
\[c_{x} = \frac{\sigma_{x}N_{A}}{10^{-3}M_{x}}\]column number density of total air from column mass 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.
\[c = \frac{\sigma N_{A}}{10^{-3}M_{air}}\]column number density from column volume mixing ratio
symbol description unit variable name \(c\) total column number density of total air \(\frac{molec}{m^2}\) column_number_density {:} \(c_{x}\) total column number density of air component x (e.g. \(c_{O_{3}}\)) \(\frac{molec}{m^2}\) <species>_column_number_density {:} \(\nu_{x}\) 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.
\[c_{x} = \nu_{x}c\]column number density from column volume mixing ratio dry air
symbol description unit variable name \(c_{dry\_air}\) total column number density of dry air \(\frac{molec}{m^2}\) dry_air_column_number_density {:} \(c_{x}\) total column number density of air component x (e.g. \(c_{O_{3}}\)) \(\frac{molec}{m^2}\) <species>_column_number_density {:} \(\bar{\nu}_{x}\) 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.
\[c_{x} = \bar{\nu}_{x}c_{dry\_air}\]column number density of air component from volume mixing ratio:
symbol description unit variable name \(a\) WGS84 semi-major axis \(m\) \(b\) WGS84 semi-minor axis \(m\) \(c_{x}\) column number density of air component x (e.g. \(c_{O_{3}}\)) \(\frac{molec}{m^2}\) <species>_column_number_density {:} \(f\) WGS84 flattening \(m\) \(g\) normal gravity at sea level \(\frac{m}{s^2}\) \(g_{0}\) mean earth gravity \(\frac{m}{s^2}\) \(g_{h}\) gravity at specific height \(\frac{m}{s^2}\) \(GM\) WGS84 earth’s gravitational constant \(\frac{m^3}{s^2}\) \(M_{air}\) molar mass of total air \(\frac{g}{mol}\) molar_mass {:} \(N_A\) Avogadro constant \(\frac{1}{mol}\) \(p\) pressure \(Pa\) \(p_{0}\) standard pressure \(Pa\) \(p^{B}(l)\) pressure boundaries (\(l \in \{1,2\}\)) \(Pa\) pressure_bounds {:,2} \(R\) universal gas constant \(\frac{kg m^2}{K mol s^2}\) \(T_{0}\) standard temperature \(K\) \(z\) altitude \(m\) \(\nu_{x}\) volume mixing ratio of quantity x with regard to total air \(ppv\) <species>_volume_mixing_ratio {:} \(\phi\) latitude \(degN\) latitude {:} \(\omega\) WGS84 earth angular velocity \(rad/s\) 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.
\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)}} \\ m & = & \frac{\omega^2a^2b}{GM} \\ p & = & e^{\frac{\ln(p^{B}(2)) + \ln(p^{B}(1))}{2}} \\ z & = & -\frac{RT_{0}}{10^{-3}M_{air}g_{0}}\ln(\frac{p}{p_{0}}) \\ g_{h} & = & g \left(1 - \frac{2}{a}\left(1+f+m-2f{\sin}^2(\frac{\pi}{180}\phi)\right)z + \frac{3}{a^2}z^2\right) \\ c_{x} & = & -\nu_{x}\frac{N_A}{10^{-3}M_{air}g_{h}}(p^{B}(2)-p^{B}(1)) \end{eqnarray}