column mass density derivations

column mass density for 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 for 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 for 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 for 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 for 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 for 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 for 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 for air component x (e.g. \(\rho_{O_{3}}\))	\(\frac{kg}{m^3}\)	<species>_density {:}
\(\sigma_{x}\)	column mass density for 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 for 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 for total air	\(\frac{kg}{m^3}\)	density {:}
\(\sigma\)	column mass density for 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 for 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 for air component x (e.g. \(c_{O_{3}}\))	\(\frac{molec}{m^2}\)	<species>_column_number_density {:}
\(M_{x}\)	molar mass for air component x	\(\frac{g}{mol}\)
\(N_A\)	Avogadro constant	\(\frac{1}{mol}\)
\(\sigma_{x}\)	column mass density for 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 for 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 for total air	\(\frac{molec}{m^2}\)	column_number_density {:}
\(M_{air}\)	molar mass for total air	\(\frac{g}{mol}\)	molar_mass {:}
\(N_A\)	Avogadro constant	\(\frac{1}{mol}\)
\(\sigma\)	column mass density for 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 for total air from pressure profile and surface pressure:

symbol	description	unit	variable name
\(\bar{g}\)	mean gravity for profile	\(\frac{m}{s^2}\)
\(g_{wgs84}\)	gravity at WGS84 ellipsoid	\(\frac{m}{s^2}\)
\(p^{B}(i,l)\)	pressure boundaries (\(l \in \{1,2\}\))	\(Pa\)	pressure_bounds {:,vertical,2}
\(p_{surf}\)	surface pressure	\(Pa\)	surface_pressure {:}
\(R_{wgs84}\)	local earth curvature radius at WGS84 ellipsoid	\(m\)
\(z(i)\)	altitude	\(m\)	altitude {:,vertical}
\(\phi\)	latitude	\(degN\)	latitude {:}
\(\sigma\)	column mass density for 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_{wgs84} & = & 9.7803253359 \frac{1 + 0.00193185265241{\sin}^2(\frac{\pi}{180}\phi)} {\sqrt{1 - 0.00669437999013{\sin}^2(\frac{\pi}{180}\phi)}} \\ g(i) & = & g_{wgs84}\left(\frac{R_{wgs84}}{R_{wgs84} + 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(i)}}} \\ \sigma & = & \frac{p_{surf}}{\bar{g}} \end{eqnarray}