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}