# 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

$$\frac{g}{mol}$$

$$N_A$$

$$\frac{1}{mol}$$

$$\sigma_{x}$$

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 {:}

$$N_A$$

$$\frac{1}{mol}$$

$$\sigma$$

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 {:}

$$R$$

$$m$$

$$z(i)$$

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}