# column number density derivations

1. 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.

$c_{x} = \sum_{i}{c_{x}(i)}$
2. 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.

$c_{x} = \sum_{i}{c_{x}(i)}$
3. 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.

$\begin{split}c_{x} = \sum_{i}{\begin{cases} z^{B}(2) \leq z_{TP}, & c_{x}(i) \\ z^{B}(1) < z_{TP} < z^{B}(2), & c_{x}(i) \frac{z_{TP} - z^{B}(1)}{z^{B}(2) - z^{B}(1)} \\ z_{TP} \leq z^{B}(1), & 0 \end{cases}}\end{split}$
4. 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.

$\begin{split}c_{x} = \sum_{i}{\begin{cases} z^{B}(2) \leq z_{TP}, & 0 \\ z^{B}(1) < z_{TP} < z^{B}(2), & c_{x}(i) \frac{z^{B}(2) - z_{TP}}{z^{B}(2) - z^{B}(1)} \\ z_{TP} \leq z^{B}(1), & c_{x}(i) \end{cases}}\end{split}$
5. 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.

$\begin{split}c_{x} = \sum_{i}{\begin{cases} p^{B}(2) \geq p_{TP}, & c_{x}(i) \\ p^{B}(1) > p_{TP} > p^{B}(2), & c_{x}(i) \frac{\ln(p^{B}(1)) - \ln(p_{TP})}{\ln(p^{B}(1)) - \ln(p^{B}(2))} \\ p_{TP} \geq p^{B}(1), & 0 \end{cases}}\end{split}$
6. 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.

$\begin{split}c_{x} = \sum_{i}{\begin{cases} p^{B}(2) \geq p_{TP}, & 0 \\ p^{B}(1) > p_{TP} > p^{B}(2), & c_{x}(i) \frac{\ln(p_{TP}) - \ln(p^{B}(2))}{\ln(p^{B}(1)) - \ln(p^{B}(2))} \\ p_{TP} \geq p^{B}(1), & c_{x}(i) \end{cases}}\end{split}$
7. 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.

$c = c_{dry\_air} + c_{H_{2}O}$
8. 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.

$c_{dry\_air} = c - c_{H_{2}O}$
9. 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.

$c_{H_{2}O} = c - c_{dry\_air}$
10. 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.

$c_{x} = n_{x} \lvert z^{B}(2) - z^{B}(1) \rvert$
11. 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.

$c = n \lvert z^{B}(2) - z^{B}(1) \rvert$
12. 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

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

$$N_A$$

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

$$\sigma_{x}$$

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.

$c_{x} = \frac{\sigma_{x}N_{A}}{10^{-3}M_{x}}$
13. 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 {:}

$$N_A$$

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

$$\sigma$$

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.

$c = \frac{\sigma N_{A}}{10^{-3}M_{air}}$
14. 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.

$c_{x} = \nu_{x}c$
15. 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.

$c_{x} = \bar{\nu}_{x}c_{dry\_air}$
16. 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 earthâ€™s gravitational constant

$$\frac{m^3}{s^2}$$

$$M_{air}$$

molar mass of total air

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

molar_mass {:}

$$N_A$$

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

$$p$$

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}