# volume mixing ratio averaging kernel derivations

1. volume mixing ratio AVK from number density AVK

symbol

description

unit

variable name

$$A^{n}_{x}(i,j)$$

AVK of number density profile of air component x (e.g. $$A^{n}_{O_{3}}(i,j)$$)

$$\frac{molec/m^3}{molec/m^3}$$

<species>_number_density_avk {:,vertical,vertical}

$$A^{\nu}_{x}(i,j)$$

AVK of volume mixing ratio profile of air component x (e.g. $$A^{\nu}_{O_{3}}(i,j)$$)

$$\frac{ppv}{ppv}$$

<species>_volume_mixing_ratio_avk {:,vertical,vertical}

$$n(i)$$

number density profile of total air

$$\frac{molec}{m^3}$$

number_density {:,vertical}

The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

$\begin{split}A^{\nu}_{x}(i,j) = \begin{cases} n(i) \neq 0, & A^{n}_{x}(i,j) \frac{n(j)}{n(i)} \\ n(i) = 0, & 0 \end{cases}\end{split}$
2. volume mixing ratio dry air AVK from number density AVK

symbol

description

unit

variable name

$$A^{n}_{x}(i,j)$$

AVK of number density profile of air component x (e.g. $$A^{n}_{O_{3}}(i,j)$$)

$$\frac{molec/m^3}{molec/m^3}$$

<species>_number_density_avk {:,vertical,vertical}

$$A^{\bar{\nu}}_{x}(i,j)$$

AVK of volume mixing ratio profile of air component x with regard to dry air (e.g. $$A^{\bar{\nu}}_{O_{3}}(i,j)$$)

$$\frac{ppv}{ppv}$$

<species>_volume_mixing_ratio_dry_air_avk {:,vertical,vertical}

$$n_{dry_air}(i)$$

number density profile of dry air

$$\frac{molec}{m^3}$$

dry_air_number_density {:,vertical}

The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all.

$\begin{split}A^{\bar{\nu}}_{x}(i,j) = \begin{cases} n_{dry_air}(i) \neq 0, & A^{n}_{x}(i,j) \frac{n_{dry_air}(j)}{n_{dry_air}(i)} \\ n_{dry_air}(i) = 0, & 0 \end{cases}\end{split}$