number density averaging kernel derivations
- number density AVK for air component from column number density AVK: - symbol - description - unit - variable name - \(A^{c}_{x}(i,j)\) - AVK for column number density profile for air component x (e.g. \(A^{c}_{O_{3}}(i,j)\)) - \(\frac{molec/m^2}{molec/m^2}\) - <species>_column_number_density_avk {:,vertical,vertical} - \(A^{n}_{x}(i,j)\) - AVK for number density profile for air component x (e.g. \(A^{n}_{O_{3}}(i,j)\)) - \(\frac{molec/m^3}{molec/m^3}\) - <species>_number_density_avk {:,vertical,vertical} - \(z^{B}(i,l)\) - altitude boundaries (\(l \in \{1,2\}\)) - \(m\) - altitude_bounds {:,vertical,2} - The pattern : for the first dimensions can represent {latitude,longitude}, {time}, {time,latitude,longitude}, or no dimensions at all. \[\begin{split}A^{n}_{x}(i,j) = \begin{cases} z^{B}(i,1) \neq z^{B}(i,2), & A^{c}_{x}(i,j) \frac{\lvert z^{B}(j,2) - z^{B}(j,1) \rvert}{\lvert z^{B}(i,2) - z^{B}(i,1) \rvert} \\ z^{B}(i,1) = z^{B}(i,2), & 0 \end{cases}\end{split}\]
- number density AVK from volume mixing ratio AVK: - symbol - description - unit - variable name - \(A^{n}_{x}(i,j)\) - AVK for number density profile for 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 for volume mixing ratio profile for 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. \[\begin{split}A^{n}_{x}(i,j) = \begin{cases} n(j) \neq 0, & A^{\nu}_{x}(i,j) \frac{n(i)}{n(j)} \\ n(j) = 0, & 0 \end{cases}\end{split}\]
- number density AVK from volume mixing ratio dry air AVK: - symbol - description - unit - variable name - \(A^{n}_{x}(i,j)\) - AVK for number density profile for 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 for volume mixing ratio profile for air component x (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. \[\begin{split}A^{n}_{x}(i,j) = \begin{cases} n_{dry_air}(j) \neq 0, & A^{\bar{\nu}}_{x}(i,j) \frac{n_{dry_air}(i)}{n_{dry_air}(j)} \\ n_{dry_air}(j) = 0, & 0 \end{cases}\end{split}\]