Size Distributions

MIAM supports two families of aerosol size distributions: modal (log-normal) and sectional (uniform bins).

Log-Normal (Modal)

A log-normal distribution is parameterized by the geometric mean radius \(r_g\) and geometric standard deviation \(\sigma_g\):

\[\frac{dN}{d\ln r} = \frac{N}{\sqrt{2\pi}\,\ln\sigma_g} \exp\!\left(-\frac{(\ln r - \ln r_g)^2}{2\ln^2\sigma_g}\right)\]

Effective radius

For use in condensation rate calculations, the effective radius accounts for the surface-area weighting of the distribution:

\[r_\text{eff} = r_g \cdot \exp(2.5\,\ln^2\sigma_g)\]

This moment relation applies to both SingleMomentMode (\(r_g\) fixed) and TwoMomentMode (\(r_g\) derived from total volume and number).

Single-particle volume

For SingleMomentMode, the volume of a single “average” particle:

\[V_\text{single} = \frac{4}{3}\pi\,r_g^3\,\exp(4.5\,\ln^2\sigma_g)\]

Number concentration is diagnosed as \(N = V_\text{total}/V_\text{single}\).

Two-moment mean radius

For TwoMomentMode, the geometric mean radius is derived from volume and number:

\[r_\text{mean} = \left(\frac{3\,V_\text{total}}{4\pi\,N}\right)^{1/3}\]

\[r_\text{eff} = r_\text{mean}\cdot\exp(2.5\,\ln^2\sigma_g)\]

Sectional (UniformSection)

A sectional bin assumes particles are uniformly distributed between \(r_\min\) and \(r_\max\):

\[r_\text{eff} = \frac{r_\min + r_\max}{2}\]

\[V_\text{single} = \frac{4}{3}\pi\,r_\text{eff}^3\]

Number concentration is diagnosed from total volume: \(N = V_\text{total}/V_\text{single}\).

Volume Calculations

All representations compute total volume from species concentrations:

\[V_\text{total} = \sum_p \sum_i [\text{species}_{p,i}]\cdot\frac{M_{w,i}}{\rho_i}\]

Species must carry molecular weight [kg mol-1] and density [kg m-3] properties for this calculation.