# Single Moment Microphysics Model

The conversion rates among the moist hydrometeors are parameterized assuming that

$\frac{\partial N_{m}}{\partial D} = n_{m}\left(D_{m}\right) = N_{0m} exp \left(-\lambda _{m} D_{m}\right)$

where $$N_{0m}$$ is the intercept parameter, $$D_{m}$$ is the diameters, and

$\lambda_{m} = (\frac{\pi \rho_{m} N_{0m}}{q_{m}\rho})^{0.25}$

where $$\rho_{m}$$ is the density of moist hydrometeors. Assuming that the particle terminal velocity

$v_{m} \left( D_{m},p \right) = a_{m}D_{m}^{b_{m}}\left(\frac{\rho_{0}}{\rho}\right)^{0.5}$

The total production rates including the contribution from aggregation, accretion, sublimation, melting, bergeron process, freezing and autoconversion are listed below without derivation. For details, please refer to Yuh-Lang Lin et al (J. Climate Appl. Meteor, 22, 1065, 1983) and Marat F. Khairoutdinov and David A. Randall’s (J. Atm Sciences, 607, 1983). The implementation of microphysics model in ERF is similar to the that in the SAM code (http://rossby.msrc.sunysb.edu/~marat/SAM.html)

## Accretion

There are several different type of accretional growth mechanisms that need to be included; these describe the interaction of water vapor and cloud water with rain water.

The accretion of cloud water forms in either the dry or wet growth rate can be written as:

$Q_{gacw} = \frac{\pi E_{GW}n_{0G}q_{c}\Gamma(3.5)}{4\lambda_{G}^{3.5}}(\frac{4g\rho G}{3C_{D}\rho})^{0.5}$

The accretion of raindrops by accretion of cloud water is

$Q_{racw} = \frac{\pi E_{RW}n_{0R}\alpha q_{c}\Gamma(3+b)}{4\lambda_{R}^{3+b}}(\frac{\rho_{0}}{\rho})^{1/2}$

## The bergeron Process

The cloud water transform to snow by deposition and rimming can be written as

$Q_{sfw} = N_{150}\left(\alpha_{1}m_{150}^{\alpha_{2}}+\pi E_{iw}\rho q_{c}R_{150}^{2}U_{150}\right)$

## Autoconversion

The collision and coalescence of cloud water to from raindrops is parameterized as following

$Q_{raut} = \rho\left(q_{c}-q_{c0}\right)^{2}\left[1.2 \times 10^{-4}+{1.569 \times 10^{-12}N_{1}/[D_{0}(q_{c}-q_{c0})]}\right]^{-1}$

## Evaporation

The evaporation rate of rain is

$Q_{revp} = 2\pi(S-1)n_{0R}[0.78\lambda_{R}^{-2}+0.31S_{c}^{1/3}\Gamma[(b+5)/2]a^{1/2}\mu^{-1/2}(\frac{\rho_{0}}{\rho})^{1/4}\lambda_{R}^{(b+5)/2}](\frac{1}{\rho})(\frac{L_{v}^{2}}{K_{0}R_{w}T^{2}}+\frac{1}{\rho r_{s}\psi})^{-1}$