Microphysics model¶

Kessler Microphysics model¶

The Kessler microphysics model is a simple version of cloud microphysics which has precipitation only in the form of rain. Hence \(q_p = q_r\). Governing equations for the microphysical quantities for Kessler microphysics from gabervsek2012dry are

\[\frac{\partial q_v}{\partial t} = -C_c + E_c + E_r\]

\[\frac{\partial q_c}{\partial t} = C_c - E_c - (A_c + K_c)\]

\[\frac{\partial q_p}{\partial t} = \frac{1}{\overline{\rho}}\frac{\partial}{\partial z}(\overline{\rho}Vq_p) + (A_c + K_c) - E_r\]

\[\frac{\partial q_t}{\partial t} = \frac{\partial q_v}{\partial t} + \frac{\partial q_c}{\partial t} = E_r - (A_c + K_c)\]

where \(C_c\) is the rate of condensation of water vapor to cloud water, \(E_c\) is the rate of evaporation of cloud water to water vapor, \(A_c\) is the autoconversion of cloud water to rain, \(K_c\) is the accretion of cloud water to rain drops, \(E_r\) is the evaporation of rain to water vapor and \(F_r\) is the sedimentation of rain. The parametrization used is given in klemp1978simulation, and is given below. Note that in all the equations, \(p\) is specified in millibars and \(\overline{\rho}\) is specified in g cm \(^{-3}\). The parametrization of the source terms are given below.

Rate of condensation of water vapor/evaporation of cloud water¶

From klemp1978simulation, we have the following expressions.

If the air is not saturated, i.e. \(q_v > q_{vs}\)

\[C_c = \frac{q_v - q_{vs}}{1 + \cfrac{q_{vs}^*4093L}{C_p(T-36)^2}}\]

If the air is not saturated, i.e. \(q_v < q_{vs}\), then cloud water evaporates to water vapor at the rate

\[E_c = \frac{q_{vs} - q_v}{1 + \cfrac{q_{vs}^*4093L}{C_p(T-36)^2}}\]

Rate of autoconversion of cloud water into rain¶

The rate of autoconversion of cloud water into rain is given by

\[A_c = k_1(q_c - a)\]

where \(k_1 = 0.001\) s^-1, \(a = 0.001\) kg kg^-1.

Rate of accretion of cloud water into rain water drops¶

The rate of accretion of cloud water into rain water drops is given by

\[K_c = k_2q_cq_r^{0.875}\]

where \(k_2= 2.2\) s^-1.

The rate of evaporation of rain into water vapor¶

The rate of evaporation of rain into water vapor is given by

\[E_r = \cfrac{1}{\overline{\rho}}\cfrac{(1- q_v/q_s)C(\overline{\rho}q_r)^{0.525}}{5.4\times10^5 + 2.55\times10^6/(\overline{p}q_s)},\]

where the ventilation factor \(C = 1.6 + 124.9(\overline{\rho}q_r)^{0.2046}\).

Terminal fall velocity of rain¶

The terminal fall velocity of rain is given by

\[V = 3634(\overline{\rho}q_r)^{0.1346}\Bigg(\cfrac{\overline{\rho}}{\rho_0}\Bigg)^{-\frac{1}{2}}~\text{[cm/s]}\]

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

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