Diagnostics

CloudMicrophysics.jl offers a couple of options to compute cloud and precipitation radiative properties based on different available parameterizations and their underlying assumptions about the size distribution and properties of particles.

Available diagnostics are:

• Radar reflectivity
• Effective radius

Radar reflectivity

The radar reflectivity factor $Z$ is used to measure the power returned by a radar signal when it encounters particles (cloud, rain droplets, etc), and is defined as the sixth moment of the particles distributions $n(r)$:

\[\begin{equation} Z = {\int_0^\infty r^{6} \, n(r) \, dr}. \label{eq:Z} \end{equation}\]

$Z$ is typically normalized by radar reflectivity factor $Z_0$ of a drop of radius $1 mm$ in a volume of $1 m^3$, and is reported as a decimal logarithm to obtain the normalized logarithmic rain radar reflectivity $L_Z$.

\[\begin{equation} L_Z = {10 \, \log_{10} \left( \frac{Z}{Z_0} \right)}. \end{equation}\]

The resulting logarithmic dimensionless unit is decibel relative to $Z_0$, or $dBZ$.

1-moment microphysics

For the 1-moment scheme we only consider the rain drop size distribution. Integrating over the assumed Marshall-Palmer distribution leads to

\[\begin{equation} Z = {\frac{6! \, n_{0}^{rai}}{\lambda^7}}, \end{equation}\]

where:

• $n_{0}^{rai}$ and $\lambda$ - rain drop size distribution parameters.

2-moment microphysics

For the 2-moment scheme we take into consideration the effect of both cloud and rain droplets. Integrating over the assumed cloud droplets Gamma distribution leads to

\[\begin{equation} Z_c = A_c C^{\nu_c+1} \frac{ (B_c C^{\mu_c})^{-\frac{3+\nu_c}{\mu_c}} \, \Gamma \left(\frac{3+\nu_c}{\mu_c}\right)}{\mu_c}, \end{equation}\]

where:

• $\Gamma \,(x) = \int_{0}^{\infty} \! t^{x - 1} e^{-t} \mathrm{d}t$ is the gamma function,
• $C = \frac{4}{3} \pi \rho_w$.

Similar for rain drop exponential distribution

\[\begin{equation} Z_r = A_r C^{\nu_r+1} \frac{ (B_r C^{\mu_r})^{-\frac{3+\nu_r}{\mu_r}} \, \Gamma \left(\frac{3+\nu_r}{\mu_r}\right)}{\mu_r}, \end{equation}\]

The final radar reflectivity factor is a sum of $Z_c$ and $Z_r$.

Effective radius

The effective radius of hydrometeors ($r_{eff}$) is defined as the area weighted radius of the population of particles. It can be computed as the ratio of the third to the second moment of the size distribution.

2-moment microphysics

We compute the total third and second moment as a sum of cloud condensate and precipitation moments:

\[\begin{equation} r_{eff} = \frac{M_{3}^c + M_{3}^r}{M_{2}^c + M_{2}^r} = \frac{{\int_0^\infty r^{3} \, (n_c(r) + n_r(r)) \, dr}}{{\int_0^\infty r^{2} \, (n_c(r) + n_r(r)) \, dr}}. \label{eq:reff} \end{equation}\]

After integrating we obtain

\[\begin{equation} M_{3}^c + M_{3}^r = A_c C^{\nu_c+1} \frac{ (B_c C^{\mu_c})^{-\frac{2+\nu_c}{\mu_c}} \, \Gamma \left(\frac{2+\nu_c}{\mu_c}\right)}{\mu_c} + A_r C^{\nu_r+1} \frac{ (B_r C^{\mu_r})^{-\frac{2+\nu_r}{\mu_r}} \, \Gamma \left(\frac{2+\nu_r}{\mu_r}\right)}{\mu_r}. \end{equation}\]

\[\begin{equation} M_{2}^c + M_{2}^r = A_c C^{\nu_c+1} \frac{ (B_c C^{\mu_c})^{-\frac{5+3\nu_c}{3\mu_c}} \, \Gamma \left(\frac{5+3\nu_c}{3\mu_c}\right)}{\mu_c} + A_r C^{\nu_r+1} \frac{ (B_r C^{\mu_r})^{-\frac{5+3\nu_r}{3\mu_r}} \, \Gamma \left(\frac{5+3\nu_r}{3\mu_r}\right)}{\mu_r}. \end{equation}\]

Liu and Hallett 1997

For 1-moment microphysics scheme the effective radius is parameterized following [54]:

\[\begin{equation} r_{eff} = \frac{r_{vol}}{k^{\frac{1}{3}}}, \end{equation}\]

where:

• $r_{vol}$ represents the volume-averaged radius,
• $k = 0.8$.

Where the volume-averaged radius is computed using

\[\begin{equation} r_{vol} = \left(\frac{3}{4 \pi \rho_w}\right)^{\frac{1}{3}} \, \left(\frac{L}{N}\right)^{\frac{1}{3}} = \left(\frac{3 \rho (q_{liq} + q_{rai})}{4 \pi \rho_w (N_{liq} + N_{rai})}\right)^{\frac{1}{3}}, \end{equation}\]

where:

• $L = \rho q$, is the liquid water content,
• $N = N_{liq} + N_{rai}$.

By default for the 1-moment scheme we don't consider precipitation and assume a constant cloud droplet number concentration of 100 $cm^{-3}$.

Constant

For testing purposes we also provide a constant effective radius option. The default values are 14 $\mu m$ for liquid clouds and 25 $\mu m$ for ice clouds, and can be easily overwritten via ClimaParams.jl.

Example figures

Below we show effective radius and radar reflecivity plots as a function of cloud water and rain water. The effective radius is computed assuming a constant cloud droplet number concentration of 100 or 1000 per cubic centimeter. The radar reflectivity is computed assuming a constant rain drop number concentration of 10 or 100 per cubic centimeter. Note the effect of using the limiters in SB2006 scheme on the radar reflectivity.

include("plots/CloudDiagnostics.jl")

CairoMakie.Screen{SVG}