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The Gamma size distribution function used to describe the size distribution of hydrometeors has the following form:
where m is the mass of a single cloud droplet,
$ f(m) $ is the number density of the cloud droplets with mass m,$ \alpha $ is the shape parameter,$ \beta $ is the slope parameter and$ {N}_{0} $ is the intercept parameter.The double-moment schemes apply two moments, representing the number concentration (
$ {N}_{\mathrm{c}} $ ) and the cloud water content ($ {q}_{\mathrm{c}} $ ). With Eq. (1), the analytical solutions of the number concentration and the cloud water content can be derived as follows:The slope parameter and intercept parameter can be obtained by the combination of Eqs. (2) and (3):
The above equations are used to diagnose the slope and intercept parameters in the bulk double-moment scheme with a fixed shape parameter. Still, there are other double-moment schemes in which the shape parameters are also diagnosed by empirical formulas, so we chose the method of Morrison and Grabowski (2007) (hereafter MG), given the simulations of spectra and moments of the mass distribution.
In the MG scheme, the shape parameter is diagnosed by the number concentration of cloud droplets as follows (Morrison and Grabowski, 2007):
To facilitate comparison with the new scheme, the spectrum distribution is expressed by Eq. (1).
The mass condensation growth rate of a single cloud droplet can be given as:
where s represents the ambient supersaturation,
$ {\rho }_{\mathrm{w}} $ is the density of water, and$ \xi $ is related to the vapor diffusion and the latent heat release, as given by Eq. (8):Here
$ {R}_{\mathrm{v}} $ is the individual gas constant for water vapor, T represents the ambient temperature, L(T) is the latent heat,$ {K}_{\mathrm{a}} $ is the coefficient of thermal conductivity of air,$ {e}_{\mathrm{w}} $ is the saturation vapor pressure, and$ {D}_{\mathrm{v}} $ represents the molecular diffusion coefficient.The change rate of cloud water content (
$ {q}_{\mathrm{c}} $ ) during condensation can be obtained by the combination of Eqs. (1), (3), and (7):The research reveals that the empirical formulas like Eq. (9) lack universality (Loftus et al., 2014). So based on another part of our research, the diagnosis equation of shape parameter is obtained by introducing a reflectivity factor in the new scheme. The reflectivity factor (
$ {Z}_{\mathrm{c}} $ ) can be calculated by its definition (proportional to the second moment of the mass distribution) and through Eq. (1):Based on previous research (Yang and Yau, 2008; Chen and Tsai, 2016), the change rate of
$ {Z}_{\mathrm{c}} $ during condensation can be obtained by the combination of Eqs. (1), (7), and (10):The shape parameter and the slope parameter can be calculated by the combination of Eqs. (2), (3), and (10):
The above parameterization schemes will be tested in a parcel model with constant supersaturation values and a 1.5D Eulerian model with the real sounding profile of Rain in Shallow Cumulus over the Ocean (RICO) (Rauber et al., 2007; Wang et al., 2016). The 1.5D model consists of two cylinders. The updraft area is located in the inner cylinder, and the downdraft area is the space between the inner and outer cylinders. The exchanges of cloud droplets and aerosols due to the entrainment and detrainment between the cloud and environment are described by the continuity equation. The details of the model were introduced in the paper of Sun et al. (2012a).
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To verify the advantages and disadvantages of the new parameterization scheme, the analytical solution of the CDS evolution is calculated based on Rogers and Yau (1989) as follows:
Or
where
$ {f}_{m}(m,t) $ is the distribution of cloud droplets with mass$ m $ at time$ t $ .Since
where
$ {f}_{\mathrm{l}\mathrm{n}\left(m\right)}(m,t) $ is the distribution of cloud droplets with logarithmic mass$ \mathrm{l}\mathrm{n}\left(m\right) $ at time$ t $ .So the change rate of the CDS with logarithmic mass based on the Lagrangian view can be given during condensation as follows:
upon substituting Eq. (7) into Eq. (17), we obtain:
and integrating Eq. (18) from
$ {t}_{0} $ to$ {t}_{0}+\Delta t $ yields:We further discretized the CDS into 3000 mass bins with the following relationship:
where
$ m\left(i\right) $ represents the mass of cloud droplets in the i-th mass bin.If the influences of solution and curvature on droplet equilibrium vapor pressure are ignored, the radius of the cloud droplet in each bin can be obtained analytically:
where
$ r\left(i,j\right) $ is the cloud droplet radius of the i-th mass bin at the time step$ j $ .At the same time, if only condensation is taken into account for the microphysical processes of droplets, the number concentration of cloud droplets in each bin is conserved in the Lagrangian view:
and the analytical cloud water content (
$ {q}_{\mathrm{c}} $ ) and volume-mean radius ($ {r}_{\mathrm{c}\_\mathrm{v}} $ ) per moment can also be obtained:where
$ {r}_{\mathrm{c}\_v} $ is in centimeters. -
An obvious monotonic relationship exists between the shape parameter and the volume-mean radius based on the simulation of the Lagrangian analytical solution based on Case 1 in Table 1, so the fitting function for the relationship between the shape parameter and the volume-mean radius with a correlation coefficient of 0.99 is obtained to modify the double-moment scheme as follows:
Cases $ s $ $ {N}_{\mathrm{c}} $ (cm–3) $ {q}_{\mathrm{c}} $ (g cm–3) $ {Z}_{\mathrm{c}} $ (cm6 cm–3) $ \alpha $ $ \beta $ (g–1) Case 1 0.1% 150 7.5 × 10−7 2.05 × 10−14 2 4 × 108 Case 2 0.3% 150 7.5 × 10−7 2.05 × 10−14 2 4 × 108 Case 3 0.03% 150 7.5 × 10−9 1.82 × 10−18 3 6 × 1010 Case 4 0.01% 150 7.5 × 10−9 1.82 × 10−18 3 6 × 1010 Table 1. The initial conditions.
In the modified double-moment scheme, the shape parameter can be calculated by Eq. (25), while the slope and intercept parameters are still obtained by Eqs. (4) and (5).
Cases | $ s $ | $ {N}_{\mathrm{c}} $ (cm–3) | $ {q}_{\mathrm{c}} $ (g cm–3) | $ {Z}_{\mathrm{c}} $ (cm6 cm–3) | $ \alpha $ | $ \beta $ (g–1) |
Case 1 | 0.1% | 150 | 7.5 × 10−7 | 2.05 × 10−14 | 2 | 4 × 108 |
Case 2 | 0.3% | 150 | 7.5 × 10−7 | 2.05 × 10−14 | 2 | 4 × 108 |
Case 3 | 0.03% | 150 | 7.5 × 10−9 | 1.82 × 10−18 | 3 | 6 × 1010 |
Case 4 | 0.01% | 150 | 7.5 × 10−9 | 1.82 × 10−18 | 3 | 6 × 1010 |