1. Introduction
The relation between Z and cloud microphysical properties is explained by two mechanisms: incoherent scattering and coherent scattering. Incoherent scattering occurs when the cloud droplets are dispersed randomly and uniformly (Bohren and Huffman 1983). The radar reflectivity factor for the incoherent scattering case is proportional to the sum of the Rayleigh scattering intensity from each droplet and independent of the microwave frequency fm. On the other hand, coherent scattering—often referred to as Bragg scattering—occurs when the droplets are distributed nonuniformly. The nonuniform distribution causes the interference of scattered microwaves, which in turn increases the radar reflectivity factor obtained from Eq. (1). This coherent scattering by discrete particles is more specifically referred to as “particulate” Bragg scattering (Kostinski and Jameson 2000). Coherent scattering can also be caused by a nonuniform distribution of the refractive index of clear air—which may be referred to as “clear-air Bragg scattering.” Most studies assume that particulate Bragg scattering is insignificant in atmospheric clouds (Gossard and Strauch 1983). However, this assumption is contradicted by the observations of developing cumulus clouds by Knight and Miller (1993) and Knight and Miller (1998). They observed significant differences between the radar reflectivity factors for 10- and 3-cm microwaves, which are classified in the S and X bands, respectively. A similar wavelength dependency of the radar reflectivity factor was found for the case of smoke plumes from an intense industrial fire by Rogers and Brown (1997), who compared the data observed by a UHF wind profiler (wavelength 32.8 cm) and an X-band radar (3.2 cm). Knight and Miller (1998) explained that these differences resulted from coherent scattering by nonuniform cloud droplet concentrations created by the turbulent mixing of cloud with environmental clear air (i.e., turbulent entrainment). That is, they attributed the differences to the large-scale nonuniform distribution of cloud droplets. Erkelens et al. (2001) investigated the influence of turbulent entrainment on the observations of Knight and Miller (1998). They analyzed the observational data using an equation for clear-air Bragg scattering based on the −
This study, therefore, aims to investigate the influence of microscale turbulent clustering on the radar reflectivity factor and construct a reliable model for estimating it. A three-dimensional direct numerical simulation (DNS) of particle-laden isotropic turbulence is performed in order to obtain turbulent clustering data, and then the influence of turbulence is analyzed and modeled. The model is then applied to two idealized radar observation scenarios to assess the influence quantitatively.
2. Computational method
a. Air turbulence
The fourth-order central-difference scheme (Morinishi et al. 1998) was used for the advection term and the second-order Runge–Kutta scheme was used for time integration. The velocity and pressure were coupled by the highly simplified marker and cell (HSMAC) method (Hirt and Cook 1972). Statistically steady-state turbulence was formed by applying an external forcing using the reduced-communication forcing (RCF) method of Onishi et al. (2011), which maintains the intensity of large-scale eddies while keeping a high parallel efficiency.
It should be noted that atmospheric turbulence is typically neither homogeneous nor isotropic. However, the assumptions of homogeneity and isotropy are reasonable for the small scales corresponding to the wavenumber range relevant to radar observations (see section 4a). Although energy-containing large-scale eddies generate large-scale inhomogeneity and anisotropy, dissipative small-scale eddies work to flatten the inhomogeneities, leading to local homogeneity and isotropy. This local homogeneity assumption is the basis of most turbulence models.
b. Droplet motions
c. Computational conditions
The computational domain was set to a cube with edges of length 2πL0, where L0 is the representative length scale. Periodic boundary conditions were applied in all three directions. The domain was discretized uniformly into 
Numerical conditions and flow properties: urms is the RMS value of velocity fluctuation, Re is the Reynolds number defined as Re = L0U0/ν, Reλ is the turbulent Reynolds number defined as Reλ = lλurms/ν, lλ is the Taylor microscale, kmax is the maximum wavenumber given by kmax = Ng/(2L0), and lη is the Kolmogorov scale.
The droplet radius rp was varied so that the Stokes number, defined as St = τp/τη, where τη = (ν/ϵ)1/2 is the Kolmogorov time scale, took values of 0.05, 0.1, 0.2, 0.5, 1.0, 2.0, and 5.0. The droplet radii for St = 1.0 were 22.9, 23.1, 20.2, and 23.4 μm for Reλ = 127, 204, 322, and 531, respectively. The number of droplets was set to 8 × 106, 1.5 × 107, 5 × 107, and 5 × 107 for Reλ = 127, 204, 322, and 531, respectively. For most of the simulations, the gravitational accelerations gi were set to zero in order to focus on the Reλ and St dependencies of turbulent clustering. However, we have also performed DNS experiments with (g1, g2, g3) = (0, g, 0), where g = 9.8 m s−2, to investigate the influence of gravitational droplet settling. Details of the numerical conditions are described in section 4d.
The code is fully parallelized for a three-dimensional domain decomposition using a Message Passing Interface (MPI) library (Onishi et al. 2013). The largest simulation (i.e., the case Reλ = 531) was performed on 32 nodes of the Earth Simulator 2 supercomputer operated by the Japan Agency for Marine-Earth Science and Technology (JAMSTEC).
3. Radar reflectivity factor
4. Results and discussion
a. Droplet distribution in turbulence
Figure 1 shows the spatial distributions of droplets within the range 0 < z < 4lη, where lη = (ν3/ϵ)1/4 is the Kolmogorov scale, for St = 0.05, 0.2, 1.0, and 5.0 at Reλ = 204. The number of the particles in each figure is similar; about 3.5 × 104. Void areas due to turbulent clustering are clearly observed for St = 1.0. For St < 1.0, the void areas are less clear. For St > 1.0, small void areas with dimensions less than 40lη are less clear than for St = 1.0, but void areas larger than 40lη are more prominent.
Spatial distributions of droplets obtained by DNS for St = (a) 0.05, (b) 0.2, (c) 1.0, and (d) 5.0 at Reλ = 204. Only droplets in the range 0 < z < 4lη are drawn.
Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0368.1
Figure 2 shows Enp(k) for St = 1.0, for different values of Reλ. The arrow indicates the range of the nondimensional wavenumber relevant to actual radar observations, which we can estimate from the range of fm used, and the typical lη that apply in atmospheric clouds. The microwave frequencies used for radar observations of clouds or precipitation range from the S band (fm ~ 2 GHz) to the W band (fm ~ 100 GHz). The typical lη in atmospheric clouds ranges from 5 × 10−4 to 1 × 10−3 m, which we estimate based on the energy dissipation rate ϵ ~ 10−3–10−2 m2 s−3 and ν ~ 10−5 m2 s−1. Since Enp(k = 2km) is used for estimating Zcluster for fm = kmcm/2π, where cm is the speed of light, the relevant wavenumber range for radar observations is estimated to be 0.05 < klη < 4.0.
Power spectra of droplet number density fluctuation obtained from DNS data for St = 1.0 at Reλ = 127, 204, 322, and 531.
Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0368.1
In atmospheric clouds, Reλ ranges from 103 to 104, higher than the maximum Reλ value (=531) used within our simulations. However, for the wavenumber range 0.05 < klη < 4.0, the maximum difference between Enp(k) values for Reλ = 204 and 531 in Fig. 2 is 11%, while for Reλ = 127 and 531, the maximum difference is 22%. These differences correspond to differences of 0.47 and 1.1 dB in the increment to Z given by Eq. (7), respectively, where a value in units of decibels is defined as AdB = 10 log10A for a given value of A. Since errors of around 1 dB are unavoidable in radar observations (Bringi et al. 1990; Carey et al. 2000), the dependency of Enp(k) on Reλ is sufficiently small for Reλ > 200 and thus for the wavenumber range relevant for radar observations. Thus, this study uses Enp(k) at Reλ = 204 to estimate Z for radar observations of atmospheric clouds.
Figure 3 shows Enp(k) of droplet number density fluctuations for different values of St at Reλ = 204. The horizontal and vertical axes are normalized using lη and the average number density 〈np〉. It is clear that Enp(k) depends strongly on St. For St ≤ 1.0, the peak values of Enp(k) are located around klη = 0.2 [i.e., (klη)peak ≈ 0.2] and become higher as St becomes closer to 1. This indicates that the representative void scale is almost constant, but that the number density difference between sparse (void) and dense (cluster) areas increases as St increases. This is because the number density of inertial particles tends to concentrate more in high-strain-rate and low-vorticity regions as τp increases (Maxey 1987). Since the Kolmogorov-scale eddies have the largest effect on the motions of St < 1 droplets, (klη)peak is almost fixed at about 0.2. On the other hand, for St ≥ 1.0 the peak location moves toward lower wavenumbers as St increases, indicating that the representative void scale becomes larger as St increases. This is because large-scale eddies preferentially concentrate large St droplets, and small-scale eddies tend to destroy this preferential concentration by uncorrelated stirring. This scale dependent clustering mechanism is explained by Goto and Vassilicos (2006) and Yoshimoto and Goto (2007). These features for St ≤ 1.0 and St ≥ 1.0 are consistent with what is observed in Fig. 1. Jin et al. (2010) examined the St dependency of Enp(k). Their power spectra show generally good agreement with ours, confirming the reliability of our simulation. It should be noted that Jin et al. (2010) used Reλ = 102, which is too small for us to use their spectra to estimate the influence of turbulent clustering on radar observations.
Power spectra of droplet number density fluctuation obtained from DNS data for (a) St ≤ 1 and (b) St ≥ 1 at Reλ = 204. The small arrows indicate the peak location of each spectrum.
Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0368.1
b. Influence of turbulent clustering on the radar reflectivity factor
Figure 4 shows clustering coefficients ζ for Reλ = 204. The horizontal axis is the microwave wavenumber difference normalized by lη. The vertical axis is normalized by 
Clustering coefficients obtained from the Enp(k) data for Reλ = 204 shown in Fig. 3 (thick lines) and those estimated by Dombrovsky and Zaichik (2010) (thin lines), which are only applicable for St < 0.6.
Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0368.1
c. Modeling of the turbulent clustering influence on radar reflectivity factor
Model parameters in Eq. (22) vs the Stokes number: (a) c1, (b) α, (c) c2, and (d) β. The solid lines show the best-fit curves given by Eqs. (23) and (24).
Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0368.1
Model parameter γ in Eq. (22) vs the Stokes number. The solid line shows the average γ value (1.6) obtained when the rightmost point is excluded.
Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0368.1
RMS error of the proposed power spectrum model Enp,model(k). The error is evaluated within the wavenumber range relevant for radar observations: 0.05 < klη < 4.0.
Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0368.1
d. Influence of gravitational settling on the power spectra of number density fluctuations
The influence of gravitational settling on the power spectra of number density fluctuations has been investigated by performing additional DNSs with gravity included. Nondimensional parameters relevant for gravitational effects are Sυ = υT/uη—where υT is the terminal velocity given by τpg and uη is the Kolmogorov velocity (Wang and Maxey 1993; Grabowski and Vaillancourt 1999)—and the Froude number (Fr = υT/urms); Sυ measures the settling influence on small scales and Fr measures it on large scales. Strictly speaking, we need multiple parameters covering the wide range of clustering scales. However, we consider these two parameters—covering the two ends of the scale range—to be sufficient for our analysis. Table 2 shows the values of Sυ and Fr in the additional DNS runs. Reλ was set to 204 and St to unity.
Numerical conditions for the cases with gravity included: Sυ is the nondimensional terminal velocity defined by Sυ = υT/uη and Fr is the Froude number defined by Fr = υT/urms. Reλ and St were set to 204 and 1.0, respectively.
Figure 8 shows the settling influence on Enp(k). As the particle settling becomes stronger, Enp(k) decreases at small scales and increases at large scales. The decrease at small scales corresponds to the increase of Sυ and indicates that settling weakens small-scale clustering (Ayala et al. 2008a,b; Woittiez et al. 2009). The increase at large scales, on the other hand, corresponds to the increase of Fr and indicates that anisotropies generated by settling lead to large-scale clustering. [Woittiez et al. (2009) observed a nearly-two-dimensional “curtain shape” clustering.] However, the increase at large scales is outside of the wavenumber range relevant for radar observations, so we need only consider Sυ. The maximum differences between Enp(k) for Sυ > 0 and Sυ = 0 are 0.35, 0.68, 1.4, and 2.2 dB for Sυ = 1.37, 2.71, 6.88, and 11.1, respectively. That is, the errors of the proposed model are smaller than 1 dB for Sυ ≤ 2.7 ≈ 3. Thus the proposed model is reliable for Sυ < 3.
Influence of gravitational settling on the power spectra of droplet number density fluctuation for St = 1.0 at Reλ = 204.
Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0368.1
e. Turbulent clustering influence in radar observations estimated by the proposed model
Recent radar observations of clouds and precipitation have been conducted using microwaves in six frequency bands: the S, C, X, Ku, Ka, and W bands, with typically used frequencies of 2.8, 5.3, 9.4, 13.8, 35, and 94 GHz, respectively. The S-, C-, X-, and Ku-band radars are often used for observing precipitation, while Ka- and W-band radars are used only for clouds. This is because the Rayleigh scattering approximation is invalid when dp/λm is larger than about 
Figure 9 shows values of 
Influence of turbulent clustering on the radar reflectivity factor (i.e., 
Citation: Journal of the Atmospheric Sciences 71, 10; 10.1175/JAS-D-13-0368.1
For Sυ > 3 however, the influence of turbulent clustering would be overestimated significantly owing to the influence of gravitational settling, as discussed in section 4d. The threshold Sυ = 3 corresponds to rp ≈ 20 μm for the stratocumulus case and to rp ≈ 30 μm for the cumulus case. The possible overestimates for large rp cannot be ignored, but they do not affect our main argument that the influence of turbulence can cause a significant error in radar observations using the S, C, X, and Ku bands.
Figure 9 indicates that the radar reflectivity factor becomes larger as fm becomes lower and that the maximum difference between the S and X bands is approximately 8 dB. These characteristics are in good agreement with the observations of developing cumulus clouds by Knight and Miller (1998), in which the reflectivity factor for the S band is about 10 dB larger than for the X band. As mentioned in the introduction, a similar frequency dependency was found for the case of smoke plumes of an industrial fire by Rogers and Brown (1997). Although the constituents and sizes of smoke particles are different from cloud droplets, turbulence could also influence the radar reflectivity factor. Thus, as speculated by Kostinski and Jameson (2000) based on the theory of particulate Bragg scattering, turbulent clustering may influence the radar observation of cumulus clouds and smoke plumes.
5. Conclusions
This study has investigated the influence of microscale turbulent clustering of cloud droplets on the radar reflectivity factor and proposed an empirical parameterization to account for it. Three-dimensional direct numerical simulations (DNS) of particle-laden isotropic turbulence were performed in order to obtain turbulent clustering data, from which power spectra of droplet number density fluctuations were calculated. The calculated power spectra show dependencies on the Taylor microscale-based Reynolds number (Reλ) and the Stokes number (St). To begin, we investigated the dependency of the turbulent clustering influence on Reλ. Results for a wide range of Reλ values (up to 531) reveal that Reλ = 204 is large enough to be representative of the whole wavenumber range relevant to radar observations of atmospheric clouds (0.05 < klη < 4, where k is the wavenumber and lη is the Kolmogorov scale). (Smaller Reλ values were found to be unable to represent the power spectrum for low wavenumbers.) Setting Reλ = 204, we then investigated the dependency on St. We observed that for St < 1 the peak of the power spectrum is located at around klη = 0.2 with the peak value increasing as the Stokes number increases toward unity. For St > 1, the peak location moves to lower wavenumbers as St increases. Based on these observations, and assuming that the power spectrum follows distinct power laws in the small- and large-wavenumber regions, we proposed an empirical model that approximately fits the power spectrum of number density fluctuations Enp(k). From this model, it was then possible to calculate the clustering coefficient ζ (i.e., the influence of turbulence on the radar reflectivity factor). A comparison between the model estimates and the DNS results for Enp(k) confirms the reliability of the model for droplets with Stokes number smaller than 2. For larger Stokes number droplets, the model estimate has larger errors, but the influence of turbulence of such large droplets is likely negligible in typical clouds. The proposed model has been applied to two idealized radar-observation scenarios: (i) a stratocumulus case, where lη = 1 × 10−3 m, and (ii) a cumulus case, where lη = 5 × 10−4 m. In both cases, the droplet volume fraction was 10−6 and the microwave frequency fm ranged from 2.8 to 94 GHz. The results show that the influence of microscale turbulent clustering on the radar reflectivity factor is significant for droplets with radius smaller than 100 μm for fm ≤ 9.4 GHz in the stratocumulus case and for fm ≤ 13.8 GHz in the cumulus case. That is, the influence of turbulent clustering can cause a significant error in retrieving cloud liquid water content from radar observations with microwave frequencies less than 13.8 GHz (S, C, X, and Ku bands). Additional DNSs with gravitational effects included reveal that the influence of gravitational settling causes significant errors in the model estimates when the nondimensional terminal velocity Sυ is larger than 3. These errors for large particles cannot be ignored but do not alter our main conclusions.
Acknowledgments
The authors thank Dr. Masaki Katsumata of the Japan Agency for Marine-Earth Science and Technology for his helpful comments on radar observations. The numerical simulations presented here were carried out on the Earth Simulator 2 and ICE X supercomputer systems operated by the Japan Agency for Marine-Earth Science and Technology. This study was supported by Grant-in-Aid for Young Scientists (A) (20686015) and by Grant-in-Aid for JSPS Fellows (21·241).
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