Influence of Turbulent Droplet Clustering on Microwave Scattering
Keigo Matsuda 1, Masaaki Hirahara 2, Ryo Onishi 1, Keiko Takahashi 1, Ryoichi Kurose 2, and Satoru Komori 2
12

1

Earth Simulator Center, JAMSTEC, Japan
Department of Mechanical Engineering and Science, Graduate School of Engineering, Kyoto University, Japan

Introduction

Clouds play crucial roles in the heat and water systems ofthe Earth. A large number of observations have been conducted
to understand the cloud physics. The radar observation is one ofthe powerful tools since it can provide two- or threedimensional data regarding the cloud microphysical properties
such as the cloud water mixing ratio and effective droplet radius[1, 2]. In the radar observation, microwave is transmitted froman antenna toward a target cloud and the reflected microwave isreceived and analysed. The relation between the transmitted
power, Pt, and the received power, Pr, of the microwave is given
as the following radar equation:
22
Pt G 2 k MW K V(1)
Pr =Z,
45 R 4
where G is the antenna gain, kMW the wavenumber ofmicrowave, R the distance between the antenna and the cloud, K
the dielectric coefficient of a water droplet, and V themeasurement volume. Z is the radar reflectivity factor. An
important point is that Z is dependent on the cloud
microphysical properties. This means that the properties can be
estimated from Z under some assumptions: homogeneity andrandomness of cloud droplet distribution are usually assumed.
However, cloud turbulence forms inhomogeneous spatial
distribution of droplets, which is often referred to turbulent
clustering or preferential concentration [3, 4, 5]. Turbulent
clustering obviously contradicts the homogeneity assumption for
the analysis of Z. This may cause significant observational error.
This study, therefore, aims to investigate the effect ofturbulent clustering of droplets on the radar reflectivity factor,
i.e. microwave scattering. At first, a radar reflectivity factor
model, which takes account of turbulent clustering, is developed
based on the data of a three-dimensional direct numericalsimulation (DNS) of particle-laden isotropic turbulence. The
model is then applied to high-resolution cloud-simulation data inorder to estimate the possible error due to turbulent clustering ina radar observation.
2DNS for Isotropic Turbulence

Turbulent air flow was simulated by using a pseudo-spectral
method based on the Fourier–Galerkin method. The detail ofnumerical method is described in Onishi et al. [7]. Statisticallysteady-state turbulence was formed by applying an external
force to maintain the intensity of large-scale eddy.
Droplet motions were tracked by the Lagrangian method.
The governing equation of the motion is dvi/dt = −(vi−ui)/τp,
where ui is the fluid velocity, vi the droplet velocity, and τp thedroplet relaxation time. Droplets were assumed as Stokes
particles, whose relaxation time τp is given as τp =(ρp/ρg)(2r2/9ν), where r is the droplet radius, ρp the waterdensity, ρg the air density, and ν the kinematic viscosity. Effects
of gravity and collisions between droplets were neglected [8].
The computational domain was set to a cube with length of
2πL0, where L0 is the representative length. Periodic boundary
conditions were applied in all three directions. Table 1 shows
computational conditions for air turbulence and statisticalproperties obtained from DNS. Droplet radius was varied so that
the Stokes number, St, which is the ratio of the droplet
relaxation time, τp, to the Kolmogorov time scale, τη, becomes
0.05, 0.1, 0.2, 0.5, 1.0, 2.0 and 5.0.
Table 1. Numerical conditions and flow properties. urms is the RMS
value of velocity fluctuation, Re the Reynolds number defined as Re =L0urms/ν, lη the Kolmogorov scale, lλ the Taylor’s microscale, Reλ theturbulent Reynolds number defined as Reλ = lλurms/ν.
L0 [m]

urms [m/s]Relη [10-4 m]

lλ [10-3 m]

Reλ

0.020

0.32

427

2.65

6.45

152

3

Radar Reflectivity Factor

Radar reflectivity factor, Z, is calculated by the following
equation [6, 9]:
2π 2 (2r ) 6(2)Z = np (2r ) 6 +E np (κ ) ,2κ

where np is the droplet number density and κ is given as κ=2kMW.
Enp(k) is the power spectrum of droplet number density
fluctuations. The second term in the right-hand side describes
the increment of Z due to droplet clustering. That is, Enp(k)represents the effect of turbulent clustering. Unfortunately, there
has not been a widely accepted analytical model for Enp(k) [10,11, 12]. Thus, in this study, Enp(k) was calculated from the DNS
data as1~(3)
E np (k ) =∑ Φ (k ) ,∆k k − 12 ∆k ≤ k ≤k + 12 ∆k~where Φ (k ) is the spectral density function of droplet number
density fluctuation given as~
Np
NpΦ (k )
1=exp (− ik ⋅ x p,i ) ∑ exp (ik ⋅ x p, j ) , (4)∑22
3
N p i =1
j =1, j ≠ i
np L0
where xp,i is the i-th droplet position vector, np the droplet
number density, Np the number of droplets. Brackets, < >,represents the ensemble average. Eq. (4) was calculated fordiscrete wavenumber vector, k, which satisfies k = (l/L0, m/L0,n/L0) for arbitral integer numbers l, m and n.
44.1

Results and Discussion

Droplet Distributions in TurbulenceFig. 1 shows the spatial distributions of droplets within therange of 0<z<4lη for St=0.05, 0.2, 1.0 and 5.0. Turbulent
clustering is clearly observed for St=1.0, and less clear forsmaller and larger St. It is obvious that the cluster scale for
St=5.0 is larger than that for St=1.0.
Fig. 2 shows the power spectra of droplets number density
fluctuation, Enp(k). The horizontal and vertical axes arenormalized by the Kolmogorov scale, lη, and the average
number density, <np>. Enp(k) strongly depends on St, and does
not show the −5/3 power law, which is often observed in a scalar
concentration spectrum. This is because the fluctuation ofZ [dBZ]
20100-10-20-30(a)-40(b)
(a)

(b)

(c)

Fig. 3. Distributions of radar reflectivity factor, Z [dBZ], onx-z plane: (a) visualization of clouds; (b) Z without clustering
effect; (c) Z with clustering effect.

(c)

(d)

101

1002

Enp(k)/(〈np〉 lη)

Fig. 1. Spatial distributions of droplets obtained by DNS for(a) St=0.05, (b) 0.2, (c) 1.0, (d) 5.0. Droplets located within
the thin layer, whose z ranges 0<z<4lη, are drawn.

turbulence has been performed to obtain clustering data forconstructing a radar reflectivity factor model. The model isbased on empirical fitting to the power spectrum of droplet
number density fluctuation obtained from the DNS. One
significant feature of the model is that it considers the Stokes
number dependency. The model has been applied to a 2.8 GHz
microwave emitted on the high-resolution cloud-simulation dataobtained by the MSSG model. The result has revealed that theturbulent clustering significantly increases the radar reflectivity
factor for the 2.8 GHz microwave. This indicates that turbulent
clustering may well cause a significant error in microwave radarobservations.

St=0.05
St=0.1
St=0.2
St=0.5St=1.0St=2.0St=5.0-1

10-210-1

01010klη
Fig. 2. Power spectra of droplet number density fluctuation
obtained by DNS.
droplet number density is mainly produced by small-scaleeddies. In order to evaluate Z by Eq. (2), an empirical model ofEnp(k) has been developed as a function of klη and St by fitting
curves to the power spectra in Fig. 2.
4.2

Influence on Radar Observation of Clouds
In order to investigate the influence of turbulent clustering
on a radar observation, this study evaluated the radar reflectivity
factor for the high-resolution cloud-simulation data of Onishi
and Takahashi [13]. They have performed a mesoscale
simulation of convective clouds over the ocean by using theMSSG (Multi-Scale Simulator for the Geoenvironment) modelof the Earth Simulator Center (ESC). The domain size is 12.8 ×
12.8 × 4.0 km. A spectral-bin scheme is used for water droplets
to take account of droplet size distributions explicitly.
In this study, the droplet size distribution in each grid cell
was taken into account for calculating Z. The frequency ofmicrowave is set to 2.8 GHz in S-band, which is often used forradar observations. Fig. 3 shows the volume rendering of clouds
and radar reflectivity factor with and without the effect of
turbulent clustering in the domain with 2.4 × 0.1 × 4.0 km.
Comparison between Fig. 3(b) and (c) clearly shows that
turbulent clustering dramatically increases the radar reflectivityfactor. This result indicates that turbulent clustering may wellcause a significant error in the microwave radar observations.
5Conclusions

This study investigated the effect of turbulent clustering ofdroplets on radar reflectivity factor. A three-dimensional direct
numerical simulation (DNS) for particle-laden isotropic

Acknowledgements
This study was supported by Grant-in-Aid for Young
Scientists (A) (No. 20686015) and for JSPS Fellows (No.
21·241).
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