The radar reflectivity factor is important for estimating cloud
microphysical properties; thus, in this study, we determine the quantitative
influence of microscale turbulent clustering of polydisperse droplets on the
radar reflectivity factor. The theoretical solution for particulate Bragg
scattering is obtained without assuming monodisperse droplet sizes. The
scattering intensity is given by an integral function including the cross
spectrum of number density fluctuations for two different droplet sizes. We
calculate the cross spectrum based on turbulent clustering data, which are
obtained by the direct numerical simulation (DNS) of particle-laden
homogeneous isotropic turbulence. The results show that the coherence of the
cross spectrum is close to unity for small wave numbers and decreases almost
exponentially with increasing wave number. This decreasing trend is dependent
on the combination of Stokes numbers. A critical wave number is introduced to
characterize the exponential decrease of the coherence and parameterized using
the Stokes number difference. Comparison with DNS results confirms that the
proposed model can reproduce the

Radar remote sensing is widely used for observing a spatial distribution of
cloud and precipitation particles because it can also provide estimates of
cloud microphysical properties. The remote-sensing data are analyzed and
displayed using the radar reflectivity factor (mm

Therefore, this study aims to investigate the influence of turbulent clustering of polydisperse droplets on particulate Bragg scattering. Firstly, the theoretical formulation of particulate Bragg scattering is extended for polydisperse particles and expressed using the cross spectrum of number density fluctuations for two different droplet sizes. Secondly, the three-dimensional DNS of particle-laden homogeneous isotropic turbulence is performed to obtain turbulent droplet clustering data, which are used to calculate the power spectrum and the cross spectrum of number density fluctuations. A parameterization for the cross spectrum is then proposed considering the dependence of the Stokes number combination, and the influence of gravitational settling is discussed and incorporated. Finally, in order to investigate the impact of turbulent clustering on radar observations of realistic clouds, the proposed model is applied to high-resolution cloud-simulation data obtained by a spectral-bin cloud microphysics simulation.

Here, we aim to formulate the radar reflectivity factor

Similarly to

In order to decompose the spatial correlation function

It should be noted that Eq. (

In order to obtain turbulent clustering data for calculating the cross
spectrum, we have performed a three-dimensional DNS for particle-laden
homogeneous isotropic turbulence. Three-dimensional incompressible turbulent
airflows were calculated by solving the continuity and Navier–Stokes equations:

Droplet motions were simulated by Lagrangian point-particle tracking. Here,
we assumed that the droplet density

The computational domain was set as a cube with edge lengths of

Computational settings of the DNS.

For this study, we performed the DNS for monodisperse and polydisperse
droplets. Table

Spatial distribution of droplets for (orange)

The power spectral density function

The spectral density functions,

Normalized cross spectra

Coherence of cross spectra for combinations of

Figure

Critical wave number

According to the above discussion, we can estimate the increase in the radar
reflectivity factor due to turbulent clustering of polydisperse droplets in
Eq. (

Critical wave number

In order to evaluate the reliability of the proposed cross spectrum model,
the

Comparisons of

The parameterization summarized in the previous subsection was obtained under
the condition without gravitational droplet settling. The settling influence for the monodispersed cases was discussed by

For the cases of polydisperse particles, the settling influence on the
coherence term must be considered as well as the influence on

The reliability of the modified parameterization for the case with
gravitational settling has been evaluated in the same way as the previous
subsection. Figure

We have applied the proposed model to the high-resolution cloud-simulation
data of

The radar reflectivity factor

The spectrum

The spectrum

Three-dimensional visualization of cloud simulation data:

The contribution of clear-air Bragg scattering was calculated by

The scalar dissipation rates for

Figure

Liquid water content (LWC), energy dissipation rate

This study focused on a vertical cross section that slices the cumulus cloud
with the largest upward velocity. Figure

Figure

In Fig.

This study has investigated the influence of microscale
turbulent clustering of polydisperse cloud droplets on the radar reflectivity
factor. Firstly, the theoretical solution for particulate Bragg scattering
for polydisperse droplets has been obtained considering the droplet size
distribution in the measurement volume and the droplet size dependence of
turbulent clustering. The obtained formula shows that the particulate Bragg-scattering part of the radar reflectivity factor is given by a double
integral function including the cross spectrum of number density fluctuations
for bidisperse droplets. Secondly, the wave number and Stokes number
dependence of the cross spectrum has been investigated using the turbulent
droplet clustering data obtained from a direct numerical simulation (DNS) of
particle-laden homogeneous isotropic turbulence without gravitational
settling. The result shows that the cross spectrum for a combination of
Stokes numbers,

In order to develop a cross spectrum model for estimating the clustering
influence on the radar reflectivity factor, we have proposed an exponential
model for the wave number dependence of the coherence and introduced the
critical wave number (i.e., the decay constant for the model) to consider the
dependence of the coherence on the Stokes number combination. The coherence
data for all combinations of six Stokes numbers ranging from 0.05 to 2.0
reveal that the critical wave number is inversely proportional to the Stokes
number difference,

The proposed model has been further extended for the case with gravitational
settling. We have assumed

Finally, the proposed model has been applied to high-resolution
cloud-simulation data of

Access to the simulation and analysis results can be granted upon request under a collaborative framework with JAMSTEC.

KM designed the study, conducted the DNSs, developed the parameterization based n the DNS data, and analyzed the cloud simulation data. RO developed the DNS program, and provided the cloud simulation data. Both authors contributed to the discussion, interpretation of the results, and to writing the paper.

The authors declare that they have no conflict of interest.

This research was supported by JSPS KAKENHI grant number JP17K14598. The numerical simulations presented were carried out on the Earth Simulator supercomputer system of the Japan Agency for Marine-Earth Science and Technology (JAMSTEC). Edited by: Graham Feingold Reviewed by: two anonymous referees