Turbulent enhancement of radar reflectivity factor for polydisperse cloud droplets

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 r(p)(3)-weighted power spectrum, which is proportional to the clustering influence on the radar reflectivity factor to a sufficiently high accuracy. Furthermore, the proposed model is extended to incorporate the gravitational settling influence by modifying the critical wave number based on the analytical equation derived for the bidisperse radial distribution function. The estimate of the modified model also shows good agreement with the DNS results for the case with gravitational droplet settling. The model is then applied to high-resolution cloud-simulation data obtained from a spectral-bin cloud simulation. The result shows that the influence of turbulent clustering can be significant inside turbulent clouds. The large influence is observed at the near-top of the clouds, where the liquid water content and the energy dissipation rate are sufficiently large.