To consider the growth of cloud droplets by condensation in turbulence, the Fokker-Planck equation is derived for the droplet size distribution (droplet spectrum). This is an extension of the statistical theory proposed by Chandrakar and coauthors in 2016 for explaining the broadening of the droplet spectrum obtained from the "Π-chamber", a laboratory cloud chamber. In this Fokker-Planck equation, the diffusion term represents the broadening effect of the supersaturation fluctuation on the droplet spectrum. The aerosol (curvature and solute) effects are introduced into the Fokker-Planck equation as the zero flux boundary condition at R 2 = 0, where R is the droplet radius, which is mathematically equivalent to the case of Brownian motion in the presence of a wall. The analytical expression for the droplet spectrum in the steady state is obtained and shown to be proportional to R exp (-cR 2), where c is a constant. We conduct direct numerical simulations of cloud droplets in turbulence and show that the results agree closely with the theoretical predictions and, when the computational domain is large enough to be comparable to the Π-chamber, agree with the results from the Π-chamber as well. We also show that the diffusion coefficient in the Fokker-Planck equation should be expressed in terms of the Lagrangian autocorrelation time of the supersaturation fluctuation in turbulent flow.

In this paper, we report on the successful and seamless simulation of turbulence and the evolution of cloud droplets to raindrops over 10 minutes from microscopic viewpoints by using direct numerical simulation. Included processes are condensation-evaporation, collision-coalescence of droplets with hydrodynamic interaction, Reynolds number dependent drag, and turbulent flow within a parcel that is ascending within a self-consistently determined updraft inside a cumulus cloud. We found that the altitude and the updraft velocity of the parcel, the mean supersaturation, and the liquid water content are insensitive to the turbulence intensity, and that when the turbulence intensity increases, the droplet number density swiftly decreases while the spectral width of droplets rapidly increases. This study marks the first time the evolution of the mass density distribution function has been successfully calculated from microscopic computations. The turbulence accelerated to form a second peak in the mass density distribution function, leading to the raindrop formation, and the radius of the largest drop was over 300 μm at the end of the simulation. We also found that cloud droplets modify the turbulence in a way that is unlike the Kolmogorov-Obukhov-Corrsin theory. For example, the temperature and water vapor spectra at low wavenumbers become shallower thank 5 3 in the inertial-convective range, and decrease slower than exponentially in the diffusive range. This spectra modification is explained by nonlinear interactions between turbulent mixing and the evaporationcondensation process associated with large numbers of droplets.

A new time scale for turbulence modulation by particles is introduced. This time scale is inversely proportional to the number density and the radius of particles, and can be regarded as a counterpart of the phase relaxation time, an important time scale in cloud physics, which characterizes the interaction between turbulence and cloud droplets by condensation–evaporation. Scaling analysis and direct numerical simulations of dilute inertial particles in homogeneous isotropic turbulence suggest that turbulence modulation by particles with a fixed mass-loading parameter can be expressed as a function of the Damköhler number, which is defined as the ratio of the turbulence large-eddy turnover time to the new time scale.

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The time-evolution of two-dimensional decaying turbulence governed by the long-wave limit, in which LD/L → 0, of the quasi-geostrophic equation is investigated numerically. Here, LD is the Rossby radius of deformation, and L is the characteristic length scale of the flow. In this system, the ratio of the linear term that originates from the β-term to the nonlinear terms is estimated by a dimensionless number, \documentclass[12pt]{minimal}\begin{document}$\gamma =\beta L_{\rm D}^2/U$\end{document}γ=βLD2/U, where β is the latitudinal gradient of the Coriolis parameter, and U is the characteristic velocity scale. As the value of γ increases, the inverse energy cascade becomes more anisotropic. When γ ⩾ 1, the anisotropy becomes significant and energy accumulates in a wedge-shaped region where \documentclass[12pt]{minimal}\begin{document}$|l|>\sqrt{3}|k|$\end{document}|l|>3|k| in the two-dimensional wavenumber space. Here, k and l are the longitudinal and latitudinal wavenumbers, respectively. When γ is increased further, the energy concentration on the lines of \documentclass[12pt]{minimal}\begin{document}$l=\pm \sqrt{3}k$\end{document}l=±3k is clearly observed. These results are interpreted based on the conservation of zonostrophy, which is an extra invariant other than energy and enstrophy and was determined in a previous study. Considerations concerning the appropriate form of zonostrophy for the long-wave limit and a discussion of the possible relevance to Rossby waves in the ocean are also presented.