Supersaturation in clouds was modeled as a passive scalar with the phase-relaxation time under mean uniform vertical gradient. The theoretical prediction that there exist two $-5/3$ power-law ranges in the supersaturation variance spectrum was examined using direct numerical simulation (DNS) and large-eddy simulation (LES). Although DNS at a Taylor microscale Reynolds number of 946 suggested the existence of two $-5/3$ ranges, the results were not conclusive because of the limited extent of the inertial-convective range. To achieve a wider inertial-convective range, LES was employed, and a modified eddy diffusivity model for a passive scalar without a phase-relaxation term was developed to yield a $k^{-5/3}$-spectrum across a broad range of wave numbers. LES results obtained by varying the phase-relaxation time demonstrated the coexistence of two $k^{-5/3}$ spectral ranges with different amplitudes. As the phase-relaxation time decreased, the transition between the two ranges shifted toward higher wave numbers, which is consistent with theoretical predictions. Furthermore, the probability density function (PDF) of the band-pass-filtered supersaturation indicated that the PDF tails at small scales became increasingly elongated depending on the ratio of the turbulence timescale to the phase-relaxation time.

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• Isotropic turbulence
• Turbulence modeling
• Turbulence simulations
• Direct numerical simulations
• Large eddy simulations