Spectra of supersaturation and liquid water content (LWC) fluctuations in cloud turbulence are theoretically studied. Equations for the variance spectra are derived using the Lagrangian renormalized approximation and examined through asymptotic analysis. Our results show that the wave-number-dependent Damköhler number, defined as the ratio of the turbulent eddy turnover time to the phase relaxation time of the supersaturation, controls the scalar transfer functions and the supersaturation excitation by turbulence. Additionally, the supersaturation spectrum $E_s\left(k\right)$ has three power-law ranges. Here two ranges follow $k^{-5/3}$; however, their amplitudes differ depending on the large-scale Damköhler number and the input ratio of the supersaturation excitations in the associated range. The third range of $E_s\left(k\right)\propto k^{-1-2C_B{D}_{\mathrm{K}}}$ includes $C_{\mathrm{B}}$ and $D_{\mathrm{K}}$ representing the Bachelor constant and Damköhler number corresponding to the Kolmogorov time, respectively. The LWC spectrum $E_{\varphi }\left(k\right)$ reveals two power-law ranges: $k^{-5/3}$ for $k<k_{\varphi *}$ and $k^{-1}$ for $k_{\varphi *}<k$, in which the transition wave number $k_{\varphi *}\overline{\eta }$ is given by the input ratio of the supersaturation excitations in the two $k^{-5/3}$ ranges by the turbulence and is consistent with field measurement results.

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• Atmospheric fluid dynamics
• Turbulence
• Turbulent mixing