A Spectral Model for the Rainfall Variability Production

Large variability in precipitation across time scales ranging from seconds to multiple decades makes its representation a daunting challenge. While progress has been made in characterizing the temporal structure of precipitation, particularly its scaling property, linking its variance production mechanisms to the spectral shape of rainfall energy, $E(f)$, where $f$ is frequency, remains unaddressed. To bridge this gap, a scale-wise spectral energy transfer model based on a nonlinear advection-diffusion of energy is proposed. This model allows a representation of precipitation variance production that can assess the relative contributions of energy transfer internal to the cascade process and external stochastic forcing from different rainfall generation mechanisms. We apply this representation to long term high-resolution (5-minute) rainfall time series collected at multiple locations characterized by distinct dominant precipitation formation regimes, including orographic, frontal, and convective systems. The model results show that the energy spectrum of precipitation follows a quasi-Lorentzian shape with parameters directly linked to the precipitation formation process. The variance production term at $f$ is primarily controlled by external stochasticity associated with the rainfall formation mechanism rather than the specifics of the internal energy routing process. Guided by these findings, a predictability measure based on a normalized spectral Shannon entropy is derived. These measures demonstrate that orographic precipitation exhibits the highest temporal predictability among the studied precipitation formation mechanisms.