Data-driven Modeling of Stratiform and Convective Rain Area

Understanding how the fractional area of different rain types—shallow, deep convective, and stratiform—varies with large-scale environmental conditions is fundamental to advancing our knowledge of moist atmospheric processes. Yet this relationship remains poorly quantified, especially at instantaneous timescales where observational uncertainty and inherent variability pose challenges. In this study, we tackle this problem by integrating satellite-derived rain area data from TRMM with environmental variables from ERA5, using a two-step framework that combines conditional normalizing flows (CNFs) with symbolic equation discovery. The CNF model reconstructs the full conditional joint distribution of three rain area types, capturing their co-dependence and reproducing observed features such as the bimodal relationship between convective and stratiform area. From these distributions, we extract the conditional means and then apply equation discovery to derive compact analytical expressions that relate rain area fractions to large-scale thermodynamic variables. The resulting equations reveal interpretable and physically consistent predictors—such as lower-tropospheric stability, free-tropospheric moisture deficit, and boundary layer buoyancy—that act as both triggering thresholds and predictors of rain area magnitude. These expressions capture well-known mechanisms in closed form, opening a path toward stochastic, physically grounded parameterizations.  Our phase-space analysis clarifies how rain types evolve across sea surface temperatures and stability regimes, linking environmental variability to the spatial extent of convective systems—crucial for precipitation prediction and cloud–radiation interactions. Together, this framework provides a probabilistically robust, physically based approach for modeling rain area, with implications for both process understanding and parameterization.