Unified PDF Parameterization of Subgrid Moist Convection and Turbulence Using a Lagrangian Stochastic Approach

A unified probability density function (PDF) parameterization for subgrid moist convection and turbulence is developed using a Lagrangian stochastic modeling (LSM) approach. The model solves the transport equations of the joint PDF of turbulent velocity and passive scalars by tracking an ensemble of stochastic particles governed by coupled stochastic differential equations (SDEs). Building on previously developed SDEs for particle velocity and temperature, the LSM is extended to represent inhomogeneous stratified turbulence and its entrainment process. Furthermore, using Lagrangian particle tracking data obtained from large-eddy simulations (LES) of boundary layer and moist convection cases, the SDEs are refined and their parameters are optimized to reproduce the Lagrangian statistics diagnosed from the LES. In the proposed model, turbulence statistics and turbulent fluxes are obtained directly from particle ensembles, providing a full representation of the turbulence PDF without invoking traditional closure assumptions for turbulent transport. The proposed model is evaluated against LES results for convective and stable atmospheric boundary layer (ABL) cases, including shallow convection cases. In convective regimes, the LSM realistically captures entrainment processes and reproduces mean thermodynamic profiles and turbulent fluxes that closely agree with LES results. The simulated joint PDFs exhibit pronounced non-Gaussian features and PDF separation in the entrainment zone. In stable ABL simulations, the LSM predicts realistic turbulence intensities and mean profiles, with near-Gaussian PDFs consistent with LES results. In the shallow convection case, the model simulates realistic vertical structures and variability of convection in the cloud layer. These results demonstrate that the proposed LSM framework provides a physically consistent and flexible approach for simulating both moist convection and turbulence with a full representation of the subgrid-scale PDF.