An Improved Perturbation Pressure Closure in the Extended EDMF Framework

The extended eddy-diffusivity mass-flux (EDMF) framework is developed as a unified parameterization for all subgrid-scale turbulence and convection in global climate models. The prognostic equations for vertical momentum and turbulence kinetic energy (TKE) in the EDMF framework require a consistent closure formulation for perturbation pressure, which has long been a challenge in convective parameterizations. This work derives an analytical solution for the perturbation pressure from idealized 2-D and 3-D rising bubbles. The derivation leads to a new closure, which combines modified pressure drag and virtual mass effects with a new momentum advection term. This momentum advection is an important source at the lower half of the thermal bubble and at cloud base levels in convective systems. It represents the essential physics of the perturbation pressure, that is, to ensure the 3D non-divergent properties of the flow. Moreover, the new formulation modifies the pressure drag to be inversely proportional to updraft depth. This is found to significantly improve simulations of the diurnal cycle of deep convection, without compromising the simulation of shallow convection, and thus is a key step towards a unified scheme. By assuming that the pressure only re-distributes TKE between subdomains, rather than doing work on the grid mean, the velocity pressure-gradient correlation closure is obtained from the perturbation pressure closure. This novel pressure closure is implemented in the extended EDMF, which successfully simulates a rising bubble test case in a single column model.