Impacts of Updraft Size and Dimensionality on the Perturbation Pressure and Vertical Velocity in Cumulus Convection. Part II: Comparison of Theoretical and Numerical Solutions and Fully Dynamical Simulations

Abstract This paper compares simple theoretical expressions relating vertical velocity, perturbation pressure, updraft size, and dimensionality for cumulus convection, derived in Part I, with numerical solutions of the anelastic buoyant perturbation pressure Poisson equation and vertical velocity w. A range of thermal buoyancy profiles representing shallow to deep moist convection are tested for both two-dimensional (2D) and three-dimensional (3D) updrafts. The theoretical expressions give similar results for w and perturbation pressure difference from updraft top to base Δp compared to the numerical solutions over a wide range of updraft radius R. The theoretical expressions are also consistent with 2D and 3D fully dynamical updraft simulations initiated by warm bubbles of varying width. Implications for nonhydrostatic modeling in the “gray zone,” with a horizontal grid spacing Δx of O(1–10) km where convection is generally underresolved, are discussed. The theoretical and numerical solutions give a scaling of updraft velocity with R (~Δx) consistent with fully dynamical 2D and 3D simulations in the gray zone, with a rapid decrease of maximum w at relatively small R and a slower decrease at large R. These results suggest that an incorrect representation of perturbation pressure may be an important contributor to biases in convective strength at these resolutions. The theoretical solutions also provide a concise physical interpretation of the “virtual mass” coefficient in convection parameterizations and can be easily incorporated into these schemes to provide a consistent scaling of perturbation pressure effects with R, updraft height, and the buoyancy profile.