Impacts of Updraft Size and Dimensionality on Cumulus Vertical Velocities: Implications for “Convection-permitting” Models and Convection Schemes

English Abstract:

The importance of updraft size and dimensionality (2D versus 3D) via perturbation pressure effects on cumulus dynamics has been long known. However, quantitative understanding is lacking and a simple, generalized theoretical framework describing these effects has not yet been developed and rigorously tested. To address this issue, approximate analytic solutions to the governing equations describing relationships between perturbation pressure, vertical velocity, updraft size, and buoyancy for 2D and 3D flows will be presented. They show a reduction of the updraft vertical velocity as a function of the square of a nondimensional length L_c ~ (R/H), where R and H are the updraft radius and height, respectively. Updrafts are also weaker in 2D than 3D because of inherent differences in mass continuity, with up to a factor of 2 difference in vertical velocity for L_c >> 1. These theoretical expressions are compared with numerical solutions calculated directly from the perturbation pressure and vertical momentum equations, as well as those from other approaches such as single normal mode expansion, for different buoyancy distributions representing shallow to deep convection. The theoretical expressions are also compared to 2D and 3D fully dynamical simulations with convection initiated by warm bubbles of varying size. Implications for nonhydrostatic “convection-permitting” models in the gray zone, with a horizontal grid spacing of ~1-10 km where updrafts are generally under-resolved, will be discussed. Results suggest that an incorrect representation of perturbation pressure because of updrafts that are too wide at these resolutions are an important contributor to biases in updraft vertical velocity. 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 an improved representation of perturbation pressure effects on updraft velocities.