Abstract
Convection parameterizations such as eddy-diffusivity mass-flux (EDMF)
schemes require a consistent closure formulation for the perturbation
pressure, which arises in the equations for vertical momentum and
turbulence kinetic energy (TKE). Here we derive an expression for the
perturbation pressure from approximate analytical solutions for 2D and
3D thermal bubbles. The new closure combines modified pressure drag and
virtual mass effects with a new momentum advection damping term. This
advection damping is an important source in the lower half of the
thermal bubble and at cloud base levels in convective systems. It
represents the effect of the perturbation pressure to ensure the
non-divergent properties of the flow. The new formulation represents 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 simulations of shallow convection. It
is thus a key step toward a unified scheme for a range of convective
motions. By assuming that the pressure only redistributes TKE between
updrafts and the environment laterally, a closure for the velocity
pressure-gradient correlation is obtained from the perturbation pressure
closure. This novel pressure closure is implemented in an extended EDMF
scheme and is shown to successfully simulate a rising bubble as well as
shallow and deep convection in a single column model.