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.