Toward a mode reduction strategy in shallow moist convection
The insufficient parameterization of low clouds which are caused by shallow convection remains one of the biggest sources of uncertainty in large-scale models of global atmospheric motion. One way to overcome this lack of understanding is to develop Boussinesq models of moist convection with simplified thermodynamics which allow for systematic studies of the cloud formation in different dynamical regimes and depend on a small set of system parameters only. This route makes the problem accessible to direct numerical simulations of turbulence without subgrid-scale modeling and provides an ideal testing bed for systematic and stepwise reductions of degrees of freedom. Such systematic reductions are studied here for a recently developed moist Rayleigh–Bénard convection model in the conditionally unstable regime. Our analysis is based on the proper orthogonal decomposition (POD) and determines the corresponding modes by a direct solution of the eigenvalue problem in form of an integral equation. The resulting reduced-order dynamical systems are obtained by a projection of the original equations of motion onto a finite set of POD modes. These modes are selected with respect to their energy as well as their ability to transport energy from large to small scales and to dissipate the energy at smaller scales efficiently such that an additional modal viscosity can be omitted for most cases. The reduced models reproduce important statistical quantities such as cloud cover, liquid water flux and global buoyancy transport to a very good degree. Furthermore we investigate different pathways to reduce the number of degrees of freedom in the low-dimensional models. The number of degrees of freedom can be compressed by more than two orders of magnitude until the models break down and cause significant deviations of essential mean transport quantities from the original fully resolved simulation data.