Shallow cumulus convection large-eddy simulations and Galilean invariance

The effects of clouds forming near the surface are one of the largest sources of uncertainty in climate projections. Atmospheric boundary layer clouds have a significant impact on the planetary energy balance. A small change in the cloud area coverage affects the global energy budget similar in magnitude to that of the anthropogenic emissions of greenhouse gases. Large-eddy simulation (LES) is currently the best available cloud modeling technique because the range of flow scales is too large for direct numerical simulation (DNS) methods. We discuss the role of LES in the development and evaluation of turbulent convection parameterizations for weather and climate models and present a study of numerical model error in cumulus convection LES. A computational-domain translation velocity can be used to improve the LES performance by allowing longer time-step intervals. The continuous equations are Galilean invariant. However, standard finite-difference-based discretizations are not discretely invariant. Even though such numerical errors are expected to be small, it is shown that in LES of buoyant convection, the turbulent large-scale flow organization can modulate and amplify the error. In LES of single-phase convection under an inversion, flow statistics are nearly Galilean invariant and do not depend on the order of accuracy of the finite difference approximation. In contrast, in LES of cloudy convection, flow statistics show a strong dependence on the frame of reference and the order of approximation. Schemes with low resolving power can produce large dispersion errors in the surface-fixed frame that can be amplified by large-scale flow anisotropies, such as strong updrafts rising in a non-turbulent free troposphere in cumulus-cloud layers.