Cloud Modeling Tests of the ULTIMATE-MACHO Scalar Advection Scheme

Numerical diffusion can be minimized using fine grid spacing and/or higher-order numerical schemes. In this study, the authors focus on higher-order scalar advection schemes and their effects on simulated cloud fields. A monotonic multidimensional odd-order conservative advection scheme has been implemented, following the approach of Leonard. It has been tested in simulations of idealized scalar fields advected by simple prescribed motion, as well as turbulence fields; large-eddy simulations of turbulent stratocumulus clouds; and simulations of deep convective clouds. New third-, fifth-, and seventh-order schemes are compared with the second-order scheme originally used in the model. For the deep cumulus case, a high-resolution large-eddy simulation with the same domain size is used as a benchmark. The fifth-order scheme shows much less numerical diffusion than the lower-order scheme. The additional improvement with the seventh-order scheme is minor. The higher-order scheme generally produces simulated cloud fields similar to those obtained with a lower-order scheme with a finer grid spacing. This effect is especially noticeable for the updraft-core statistics of the deep cumulus simulation, as compared with the benchmark simulation. The fifth-order scheme with coarse horizontal resolution produces results close to those of the benchmark simulation. Compared to a high-resolution simulation with the low-order scheme, the numerical cost of the fifth-order simulation is smaller than a factor of 10.