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1. Introduction
Cloud models are essential tools for the study of cloud processes. Cloud models can be split into two major components: the dynamical core and the physics. The dynamical core includes the numerical representation of the governing equations for momentum and scalars, including time-stepping and advection. Model physics represents parameterized physical processes such as microphysics, radiation, and diffusion.
Since diffusion is a physical process, a subgrid-scale physical parameterization should be used to represent diffusion in a model. In reality, however, numerical diffusion exists in virtually all cloud models, as a result of the numerical schemes used for advection. Numerical diffusion can cause numerical entrainment, which dilutes and evaporates clouds.
One way to reduce numerical diffusion is to use a small grid spacing. This method works but is expensive. For instance, Khairoutdinov et al. (2009) simulated deep convective clouds with a 100-m horizontal grid spacing for a 204.8-km domain width with 256 layers. This simulation was 256 times as expensive as a run with a 1.6-km horizontal grid spacing, if the other parameters are unchanged.
For cloud models, both momentum and scalars must be advected. The momentum advection scheme is often formulated so as to satisfy momentum and kinetic energy conservation. The scalar advection scheme is often designed to satisfy scalar conservation, shape preservation, and monotonicity.
The second- and third-order advection schemes have been widely used for cloud models [e.g., the multidimensional positive definite advection transport algorithm (MPDATA; Smolarkiewicz and Grabowski 1990), the piecewise parabolic method (PPM; Carpenter et al. 1990), the Uniformly Third-Order Polynomial Interpolation Algorithm (UTOPIA; Leonard et al. 1993), and the Monotone Upstream-centered Schemes for Conservation Laws (MUSCL; van Leer 1997). Recently higher-than-third-order schemes have been developed and tested (e.g., Skamarock 2006; Blossey and Durran 2008; Wang et al. 2009)].
In this study, we focus on the effects of numerical accuracy on scalar advection in the context of cloud simulations. We have developed a monotonic multidimensional higher-order advection scheme by utilizing the methods created by Leonard (1991) and Leonard et al. (1996). For this new scheme, only odd-order schemes were used; odd-order schemes are space-uncentered schemes, which minimize numerical dispersion. Leonard (1991) showed that even-order schemes gives highly oscillatory solution, and they are much more sensitive to Courant number than odd-order schemes....