An Analysis of Klemp–Wilhelmson Schemes as Applied to Large-Scale Wave Modes

Abstract

Abstract The use of Klemp–Wilhelmson (KW) time splitting for large-scale and global modeling is assessed through a series of von Neumann accuracy and stability analyses. Two variations of the KW splitting are evaluated in particular: the original acoustic-mode splitting of Klemp and Wilhelmson (KW78) and a modified splitting due to Skamarock and Klemp (SK92) in which the buoyancy and vertical stratification terms are treated as fast-mode terms. The large-scale case of interest is the problem of Rossby wave propagation on a resting background state. The results show that the original KW78 splitting is surprisingly inaccurate when applied to large-scale wave modes. The source of this inaccuracy is traced to the compressible vertical adjustment—and more precisely, to the splitting of the hydrostatic balance terms between the small and large time steps. The splitting errors can be reduced somewhat through implicit biasing, but large biasing coefficients are needed for acceptable error levels—and even then the time steps are limited to moderate values. The errors in the KW78 splitting are shown to be largely absent from the SK92 scheme. Two versions of the SK92 splitting are considered in particular: the original leapfrog splitting (SK92-LF) of Skamarock and Klemp and the third-order Runge–Kutta splitting (SK92-RK) proposed by Wicker and Skamarock. The mixed cubic (on the large time step) and quadratic (on the small step) behavior of the SK92-RK scheme is described in detail and is compared with the strictly quadratic behavior of the SK92-LF method.