We explore some numerical issues arising in two different situations mostly due to unexpected bad choices of the numerical scheme. The first experiment explores a bad performance of the popular central scheme for conservation laws of Nessyahu and Tadmor (1990), in two spatial dimensions, when applied to equatorially trapped waves. We show that due to the excessive averaging across the direction of propagation, the central scheme distorts the dispersion relations of the underlying waves and results in significant deformations of the wave structure that amplify with time. The second experiment concerns one of the most popular cumulus parametrizations used in operational models for climate predictions, namely the Zhang-McFarlane (1995) scheme for cumulus convection. Numerical tests demonstrate that the standalone version of the Zhang-McFarlane (ZM) scheme exhibits steady grid-scale oscillations in the vertical that do not amplify or damp with time.
Because one of our future goals is to couple the ZM scheme to a stochastic model for convective inhibition and/or moisture preconditioning, it is important to be able to control such oscillation in order to distinguish the physically relevant noise induced by the stochastic model and the one that is due to pure numerical and/or model artifacts. To help elucidate this issue we introduce an idealized version of the scheme that is simple enough for facilitate analysis. We show that the idealized model exhibits weakly unstable modes that oscillate and grow with height at all vertical wavenumbers, which are believed to be reminiscent of the grid-scale oscillations seen in the ZM scheme. In fact, while the grid-scale oscillations are still present in the idealized model, the numerical scheme inherited from the original ZM model--a partially implicit staggered-grid scheme to conserve some form of column moist-static energy--is stable in the von Neumann sense though it is only slightly damped. Due to the weakness of the instability, the addition of a small amount of numerical diffusion reveals to be sufficient to damp the grid scale oscillations and interestingly the application of a MUSCL-type scheme, that are know to implicitly induce a small amount of numerical viscosity, has the same effect. Therefore, one can speculate that such oscillations may not be present in the full version of the ZM scheme as implemented is GCMs due to numerical diffusion induced through various other processes.
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