Determining the effective resolution of advection schemes. Part I: Dispersion analysis

The effective resolution of a numerical scheme describes the smallest spatial scale (largest wavenumber) that is completely resolved by that scheme. Using dispersion relation analysis allows the effective resolution of a numerical scheme for the advection equation to be calculated. The advection equation is a fundamental building block of dynamical cores of atmospheric and ocean models, and this analysis provides an indication of the effective resolution of the numerical methods used by dynamical cores. Using a variety of finite-difference schemes, the effect on effective resolution of using explicit diffusion and hyper-diffusion terms is examined. The choice of order-of-accuracy, and the time-stepping of the numerical scheme is also investigated with regard to effective resolution. Finally, we apply this analysis to methods that are commonly used in dynamical cores of atmospheric general circulation models, namely semi-Lagrangian and finite-volume methods.