Importance of temporal symmetry in spatial discretization for geostrophic adjustment in semi‐implicit Z‐grid schemes

Among the dynamical cores of numerical weather prediction communities, many different discretization methods can be distinguished to solve the equations governing the motions in the atmosphere numerically. One of them, the Z-grid approach, is based on solving the equations formulated in terms of divergence and vorticity on an Arakawa A-grid, a grid where all the variables are defined at the same grid points. To permit an efficient semi-implicit (SI) treatment, Z-grid schemes were proposed in the literature that first perform SI time discretization on the momentum equations formulated in terms of velocity components in order to construct from this a discretized divergence equation. This publication shows that a careful formulation of such SI Z-grid schemes is required to conserve appropriate dispersion relations for inertia–gravity, inertia–Lamb and Rossby waves. It is proven analytically for a two time-level (2TL) SI Z-grid scheme of the 1D shallow-water equations that the spatial discretization must respect temporal symmetry, meaning that the spatial discretization must be identical in the implicit and explicit parts of the scheme. If not, the discretized waves are damped or amplified and their phase and group velocity may be seriously distorted. These findings are discussed in detail and both 1D and 2D numerical tests are carried out to demonstrate that a symmetric formulation is an important modelling constraint in order to obtain an appropriate geostrophic adjustment.