Impacts of the horizontal and vertical grids on the numerical solutions of the dynamical equations – Part 1: Nonhydrostatic inertia–gravity modes

Abstract

Abstract. We have used a normal-mode analysis to investigate the impacts of the horizontal and vertical discretizations on the numerical solutions of the nonhydrostatic anelastic inertia-gravity modes on a midlatitude f plane. The dispersion equations are derived from the linearized anelastic equations that are discretized on the Z, C, D, CD, (DC), A, E and B horizontal grids, and on the L and CP vertical grids. The effects of both horizontal grid spacing and vertical wavenumber are analyzed, and the role of nonhydrostatic effects is discussed. We also compare the results of the normal-mode analyses with numerical solutions obtained by running linearized numerical models based on the various horizontal grids. The sources and behaviors of the computational modes in the numerical simulations are also examined.Our normal-mode analyses with the Z, C, D, A, E and B grids generally confirm the conclusions of previous shallowwater studies for the cyclone-resolving scales (with low horizontal wavenumbers). We conclude that, aided by nonhydrostatic effects, the Z and C grids become overall more accurate for cloud-resolving resolutions (with high horizontal wavenumbers) than for the cyclone-resolving scales.A companion paper, Part 2, discusses the impacts of the discretization on the Rossby modes on a midlatitude β plane.

Implementation of a brittle sea-ice rheology in an Eulerian, finite-difference, C-grid modeling framework: Impact on the simulated deformation of sea-ice in the Arctic

“…at F-points. This grid staggering arrangement corresponds to that of the Arakawa E-grid (Arakawa and Lamb, 1977;Janjić, 1984;Konor and Randall, 2018), in which tracers are defined at both the center and the four corners of the grid cell, while the two components of the velocity vector are defined at the center of the four edges of the grid cell (Fig. 2…”

“…Various treatments and methods have been proposed, from filtering approaches to more advanced ones such as the introduction of auxiliary velocity points, midway between the neighboring tracer points (Mesinger, 1973;Janjić, 1974). Recently, Konor and Randall (2018) have mentioned the need to introduce a "horizontal mixing process" to avoid the "separation of solutions" when using the E-grid.…”

Dispersion analysis of compatible Galerkin schemes on quadrilaterals for shallow water models

“…However, the Qń Λ k family has a significant spectral gap in the shortest wavelength part of the spectrum that the M GD n family avoids. It is unclear which of the two families will perform better in practice, and it seems likely that both will do poorly, just like C-grid finite difference models with a poorly resolved Rossby radius [17,18,30,32,36].…”

Impacts of the horizontal and vertical grids on the numerical solutions of the dynamical equations – Part 2: Quasi-geostrophic Rossby modes

“…In a companion paper (Konor and Randall, 2018; hereafter Part 1), we discuss the horizontal discretization of the linearized anelastic equations on the Z, C, D, CD, (DC), A, E and B grids, and vertical discretization on the L and CP grids. We introduced the DC grid in Part 1 to test the hypothesis that the CD-grid (and DC-grid) solutions are dominated by the corrector step and the grid used with it.…”

“…The present paper gives a corresponding analysis of the dispersion of three-dimensional Rossby modes on a midlatitude β plane. Previous studies (e.g., Neta and Williams, 1989;Dukowicz, 1995) have mostly used the discrete shallow-water equations on a midlatitude β plane. Thuburn (2008) analyzed the inaccuracies of the Rossby modes on the hexagonal C grid and proposed a discretization that minimizes these inaccuracies.…”