## 1. Introduction

Previous studies have shown that the dispersion properties of various kinds of waves in numerical models of stratified, rotating fluids, such as the atmosphere and ocean, can be improved by storing the prognostic variables on a staggered grid (e.g., Arakawa and Lamb, 1977; Williams, 1981; Arakawa and Moorthi, 1988; Fox-Rabinovitz, 1991, 1996; Randall, 1994; Ringler and Randall, 2002; Ničković *et al.*, 2002; Thuburn and Woollings, 2005; Thuburn, 2006). Various grid staggerings are possible. In the horizontal, the C-grid arrangement of variables (Fig. 1) is known to capture well the propagation of inertio-gravity waves, and hence the process of geostrophic adjustment, provided the Rossby radius is well resolved (Arakawa and Lamb, 1977). The disadvantage of the C-grid is that the *u* and *v* wind components are not colocated, so some averaging is unavoidable in the Coriolis terms. For this reason the C-grid poorly represents near-grid-scale inertio-gravity waves when the grid spacing is greater than the Rossby radius, for which the Coriolis terms are dominant.

The Coriolis terms, in particular, the northward gradient β of the Coriolis parameter *f*, are also crucial for the dynamics of Rossby waves. It might therefore be expected that the unavoidable averaging of Coriolis terms on the C-grid will inevitably lead to a poor representation of the propagation of near-grid-scale Rossby waves. Indeed, previous theoretical studies (Wajsowicz, 1986; Dukowicz, 1995; Gavrilov and Tošić, 1998, 1999) seem to support this, predicting that Rossby wave frequencies will rapidly approach zero as the wavelength approaches twice the grid spacing. However, some numerical calculations of normal mode frequencies for C-grid discretizations of the shallow water equations (Thuburn and Staniforth, 2004) found quite good representation of Rossby wave propagation, with none of the predicted retardation for short-meridional wavelengths. There is a strong motivation for trying to understand this apparent improvement of the numerical results over the exisiting theory, as this could allow the systematic construction of numerical models with improved Rossby wave propagation properties; that is the topic of this article.

There were several differences between the previous theoretical studies cited and the Thuburn and Staniforth (2004) numerical study. The theoretical studies all used quasigeostrophic theory in β-plane geometry, whereas the numerical study did not make the quasigeostrophic approximation and used spherical geometry. However, as discussed in detail below, the crucial difference is in the details of the way the Coriolis terms are discretized: The Thuburn and Staniforth (2004) work was based on energy-conserving discretizations of the Coriolis terms, whereas the theoretical studies were not. In section 2 below an analytical approximation to the dispersion relation for the linearized β-plane shallow water equations is derived, from which approximate frequencies for the inertio-gravity and Rossby wave frequencies are obtained. The derivation is then repeated in section 3 for various C-grid discretizations, showing that the numerical Rossby wave propagation is indeed sensitive to the details of the way the Coriolis terms are discretized. Some of the ramifications are discussed in section 4.