## 1. Introduction

Many operational weather and climate models are based on a latitude–longitude (lat–lon) spherical grid. Its logically rectangular structure, orthogonality, and symmetry properties, make it relatively straightforward to obtain various desirable, accuracy-related, properties (section 1.2). However, due to the convergence of the meridians at the poles, the lat–lon grid is expected to lead to a severe scalability bottleneck on the massively parallel computing platforms that are becoming available, and many modelling groups are now investigating alternatives. It is therefore timely to review the spherical grids that have been proposed in the literature. However, when considering the choice of grid, it is important not to focus exclusively on the massively parallel scaling issue since it is only one, albeit an important one, of a number of issues that require examination *in toto*. The geometrical properties of a grid, in combination with the numerical methods used, have direct consequences for the accuracy of numerical simulations. In particular, it is possible that the structure of the underlying grid becomes visible in the solution in the form of noise or systematic errors (‘grid imprinting’), or the numerical model might support spurious dynamical modes (‘computational modes’) that have no physical counterparts and degrade the fidelity of the numerical solution. Perfect scalability of a numerical model on a massively parallel computer is of no benefit if it is achieved by unacceptably compromising accuracy or stability. Also, the grid structure can constrain the types of numerical scheme that are feasible. This review surveys the various approaches that have been proposed for gridding the sphere, and some of the likely implications of grid structure for numerical accuracy.

### 1.1. Limitations of the full lat–lon grid

On a full lat–lon grid, the convergence of the meridians leads to resolution clustering at the poles. (Here we use the term ‘full’ to signify the standard lat–lon grid, and thereby distinguish it from the ‘reduced’ lat–lon grids discussed in section 3.1.) For an explicit time integration scheme with Eulerian advection, this convergence places an unbearable restriction on the size of time step as resolution is increased. This restriction, and other problems related to the poles, stimulated some early research into quasi-uniform spherical grids (Sadourny *et al.*, 1968; Williamson, 1968, 1969, 1970, 1971; Sadourny, 1972; Cullen, 1974). However, the development of semi-implicit time integration schemes in combination with semi-Lagrangian advection (e.g. Staniforth and Côté, 1991; Staniforth, 1997; Williamson, 2007), allowed the severe time-step restriction to be by-passed whilst retaining the lat–lon grid, thereby greatly enhancing the viability of lat–lon grids. Such schemes have proven very popular in operational and research models worldwide (e.g. Williamson, 1997; Côté al., 1998; Tolstykh, 2003; Davies *et al.*, 2005; Zhang and Shen, 2008).

However, both semi-implicit time integration (which requires the solution of a global elliptic problem at each time step) and semi-Lagrangian advection require significant data communication among the grid points clustered around the poles. When the grid is decomposed across a number of processors on a parallel computer, these schemes then require significant data communication between processors. Communication between processors is slow and can leave the processors waiting for data; this then becomes the bottleneck to performance, and to the reduction of computing time with increasing number of processors (scalability). Current thinking on how computer architectures will evolve is that increased computer power will be achieved by harnessing hundreds of thousands of processors which are individually little or no faster than current ones. If true, it is anticipated that the resolution-clustering property of the lat–lon grid around its poles will ultimately make it unfeasible for high-resolution modelling, since increased resolution will only worsen the scaling problem. There is therefore renewed impetus to investigate alternative, quasi-uniform spherical grids as a basis for global weather and climate models (Williamson, 2007). In this regard, a valuable survey of recent developments in numerical techniques for global atmospheric models, based upon lectures given at a symposium held in 2008 at the National Center for Atmospheric Research, Boulder, Colorado, may be found in Lauritzen *et al.* (2011).

### 1.2. Essential and desirable properties of a dynamical core

The *dynamical core* may be loosely defined as the component of a numerical model that solves the adiabatic and frictionless governing equations on resolved scales. (There are some subtleties, such as whether scale-selective dissipation terms should be considered part of the dynamical core or a parametrisation of subgrid-scale processes (Williamson *et al.,* 1992; Thuburn, 2008b); these need not concern us here.) Certain properties are considered essential, or at least highly desirable, for a dynamical core. They affect the stability and accuracy of the numerical solution, particularly the behaviour of marginally resolved scales. The ease with which we can obtain these properties is strongly influenced by the geometric properties of the chosen grid.

1. Mass conservation.

Conservation of mass of dry air and mass of trace species is considered highly desirable for climate research and climate prediction. (For the much shorter integration times used in weather prediction, however, conservation is not usually considered essential.) Mass conservation is straightforward to achieve with an Eulerian finite-volume discretisation of the mass continuity equation on any grid structure. However, mass-conserving semi-Lagrangian discretisations are more complex (Zerroukat *et al.,* 2004; Lauritzen *et al.,* 2010b) and typically depend on being able to exploit some underlying grid structure, such as a logically-rectangular structure, to avoid being excessively expensive. Because of the complexity of conservatively mapping from the Eulerian grid cells to arbitrary Lagrangian departure cells, it seems unlikely that mass-conserving semi-Lagrangian schemes can be competitive on more general kinds of grid.

2. Accurate representation of balanced flow and adjustment.

The large-scale atmospheric flow is dominated energetically by motions close to hydrostatic and geostrophic balance. It is crucial, therefore, for a dynamical core to accurately represent the slow, balanced component of the flow: the Rossby waves and nonlinear vortical motions. It is also crucial for a dynamical core to represent the *adjustment* process by which the flow returns towards balance after perturbation away from balance. Adjustment occurs through the radiation, dispersion, and ultimate dissipation of fast acoustic and inertio-gravity waves. To capture the dispersion of the full spectrum of fast waves sufficiently accurately, three approaches are currently known: (i) a *C*-grid or *Z-*grid placement of the predicted variables (section 5) with centred-difference approximations to the relevant spatial derivatives, (ii) a spectral representation of the predicted variables with spatial derivatives calculated directly from the basis functions, or (iii) a collocated placement of predicted variables with a Riemann solver based numerical method, e.g. Ullrich *et al.* (2010). Spectral methods require global rearrangement of data in transforming between the spectral and grid-space representation of model fields at each time step; therefore there is growing concern regarding the parallel efficiency of spectral methods on future exascale supercomputing architectures. Riemann solver methods are at an early stage of development for atmospheric modelling; they are not yet widely used and, to date and to our knowledge, are exclusively based on directional splitting, restricting their use to grids with a local logically-rectangular structure. The *Z*-grid requires the solution of Poisson equations on each model level at each time step to recover the wind field from the vertical vorticity and horizontal divergence; its use, therefore, is contingent on the availablity of a sufficiently fast and scalable Poisson solver. For these reasons we concentrate mainly on *C*-grid discretisations in the rest of this review.

3. Computational modes should be absent or well controlled.

When linearised about some simple basic state, such as a state of rest, the discretisation may be expected to support a spectrum of wave modes that are analogous to those supported by the continuous governing equations. In some cases, however, the discrete modes may have no continuous analogues, or they may exhibit unphysical behaviour such as failure to propagate (zero frequency); they are then called *computational modes*. Some schemes have one or more families of computational modes in their discrete dispersion relation (section 5). Computational modes often have spatial structure close to the grid scale. Their existence can lead to unphysical behaviour in response to forcing by boundary conditions, physical parametrisations, or nonlinearity, or to the insertion of data through data assimilation. The problem often manifests itself as near-grid-scale noise in one or more fields, but it can also appear, for example, when a necessary condition for a physical instability is spuriously satisfied by the numerics (Arakawa and Moorthi, 1988). For these reasons, computational modes are considered highly undesirable. If a scheme supports them, then a mechanism for controlling them is essential. The existence of (branches of) computational modes in the discrete dispersion relation is related to the relative numbers of degrees of freedom in the mass and velocity fields. The number of degrees of freedom is in turn related to the chosen grid staggering, and to the associated number of vertices, faces, and edges. Much of the discussion below therefore focuses on these numbers of degrees of freedom for different grids, assuming a *C*-grid placement of variables.

4. The geopotential gradient and pressure gradient should produce no unphysical source of vorticity.

5. Terms involving the pressure should be energy conserving.

6. Coriolis terms should be energy conserving.

7. There should be no spurious fast propagation of Rossby modes; geostrophic balance should not spontaneously break down.

8. Axial angular momentum should be conserved.

These last five properties are all related to the ability of the discretisation to mimic basic geometrical or mathematical properties of the continuous equations. Schemes with such properties are sometimes called *mimetic*, e.g. Hyman and Shashkov (1997) and Taylor and Fournier (2010).

The geopotential-gradient and pressure-gradient terms are the two largest terms in the governing equations, so it is crucial that these terms are approximated accurately. It is especially important that there should be no spurious forcing of vortical motions that could project onto the energetically-dominant meteorological signal. The requirement here is that the discretisation should possess a discrete analogue of the vector calculus identity ∇ × ∇Ψ ≡ 0 for any scalar field Ψ.

The adiabatic frictionless continuous governing equations conserve energy. However, they do permit transfer of energy between scales, and there is evidence of a small but systematic nonlinear downscale energy transfer in the real atmosphere at scales near the resolution limit of typical weather and climate models. There is room for debate, therefore, over whether a dynamical core should conserve energy on resolved scales or permit a transfer to unresolved scales (Thuburn, 2008b). However, there is a strong argument for requiring the large, and essentially linear, pressure-gradient and Coriolis terms to be individually energy conserving. Energy conservation by the terms involving pressure typically reduces to the requirement that the discretisation have a discrete analogue of the vector calculus identity **v** · ∇Ψ + Ψ∇ · **v** ≡ ∇ · (Ψ**v**) for any scalar field Ψ and vector field **v**. Energy conservation by the Coriolis terms can be achieved straightforwardly when all components of the velocity vector are stored at the same location. However, for a *C*-grid staggering it becomes non-trivial, even on a lat–lon grid (Arakawa and Lamb, 1977).

As discussed above, the discretisation should support a spectrum of waves analogous to those supported by the continuous equations. The Rossby waves should be slowly propagating. In the limit of constant Coriolis parameter *f*, their frequency should become zero, and for non-constant *f* their frequency should be of order *β*, the gradient of *f*. Failure to satisfy these properties means that flows that ought to be geostrophically balanced spontaneously break down on a short timescale. The first *C*-grid scheme to be analysed for a hexagonal grid (Ničković *et al.,* 2002) suffered from this spontaneous breakdown of balance, though a solution was subsequently found (Ringler *et al.,* 2010; Thuburn, 2008a). For linear shallow-water flow with constant *f*, the solution requires the scheme to possess a discrete analogue of the vorticity equation *∂*_{t}*ζ* + *f*∇ · **v** = 0, so that the vorticity *ζ* is steady when the divergence ∇ · **v** vanishes.

Conservation of axial angular momentum is desirable for an accurate representation of the zonal jets that dominate the large-scale circulation of the troposphere and middle atmosphere. However, as far as we are aware, to date this has only been achieved for discretisations on a lat–lon grid (in this context, the Gaussian grid of a spectral model is considered to be a lat–lon grid). To achieve it on a grid other than a lat–lon grid appears to be particularly challenging.

It is far from trivial to obtain all of the properties 1 to 8 in a single scheme. One of the reasons for the sustained popularity of the lat–lon grid is that its logically rectangular structure, orthogonality and symmetry can be exploited to obtain all eight properties. Obtaining the same properties on alternative grid structures is a challenging current area of research. Some recent progress has been made for horizontal finite-difference *C*-grid approximations (Ringler *et al.,* 2010; Thuburn *et al.,* 2009) provided the grid has a dual that is orthogonal to it–for example a Voronoi diagram and its associated Delaunay triangulation (Augenbaum and Peskin, 1985). This scheme gives properties 1, 2, and 4–7; orthogonality is relied upon to obtain 4–7. Whether or not property 3 is obtained depends on the numbers of mass and velocity degrees of freedom.

9. Accuracy approaching second order.

10. Minimal grid imprinting.

Only five truly homogeneous spherical grids are possible, and these are all far too coarse for practical use (section 2). Any practical grid will be, at best, quasi-uniform, and will have a number of special points, or special lines, where the grid structure is locally different from the structure elsewhere. It may be difficult to achieve a discretisation that is second-order accurate near these special points and lines. Even if second-order accuracy can be achieved, the dependence of the truncation error on the local grid structure is likely to leave some signal of the grid structure in the numerical solution (section 6). It is desirable to choose the grid and discretisation to minimise such grid imprinting.

### 1.3. Scope of the review and outline

Because grid geometry is so important, some of the mathematics of polyhedra is reviewed and discussed in section 2. This provides some important information on the relative number of degrees of freedom that result from where one places dynamical variables on the differently-shaped polygonal faces of a polyhedron.

Many quadrilateral grids have been proposed in the past, and these are reviewed in section 3. A brief overview of triangular and hexagonal/pentagonal grids that have been used for atmospheric models–there are relatively few such grids–is presented in section 4. Atmospheric models have also been developed using unstructured triangular and pentagonal-hexagonal grids (e.g. Bacon *et al.*, 2000; Gopalakrishnan *et al.*, 2002; Ford *et al.*, 2004a,b; Pain *et al.*, 2005; Piggott *et al.*, 2005; Lauter *et al.*, 2007; Weller and Weller, 2008; Bernard *et al.*, 2009b; Weller *et al.*, 2009, 2010; Ringler *et al.*, 2011) and are similar in many ways to those for oceanic applications. An important virtue of unstructured grids is their ability to match a grid to awkward-shaped boundaries, such as the land boundaries of oceanic models. This is however much less important for atmospheric modelling since the lower boundary can generally be handled by terrain-following coordinates. A further advantage of unstructured grids is for static or dynamically adaptive mesh refinement. Although very attractive in principle, in practice there are problems with this approach, principally due to spurious dispersion issues; Slingo *et al.* (2009) provide a recent review of this area of research. Any model that uses an unstructured grid must, as a minimum, perform well for the special case when the grid is configured to have structure. For this reason, and in the present context, unstructured grid methods are only considered peripherally.

Analyses of Rossby-wave dispersion and geostrophic adjustment for discretisations of the shallow-water equations are discussed in section 5. This kind of analysis is a powerful tool for exposing the existence of spurious or poorly represented wave modes for any proposed grid and discretisation, and hence for predicting possible poor behaviour in more complete models. Previous investigations into grid imprinting are reviewed in section 6.

Conclusions are drawn in section 7, and some open questions that merit further investigation are identified.