## 1. Introduction

An important challenge in numerical weather and climate prediction is to obtain accurate coupling between models formulated for the parametrised physics and the models formulated for resolved large-scale dynamics. This is a difficult problem due to the wide range of both spatial scales and temporal scales supported.

A number of authors have considered the appropriate time-stepping scheme for the coupled problem (Wedi, 1999; Staniforth *et al.*, 2002a,b; Williamson, 2002; Cullen and Salmond, 2003; Dubal *et al.*, 2004, 2005, 2006). However, so far little emphasis has been placed on the spatial aspects of this so-called physics–dynamics coupling problem. This article addresses the vertical spatial aspects for coupling large-scale dynamics to an eddy viscosity model of the planetary boundary layer.

There is considerable choice for how to construct the model in space, namely the three connected issues of vertical coordinate system, choice of predicted variable and variable arrangement. The vertical coordinate may be chosen from, among others, height-based, mass-based or pressure-based. For a fully compressible system, the variables required in order to obtain a closed calculation of the governing equations include the three components of velocity and two thermodynamic variables. The two thermodynamic variables can be chosen from, for example, potential temperature, temperature, pressure, density or entropy. The arrangement of the variables divides into choice of horizontal staggering and choice of vertical staggering. In order to minimise spatial averaging of terms in the governing equations, it may be beneficial to store model variables at different places in space. Examples of the horizontal staggering include the classic Arakawa A- to E-grids (Arakawa and Lamb, 1977). Popular in atmospheric modelling, due to good representation of the dispersion relation, and in operational use at the Met Office (Davies *et al.*, 2005), is the C-grid. In the vertical there are two common choices of staggering, the Lorenz grid (Lorenz, 1960) and the Charney–Phillips grid (Charney and Phillips, 1953). The configuration of variables for these grids are shown in Figure 1. There is ongoing debate as to which staggering is the best option; for example the Met Office now employs the Charney–Phillips configuration (Davies *et al.*, 2005) whereas the European Centre for Medium-range Weather Forecasts (ECMWF) uses a variation of the Lorenz configuration (Beljaars, 1992; Untch and Hortal, 2004). These issues of spatial arrangement are inextricably connected with each other. For a particular vertical coordinate, it may be preferable to use certain thermodynamic variables and as a result a certain grid staggering.

Considering the breadth of choice available, it is possible that some heuristic argument could be applied, based on the features that a model should be capable of capturing, in order to make a decision on how to spatially construct a model. However, the issue here is that the full equation set is highly complex and so differences between two rival configurations may not be immediately apparent or clear-cut. In order to address the question for the ‘dynamics only’ case, Thuburn and Woollings (2005) constructed 168 test cases covering three types of vertical coordinate, every combination of two from a choice of five thermodynamic variables and a number of different vertical staggerings, including the Lorenz and Charney–Phillips grids. Thuburn and Woollings (2005) concentrated on the vertical configuration; the horizontal staggering has been well studied (e.g. Fox-Rabinovitz, 1994). The methodology used was one of normal mode analysis, to allow systematic checking of every combination and to grade any configuration from optimal to problematic based on its ability to represent the dispersion relation. For the height-based and mass-based coordinate, the optimal configurations are those which have pressure and potential temperature as the thermodynamic variables and use the Charney–Phillips grid for the arrangement of variables. It was later identified that, when using the Exner form of pressure gradient, density can be used in place of pressure as a prognostic variable whilst retaining optimal representation of the dispersion relation (Thuburn, 2006; Toy and Randall, 2007). For the same vertical coordinate and choice of thermodynamic variables that is optimal when using the Charney–Phillips grid, the Lorenz grid is only near-optimal; it gives quite good dispersive properties but supports a zero frequency computational mode (Schneider, 1987; Arakawa and Moorthi, 1988; Hollingsworth, 1995; Arakawa and Konor, 1996; Cullen *et al.*, 1997). The computational mode is a spurious solution resulting from having one too many degrees of freedom; it can interact with other modes and reduce overall accuracy. On the other hand, the Lorenz grid provides a cleaner route to ensuring conservation of energy than the Charney–Phillips grid.

The work of Thuburn and Woollings (2005) was for the inviscid case, i.e. one where only the resolved-scale dynamics are captured in the model but without any physics. Different conclusions may be reached regarding optimal vertical configuration when physics are included in the equations. Thuburn and Woollings (2005) show the Charney–Phillips grid to give the optimal configuration for the dynamics, however it is likely (section 3) that the Lorenz grid will offer the most accurate representation of the stably stratified boundary-layer parametrisation (e.g. Cullen *et al.*, 1997).. In addition to the conflict that arises when modelling physics and dynamics separately, it is not clear how the coupling will influence the choice of configuration. The boundary layer will likely distort certain features of the dynamics, for example. An important question here is what happens to the computational mode. For example, suppression of the computational mode by the boundary layer would be of significant benefit to the Lorenz grid. This question is addressed in the second part of this study (Holdaway *et al.*, 2012, hereafter Part II).

Given the apparent conflict that arises between the dynamics-only case and the boundary layer-only case, the coupled version provides an interesting way to develop some understanding of the spatial aspects of the physics–dynamics coupling, particularly the choice of Lorenz or Charney–Phillips grid.

A popular method for comparing competing numerical configurations, and seen in a number of studies (e.g. Arakawa and Konor, 1996; Zhu and Smith, 2003) relies on examination of model output after simulations with the competing configurations. Although this technique provides useful testing methodologies, it does not offer particularly general results. A more systematic methodology that also allows for more general conclusions involves expanding the system into its steady and time-dependent (transient) parts and examining the two separately; this is the methodology used by Thuburn and Woollings (2005). Examining the steady state first allows one to examine the overall structure of the solution and identify any particularly problematic configurations. The transient part of the solution can be examined by studying the modes of variability (Part II). This methodology enables the study of the individual scales of motion supported by the time-dependent equations and so reveals why differences between particular configurations arise.

Despite the simplifications afforded by eliminating the time-dependent terms in the equations, it is shown below that examination of the steady state is non-trivial. Complexity in the equations results in difficulty obtaining good convergence properties. Further, additional terms are required in order to produce a realistic potential temperature profile. Part I examines the methodology that is required to examine the steady state and presents a comparison of the ability of the Lorenz and Charney–Phillips grids for capturing the steady state. The methodology required to examine the transients is also non-trivial to apply. This methodology, as well as results of the comparison, are presented in Part II of this study.

Part I is arranged as follows: section 2 outlines the equation set which will be used to simultaneously capture the boundary layer and dynamics. Section 3 provides discussion of the Lorenz and Charney–Phillips grids and why a conflict arises in which grid will be optimal. Section 4 outlines the equations required for obtaining a steady state of the problem. Section 5 compares the use of the Lorenz and Charney–Phillips grids for computing the steady state for the boundary layer together with the dynamics, and for the boundary layer on its own. Section 6 offers some concluding remarks.