1. Introduction
An integrated forecasting and data assimilation system is under continual development at Recherche en Prévision Numérique Atmosphérique (RPN) in partnership with the Canadian Meteorological Centre (CMC) of Environment Canada. The initial basic hydrostatic formulation of the CMC-RPN Global Environmental Multiscale (GEM) model used in the forecasting system is described by Côté et al. (1998). A nonhydrostatic extension using the hydrostatic pressure coordinate of Laprise (1992) is given by Yeh et al. (2002). Originally designed to operate solely in the global mode while using variable resolution for regional modeling applications, the model (GEM3) is now equipped with a limited-area model (LAM) configuration due to a technology transfer from the Mesoscale Compressible Community (MC2) model (Thomas et al. 1998). Presently, the option of making global integrations on a dual-LAM Yin–Yang grid is being developed (Qaddouri and Lee 2011).
Perhaps the most characteristic feature of the model is its implicit semi-Lagrangian iterative space–time integration scheme. Due to the fourth-order accuracy of its treatment of nonlinear advection, finite differences are found to provide sufficient accuracy for the spatial discretization of the remaining terms. This in fact constitutes a radical change of paradigm for CMC-RPN from the previous forecasting system. In effect, GEM3 has replaced, for global integrations, a spectral model with finite elements in the vertical (Ritchie 1991) and, for regional integrations, a model with finite elements in the horizontal and finite differences in the vertical (Tanguay et al. 1989). Both models were defined on regular1 (unstaggered) grids in the horizontal as well as in the vertical. On the other hand, the finite-difference horizontal discretization of GEM3 employs the Arakawa C grid (Arakawa 1988) in spite of the added complexity for the semi-Lagrangian treatment of horizontal wind components (Côté et al. 1998). In the vertical, the grid remains regular (R grid).
Noise problems were diagnosed in GEM3, related to vertical discretization on the R grid. In particular, the presence of numerical (2 Δz) modes was clearly detected by singular vector analysis (Zadra et al. 2004). Similar, though less serious, problems with stationary computational modes have been diagnosed by Arakawa and Moorthi (1988), Hollingsworth (1995), and Arakawa and Konor (1996) on the Lorenz (1960) (L) grid and simultaneously shown to be absent on the Charney and Phillips (1953) (CP) grid. The findings of Zadra et al. (2004), along with a model behavior in terms of noise very much resembling the ones documented in the above-mentioned studies, prompted us to reexamine the issue of vertical discretization in GEM3, particularly since it seemed to be the only operational NWP model using the R grid. While advocating for the CP grid, Arakawa and Konor (1996) noted that it was “not widely used in the primitive equation models because it is easier with the L grid to maintain some of the integral properties of the continuous system”. Nevertheless, there are a few studies that demonstrate the utility (Robert et al. 1972, Yakimiw and Girard 1987, Tanguay et al. 1990, Girard et al. 2005) as well as the potential for conservation (Laprise and Girard 1990, Arakawa and Konor 1996) of the CP grid.
The grid issue is usually considered to be one of positioning of the discrete variables and its impacts on computational stability and accuracy. Computational dispersion properties of the linear equations are also an important consideration. In the vertical, with four variables and two positions, eight different grid-variable configurations may be constructed (Tokioka 1978). In reality, the grid issue is more complex since, as will be shown, with four quite simple equations in four variables on two positions, up to 41 grid-equation-variable configurations are possible and indeed more configurations can be considered if we add the time dimension (Fox-Rabinovitz 1994). Surprisingly, the R grid as defined and used in GEM3 has not been previously discussed even though it is comparable to the CP and L grids in terms of stability, accuracy, and dispersion properties.
Even once a grid is decided upon, there remains flexibility in the choice of coordinate and history-carrying variables within the context of the Euler equations (Thuburn and Woollings 2005). A height-based coordinate, for example, is certainly a valid option. An incremental approach to development suggested, however, a straightforward introduction of staggering (the strict conversion from the R to the CP grid) as a first step. Many arguments, though, played in favor of at least a “mild” change of coordinate, from the original terrain following of hydrostatic-pressure type (Yeh et al. 2002) of GEM3 to one of log-hydrostatic-pressure type for GEM4 (this presently described version of GEM), which we are calling the ζ coordinate.
The grid issue is addressed in section 2 with emphasis on the unique properties of the CP grid. The derivation of the Euler equations in the ζ coordinate is made in section 3. They are vertically discretized on the CP grid in section 4. In section 5, the main elements of the solution method are presented and we end with a summary in section 6.
2. Unique properties of the Charney–Phillips grid
a. Equations: Full and linearized
b. Vertical structures: Analytic and discrete
Table 1 lists the 12 semidiscrete equations that can thus be formed, three sets of four, labeled according to averaging requirements: x if no averaging is needed, a if averaging is needed and applied on the first term, and b if averaging is needed but applied on the second term. Table 2 lists the eight possible grids. Table 3 lists all of the 41 possible grid-equation configurations (i.e., combinations of equations that can be formed) along with the resulting discrete vertical structure equations. Note that, unlike the unique analytic equation, there may be up to four discrete equations per configurations differing by one or even two averaging operators, depending on the variable, u, w, P, or b, chosen in the elimination process.
Twelve semidiscrete structure equations, classified according to their vertical averaging requirements: no averaging x, first term averaging a, and second term averaging b.
Eight discretization strategies for positioning four variables in two grid positions and associated grid numbers (g1–g8).
Forty-one [1 + 6 × 4 + 16] resulting grid-equation configurations, i.e., ways to discretize four equations (Table 1) involving four variables and their two grid positions (Table 2). In the first column are the grid numbers, in the second the configuration identifier, in the third the names given to the four main configurations discussed in the text, in the fourth column the names originally given by Tokioka (1978) to eight configurations, and in the last column the corresponding numerical structure equations.
Looking at Table 3 heuristically, the presence of an averaging operator must be seen as reducing accuracy as well as potentially providing support for computational modes. We highlight four main configurations: CP, R, and a pair of L-type grids, La and Lb. We also identify the eight configurations as analyzed by Tokioka (1978): A, B, C, D, A′, B′, C′, and D′. Grid g1, the CP grid, admits a single configuration, additional evidence of its unique properties. Grids g2–g7 each allow four types of configurations. The first type, with two equations taken from set a, simply leads to inaccurate systems since derivatives are evaluated over two grid lengths instead of a single one. Fox-Rabinovitz (1994) chose g2 (1x, 2a, 3a, 4x) as his regular grid configuration, which is characterized by the absence of staggering in both variables and equations. The second and third types, with one equation taken from set a, suffer from special computational solutions resulting from the presence of an additional averaging operator appearing on some of the discrete structure equations. Tokioka (1978) demonstrated the presence of such modes (nΔz = 
c. Configurations CP, R, La, and Lb: Their properties
Restricting our attention to these four configurations, only two discrete vertical structure equations are involved. The first one corresponds to the CP configuration, the second to the others. Horizontal and vertical phase speeds (cx and cz), as well as group velocities (cg,x and cg,z), have often been compared between the CP and L grids. Here, the results are summarized in Tables 4 and 5, respectively. We emphasize the fact that configuration R has the same basic dispersive properties as the L configurations, according to the limited analysis framework adopted here (pure gravity wave modes). The maximum wavenumber in the vertical is given by nmax = 2π/λmin (λmin = 2Δz). In the horizontal, phase speeds and group velocities are identically defined analytically. Numerically, they are overestimated with the CP configuration, underestimated with the others. The CP configuration is slightly more accurate. In the vertical, analytic phase speeds and group velocities have the same amplitude but opposite sign. Numerically, amplitude differs but signs remain correct, an important feature for group velocities. Ratios of numerical to analytical phase speeds and vertical group velocities for the CP (thick solid lines) and the other three configurations (dashed lines) are depicted in Figs. 1 and 2, respectively. Fundamentally, though, the four configurations may be said to be fairly equivalent in these respects.
Exact and approximate discrete phase speeds in the horizontal cx and vertical cz directions as functions of wavenumber n.
Exact and approximate discrete group velocities in the horizontal cg,x and vertical cg,z directions as functions of wavenumber n.
Ratio to exact wave speed as a function of wavenumber n and vertical grid length Δz: CP grid (thick solid line), exact (thin solid line), and R and L grids (dashed line).
Citation: Monthly Weather Review 142, 3; 10.1175/MWR-D-13-00255.1
Ratio to exact wave speed as a function of wavenumber n and vertical grid length Δz: CP grid (thick solid line), exact (thin solid line), and R and L grids (dashed line).
Citation: Monthly Weather Review 142, 3; 10.1175/MWR-D-13-00255.1
Ratio to exact wave speed as a function of wavenumber n and vertical grid length Δz: CP grid (thick solid line), exact (thin solid line), and R and L grids (dashed line).
Citation: Monthly Weather Review 142, 3; 10.1175/MWR-D-13-00255.1
Ratio to exact vertical group velocity as a function of wavenumber n and vertical grid length Δz: CP grid (thick solid line), exact solution (thin solid line), and R and L grids (dashed line).
Citation: Monthly Weather Review 142, 3; 10.1175/MWR-D-13-00255.1
Ratio to exact vertical group velocity as a function of wavenumber n and vertical grid length Δz: CP grid (thick solid line), exact solution (thin solid line), and R and L grids (dashed line).
Citation: Monthly Weather Review 142, 3; 10.1175/MWR-D-13-00255.1
Ratio to exact vertical group velocity as a function of wavenumber n and vertical grid length Δz: CP grid (thick solid line), exact solution (thin solid line), and R and L grids (dashed line).
Citation: Monthly Weather Review 142, 3; 10.1175/MWR-D-13-00255.1
In Table 6 the numerical schemes and corresponding computational grids for these four configurations are explicitly shown. It is not possible to have all dependent variables collocated in the L configurations. This is a property of g2 configurations (Table 3), of which R (GEM3) is a member. The g2 configurations only differ by staggering of the equations. In this case, R is characterized by staggering of the vertical momentum (hydrostatic) and continuity (divergence) equations with respect to the horizontal momentum and thermodynamic equations, staggering necessary to achieve second-order accuracy. Note the complete absence of staggering of equations in Lb, a property that facilitates the development of conservative schemes.
The four main discrete grid-equation configurations as selected from Table 3 and discussed in the text.
There is yet another very important difference between these four configurations, namely the absence or presence of stationary numerical modes (ν = 0). Such modes also occur in the presence of averaging operators. Setting ν = 0 in the horizontal momentum equation 
3. Equations: Vertical coordinate and history-carrying variables
Vertical model-level structure showing ζ levels as functions of pressure for values below 200 hPa (where the surface is at sea level) in a typical operational configuration. A sinusoidal mountain extending to 600 hPa in the center of the domain demonstrates the compaction of levels on high terrain. (left) The rectification coefficients are rmin = rmax = 4.5; (right) rmin = 2 and rmax = 100 provide a much faster rectification.
Citation: Monthly Weather Review 142, 3; 10.1175/MWR-D-13-00255.1
Vertical model-level structure showing ζ levels as functions of pressure for values below 200 hPa (where the surface is at sea level) in a typical operational configuration. A sinusoidal mountain extending to 600 hPa in the center of the domain demonstrates the compaction of levels on high terrain. (left) The rectification coefficients are rmin = rmax = 4.5; (right) rmin = 2 and rmax = 100 provide a much faster rectification.
Citation: Monthly Weather Review 142, 3; 10.1175/MWR-D-13-00255.1
Vertical model-level structure showing ζ levels as functions of pressure for values below 200 hPa (where the surface is at sea level) in a typical operational configuration. A sinusoidal mountain extending to 600 hPa in the center of the domain demonstrates the compaction of levels on high terrain. (left) The rectification coefficients are rmin = rmax = 4.5; (right) rmin = 2 and rmax = 100 provide a much faster rectification.
Citation: Monthly Weather Review 142, 3; 10.1175/MWR-D-13-00255.1
4. Vertical discretization on the Charney–Phillips grid
The CP grid, giving the position occupied by each variable in the vertical domain. The model is composed of N layers, inside of which (in the middle of which only if the layers are equal) are the momentum levels [1, 2, …, N], where U, V, ϕ, and q are positioned but note that ϕ and q can also be defined on the boundaries. These N layers are delimited by N − 1 interfaces corresponding to the thermodynamic levels [3/2, …, N − 1/2], where T, w, μ, and 
Citation: Monthly Weather Review 142, 3; 10.1175/MWR-D-13-00255.1
The CP grid, giving the position occupied by each variable in the vertical domain. The model is composed of N layers, inside of which (in the middle of which only if the layers are equal) are the momentum levels [1, 2, …, N], where U, V, ϕ, and q are positioned but note that ϕ and q can also be defined on the boundaries. These N layers are delimited by N − 1 interfaces corresponding to the thermodynamic levels [3/2, …, N − 1/2], where T, w, μ, and
Citation: Monthly Weather Review 142, 3; 10.1175/MWR-D-13-00255.1
The CP grid, giving the position occupied by each variable in the vertical domain. The model is composed of N layers, inside of which (in the middle of which only if the layers are equal) are the momentum levels [1, 2, …, N], where U, V, ϕ, and q are positioned but note that ϕ and q can also be defined on the boundaries. These N layers are delimited by N − 1 interfaces corresponding to the thermodynamic levels [3/2, …, N − 1/2], where T, w, μ, and
Citation: Monthly Weather Review 142, 3; 10.1175/MWR-D-13-00255.1
The placement of horizontal momentum, thermodynamic, continuity, and hydrostatic equations, as well as horizontal wind, temperature, geopotential, and vertical motion 
Some readers may wonder about the choices made near boundaries and appearing in Fig. 4. In particular, why the extra degrees of freedom for temperature afforded by ¼ and N + ¼ thermodynamic levels? This has justly been argued against (Thuburn and Woollings 2005) and, perhaps, signals the return of problematic numerical modes. Models in height coordinates have been devised without such levels (Davies et al. 2005). The rationale for their retention in the present case is mostly based on the need for reliable near-surface and near-model-top temperatures in the assimilation component of the modeling system and the fact that such near-boundary levels are present in the previous model version, GEM3. The forecast model vertical resolution near the boundaries is still rather poor. In GEM4, the last momentum level corresponds to the second from last momentum level in GEM3 and sits at ~40 m above ground. Without the extra thermodynamic level, the last prognostic temperature would occur at approximately 80 m, clearly an unsatisfactory situation. Here, to allow for a fair comparison between the two model versions, we chose to adopt the strategy of making momentum levels coincide except at the surface. Thus, GEM4 ends up with the same number of staggered thermodynamic levels as GEM3 but with one fewer momentum level. Similar arguments do apply to the first thermodynamic level, as GEM4 cannot afford to lose this degree of freedom near the top when performing a strict comparison, but there are arguments in favor of its elimination, which are discussed in the next section.
5. The solution method
This discretized procedure of variable elimination leading to the elliptic equation precisely mimics the analytic procedure, replacing differences by derivatives and ignoring averages. In spite of this, given the nature of the top boundary condition, the presence of temperature (Fig. 4) and the thermodynamic equation (34) at the top level is problematic. In effect, the equation is highly simplified due to the boundary condition: d lnT/dt = Q/cpT since by construction d lnp/dt = 0. Considering that vertical advection also vanishes, since 
6. Summary
This study was motivated by the fact that numerical noise detected in GEM3 was found to be in large part related to vertical discretization on a regular grid. While examining this issue of grid type within the context of vertically discretizing the Euler equations, we found that the R grid used by the GEM model was not only unpopular but also unfamiliar to the meteorological community. This despite the fact that its main properties, including accuracy, appeared comparable to those of the other main grids used: the Lorenz and Charney–Phillips grids. While comparing these grids, only one aspect seemed pertinent to the problem at hand, namely the presence or absence of stationary numerical modes as diagnosed from a simple eigenmode analysis. In this respect the R grid seemed to be unique in suffering from the presence of two such modes (the presence of one of them on the L grids being well known), both of which are absent from the CP grid. The elimination of such stationary numerical modes then seemed enough justification for modifying the dynamical core of GEM to discretize the equations on the CP grid.
Forecast models are known to concentrate their vertical resolution in the troposphere for obvious efficiency reasons and GEM is no exception. One known deficiency of GEM was its discretization of the hydrostatic equation using linear differencing. This, coupled with rapidly degrading resolution in the stratosphere, amounts to a loss of accuracy that can be at least partially alleviated by adopting logarithmic differencing where required. This prompted us to investigate a slightly different vertical coordinate of the log-hydrostatic-pressure type. Already the model’s pressure-related variables, s and q, were logarithms of pressure ratios, a form that greatly simplifies the linearization process. Although well known to theorists, this is the first time such a coordinate is used in NWP as far as we know.
The discretization of the Euler equations transformed to log-hydrostatic-pressure coordinate using simple difference and average operators is a fairly straightforward process as shown, and the solution procedure using an implicit iterative semi-Lagrangian scheme is outlined. A distinguishing feature of the CP grid is the staggering of temperature and vertical motion with respect to horizontal motion. Assuming that vertical motion is defined at the boundaries (after all it vanishes in closed-top conditions and at the surface), this poses the problem of what to do with the temperature. For the open-top condition, carrying temperature on the boundary was shown to be useful by McTaggart-Cowan et al. (2011). We believe the issue of carrying temperature on, or near to, closed boundaries for NWP applications is still not well resolved. We have argued against such a level near the top, suggesting a truncated closed-top boundary condition, and in favor of maintaining one near the surface when vertical resolution is rather poor.
The GEM4 model described here has been operational in most of the forecasting applications at the CMC since 2012. In a follow-up study, we plan to discuss the implementation problems encountered, in particular issues involving the adaptation of the semi-Lagrangian advection scheme and physical parameterization schemes to a staggered vertical grid. The GEM3 and GEM4 models will be compared in a controlled situation involving actual forecasts as well as idealized model integrations. The impacts of the changes will be presented, demonstrating that the Euler equations as formulated and discretized in GEM4 indeed lead to improved model stability (reduction of noise) and accuracy, and ultimately to improved operational forecasts.
Acknowledgments
The authors wish to thank Celal Konor and an anonymous reviewer for their careful reviews of this paper.
APPENDIX
Stability and the Definition of Hydrostatic Pressure
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Here, we use the term “regular,” following Fox-Rabinovitz (1994), to describe a grid system in which all variables are collocated.
