1. Introduction

Global models admitting the use of variable resolution over the globe are interesting since they may be adjusted to global, regional, or local forecasts. They also present the great advantage over local models of requiring no lateral boundary conditions. Variable-resolution global models have been developed in conjunction with finite elements (Paegle 1989; Côté et al. 1993, 1997; Staniforth et al. 1998), finite differences (Sharma et al. 1987; Barros et al. 1990; Fox-Rabinovitz et al. 1997), and spectral discretizations (Courtier and Geleyn 1988; Geleyn 1998). The variable-resolution spectral models are based on global conformal coordinate transforms. The finite-element and finite-difference models are mainly based on stretched (possibly rotated) grids. Grids generated by stretching the coordinates of a global uniform grid are attractive, since they are logically rectangular, allowing for computationally efficient implementations of the algorithms. Also, the discretization order may be preserved through the use of smooth stretching maps (e.g., Kalnay de Rivas 1972; Barros et al. 1990; Fox-Rabinovitz et al. 1997). The possible disadvantage of stretched grids comes from the fact that the aspect ratios between meridional and latitudinal mesh sizes may vary a lot over the grid. This causes changes in the direction of anisotropy in the elliptic equations resulting from semi-implicit discretizations. Consequently, these equations will be more costly to solve than those on uniform grids.

In this paper, we consider an alternative approach to a variable-resolution finite-difference global model, based on local refinement techniques. In this approach, we employ a relatively coarse global uniform basic grid that is successively refined toward a region of interest, covered by a uniform high-resolution mesh. The successive refinement levels employ mesh sizes varying by a factor of 2; in this way we obtain a smooth transition from the global coarse mesh to the high-resolution area. Local refinement techniques are commonly employed in computational fluid dynamics applications (e.g., Berger and Oliger 1984) and have also been used in numerical weather prediction (Dietachmayer and Droegemeir 1992; Skamarock and Klemp 1993; Fulton 2001). In the context of a global shallow-water model, their use has been investigated by Ruge et al. (1995) [their studies were based on a global semi-implicit model developed by Barros et al. (1990)]. Ruge et al. (1995) have shown the feasibility of the approach but did not demonstrate its efficiency. This was related to the fact that the Barros et al. (1990) model was an Eulerian model on an A grid that required Fourier filtering close to the poles—to avoid prohibitive Courant–Friedrich–Lewy (CFL) conditions—and a Shapiro filter to suppress short modes. Both filters involve global aspects (in the sense that periodic boundary conditions favor their performance) and are not ideally suited to locally refined grids. Although using local refinements, Ruge et al. (1995) kept global functions on all levels in order to apply the filters. Their implementation was rather based on convenience.

In the present paper we consider a global two-time-level semi-implicit semi-Lagrangian shallow-water model, in which we combine algorithmic aspects of the models by Temperton and Staniforth (1987), Ritchie (1988), and Bates et al. (1990). The use of a semi-Lagrangian time discretization (with a decentering parameter) makes the model stable and is very convenient to deal with in conjunction with locally refined grids because it presents no CFL-type restrictions and also because interpolating on locally refined meshes is not more complicated than on uniform grids. The Lagrangian trajectory computations can be handled in the same efficient way as in uniform grids. In our two-time-level scheme the mass divergence term is treated in a fully implicit manner, making the resulting elliptic equation nonlinear. This equation is solved by a multigrid method, adapted from Barros (1991). The solver is formulated in the full approximation storage (FAS) mode (cf. Brandt 1977), which is the most appropriate formulation to deal with nonlinear equations and locally refined grids. It requires no global linearization: a single local Newton step is employed only within the relaxation routine. The performance of the multigrid solver is similar to the performances achieved on uniform grids (as in Barros et al. 1990; Bates et al. 1990). The total computational cost of the method is roughly proportional to the total number of grid points of the locally refined grid. Since those are mainly concentrated over the region of interest, we obtain an effective way of obtaining local predictions, keeping the advantages of having a global model.

We have carried out numerical experiments with real data, comparing the accuracy obtained with uniform high-resolution grids and with the locally refined grids, having the same resolution only over the region of interest. At considerably lower computational costs we obtain similar precision over the high-resolution area, in the range of up to 48 h, with the locally refined scheme. The results indicate that our approach is well suited to obtain efficient variable-resolution finite-difference global models.

This paper is structured as follows: in section 2 we describe the shallow-water model developed. Section 3 is dedicated to the local refinement aspects of the algorithms. Numerical results are presented in section 4, followed by some conclusions.

2. The shallow-water model

We developed a two-time-level semi-Lagrangian semi-implicit global shallow-water model, combining ideas of the models by Temperton and Staniforth (1987), Ritchie (1988), and Bates et al. (1990). (In fact, all terms are treated implicitly in the model. We still call it a semi-implicit model since the advection terms are treated in a Lagrangian way.) The equations are written in vector formulation as

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where V = (u, υ) is the horizontal velocity, ∇_H is the horizontal gradient, Φ is the geopotential, f is the Coriolis parameter, and the total derivative is given by d/dt = ∂/∂t + V · ∇_H.

a. Time-stepping scheme

The temporal discretization includes the possibility of employing a decentering parameter ϵ for noise suppression. It is formulated as

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with the superscripts referring to time instants t_n+1 = (n + 1)Δt and t_n = nΔt. The departure point of a Lagrangian trajectory is denoted by the subscript *. With ϵ = 0.5, we have a totally centered scheme, O(Δt²) accurate. Some dissipation is introduced while choosing ϵ slightly less than 0.5. We follow Ritchie's (1988) treatment of the vector momentum equations and Lagrangian trajectory computations [similar treatments could be employed as in Côté and Staniforth (1988) and Bates et al. (1990)]. After manipulating the momentum equations [as in Ritchie (1988), Eqs. (32)–(45)] we arrive at the equations for the wind components u and υ. The differences in the resulting equation come from the fact that we treat the Coriolis terms implicitly and that our scheme is two time level. With the notation F = (1 − ϵ)Δtf, we arrive at the following equation:

Vⁿ⁺¹ϵt_Hⁿ⁺¹A_v(5)

with

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where the right-hand-side terms A_u and A_υ involve terms from the previous time step evaluated on departure points. We still have the velocity components u and υ coupled by the Coriolis terms. They are decoupled by the step

Vⁿ⁺¹⁻¹A_vϵt_Hⁿ⁺¹(6)

where

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with G = 1/(1 + F²). The mass equation (4) is written as

ⁿ⁺¹ϵt_HVⁿ⁺¹A_Φ(8)

with the right-hand-side A_Φ defined by the terms from the previous step at the departure point. Now, building the divergence from Eq. (6) and employing it in Eq. (8), we arrive at the following nonlinear scalar elliptic equation for the geopotential:

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where

B_ΦA_Φϵt_H⁻¹A_v(10)

Once Eq. (9) is solved the time step is completed through evaluation of uⁿ⁺¹ and υⁿ⁺¹ from Eq. (6).

b. Spatial discretization

Our spatial discretization is based on second-order finite differences on a uniform (λ, θ) latitude–longitude Arakawa C grid (cf. Fig. 1), with mesh size h = π/N_θ, where N_θ + 1 is the number of grid latitudes (including the poles), and N_λ = 2N_θ is the number of longitudes. The C-grid distribution of variables is very convenient for centered differences. For instance, short (one mesh size) differences can be employed when discretizing the geopotential gradient. On the other hand, for the evaluation of the Coriolis terms we will need to average the velocity values from the closest surrounding grid points since the velocity components do not lie on the same grid points.

We now provide some more details about the discretization of the equation for the geopotential (9). In spherical coordinates it can be put in the form

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with μ = a²/[(1 − ϵ)²Δt²], where a is the earth's radius. We note that F and G depend only on θ. The discrete form of Eq. (11) on a mesh point (λ_i, θ_j) = (ih, −π/2 + jh), with 1 ≤ j ≤ N_θ − 1, is given by

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with

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At the poles (mesh points corresponding to j = 0 and j = N_θ), the differential operator from Eq. (11) has singular terms, and finite differencing cannot be applied directly. We proceed as in Bates et al. (1990), integrating (11) over a polar cap of radius h/2 in order to derive the discrete form of the equations. For instance, around the North Pole we have

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The integral of the other terms from (11) vanishes because of periodicity in λ. From (13) we obtain

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Equation (14) is then approximated numerically. For the left-hand side we employ the midpoint rule, while for the right-hand side we apply a multiple trapezoidal rule, combined with finite-difference approximations of the derivative of Φ on each grid longitude. Altogether we obtain the (second order) discretization:

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Analogously, at the South Pole (P_S) we get

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The evaluation of the right-hand-side B_Φ at the poles involves the computation of the divergence of certain fields. The divergence of a given field F = (F₁, F₂) at the pole will also be approximated with the help of an integral form over a polar cap Ω:

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After numerical integration we derive the second-order approximation:

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The expression at the South Pole is similar.

c. Solution of the nonlinear elliptic equation

The resulting elliptic equation is similar to the one solved in Bates et al. (1990). The principal difference concerns the nonlinear term resulting from the fully implicit treatment of the mass equation. The equation has the form lnΦ + LΦ = R, where L is a linear differential operator. A common approach to this sort of mildly nonlinear equation is to use a Picard iteration of the form LΦ^(k+1) = R − lnΦ^(k), requiring only a few iterations of a linear solver for L. With the use of a multigrid solver we do not need to employ a global linearization; only a local linearization inside the relaxation procedure will be necessary. This allows for solving the nonlinear equation at practically the same costs for solving the similar linear equation.

The multigrid we employ here has similar components to the one used in Bates et al. (1990). We describe the necessary modifications to deal with the nonlinear equation. For this, we formulate the coarse-grid correction equation in the FAS mode [for multigrid terminology see Brandt (1977) or the review by Fulton et al. (1986)]:

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In this formulation one computes a correction to the fine-grid approximation u_h not directly (as in the correction scheme), but by first computing the full coarse-grid approximation u_2h. The correction will then be built as the difference c_2h = u_2h − Ĩ^2h_hu_h. With this form L_h may be nonlinear; in case it is linear, this form is equivalent to the correction scheme. The τ-correction term τ^2h_h = L_2hĨ^2h_hu_h − I^2h_hL_hu_h can also be seen as a modification induced by the fine grid in order to improve the quality of the coarse-grid solution [see also Ruge et al. (1995); for an alternative formulation of τ^2h_h that is more convenient for estimates of the local truncation error, see Fulton (2003)]. The correction is then interpolated back and added to the fine-grid approximation. The restriction operators (from fine to coarse grid) I^2h_h and Ĩ^2h_h are normally different, with the latter commonly taken as simple injection.

The effectiveness of a multigrid solver depends on the adequate choice of its components. We employ here bilinear interpolation to transfer the correction from coarser to finer meshes, a full-weighting operator I^2h_h (as in Bates et al. 1990) to transfer the residual from finer to coarser meshes, and injection as the restriction operator Ĩ^2h_h for the fine-grid approximation. Line relaxation is fundamental to deal with the anisotropy of the equations on spherical grids (see Barros 1991). We employ a nonlinear zebra line relaxation, in which a single Newton step is performed to approximately solve the equation on each line. The costs are dominated by the solution of tridiagonal systems, related to each line. The overall cost of the scheme is practically the same as for the linear multigrid solver. The solver is used in full multigrid (FMG) mode (Brandt 1977), with a single V cycle being sufficient to approximately solve the equations on each consecutive level, from the coarsest to the finest mesh. Bicubic interpolation of the coarser solution provides an accurate first guess on the next level. Our V cycles employ only one sweep of relaxation before and one after the coarse-grid correction. The efficiency of the linear solver is preserved, the convergence rates per V cycle lie below 0.1, independently on h (see Garcia 2001). We will also see in section 3 that the use of the FAS mode is the natural formulation when extending the solver to deal with locally refined meshes.

3. Local refinement aspects

In this section we define the local refinements and describe the way they are built. We also present the necessary adaptions to the shallow-water model in order to deal with the locally refined meshes.

4. Numerical results

We first tested the correctness, stability, and efficiency of our shallow-water model. The testing was actually carried out in an incremental way, starting with the implementation of a passive advection scheme, followed by a barotropic vorticity model, and then going to the shallow-water model [see Garcia (2001) for the intermediate models]. One of the components of the scheme was the multigrid solver adapted to locally refined grids (tested already with the barotropic vorticity model) and to the nonlinear case (in the shallow-water model) from a fast Poisson solver from Barros (1991). Its second-order accuracy has been verified by solving the equations with (known) prescribed solutions. The solver proved to be efficient, maintaining practically the same convergence rates of the Poisson solver, with a computational work proportional to the number of grid points.

The intermediate models allowed us to isolate some aspects of the scheme. With just the passive advection model (we followed the tests by Ritchie 1987), we could assess the effects of interpolation on locally refined grids in a Lagrangian scheme. We could conclude that the technique was appropriate and we observed no spurious wave reflections on grid interface boundaries. Some (small) undershooting due to cubic Lagrangian interpolation was observed, but this was not more severe on locally refined meshes than on uniform ones. We did not feel the need to handle this, although shape-preserving interpolation (Williamson and Rasch 1989) could be employed. We successfully tested its use on one-dimensional locally refined grids (Garcia 2001).

For the first tests with the shallow-water scheme, we carried out long run integrations, with a wavenumber-4 Haurwitz wave, on very fine uniform meshes (up to 1536 × 768 grid points) with one-hour time steps, observing no stability problems (with no decentering). For testing the model with real data we considered the case presented in Williamson et al. (1992), with initial conditions from 21 December 1978 at 500 mb. We took the initialized data made available by them, without the use of an initialization algorithm tailored for our model. The model was integrated for several days on uniform meshes with up to 768 × 384 grid points; time steps were chosen in the range from 30 min up to 3 h. No stability problems were present, although some noise was observed in the runs on fine meshes with no decentering. This noise was suppressed by employing a decentering parameter ϵ = 0.49.

Next we have placed a uniform mesh over the northern Pacific, with a resolution of 0.46 875° (corresponding to the resolution of a uniform mesh with 768 × 384 points). We carried out tests with locally refined grids, employing a basic uniform grid of either 96 × 48 or 192 × 96 grid points. Those were successively refined (by introducing up to two or three refinement levels) toward the uniform mesh over the northern Pacific. In Fig. 4 we display the initial geopotential field and the local refinement regions. The innermost region, containing a low and a high pressure center, is the area of interest (where we will make comparisons). The second innermost region is the location of the finest refinement. It is surrounded by the coarser-refinement regions with doubled mesh sizes at each step. The outer refined region is only used in the case of a 96 × 48 basic grid; with the basic grid of 192 × 96 grid points the refinements begin on the second outermost region. In Figs. 5, 6, and 7 we present results after 24, 36, and 48 h of integration, respectively (all carried out with a 1-h time step). In each of these figures we present the results obtained with different refinements, including also a fine uniform mesh of 768 × 384 grid points, used as a reference run. The locally refined grids employ a 96 × 48 basic grid, with one refinement level and also with three refinement levels, with the latter employing the same fine uniform resolution over the region of interest as the reference run. We also consider a 192 × 96 basic grid with two refinement levels, with the same fine resolution over the area of interest. For comparison, results on a uniform 96 × 48 coarse grid are also presented. Already after 24 h, one can see in Fig. 5 the beneficial effects of the introduction of a single refinement level in the uniform coarse grid. The use of more refinement levels gives the solution about the same accuracy as the one achieved on the uniform fine grid. The same facts are observed after 36 h of integration in Fig. 6. In Fig. 7 we see that with the finer basic grid we still obtain very good agreement with the reference run, while the refined grid employing the coarser basic grid starts to present somewhat larger errors. In Table 1 we quantify this, presenting the root-mean-square errors (with respect to the reference run) over the region of interest. After a day of integration the errors are of 5 m for the 96 × 48 basic grid with three levels of refinement and of 3 m for the 192 × 96 basic grid with two refinement levels. These errors grow, respectively, to 10 and 6 m after 36 h and to 16 and 9 m after 2 days of of integration. We repeated the same experiments with a time step of 15 min; the results are presented in Table 2. We can see that the rms errors are similar to those obtained with a time step of 1 h. This indicates that the improvements obtained by a better spatial resolution in the refined areas are not significantly affected by the time truncation errors.

A global view of the effects of the use of the locally refined grid is shown in Fig. 8, where we show the solution on the Northern Hemisphere after 36 h of integration with a 96 × 48 basic grid and three levels of refinement. For comparison we show results with the reference run and with the uniform coarse basic grid. We can see that over the refined area the quality of the solution with the local refinement agrees with that of the reference run, while it is closer to the coarse grid solution outside the mesh refinement. In Fig. 9 we show the same comparisons but after 6 days of integration. (The runs were carried out with decentering activated only during the first day of integration; in the remaining days we switched to ϵ = 0.5.) Although we can still see some positive effect of the refinements in the quality of the results over the area of interest, it has been clearly contaminated by the poorer resolution outside the area of refinement. Poorly resolved features outside the refined areas tend to propagate inward after such a period, as one could expect. The technique is aimed for short-range regional forecasts.

In Table 3 we present the computational costs (CPU seconds on a Dec-Alpha 4100 workstation) corresponding to one day of integration with a time step of 1 h on the several grids used in the tests. We also include the number of grid points contained on each of these grids. We see that the computational costs are roughly proportional to the number of grid points, showing the efficiency of the approach.

We carried out similar tests placing the region of interest (of the same size as in the present case) over the United States and over the high center located at northern Africa. Results are qualitatively similar. With the 192 × 96 basic grid with two refinement levels we get rms errors of 3.5 m after 36 h and of 7.5 m after 48 h in the region of interest over the United States and of 8.2 m over northern Africa after 2 days of integration. The results show that with local refinement we can achieve accuracies comparable to that of fine uniform grids over the regions of interest, in the range of up to 2 days of integration, with significant reduction in computational costs.

5. Conclusions

We developed a variable-resolution two-time-level semi-implicit semi-Lagrangian global shallow-water model. The model employs a global uniform basic grid that is successively refined toward a region of interest, covered by a high-resolution uniform grid. Successive refinement levels employ mesh sizes varying by a factor of 2, allowing for a smooth transition between low- and high-resolution areas. We have carried out long run integrations with the model proving to be very stable. In the range of 2-day integrations the model has been quite effective in providing high-resolution accuracy over an area of interest with great efficiency. The accuracy has been verified against global uniform high-resolution runs, with the locally refined model achieving similar precision at significantly lower costs. The computational efficiency of the scheme comes from the combination of a semi-Lagrangian algorithm with the use of multigrid techniques in the solution of the nonlinear elliptic equation resulting from the semi-implicit discretization. Multigrid is very adequate to handle locally refined meshes, keeping the same efficiency it presents on uniform grids. Our total computational effort has shown itself to be proportional to the total number of grid points on the refined mesh. Our conclusion is that this is an effective way to have variable resolution on finite-difference models. The use of a semi-Lagrangian method is also well adapted to moving local refinements; tests on this direction will be reported elsewhere. Our principal interest now is to extend our investigations to a global model based on the primitive equations.