## 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

**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

*ϵ*for noise suppression. It is formulated aswith the superscripts referring to time instants

*t*

_{n+1}= (

*n*+ 1)Δ

*t*and

*t*

_{n}=

*n*Δ

*t.*The departure point of a Lagrangian trajectory is denoted by the subscript *. With

*ϵ*= 0.5, we have a totally centered scheme,

*O*(Δ

*t*

^{2}) 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*)Δ

*tf,*we arrive at the following equation:

**V**

^{n+1}

*ϵ*

*t*

_{H}

^{n+1}

**A**

_{v}

*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

**V**

^{n+1}

^{−1}

**A**

_{v}

*ϵ*

*t*

_{H}

^{n+1}

*G*= 1/(1 +

*F*

^{2}). The mass equation (4) is written as

^{n+1}

*ϵ*

*t*

_{H}

**V**

^{n+1}

*A*

_{Φ}

*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:where

*B*

_{Φ}

*A*

_{Φ}

*ϵ*

*t*

_{H}

^{−1}

**A**

_{v}

*u*

^{n+1}and

*υ*

^{n+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} = 2*N*_{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.

*μ*=

*a*

^{2}/[(1 −

*ϵ*)

^{2}Δ

*t*

^{2}], 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 bywith

*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 haveThe integral of the other terms from (11) vanishes because of periodicity in

*λ.*From (13) we obtainEquation (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:Analogously, at the South Pole (

*P*

_{S}) we getThe 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**= (

*F*

_{1},

*F*

_{2}) at the pole will also be approximated with the help of an integral form over a polar cap Ω:After numerical integration we derive the second-order approximation: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 + *L*Φ = *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 *L*Φ^{(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.

*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}

_{h}

*u*

_{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*

_{2h}

*Ĩ*

^{2h}

_{h}

*u*

_{h}−

*I*

^{2h}

_{h}

*L*

_{h}

*u*

_{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}

*I*

^{2h}

_{h}

*Ĩ*

^{2h}

_{h}

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}*Ĩ*^{2h}_{h}

### d. Aspects related to the semi-Lagrangian discretization

The semi-Lagrangian method on a C grid requires different trajectories for the velocity components and geopotential. We compute the trajectories for the Φ grid employing Ritchie's (1987) approach. The wind is linearly extrapolated to the intermediate time *t*_{n+1/2}, and the nonlinear equation for the midpoint of the trajectory is solved by a few iterations, with bilinear spatial interpolation of velocities used in the process. Once the trajectories for the Φ grid have been determined, the departure points for the *u* and *υ* grid are obtained by linear interpolation of the departure points of the two closest (to the arrival *u* or *υ* point) Φ grid points (see Fig. 2). In order to determine the *υ* departure points close to the poles we project to the tangent plane (over the pole), interpolate there, and project the result back to the sphere. The right-hand sides of the equations are evaluated on the respective *u,* *υ,* or Φ grid points. Bicubic interpolation is employed to interpolate the values to departure points.

## 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.

### a. The grid

We use a global uniform (relatively coarse) grid as the basic grid, providing the coarsest resolution employed in the global integration. We place a uniform fine-resolution (logically rectangular) grid over an area of interest located between two given latitudes and two longitudes. (We did not consider including the poles in the area of interest. This is conceptually feasible, but would bring some complications to the implementation.) We employ a certain number of intermediate refinement levels in order to have a smooth transition between the coarse and the fine resolution. We use a factor of 2 in the relation between the mesh spaces of two consecutive levels of refinement. The transition zone between refinement levels can be narrow but should contain three grid lines. The relative position between a coarser mesh and the next refined level is shown in Fig. 3. The grid points displayed in the figure represent, assuming that the finest mesh and the next coarser grid are shown, the *active* grid points of both levels. The union of all active points of all levels compose the locally refined grid where the solution is computed. From an implementation point of view, a refinement level is composed by a uniform (rectangular) mesh, also covering the area where the next refinement level is placed. It also includes two extra grid lines in each direction (we refer to these extra grid points as ghost points). These points, where the solution is not computed, are convenient for uniformity in the code and will be used to store interpolated values from coarser levels, which will be necessary when computing on the active points. Bicubic interpolation will be used to produce the values at the ghost points in order to keep the discretization order.

### b. Algorithmic details on the locally refined grids

We provide here the details of a time step on a locally refined grid, remembering that this is composed only by the active points of all refinement levels. At first the departure points of all active points are computed. The only difference with respect to the algorithm on a uniform grid is that the (bilinear) interpolation of velocities is performed on a locally refined mesh. Since the grid is locally uniform, interpolation proceeds in the same way, at least away from the mesh transition zones. The use of interpolated values to the ghost points permits a uniform treatment over the complete grid. We only need to locate the refinement level covering the point location and interpolate there.

The second step in the algorithm requires the computation and manipulation of the right-hand side of the equations, including interpolation to departure points. The values of the variables are interpolated to the ghost points. With these values available, the computations at active grid points close to interfaces between different refinement levels are done in the same way as at interior active points of the grids.

The multigrid solution of the elliptic equation needs to be explained in more detail. Let Ω_{h0}_{h1}_{hk}*h*_{k} = 2*h*_{k−1} = ⋯ = 2^{k}*h*_{0}, and the region covered by a grid Ω_{hj−1}_{hj}_{hj}_{hj−1}_{hk+1}_{hm}*h*_{k} = *h*_{k+1}, … , 2*h*_{m−1} = *h*_{m}, such that Ω_{hm}

The first step in the multigrid method will be to define the right-hand sides of Eq. (11) on all grids. This is done by restriction of the values computed on active grid points to the coarser levels below the meshes they belong to. The full multigrid process begins at the coarsest level Ω_{hm}_{hm−1}_{hm−1}_{hm}_{hk}_{hj+1}_{hj}_{hj}*τ* correction [according to Eq. (19)]. In the regions of the coarse grid not covered by the fine grid this *τ* correction is simply set to zero (see also Ruge et al. 1995). Accordingly, the coarse-grid correction will only be added where the fine grid is defined. The relaxation on these finer levels is slightly different, since the domains do not have full latitude lines. The boundary values are kept fixed and are used as known values in the line relaxation. After the V cycle is finished on level Ω_{hj}_{hj−1}_{h0}

Having the geopotential computed through the multigrid algorithm, Eq. (6) is employed to determine the new wind field (before it, we fill the new ghost values of the geopotential).

## 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.

We acknowledge support from CNPQ and Fapesp Grants 97/08187-8 and 01/13104-1. We also thank LCCA-USP for using their computer resources. We thank the referees for their helpful comments.

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Rms errors (m) with respect to the reference run over the interest region (a time step of 1 h has been used in all cases)

Rms errors (m) with respect to the reference run over the interest region (a time step of 15 min has been used in all cases).

CPU times (s) for 1 day of integration on several grids (a time step of 1 h has been used in all cases)