## 1. Introduction

In the past decade, some sophisticated numerical algorithms that were originally developed and used in the computational fluid dynamics (CFD) community have been introduced and getting an increasing popularity in global atmospheric and oceanic modeling. Among the representative ones are the spectral-element (SE) method (Thomas and Loft 2002; Giraldo and Rosmond 2004) and the discontinuous Galerkin (DG) method (Nair et al. 2005a). Although these methods usually require more computational effort than the conventional finite-difference or finite-volume methods, their superiority in numerical accuracy and convergence motivates the further exploration to implement them in geophysical fluid simulations.

We have recently developed a general numerical framework, the so-called Constrained Interpolation Profile/Multimoment Finite-Volume Method (CIP/MM FVM; Yabe et al. 2001; Xiao and Yabe 2001; Xiao 2004; Xiao et al. 2006a; Ii and Xiao 2007) for CFD applications. The basic idea of the CIP/MM method is to make use of more than one moments, which are in fact the volume-integrated average (VIA) and the point value (PV) in the present model for a physical field when constructing the spatial numerical approximations, and to treat all the moments as the model variables that are integrated forward in time separately. Based on multimoments, the high-order interpolation can be constructed over single grid cell. The CIP/MM FVM has a great flexibility in updating different moments (i.e., different moments can be computed by different numerical methods). In the present work, the PV, which is not necessarily conserved, is updated by a semi-Lagrangian and semi-implicit numerical procedure that has been widely used in meteorological modeling. The conservative moment VIA on the other hand is advanced through a flux-based finite-volume formulation (or equivalently a volume remapping). The resultant numerical algorithm can be interpreted as a combination of the semi-implicit semi-Lagrangian method and a finite-volume method, where one of the moments (VIA) is numerically conserved, and is quite simple and easy to implement. Different numerical dispersion can also be obtained from the multimoment formulation. Xiao et al. (2006b), for example, discussed the numerical dispersion of the simplest multimoment finite-volume method for the geostrophic adjustment.^{1}

Another problem associating the construction of the global circulation model comes from the numerical representation of the spherical geometry. Because of the singularity and the convergence of meridians in the polar regions of the longitude–latitude grid system, other alternatives that share more uniform grid spacing are recently explored, for instance, the icosahedral geodesic grid (Heikes and Randall 1995; Stuhne and Peltier 1999; Tomita et al. 2001; Majewski et al. 2002) and the gnomonic-cubic grid (Sadourny 1972; Rančić et al. 1996; Ronchi et al. 1996; McGregor 1997). Being a Chimera grid, the Yin–Yang grid (see Fig. 1) was suggested by Kageyama and Sato (2004) as a quasi-uniform overset grid free from the polar singularity. A Yin–Yang grid is constructed by overlapping two perpendicularly oriented longitude–latitude grid components of the low-latitude part. With each of its component being part of the conventional longitude–latitude grid, the Yin–Yang grid provides a convenient platform readily to accommodate the numerical works originally developed for the latitude–longitude grid. Li et al. (2006) developed an accurate semi-Lagrangian scheme on the Yin–Yang grid by using a high-order interpolation for the overlapping region. However, the interpolation for communicating data across the Yin–Yang border usually does not guarantee the numerical conservativeness. Peng et al. (2006) proposed an exactly conservative advection transport scheme on the Yin–Yang grid based on a piecewise constant reconstruction. For higher-order approximations or the general system of conservative equations for fluid dynamics, the complete conservativeness across the Yin–Yang overlapping region requires more complicated numerics. Nevertheless, as shown in this paper, using a conservative numerical scheme for each individual grid component and minimizing the overlapping zone, one can still obtain adequate, even if not rigorous, conservativeness for medium-range weather prediction or short-range climate simulation.

In this paper, we will implement the CIP/MM FVM on the Yin–Yang grid. All the computations on each individual grid component are exactly conserved and the halo layer in the overlapping region is minimized by the multimoment reconstruction. This paper is organized as follows. The Yin–Yang grid is briefly described in section 2. Section 3 presents the spatial discretization based on two kinds of moments (i.e., PV and VIA) in two-dimensional longitude–latitude grid. The time integration scheme for the shallow-water equation is discussed in section 4. The semi-implicit/semi-Lagrangian procedure (McDonald and Bates 1989) is adopted to compute the PVs. With the PVs updated, the VIAs are simply predicted by evaluating the numerical fluxes in terms of the PVs averaged over two time steps. The scheme for advection is briefly presented in section 5. The model is verified in section 6 by a series of widely used benchmark tests. We present our conclusions in section 7.

## 2. Yin–Yang grid

The Yin–Yang grid (Kageyama and Sato 2004) is an overset grid for spherical geometry, which consists of two notched longitude–latitude grids normal to each other (Fig. 1). The component (Yin or Yang) grid is selected to be the low-latitude part of the longitude–latitude grid. The composition of the two component grids, with one of the components being perpendicular to the other, covers the globe with an overlapping zone where the data need to be communicated between the two components during the computation.

*X*,

*Y*,

*Z*) aswhere the superscript

*n*denotes the Yang coordinate and

*e*is the Yin coordinate.

In this paper, the computational domain (Yin or Yang grid) is defined to be from 45°S to 45°N in the latitudinal direction and from 45° to 315°E in the longitudinal direction. As we can see from the next section, the spatial reconstruction based on multimoments can be done over a single computational cell, thus the overlapping zone of the Yin–Yang grid used here is minimized.

*l*represents

*n*(Yang grid) or

*e*(Yin grid). As mentioned before, the domain bounds of the two components are specified, respectively, as

*φ*

_{min}= −

*π*/4,

*φ*

_{max}=

*π*/4,

*λ*

_{min}=

*π*/4, and

*λ*

_{max}= 7

*π*/4. Here

*N*and

_{φ}*N*denote the total grid points in the latitude and longitude directions, respectively.

_{λ}As an overset grid, the interpolation is needed between the Yin and Yang grid to communicate data. For scalar variables, the data transfer between the Yin and Yang grids is straightforward as long as the interpolation in the overlapping region is properly constructed. However, extra attention must be paid when one communicates a vector field since the expressions of a vector in the Yin and Yang spherical coordinates are different.

**v**= (

*u*,

*υ*) between Yin and Yang components is given aswhere the projection matrix 𝗣 is

It is worth noting that each component grid is nothing but part of the conventional longitude–latitude grid, so numerical techniques developed for the latter can be transplanted to the Yin–Yang grid very conveniently. Moreover, since only the lower-latitude part of the longitude–latitude grid is used for both the Yin and Yang components, the grid spacing of the Yin–Yang grid is quite uniform. In fact, the ratio of the minimum and the maximum grid spacing is approximately 0.707.

Bearing in mind that the Yin and Yang components are identical to a longitude–latitude coordinate, we limit our discussions in the following sections only to the Yang grid.

## 3. Spatial reconstruction based on multimoments

The basic idea of the CIP/MM method is that one can make use of more than one kind of moment to describe the spatial variation of a given physical field and predict all the moments in time as model variables. In the present paper, we use two kinds of moments (i.e., the VIA and the PV).

Considering a general two-dimensional control volume (mesh cell) Ω* _{ij}* ≡ [

*ξ*

_{i}_{−1/2},

*ξ*

_{i}_{+1/2}] ⊗ [

*η*

_{j}_{−1/2},

*η*

_{j}_{+1/2}] shown in Fig. 2, two kinds of moments are defined, respectively, for the field variable Φ(

*ξ*,

*η*). When applying this to the longitude–latitude grid, one just needs to switch

*ξ*to

*λ*and

*η*to

*φ*.

*asEight PVs are defined at the four vertices and the middle points of the four boundary edges of Ω*

_{ij}*aswith (*

_{ij}*ζ*,

*ϑ*) being (

*i*− ½,

*j*− ½), (

*i*,

*j*− ½), (

*i*+ ½,

*j*− ½), (

*i*− ½,

*j*), (

*i*+ ½,

*j*), (

*i*− ½,

*j*+ ½), (

*i*,

*j*+ ½), and (

*i*+ ½,

*j*+ ½), respectively.

*. The coefficients of interpolation function can be determined immediately by using the constraint conditions. Further details may be found in the appendix*

_{ij}The piecewise interpolation reconstruction function *F*_{i,j}(*ξ*, *η*) will be then used in finding the departure point values in the semi-Lagrangian updating and other spatial discretizations. As will be seen later, defining two kinds moments on a single cell does not only allow us to construct high-order reconstruction interpolation with compact mesh stencil, but more importantly, it provides us with a framework to compute the PVs efficiently through the semi-Lagrangian method and to update the VIA with numerical conservativeness.

## 4. Time integration for the shallow-water equations

As discussed above, the CIP/MM FVM keeps both VIA and PV as the model variables that need to be predicted in time. We use different forms of the shallow-water equations as the governing equations for each of them (i.e., the advective form for the PV moment and the conservative form for the VIA moment).

### a. The governing equations for PV and VIA moments

*a*is the Earth radius. The horizontal wind components along the latitude and longitude are defined as

*u*≡ (

*d*/

*dt*)(

*a*cos

*λ*) and

*υ*≡ (

*d*/

*dt*)(

*aφ*), respectively. The divergence

*D*is expressed byThe substantial derivative is given by

^{6}m

^{2}s

^{−1}unless it is specified otherwise. Here

*N*and

_{u}*N*are the spherical metric terms (i.e., the curvature term) given byHere we denote

_{υ}*h*as the depth of fluid and

_{w}*h*is the total height of the free surface. In case the effect of the topography

*h*(

_{s}*λ*,

*φ*) is included, the total height of the water surface is

*h*=

*h*+

_{w}*h*.

_{s}*G*

*h*

_{w},

*G*

*h*

_{w}

*u*

^{1}, and

*G*

*h*

_{w}

*u*

^{2}are the conservative variables whose VIA values are updated by flux-based finite-volume formulations given later. The contravariant velocity components are

*u*

^{1}= (

*u*/

*a*cos

*φ*) and

*u*

^{2}= (

*υ*/

*a*) with the metric factor being

*G*

*a*

^{2}cos

*φ*.

### b. The semi-implicit semi-Lagrangian procedure for PVs

Since the pioneering work of Wiin-Nielsen (1959), the semi-Lagrangian scheme has received much attention for its superiority in computational stability and efficiency for large time steps. Robert (1981, 1982) has obtained stable integrations for the shallow-water equations of different forms with large time steps, using a semi-Lagrangian semi-implicit model. Since then, many researchers such as Ritchie (1985) and McDonald and Bates (1987, 1989) have investigated the numerical accuracy, computational stability, and efficiency of the semi-implicit semi-Lagrangian scheme. Staniforth and Côté (1991) gave a detailed review of this method and its applications to numerical models for atmospheric dynamics. In the present paper, we use the semi-implicit semi-Lagrangian solution to update the PVs.

Following the procedure used by McDonald and Bates (1989), we integrate the momentum equations [Eqs. (14) and (15)] in two half time steps of Δ*t*/2. In the first half-step (i.e., the semi-Lagrangian step) the Coriolis terms are treated implicitly while the pressure gradient and the divergence damping terms are treated explicitly. The reason is that we can easily solve the half-time-level momentum directly by ways of semi-Lagrangian method, so it is called the semi-Lagrangian step. The second half-step, which is called the semi-implicit step, starts with those terms obtained from the first half-step. The curvature terms are integrated for a single time step Δ*t* in the first step. The continuity equation [Eq. (16)] is integrated in a full step of Δ*t*.

*be a constant reference height such that*h

*h*=

*+*h

*h*′ and |

*h*′| ≪

*. In the all test cases presented in this paper, we set*h

*as*h

*h*

_{0}in the initial condition for height. Sorting all the (

*n*+ 1)th time-level quantities to the left-hand side in the resulting equations we haveandwherewith

In the above expressions, the subscript “*d*” denotes the position at the departure point where the corresponding value of the physical field should be obtained from the multimoment reconstruction described in section 3.

**B**= (

*B*,

_{u}*B*). Eliminating

_{υ}*D*

^{n+1}between Eqs. (22) and (23) leads to the Helmholtz equationwhereand

The spatial discretization for above semi-implicit semi-Lagrangian formulation is carried out completely in terms of the PVs. The great circle algorithm of McDonald and Bates (1989) is used to determine the departure location and the detailed procedures can be found in Li et al. (2006). For the eight PV moments (black circle) shown in Fig. 2, we can directly utilize the interpolation function *F*_{i,j} (*λ*, *φ*) constructed in the previous section to obtain the velocity components *u*, *υ*, and the height *h* field at the departure point. For the value at the triangle point ^{n}_{i,j} in Fig. 2, which is not a model variable, we should evaluate its value at the departure point by using the same interpolation function *F*_{i,j} (*λ*, *φ*) since we will need it in the semi-implicit semi-Lagrangian solution. For the pressure gradient term, we first utilize central differencing to evaluate the gradient in the arrival point at the current time level, then use a third-order Lagrangian interpolation function (Li et al. 2006) based on 16 grid points to obtain the gradient value at the departure point. At this step, the point value at the cell center is involved as other PVs and should be computed from the multimoment reconstruction. We have then obtained the right-hand side terms in the prognostic equations [Eqs. (20)–(22)]. With all the values at the departure points known, we can obtain the right-hand side of the Helmhotlz equation *H* in Eq. (24).

^{n}

_{i−1,j},

^{n}

_{i,j},

^{n}

_{i+1,j},

^{n}

_{i,j−1}, and

^{n}

_{i,j+1}is used to approximate the second derivative at the (

*i*,

*j*) point. Because of the uniform spacing in respect to

*λ*and

*φ*, we have

Note that the trigonometric functions cos*φ _{i}*,

_{j}_{+1/2}and cos

*φ*,

_{i}

_{j}_{−1/2}are computed at the middle points shown as the triangle marks in the

*φ*direction in Fig. 3.

The discretized Helmholtz equation has to be solved over the whole sphere for a globally converged numerical solution. Extra attention must be paid in the overlapping region of the Yin–Yang grid. As shown in Fig. 4, when we discretize the Helmholtz equation at a boundary point N in the Yang component, for example, the value of Φ at point M that falls in the Yin component (indicated by the dashed lines) is needed. The value of Φ out of the boundary (the solid line) is interpolated by a third-order Lagrangian interpolation function (Li et al. 2006) based on the 16 points from another component grid (Yin grid). The same applies when Helmholtz equation is solved on the Yin grid and requires the value at a point falling in the Yang grid.

The Helmholtz equation is solved iteratively by the classical Schwarz method on the Yin–Yang grid with the boundary data exchanged at every step through the interpolation. Thus, the equivalent overlap subdomain in context of the Schwarz method is “one”. In our computations, we require the residual for both the Yin and Yang grids to be less than 10^{−8}. It is noted that Qaddouri et al. (2007) recently introduced an optimized Schwarz methods in the Yin–Yang grid by using improved transmission conditions across the Yin–Yang boundary.

Successive overrelaxation (SOR; Press et al. 1992) is used to solve the Helmholtz equation on each component of the Yin–Yang grid, respectively, in the present model. As expected, the iteration number needed for convergence depends on the problem size. Shown in Table 1, more iteration number is required for simulation with higher resolution. Other more scalable solver, such as the multigrid methods (Wesseling 1991) should be more demanding in large-scale simulation. We summarize the procedure for getting numerical solution to the Helmholtz equation [Eq. (24)] on the Yin–Yang grid as follows:

- Calculate the values at the boundary points (e.g., point M in Fig. 4) of the Yang grid from the Yin grid by interpolation.
- Use SOR iterative method to solve the Helmholtz equation in the Yang component to update the numerical solution of the Helmholtz equation for one step.
- Interpolate the values at the boundary points for the Yin grid from the Yang grid by interpolation in a similar manner.
- Similar to step 2, iterate the Helmholtz equation in the Yin component grid to update the numerical solution for one step.
- Repeat steps 1–4 until the numerical solution converges on both the Yin and Yang grids.

Once we get the water height field *h*^{n+1}_{w} as the solution of the Helmholtz equation at the next time level, we can evaluate the velocity components *u*^{n+1} and *υ*^{n+1} at the next time level by Eqs. (20) and (21).

### c. The flux-based finite-volume method for VIA

*,*

_{i}*asThe governing equations for VIA are obtained by integrating Eq. (28) over domain Ω*

_{j}*,*

_{i}*:Given all the PVs on the control volume boundary, the integrals over the boundary segments in Eq. (30) are computed by the three-point quadrature formula:The area integral of source term is computed by the following nine-point quadrature formula:*

_{j}*n*+ 1 are already known from the semi-implicit semi-Lagrangian procedure discussed above, a second-order time integration can be used for VIA aswhere

**U**)

^{n}

_{i,j}and

**U**)

^{n+1}

_{i,j}means the right-hand sides of Eq. (30) computed with the PVs at the

*n*th and the (

*n*+ 1)th step, respectively.

Because we have to keep both PV and VIA independently as the model variables, the present model requires twice as much as the memory storage needed in a single-moment model. Relative to the conventional semi-Lagrangian/semi-implicit scheme, extra computation cost is required for the numerical fluxes in Eq. (30). Given the PV moments that are updated at an expense, which is almost the same as in a conventional semi-Lagrangian scheme, the numerical fluxes can be explicitly computed. In Table 2, we show the CPU time consumed by different parts of the model for different resolutions. Updating VIA takes only a small portion of the total CPU time.

As discussed above, adding a conserved moment (i.e., the volume integrated value) provides us a convenient way to enforce the numerical conservativeness in a semi-Lagrangian semi-implicit model. Being a general methodology, it should be able to apply to other semi-Lagrangian semi-implicit formulations as well. As another particular way to introduce the conservation to a semi-Lagrangian semi-implicit model, we note that Lauritzen et al. (2006) presented a mass-conservative semi-Lagrangian semi-implicit model by using a volume-mapping advection transport scheme.

## 5. The advection scheme

For completeness, we briefly describe the advection scheme used in the numerical tests in this paper.

*h*can be expressed ason the sphere, where

As discussed above, both the VIA and PV as the model variables need to be predicted in time.

*D*denotes the divergence as given before.

*n*, one can predict the PVs on the cell boundary bywhere subscript “up” denotes the piecewise interpolation function for the cell in which the departure point (

*λ*−

_{ζ}*δλ*,

_{ζ}*φ*−

_{ϑ}*δφ*) falls, and

_{ϑ}*C*is the trajectory connecting the departure and arrival points. The displacement (

*δλ*,

_{ζ}*δφ*) is computed according to the velocity field. In the present model the great circle algorithm of McDonald and Bates (1989) is used to determine the departure location.

_{ϑ}Given all the PVs for both the *n*th and (*n* + 1)th time steps on the control volume boundary, the VIA moment of the advected field ^{n+1}_{i,j} can be computed by a finite-volume formulation in the same exact way as the first equation of Eq. (30).

The interpolation for communicating data across the Yin–Yang boundary in the advection computations is conducted in the same manner as previously discussed.

## 6. Numerical tests

*I*asand the normalized global errors aswhere

*h*

_{yang,}

*(*

_{T}*λ*,

*φ*) and h

_{yin},

*(*

_{T}*λ*,

*φ*) are the true solutions on the Yin–Yang grids, respectively. Note that Eq. (34) indicates that the integral over the overset region of the Yin–Yang grid is computed twice.

### a. Solid-body rotation of a cosine bell

*r*=

*a*cos

^{−1}[sin

*φ*

_{0}sin

*φ*+ cos

*φ*

_{0}cos

*φ*cos(

*λ*−

*λ*

_{0})], and the center of the distribution is

*λ*

_{0}=

*π*/2 and

*φ*

_{0}= 0.

*α*is the angle between the axis of the solid-body rotation and the polar axis of the spherical coordinate system. Tests in this paper were run with

*α*= 0,

*α*=

*π*/4,

*α*=

*π*/2, and

*u*

_{0}= 2

*πa*/(12 days).

We carried out global advection tests with the resolution of 2.8125° × 2.8125° (equivalent to a 32 × 96 × 2 grid). The Courant–Friedrichs–Lewy (CFL) number is specified as 0.5. The normalized errors are shown in Tables 3 for different angles. It reveals that the Yin–Yang grid can get accurate advections along different paths on the sphere. The transported concentration and difference between the numerical solution and exact solution are shown in Fig. 5 for *α* = *π*/4. The time evolution of the corresponding ℓ_{1}, ℓ_{2}, and ℓ_{∞} errors for *α* = *π*/4 and *α* = *π*/2 are also given in Fig. 5.

We compared our numerical results with other existing global advection schemes. For the convenience of intercomparison, we referenced the numerical results from other schemes, such as the semi-Lagrangian transport on a reduced grid (RG; see Table 2 of Rasch 1994), the conservative cell-integrated semi-Lagrangian scheme (CISL; see Table 1 of Nair and Machenhauer 2002), flux-form semi-Lagrangian (FFSL; see Table 6 of Lin and Rood 1996), the semi-Lagrangian inherently conserving and efficient scheme (SLICE; see Table 1 of Zerroukat et al. 2004), and the constrained interpolation profile/conservative semi-Lagrangian scheme with rational function (CIP-CSLR; see Table 3 of Peng et al. 2006). In Table 4, the maximum and minimum norms are defined as in Rasch (1994), and YY-MM denotes our multimoment scheme in the Yin–Yang grid. YY-L and YY-C, RG2.8 and RG2.8-M, FFSL-3 and FFSL-5, CISL-M and CISL-P, and SLICE-S stand for the corresponding schemes in Peng et al. (2006), Rasch (1994), Lin and Rood (1996), Nair and Machenhauer (2002), and Zerroukat et al. (2004), respectively. The error measures show that the multimoment finite-volume scheme on Yin–Yang grid (YY-MM) has a competitive numerical accuracy.

### b. Advection test of a sine wave

A more smooth distribution of a sine wave (Li et al. 2006) was used to evaluate the convergence rate of the scheme with grid refinement. In this test, we choose the CFL equal to be 0.5.

*h*

_{0}= 1 and

*k*= 4 is the wavenumber. The velocity component fields are the same as those of solid-body rotation of a cosine bell.

In this test, three levels of grids with the grid spacing gradually refined were used. As shown in Table 5, a third-order accuracy is verified on the Yin–Yang grid.

### c. Cyclogenesis test

*λ*′,

*φ*′) with the rotated north pole at (

*π*+ 0.025,

*π*/2.2). The vortical velocity field is specified aswhere

*r*′ =

*r*

_{0}cos

*φ*′ is the radial distance to the center of the vortex and

*r*

_{0}= 3. The angular velocity is defined to be

In this test, 32 × 96 × 2 cells are employed in the computation. The numerical result after 32 time steps is given with the corresponding analytical one in Fig. 6. The structure of the vortex center is well captured by our scheme and is in good agreement with the analytical one. Table 6 shows the numerical errors of the CIP/MM FVM as well as other existing schemes, such as SLICE-S (see Table 3 of Zerroukat et al. 2004), CISL (see Table 4 of Nair and Machenhauer 2002), and the standard nonconservative bicubic semi-Lagrangian scheme (BiCubic-SL). Again, our multimoment scheme is found to be competitive.

### d. Geostrophic flow across the Pole

This test is proposed by McDonald and Bates (1989), simulating a cross-polar flow with a geostrophically balanced initial state.

*gh*

_{0}= 5.768 × 10

^{4}m

^{2}s

^{−1}and

*υ*

_{0}= 20 m s

^{−1}.

In this test, a 32 × 96 × 2 grid is employed. We have run with the time steps of 10 min (as the reference) and 30 min separately. The numerical results are plotted in Fig. 7 and Fig. 8, respectively. Both the results look very similar to McDonald and Bates’s (1989) reference solution. When the time step is 30 min, the effective CFL number in terms of the gravity wave is about 1.6. It means that the present model is robust and works well even with a large time step for the fast waves. It should also be noted that since the quadrature for numerical flux is computed from the sample points along the trajectories, more sample points are needed to maintain high accuracy and stability when a larger CFL number is used. In practice, a CFL not much larger than unity is desirable.

### e. Steady-state geostrophic flow

*a*is the Earth radius, Ω = 7.292 × 10

^{−5}s

^{−1}is the rotation rate of Earth,

*u*

_{0}= 2

*πr*/(12 days), and

_{a}*gh*

_{0}= 2.94 × 10

^{4}m

^{2}s

^{−2}. The Coriolis parameter associated with this solution is

*f*= 2Ω(−cos

*λ*cos

*φ*sin

*α*+ sin

*φ*cos

*α*).

The 5-day integration results and the corresponding differences between the numerical solution and exact solution with 32 × 96 × 2 cells are shown in Fig. 9 for different angles. The normalized error and convergence rate along different angles are shown in Tables 7 –9. It is observed that more significant numerical errors are produced when the flow is in the zonal direction. This is due to the structure of the Yin–Yang boundary. A zonal flow passes through the longest boundary of the Yang grid, thus mostly affected by the numerical errors from the interpolation for communicating data across the Yin–Yang boundary. The numerical result after 5 days with the time step of 1200 s is also shown in Fig. 10 along the equatorial direction. In this case, our model still converges with an overall rate higher than second order when the refined grid is used. Considering the second-order central differencing used to discretize the Helmholtz equation, we may accept this as a reasonable result.

### f. Zonal flow over an isolated mountain

*u*

_{0}= 20 m s

^{−1}and

*h*

_{0}= 5960 m. The surface mountain height is given bywhere

*h*

_{s0}=2000 m,

*R*=

*π*/9, and

*r*

^{2}= min[

*R*

^{2},

*λ*−

*λ*

_{c})

^{2}+ (

*φ*−

*φ*

_{c})

^{2}

*λ*,

_{c}*φ*) is the center of the mountain located at (3/2

_{c}*π*, 1/6

*π*) in this test.

With resolution of 80 × 240 × 2 and time step of 200 s, our simulated results at day 5, 10, and 15 (shown in the left panels of Fig. 11) look quite similar to those from a spectral simulation of T213 (Jakob et al. 1993, p. 37). For comparison with spectral results, we repeat the simulation on the same resolution by using the Spectral-Transform Shallow-Water Model (STSWM) of Jakob et al. (1993, p. 37). The differences with the STSWM solution of height field on days 5, 10, and 15 are shown in the right panels of Fig. 11. As expected, the largest difference is found near the mountain. In fact, it is observed that spurious oscillations in the vicinity of the mountain at all resolutions are exhibited in the STSWM (Jakob et al. 1993, p. 37). In our model, however, the topographic source term is formulated in a well-balanced way with the gradient force term as discussed in Xing and Shu (2005). There is not noticeable spurious oscillations near the mountain in our results.

The relative variation in the total mass up to 15 days is plotted (see Fig. 13). We have a relative error in the total mass over the whole sphere with a magnitude of 10^{−4}%. Nevertheless, as mentioned earlier, with the mass exactly conserved on each individual component grid by CIP/MM FVM and the minimized overlapping zone where the numerical conservation is not rigorously guaranteed, the resulting numerical model has a conservativeness adequate for at least the medium-range weather prediction.

### g. Rossby–Haurwitz wave

*ω*=

*K*= 7.848 × 10

^{−6}s

^{−1},

*h*

_{0}= 8 × 10

^{3}m, and

*R*= 4.

In this test, we used a 48 × 144 × 2 Yin–Yang grid that has a spatial resolution equivalent to T63 (1.875° × 1.875°). The numerical results of the height field at 7 and 14 days are plotted in the left panel of Fig. 12. It is observed that the structure of a four-wave chain propagates zonally. Carefully examining the fine structures of the numerical solution, we find that our numerical output looks quite close to that of Jakob et al. (1993, p. 37) with a high resolution (T213). To compare the results with spectral solution, we repeated the simulation by the STSWM of Jakob et al. (1993, p. 37) with a T63 resolution. The differences between the two models are shown in the right panels of Fig. 12.

We plot the time evolution of the total mass in Fig. 13. Although the numerical conservation is currently not guaranteed in the Yin–Yang grid, the total mass variation up to 14 days in this case is negligibly small.

### h. A more realistic case with the analyzed 500-mb height and wind field initial conditions

The last numerical test is a more realistic numerical experiment suggested as case 7 in Williamson et al. (1992). The initial conditions of height and wind are from the analysis of the 500-mb pressure surface at 0000 UTC 21 December 1978. We have computed these with both the CIP/MM FVM and the STSWM of Jakob et al. (1993, p. 37).

Figure 14 displays the height field from CIP/MM FVM after 5 days on a North Pole stereographic projection (left panel) with the resolution of T42 and time step 600s. The basic pattern is quite close to the spectral solution of STSWM (Jakob et al. 1993, p. 37) with the same resolution. The difference with the spectral solution of STSWM is shown in the right panel of Fig. 14. The simulation results of the South Pole are also presented in the Fig. 15. Again, the numerical results of CIP/MM FVM are in good agreement with the STSWM (Jakob et al. 1993, p. 37).

In this section, we have extensively conducted numerical benchmark tests to validate the present model. Shown above, the numerical results reveal that the model is computationally stable and robust, and exhibits competitive accuracy relative to other existing ones. We have never observed any computational mode from the numerical results in either the idealized or the realistic case, even when multimoments are involved in the numerical procedure.

## 7. Conclusions

We have developed a finite-volume model for shallow-water equations on the Yin–Yang spherical grid to get rid of the polar singularity and to explore the uniformity in grid spacing. A multimoment spatial discretization is used to get high-order interpolation reconstruction with compact grid stencil, hence, to minimize the overlapping region where the numerical conservativeness is not guaranteed. The PV moments are updated by a semi-implicit semi-Lagrangian approach to maintain the computational efficiency, whereas the VIA moments are computed through a flux-based finite-volume formulation to get the exact numerical conservation on each individual grid component. Even though we have only tried the semi-implicit semi-Lagrangian method of McDonald and Bates (1989) in the present paper, the concept of including an extra moment (i.e., the volume integrated value) to enforce the numerical conservativeness should be able to be applied to other semi-implicit semi-Lagrangian formulations that usually do not possess numerical conservativeness.

The numerical tests show that the present model has competitive numerical accuracy and computational efficiency even in the presence of fast wave. The conservation in total mass is adequate for at least the medium-range weather predictions or short-range climate simulations.

The numerical techniques presented in this paper constitute a promising and practical numerical framework to develop atmospheric and oceanic GCMs in spherical geometry.

## Acknowledgments

The authors gratefully acknowledge the anonymous reviewers for their constructive comments on the manuscript, particularly for the careful examination on the dispersion relation of the M grid in Xiao et al. (2006b).

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## APPENDIX

### Determination of the Coefficient in Multimoment Spatial Reconstruction

Shown in Fig. 2, we define two kinds of moments, that is the volume-integrated average and the point value, respectively, for the field variable Φ(*ξ*, *η*). Based on one VIA and eight PVs, we build a 2D quadratic polynomial for interpolation construction *F*_{i,j} (*ξ*, *η*) [given by Eq. (13)] over the single cell Ω* _{ij}*.

Iteration numbers needed for the numerical solution of the Helmholtz equation to converge (with the maximum residual on both the Yin and Yang grids being less than 10^{−8}). Displayed are the numbers on Yin grid/the numbers on Yang grid.

Elapsed time in seconds per step. The hardware platform is a single node (single CPU of 1.9 GHz, 8 GB memory) of an IBM cluster1600 (AIX, version 5). “Helmholtz Eq.,” “PV,” “VIA,” “Other,” and “Tot” denote the time for solving Helmholtz equation, updating PV moments (including the computation of Helmholtz equation), updating the VIA moment, the other part, and the total time per step, respectively.

Normalized errors of the cosine bell advection test with different orientation angles.

Error measures for the solid-body rotation of a cosine bell after one revolution (256 steps). For comparison, results of other existing global advection schemes are also presented. See the text for details.

Normalized errors and convergence rates of smooth wave advection in the Yin–Yang grid. CFL = 0.5, and *α* = 0.

Comparison of normalized errors at different times for the cyclogenesis test. A 32 × 96 × 2 grid is used. The results at *t* = 3 (after 32 time steps) and *t* = 6 (after 64 time steps) are given. For comparison, we include the numerical results of other existing schemes as well.

Normalized errors and convergence rates of case 2 in Williamson et al. (1992) for *α* = 0.0 and *c* = 3.0 × 10^{6}.

As in Table 7, but for *α* = *π* /4.

As in Table 7, but for *α* = *π* /2

^{1}

It is observed that this mode has a frequency significantly lower than that of the physical one discussed in Xiao et al. (2006b). From the numerical experiments carried out so far, we have not found a significant effect of thiscomputational mode on the numerical solutions.