## 1. Introduction

Atmospheric global models may be categorized into spectral or gridpoint models depending on the mathematical representation of model variables. Since the advent of the fast Fourier transform in the early 1970s (e.g., Orszag 1970), spectral models have gained tremendous popularity over gridpoint global models and have since been used in most operational weather forecast centers around the world. However, there are drawbacks for spectral models in terms of operational counts in high-resolution models and global communication overheads in massively parallel processors. In recent years, these have led to the active developments of a new kind of gridpoint global model formulated on the geodesic grid with various numerical local discrete schemes. The most uniformly distributed geodesic grid is the icosahedral grid, which can be configured as the icosahedral-hexagonal grid consisting of a large number of hexagonal cells with 12 embedded pentagons. This grid is particularly suitable for finite-volume numerics (e.g., van Leer 1977; Lin and Rood 1997) in which conventional finite-difference operators are replaced by numerically approximated line integrals along circular-type cell boundaries.

To investigate the novel idea of the icosahedral global model, Williamson (1968) and Sadourny et al. (1968) approximated the shallow-water model (SWM) on the icosahedral grid with finite-difference approaches. Their pioneering studies have been further extended by other researchers with different numerical schemes. Heikes and Randall (1995) and Tomita et al. (2001) solved an icosahedral-hexagonal SWM with finite-volume approaches. Thuburn (1997) developed a potential vorticity–based, icosahedral-hexagonal SWM with a shape-preserving advection scheme. Stuhne and Peltier (1996) employed a finite-element multigrid method to solve a barotropic model on the icosahedral grid, which was further extended into SWM by Stuhne and Peltier (1999). Ringler and Randall (2002) developed an icosahedral-hexagonal SWM that conserves the potential enstrophy and energy within the time truncation error. The Deutscher Wetterdienst developed a new icosahedral-hexagonal global model for operational global weather forecasts (Majewski et al. 2002). Bonaventura and Ringler (2005) analyzed the conservation properties of C-type staggering on Delaunay triangulations of the sphere. Tomita et al. (2004) developed a nonhydrostatic general circulation model (GCM) formulated on the icosahedral-hexagonal grid and successfully ran superhigh-resolution GCM simulations without cumulus parameterizations.

Even though an icosahedral grid provides a quasi-uniform coverage over the sphere, the grid distances between the two nearest grid points are by no means regular. The irregularity in grid spacing, although small, requires special numerical treatment. For example, Tomita et al. (2001, 2002) relocated each icosahedral grid point to the gravitational center of the cell after spring dynamics had been applied. Heikes and Randall (1995) and Majewski et al. (2002) included the irregular spacing into the basis functions of the Vandermonde matrix for polynomial interpolation. Heikes and Randall (1995) used six basis functions for the curved surface fitting on the sphere, and Majewski et al. (2002) used five basis functions with a least squares minimization to approximate spatial derivatives on the sphere. These basis functions are determined by the locations of the nearest six neighbors surrounding the cell to which the polynomial interpolation is approximated. However, the use of the Vandermonde matrix on spherical coordinates requires more than six basis functions for the second-order-accurate approximation because of the curved surface on the sphere. In mathematical terms, the second-order-accurate Vandermonde matrix requires six basis functions, and thus a stencil operation involving six neighboring grid points, only if these Vandermonde stencil points are coplanar. However, the icosahedral grid points on the sphere are not coplanar. Thus, Heikes and Randall (1995) designed a recursive procedure to reduce the coverage area of Vandermonde stencil points in an effort to bring the stencil points closer to a tangential plane. The procedure is repeated until these points are nearly coplanar. This recursive procedure not only requires extra computations, but also fails to make these points exactly coplanar as required for a true second-order scheme. The goal of a coplanar stencil can be achieved if one formulates the Vandermonde matrix on a locally projected plane rather than on a spherical coordinate whose surface is curved in a three-dimensional space.

The purpose of this study is to develop a finite-volume icosahedral SWM formulated on a locally projected plane with the second-order Vandermonde coplanar stencil points. The local plane is defined on the two-dimensional general stereographic coordinate (GSTC; Lee et al. 1995) whose projection center is formulated flexibly at any given point on the earth. The formulation of SWM on a locally projected plane not only allows us to correctly use the Vandermonde coplanar stencil, it also greatly simplifies the mathematic formulation of finite-volume operators from three to two dimensions. The choice of the finite-volume scheme in this study is twofold. First is its flexibility in the line integration along any irregular cells and its excellent conservation properties inherited in the method. Second, unlike finite-difference schemes, the finite-volume scheme allows discontinuity existing on cell boundaries, which accommodates slight discontinuity that may be caused by local projections. Note that the mass conservation over discontinuous interfaces is achieved by computing mass fluxes defined at cell interfaces using the piecewise upwind scheme (van Leer 1977).

Section 2 shows the GSTC formulation upon which we formulate the continuous form of SWM and finite-volume operators with the map factor. Section 3 describes numerical aspects of the model, including the icosahedral grid generation, discretizations of finite-volume operators on the locally projected grid, and the polynomial interpolation with the Vandermonde matrix. Numerical experiments based on standard test cases of Williamson et al. (1992, hereinafter W92) are given in section 4. Conclusions are presented in section 5.

## 2. SWM formulation on GSTC

Heikes and Randall (1995) show that it is more accurate to apply polynomial interpolation on a set of coplanar points. In this study, the coplanar interpolation stencil is achieved by the use of a local projection to project the stencil points from the curved earth surface onto the two-dimensional plane of the GSTC, upon which model variables are defined and numerical calculations are undertaken.

### a. GSTC coordinate

Map projections typically have been used in regional models to map the curved earth surface onto a plane (e.g., Haltiner and Williams 1980). One such mapping often used in regional atmospheric models is the polar stereographic projection. However, its mapping factor, the ratio of the distance between two points on the map and on the sphere, increases as the grid point moves away from the projection center (i.e., the North or South Poles). In the context of a large-scale *β*-plane system, Philips (1973) introduced a stereographic projection similar to the polar stereographic projection with the tangent plane located at the point of longitude *λ* = 0 and latitude *θ* = *θ*_{0}, where *θ*_{0} can be placed at midlatitude to reduce the mapping factor and, therefore, to approximate the *β*-plane system better. However, in many applications, the center of the area of interest may not be located at the Greenwich line (i.e., *λ* = 0). Lee et al. (1995) extended the prime meridian stereographic projection to a GSTC with the tangent plane located at any longitude and latitude on the earth. In their study, the GSTC was placed at the center of the wind profiler network from which data were sampled and studied.

*θ*=

*θ*

_{0}and longitude

*λ*=

*λ*

_{0}. The portion of the earth’s surface covering the target grid point and its surrounding stencil points necessary for the interpolation is projected onto the GSTC by the antipode of the tangential point at

*λ*=

*π*+

*λ*

_{0}and

*θ*= −

*θ*

_{0}. The transformation between the spherical and GSTC is given by

*m*is the map factor,

*a*is the radius of the earth,

*θ*is the latitude, and

*λ*is the longitude. These transformation formulas are the same as those in Philips (1973) when

*λ*= 0. The center of the GSTC is at the tangent point where

*x*= 0 and

*y*= 0. In the transformed system, the positive

*x*axis is directed toward the east of the origin along the latitudinal circle of

*θ*=

*θ*

_{0}and the

*y*axis is defined positive poleward of the origin along the meridian

*λ*=

*λ*

_{0}. The mathematic formulations of wind components are given as follows:

*u*,

_{m}*υ*) and (

_{m}*u*,

_{s}*υ*) are zonal and meridional wind components on GSTC and the spherical coordinate, respectively. Note that at the target icosahedral grid where

_{s}*λ*=

*λ*

_{0}and

*θ*=

*θ*

_{0}the wind components on GSTC and the spherical coordinate are identical. The wind vector transformation coefficients of cos(

*α*) and sin(

*α*) depend only on the location of winds and the central projection point and can be precalculated to save computational time. Thus, the extra computations arising from the use of GSTC are reduced to just a few addition and product operations, which are negligible.

### b. SWM on a local coordinate

SWMs have traditionally been used by the global modeling community as a vehicle for testing novel numerical approaches in the development of global models. SWMs on a sphere are typically formulated and solved on the spherical coordinate. However, as previously discussed, to have coplanar Vandermonde stencil points, we chose to formulate the SWM on a locally projected plane even though icosahedral grid points are on the spherical surface. In our solution procedure, all of the Vandermonde stencil points on the sphere are projected onto a common plane for numerical computations such as flux exchanges between the target and surrounding cells.

*t*is for time and the velocity components of

*u*and

_{m}*υ*on GSTC have been defined previously. The relative vorticity is denoted as

_{m}*ζ*,

*f*is the Coriolis parameter,

*E*is the kinetic energy, and

_{k}*ϕ*is geopotential. The kinetic energy

*E*is defined as 0.5(

_{k}*u*

^{2}

*+*

_{m}*υ*

^{2}

*). Note that the map factor is close to 1 in the control volume over which numerical calculations are operated. Thus, SWEs are essentially those on simple Cartesian coordinates. The subscript of*

_{m}*m*in

*u*and

_{m}*υ*will be omitted hereinafter for simplicity.

_{m}*χ*is expressed as an area mean as follows:

*χ*is the cellular-averaged value, the symbol

*χ̂*denotes the point value of

*χ*,

*A*is the area over the target icosahedral cell, and the subscript

*A*in the integration stands for the integration over an area of

*A*. In this study, the symbol

*δ*is the divergence flux of the scalar variable

_{χ}*χ*and the subscripts of

*A*and

*S*in the integrations stand for the area and line integrations, respectively. The line integration is carried over a closed curve

*S*around the target cell; the norm vector

**n**is defined as the unit vector normal to the line integration of

*S*. The first equality means that the divergence flux of the tracer is defined as the areal average of the divergence operator of the tracer multiplied by the velocity over the icosahedral cell. The area integration of the divergence operator can be approximated by the Gauss theorem and reduced to the line integration as in the second equality shown above. Note that the map factor squared

*m*

^{2}is omitted from the above equation because it is equal to 1 at the cellular center, which is also the projection center. The vorticity operator is written as

**l**is defined as the unit vector tangential to the line integration of

*S*. Positive

**l**is defined in the counterclockwise direction. The first equality means that the vorticity

*ζ*is defined by the areal average of a curl operator of velocity over the icosahedral cell. The area integration of the curl operator can be approximated by the Stokes theorem, which reduces the two-dimensional area integration into the one-dimensional line integration as shown by the second equality. The gradient operator is written as

*χ̂*is evaluated at the midpoint along the line integral of

_{c}*S*as shown in Tomita et al. (2001).

## 3. Numerical grid and discretization

The governing equations shown in Eqs. (1)–(3) are discretized on the icosahedral-hexagonal grid with spatial derivatives approximated by finite-volume line integrations along the cellular edges. Model variables are defined on the nonstaggered icosahedral grid (i.e., the Arakawa-A grid), and the third-order Adam–Bashforth scheme is used to discretize time-tendency terms. The explicit Adams–Bashforth scheme, which requires only one evaluation of force term per time step, is an accurate and efficient scheme for numerical weather prediction models (e.g., Durran 1991; MacDonald et al. 2000). Details of the Adams–Bashforth scheme can be found in Durran (1991) and will not be repeated here.

### a. Icosahedral grid generation

An icosahedral geodesic grid is generated from an icosahedron that has 12 vertices and 20 equilateral spherical triangles with 30 edges. Each edge is a segment of a great circle on the sphere. The icosahedral grid provides a quasi-uniform coverage of the sphere and allows hierarchical refinements of grid spacing. There are many ways to construct icosahedral grids depending on the choice of orientations, subdivisions, and so on. The icosahedral grid generator used in this study generally follows the twisted icosahedral grid described by Heikes and Randall (1995) but without the twist. In this study, one vertex is placed at the North Pole and the opposite one is placed at the South Pole; the orientation is fixed by aligning one of the edges emanating from the vertex at the North Pole with the prime meridian. Once the orientation of the original icosahedron is decided, a hierarchical-resolution, discrete grid is obtained by subdividing each planar triangle of the icosahedral into four small triangles by bisecting the edges and then connecting the new split edges into small triangles. The new vertices created from the planar triangles are projected onto the spherical surfaces.

*l*times, to create a model grid with desirable resolution. The total number of grid points

*n*and the number of divisions

*l*obey the following power-of-two law:

*l*is referred to as the icosahedral grid level in this study. For example,

*G*

_{0}refers to the lowest grid level

*l*= 0, in which the total number of grid points is

*n*= 12 (the original 12 icosahedral points) and

*G*

_{5},

*G*

_{6}, and

*G*

_{7}correspond to

*n*= 10 242, 40 962, and 163 842, respectively. The resolution is doubled when the grid level is increased by 1. Even though the construction of the icosahedral grid starts from the equilateral triangles covering the icosahedron (

*n*= 0), the resulting icosahedral triangles are no longer equilateral. This is because the process of splitting the triangles and then projecting the corners onto the spherical surface changes edges unequally so that they are no longer equal in length. Thus, the icosahedral triangles with

*l*> 0 are no longer equilateral. However it is still the most uniformly distributed geodesic grid. It can be shown analytically that the ratio of the grid maximum distance to the minimum distance is 1.19 for grid levels of

*l*> 0. If the target resolution in a particular level is defined as 1, then the grid spacing in this particular grid level spreads from about 0.9 to 1.1. The target resolutions for

*G*

_{5},

*G*

_{6}, and

*G*

_{7}are approximately 240, 120, and 60 km, respectively.

The icosahedral grids constructed as above include triangular cells, which are referred to as icosahedral-triangular grids. These triangular cells are used as the basis for the construction of an icosahedral-hexagonal grid on which typical finite-volume icosahedral models are formulated. The icosahedral-hexagonal grid is constructed by connecting the Voronoi corner points as shown in Fig. 4 of Heikes and Randall (1995). These Voronoi cells are all shaped hexagonally, except for the 12 pentagon cells surrounding the original 12 points of the icosahedron. It is important to note that each cell has five or six neighbors, all of which share an edge with it, and there is no neighbor with which it shares only a vertex. In other words, each cell center shares one and only one edge with each of its neighbors. This feature makes the icosahedral-hexagonal grid suitable for a finite-volume method to compute flux exchanges between the target cell and its surrounding cells because there is only one edge shared by two adjacent cells.

Even though the icosahedral grid provides a quasi-uniform coverage over the sphere, slight grid variations among the pentagon and hexagon cells are enough to create numerical noise around the pentagon cells where the grid variations are largest. Heikes and Randall (1995) optimized the icosahedral grid with a quadratic function of the difference between the midpoints of the cell edge and the segment connecting the two grid points perpendicular to the cell edge. Tomita et al. (2001, 2002) developed another kind of grid optimization scheme based on the spring dynamic that is designed to reduce the internal variations of standard icosahedral grids. These studies demonstrated that the use of an optimized grid lead to better numerical results. Unless stated otherwise, numerical experiments shown in this study are performed on the optimized spring dynamic grid.

### b. Discretization of finite-volume operators

*δ*shown in Eq. (4) may be discretized on the two-dimensional plane as follows:

_{χ}*k*and

*i*denote the

*k*th cell and the

*i*th edge of the cell

*k*. The area for the cell

*k*is denoted as

*A*; and

_{k}**V**

*,*

_{k,i}**n**

*, and*

_{k,i}*m*are the velocity, the normal unit vector, and the map factor, respectively, at the

_{k,i}*i*th edge of the cell

*k*. The

*n*in the summation symbol denotes the total number of edges in the

*k*th cell. The number

*n*is 5 for the pentagon cell and 6 for the hexagon cell. The Δ

*s*is the length of the

_{k,i}*i*th segment that circumscribes the cell

*k*. The tracer variable

*χ*is defined at the

_{k,i}*i*th edge of the cell

*k*. To define the divergence fluxes at cell edges, we interpolate

*χ*and

_{k,i}**V**

*using the Vandermonde matrix. The conservative transports are readily achieved by applying the second-order van Leer (1977) upwind scheme with the interpolated edge fluxes and summing the edge fluxes in and out of edges around each cell.*

_{k,i}*ζ*shown in Eq. (5) may be discretized on the two-dimensional plane as follows:

**l**

*, which is the tangential unit vector at the*

_{k,i}*i*th edge of the cell

*k*. The line integral of the gradient operator

**∇**

*shown in Eq. (6) may be discretized on the two-dimensional plane as follows:*

_{χ}**V**

*and*

_{k,i}*χ*, which have to be obtained through some kind of interpolation or surface-fitting functions. In this study, they are obtained using polynomial interpolation based on the existing variables of the surrounding icosahedral grids.

_{k,i}### c. Polynomial interpolation on a projected plane

*i*= 0, 1, … ,

*n*and solving for the unknown coefficients

*a*

_{0},

*a*

_{1}, … ,

*a*. The dimension of polynomial interpolation, in strict terms, is determined by the span of space in which the data points

_{n}*x*are distributed. For example, Lee et al. (1995) use a four-dimensional polynomial in (

_{i}*x*,

_{i}*y*,

_{i}*z*,

_{i}*t*) to estimate observational data from a spatially irregularly distributed wind profiler network collected over a period of time. In model applications, to interpolate global model variables defined on the curved earth surface, to be strict, a three-dimensional polynomial formulated on (

_{i}*x*,

_{i}*y*,

_{i}*z*) should be used.

_{i}*b*(

*x*,

*y*), defined on a locally projected two-dimensional plane is expressed as sums of two-dimensional polynomials in the form similar to that used in Lee et al. (1995):

*x*and

*y*are independent variables defined on the projected two-dimensional plane. A two-dimensional polynomial basis function is denoted as

*x*with the associated coefficients denoted as

^{i}y^{j}*c*. The degree of a multidimensional polynomial is defined as the maximum power of any independent variable in the polynomial. In this study, the interpolation uses the polynomials of degree less than or equal to 2 (i.e.,

_{i,j}*N*= 2), which includes six polynomials.

*n*×

*n*(where

*n*is the number of polynomials) matrix equation:

**C**, an

*n*× 1 column vector. The

*n*×

*n*matrix 𝗔 is typically referred to as the Vandermonde matrix, which is determined only by the locations of the surrounding grid points, and the

*n*× 1 column vector

**B**is calculated from the model variables. The number of model variables used in the fit should match the number of coefficients in the Vandermonde matrix. The second-degree polynomial used in this study includes six coefficients that are matched by the nearest six model variables surrounding the edge variable to be interpolated. The coefficients of the fit

**C**can be determined by the standard matrix solver with the precalculated weighting functions that drastically reduce the computational time.

## 4. Numerical results

The numerical accuracy of the SWM is evaluated with a series of the test cases proposed by W92. These are viewed as standard test cases for SWM developers, in part, because they provide a common platform for intercomparisons among SMWs with various numerical approximations. Note that no numerical dissipation operators are used in the following numerical exercises. Furthermore, the mathematical expressions of *l*_{1}, *l*_{2}, and *l*_{∞} error norms used in this study follow those of W92.

### a. Cosine bell advection

*h*

_{0}= 1000 m and

*r*is the great circle distance between any given point at (

*λ*,

*θ*) and the center of the cosine bell whose initial location is given as (

*λ*,

_{c}*θ*) = (3

_{c}*π*/2, 0). The radius

*R*is chosen as

*R*=

*a*/3, and the velocity parameter of

*u*

_{0}is specified as

*u*

_{0}= 2

*πa*/(12 × 86 400) to provide a 12-day period for the cosine bell to move around the sphere once. The parameter

*α*is the angle between the axis of a solid-body rotation and the positive polar axis of the spherical coordinate.

Figures 1a and 1b show, respectively, the initial and final states of a cosine bell for *α* = *π*/2 with G5 grid (i.e., Δ*x* ≈ 240-km resolution; all model results on the icosahedral grid are interpolated to a latitude–longitude grid for display purposes). The maximum height of 1000 m located initially at the center of the bell reduces to 938 m after a 12-day integration. There are no Pole problems during the 12-day advection passing through both Poles. There is a slight phase error and a small undershoot at the wake of the cosine bell shown in Fig. 1b after a 12-day integration. These errors are not uncommon to an icosahedral SWM at similar resolution without positive-definite constraint (e.g., Fig. 6 of Heikes and Randall 1995). Experiments repeated with different rotational angles (i.e., different *α* values) suggested by W92 confirm that the cosine bell solution is not sensitive to rotational angles as expected from icosahedral SWMs. The truncation errors and their convergence rates will be shown in the next numerical experiment.

### b. Steady-state nonlinear geostrophic flow

*f*is the Coriolis parameter used in the analytical solution. The parameter values used in this study are

*u*

_{0}= 2

*πa*/(12 × 86 400) and

*gh*

_{0}= 2.94 × 10

^{4}m

^{2}s

^{−2}.

In this test case, the analytic solution of *α* = 0 is used to specify the initial icosahedral grid point values for *u*, *υ*, and *h*, which are then integrated for 5 days in our SWM. The differences between the analytic and model solutions are computed in terms of *l*_{1}, *l*_{2}, and *l*_{∞} norms to quantify TEs of the model solution. Numerical experiments are undertaken with three different resolutions on G5, G6, and G7 grids. Note that the G7 grid is 2 times the resolution of the G6 grid, which is 2 times the resolution of the G5 grid. In this test case, the spring dynamic is applied to the G7 grid, and the low-resolution grid points of G5 and G6 are downsampled from the G7 grid points so that high-resolution grids include all of the points of the low-resolution grids.

Figure 2 shows the time evolution of *l*_{1} norm TE in *ϕ* at three different grid resolutions. Curves A, B, and C are TE for the G5, G6, and G7 grids, respectively. Note that these curves show oscillatory TEs with periods of less than 1 day. These oscillatory TEs exist in all three different resolutions. Of more interest is that the three curves oscillate in a coherent way with similar oscillatory periods independent of grid resolutions. These oscillatory TEs are also found in other independent studies such as Heikes and Randall (1995) and Tomita et al. (2001) with icosahedral SWMs. Since this kind of oscillatory TE does not exist in our previous TE analysis with similar analytic flow on a uniform grid (see Fig. 3 in MacDonald et al. 2000), we believe that these oscillations in TEs are caused by the quasi-uniform icosahedral grid. Indeed, Tomita et al. (2002) decomposed these oscillatory modes onto Hough harmonics and showed that these oscillatory TEs are caused by the icosahedral grid structure. The oscillatory nature of the TEs results in oscillations in the TE ratio calculated between two model resolutions. To remove the oscillation from the TE ratio, the TE ratio between two model resolutions is averaged over the whole integration period rather than having just one ratio computed at a given time.

Figure 3 shows TEs in *ϕ* for *l*_{1}, *l*_{2}, and *l*_{∞} as function of grid resolution from G5 (Δ*x* ≈ 240 km) to G7 (Δ*x* ≈ 60 km). Curve A in Fig. 3 shows the so-called perfect TE convergence rate that is used as a guideline for the evaluation of the TE convergence rate. Curves B, C, and D show *l*_{1}, *l*_{2} and *l*_{∞} TEs, respectively, as a function of grid resolution. Note that the grid resolution of G6 is 2 times that of G5, and G7 doubles the resolution of G6. For a second-order numerical scheme, the TE from G5 to G6 and G6 to G7 should be reduced by a factor of 4. The TEs in curve A are computed with a reduction factor of 4 between two successive grid resolutions. The slope of A shows the perfect TE reduction rate for a second-order numerical scheme. The TE convergence rate of our icosahedral SWM is evaluated with *l*_{1}, *l*_{2}, and *l*_{∞} represented by curves B, C, and D. These curves show that TEs represented by three error norms reduce as the resolution increases. The *l*_{1} error is smaller than that of *l*_{2}, which is smaller than the *l*_{∞} error. Most important, the slopes of curves B, C, and D are very similar to that shown in curve A, indicating that TE in our SWM achieves a second-order convergence rate in *ϕ*. Similar figures (not shown) in *u* and *υ* also show the TE reduction slopes being similar to the perfect second-order reduction slope. This indicates that *u* and *υ* also converge at a second-order TE reduction rate. These TE convergence rates demonstrate that the SWM possesses a second-order finite-volume accuracy that is consistent with the degree of approximation chosen in the interpolation.

### c. Zonal flow over an isolated mountain

*α*= 0,

*h*

_{0}= 5960 m, and

*u*

_{0}= 20 m s

^{−1}. The mountain profile is given as

*h*

_{s}_{0}= 2000 m,

*R*=

*π*/9, and

*r*

^{2}= min[

*R*

^{2}, (

*λ*−

*λ*)

_{c}^{2}+ (

*θ*−

*θ*)

_{c}^{2}]. The center of the mountain is chosen as

*λ*= 3

_{c}*π*/2 and

*θ*=

_{c}*π*/6. With

*α*= 0, the initial height field is a function of latitude only, and the maximum height field is located along the equator. The zonal wind field is in geostrophic balance with the height field. The meridional wind is initially zero.

Figures 4a–c show the zonal, meridional and height fields, respectively, on day 5. These figures show the disturbance at this stage is confined to the low and midlatitudes. Figures 5a–c are same as those shown in Fig. 4, but for the simulation on day 10. The zonal and meridional flows in Fig. 5 show complex mountain-forced waves expanding globally into high latitudes. Figure 5a shows the negative zonal wind in the northeast of the mountain, which indicates the reversed zonal flow at lee side. Figure 5c shows the deepening of the leeside trough as well as downstream disturbances that are consistent with zonal and meridional flow disturbances. Figures 6a–c are the same as those in Fig. 5, but for the simulation on day 15. Figure 6a shows the magnitude of the reversed zonal flow on the leeside increases from −3.98 m s^{−1} on day 10 to −10.8 m s^{−1} on day 15. Also, the zonal wind near the North Pole converges near 180° longitude. The intensification of the lee trough into a closed low shown in Fig. 6c correlates very well with the intensifying reversed zonal flow over the region. Also, a closed low forms in the high latitudes of the Northern Hemisphere next to the strong ridge. This high-latitude closed low is consistent with the reversed zonal flow in the region. The height field shown in Fig. 6c is found to be very similar to several other published results [e.g., Fig. 4 in Lin and Rood (1997) and Fig. 6 in Thuburn (1997)].

Because of small-scale oscillations near the upstream of the mountain in the spectral solutions, we decided not to undertake a direct error analysis against a spectral reference solution. Instead, we approached the problem in the same manner as Lin and Rood (1997). In this approach, we examined the convergence of the solutions obtained with three different resolutions. Figures 7a–c show the height fields simulated on day 15 with grids G5, G6, and G7, respectively. Comparisons of the height fields among Figs. 7a–c show the solutions as being almost identical at three different resolutions; for example, the location and magnitude of the closed lows on the lee side of the mountain, the ridge next to it, the closed low pressure system at high latitude in the Northern Hemisphere, and disturbances over the tropics are all nearly identical. Thus, we may draw the same conclusion as that of Lin and Rood (1997) that the convergence of the forced-mountain-wave solutions has been nearly achieved at these resolutions.

### d. Rossby–Haurwitz wave

Figure 8 shows the Rossby–Haurwitz wavenumber-4 height field used as the initial condition for SWM integrations with the G5 grid (i.e., Δ*x* ≈ 240-km resolution). It is well known that the Rossby–Haurwitz solution is sensitive to slight irregularities in SWM numerical approximations (Thuburn and Li (2000)). Any small amount of noise caused by inconsistent discretization is amplified quickly, which destabilizes the Rossby–Haurwitz solution. Thus, the Rossby–Haurwitz wave has been used as a test solution when debugging SWM numerical schemes. Thuburn and Li (2000) showed that grid-related TEs excite the Rossby–Haurwitz wave instability, causing the wave to break down faster in the icosahedral model than in other spectral or regular-grid models. Figures 8b and 8c show the Rossby–Haurwitz solutions at days 7 and 14, respectively. The wave moves eastward and maintains a symmetric wavenumber-4 pattern after a 2-week simulation without numerical smoothing operators. Figure 8c is similar to Fig. 20a of Tomita et al. (2001) obtained from their icosahedral SWM with the same model resolution.

The mass conservation in our SWM is also examined in this simulation integrated up to 3 weeks. Figure 9 shows the change of total mass to the initial total mass over the whole model integration time. The horizontal axis represents the model integration time, from day 0 to day 21. The vertical axis depicts the change of the total mass to the initial total mass (i.e., the ratio of total mass at a given time to the initial total mass minus 1). In the ideal case, the mass conservation is represented by a straight zero line with no changes, which means there is no loss or addition of mass into the system. Figure 9 shows that the change of total mass oscillates around the zero line between −10^{−15} and 10^{−15} during the 21-day integration. These errors are within machine runoff errors on a 32-bit machine with double-precision real numbers. Figure 9 shows that our finite-volume icosahedral SWM achieves mass conservation within round-off errors as one would expect from a finite-volume model.

## 5. Conclusions

A finite-volume icosahedral SWM formulated on a local Cartesian coordinate has been developed and evaluated with the standard test cases of W92. The SWM is discretized with the third-order Adam–Bashforth scheme in time and the second-order finite-volume operators in space. The finite-volume operators are applied to model variables defined on the nonstaggered icosahedral-hexagonal grid. Edge variables, which are necessary for the flux exchanges between two adjacent cells, are interpolated with the second-degree interpolating polynomial based on the complete set of basis functions in the Vandermonde matrix. The irregular grid distances forming the Vandermonde basis functions determine the weighting coefficients in the interpolation.

The local coordinate is defined on a general stereographic projection plane, which reduces the solution space of SWM from the three-dimensional curved spherical surface to the two-dimensional GSTC plane. As a result, the three-dimensional finite-volume operators on the sphere are reduced to two-dimensional operators that are further reduced to line integrations along the straight-line segments, rather than on a nonstraight line along curved spherical surfaces. Thus, the use of GSTC results in straightforward implementations of line integration in finite-volume operators. More important, the reduction of solution space also reduces the number of complete basis functions in the second-degree Vandermonde matrix to six and allows us to interpolate edge variables with the model variables at the nearest six neighboring cells. A previous study by Heikes and Randall (1995) showed that it is more accurate if polynomial interpolation approximates edge variables with a set of coplanar points that constitute the Vandermonde matrix. In this study, we demonstrated that the coplanar stencil points can be achieved with GSTC, which requires only a few additional fast computer operations. In all of the numerical exercises shown in this study, no noise is caused by the use of the local coordinate.

The accuracy of SWM was evaluated with the standard test cases of W92. The first case of W92 is the cosine bell advection, which demonstrated that the SWM is free of Pole problems and that the solution is not sensitive to rotational angles. The second test case of W92 provides an analytic solution to quantify SWM TEs and their convergence rates as functions of the model resolution. We demonstrated that the SWM truncation errors in *l*_{1}, *l*_{2}, and *l*_{∞} reduce by a factor of 4 when doubling resolutions from G5 to G6 to G7 grids. The truncation error convergence test demonstrated that the SWM is a second-order finite-volume model. The response of SWM to the mountain forcing was examined with case 5 of the W92 in which an initially simple field is subjected to the forcing caused by the impinging zonal flow on an isolated mountain. SWM was integrated for 15 days, and numerical solutions on days 5, 10, and 15 were examined. The solutions were found to be very similar to those shown in other SWMs with different numerical grids and discrete schemes. Also, the solutions on day 15 with three different resolutions were very similar, indicating that the numerical solutions converged with discrete resolutions.

The Rossby–Haurwitz solution, which is sensitive to small numerical errors, was used to test the stability of the SWM without numerical smoothing operators. In this study, we showed that the wavenumber-4 Rossby–Haurwitz solution moves eastward and maintains a symmetric pattern for up to 14 days in our icosahedral SWM simulation. The mass conservation in our SWM was also examined in the Rossby–Haurwitz simulation integrated up to 3 weeks. It shows that our finite-volume icosahedral SWM achieves mass conservation within round-off errors as one would expect from a finite-volume model.

## Acknowledgments

The authors thank Dr. Shian-Jiann Lin for many insightful discussions that led to the improvement of the finite-volume scheme. We are grateful to Dr. Steven Koch and the anonymous reviewers for their valuable comments that improved the quality of the paper. Thanks are also given to Drs. Ning Wang and Yuanfu Xie, who helped with the icosahedral grid generation, and to Ms. Ann Reiser for editorial reviews.

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The temporal variations of truncation errors in *ϕ* with three different resolutions at G5 (label A), G6 (label B), and G7 (label C) grids.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The temporal variations of truncation errors in *ϕ* with three different resolutions at G5 (label A), G6 (label B), and G7 (label C) grids.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The temporal variations of truncation errors in *ϕ* with three different resolutions at G5 (label A), G6 (label B), and G7 (label C) grids.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The truncation errors in *ϕ* with three different error norms for the idealized curve (label A), *l*_{1} (label B), *l*_{2} (label C), and *l*_{∞} (label D).

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The truncation errors in *ϕ* with three different error norms for the idealized curve (label A), *l*_{1} (label B), *l*_{2} (label C), and *l*_{∞} (label D).

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The truncation errors in *ϕ* with three different error norms for the idealized curve (label A), *l*_{1} (label B), *l*_{2} (label C), and *l*_{∞} (label D).

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The (a) zonal, (b) meridional, and (c) height fields simulated on day 5. The contour interval is 3 m s^{−1} for the wind fields and 50 m for the height field.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The (a) zonal, (b) meridional, and (c) height fields simulated on day 5. The contour interval is 3 m s^{−1} for the wind fields and 50 m for the height field.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The (a) zonal, (b) meridional, and (c) height fields simulated on day 5. The contour interval is 3 m s^{−1} for the wind fields and 50 m for the height field.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

As in Fig. 4, but for the simulation on day 10.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

As in Fig. 4, but for the simulation on day 10.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

As in Fig. 4, but for the simulation on day 10.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

As in Fig. 4, but for the simulation on day 15.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

As in Fig. 4, but for the simulation on day 15.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

As in Fig. 4, but for the simulation on day 15.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The height fields simulated on day 15 with three different resolutions at (a) G5, (b) G6, and (c) G7 grids. The contour interval (50 m) is the same in the height fields derived from three different resolutions.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The height fields simulated on day 15 with three different resolutions at (a) G5, (b) G6, and (c) G7 grids. The contour interval (50 m) is the same in the height fields derived from three different resolutions.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The height fields simulated on day 15 with three different resolutions at (a) G5, (b) G6, and (c) G7 grids. The contour interval (50 m) is the same in the height fields derived from three different resolutions.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The Rossby–Haurwitz wavenumber-4 height field for SWM simulations (a) at the initial time, (b) on day 7, and (c) on day 14. The contour interval is 100 m for the height fields.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The Rossby–Haurwitz wavenumber-4 height field for SWM simulations (a) at the initial time, (b) on day 7, and (c) on day 14. The contour interval is 100 m for the height fields.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The Rossby–Haurwitz wavenumber-4 height field for SWM simulations (a) at the initial time, (b) on day 7, and (c) on day 14. The contour interval is 100 m for the height fields.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The change of the ratio of total mass relative to the initial total mass.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The change of the ratio of total mass relative to the initial total mass.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1

The change of the ratio of total mass relative to the initial total mass.

Citation: Monthly Weather Review 137, 4; 10.1175/2008MWR2639.1