a. Parabolic representation of cells in 1D

First we consider the 1D case of the CISL scheme and further it will be extended to 2D, including spherical geometry. A parabola in any Eulerian cell can be uniquely defined as

hxh₀h₁xh₂x²(1)

where h₀, h₁, and h₂ are the coefficients of the parabola, and x is a normalized local variable such that

x

Let h be the cell-averaged density defined by

and let h_L = h(−1/2) and h_R = h(1/2) be the left and right cell boundary (edge) values, respectively, which are cubically interpolated from the given neighboring cell-averaged densities (Colella and Woodward 1984). Then the parabola is determined in such a way that it fits the boundary values and fulfills (2). With these constraints the density distribution function h(x) can be represented as follows:

where for convenience we have introduced the derived coefficients,

δhh_Rh_Lh_Shh_Lh_R

These coefficients can be further modified to make monotonic or positive-definite versions of h(x). In the present study, we use such filtering constraints as suggested by Carpenter et al. (1990) and LR96.

b. Cell representation in 2D

Rančić (1992) showed that the CISL method can be applied for conservative transport in a 2D context, by introducing a biparabolic function to represent the distribution of density in each Eulerian cell. A biparabolic function is a direct extension of (1) in 2D, with nine coefficients. Computation of these nine coefficients can be very expensive. However, we use a “quasi-biparabolic” function (4) with only five coefficients to represent the mass distribution in each cell. This can significantly reduce the computational cost of the remapping procedure (section 3) by simplifying the area integrals involved in the 2D remapping scheme. Although formally only second-order accurate in the two coordinate directions, our results show that the quasi-biparabolic function still yields a very accurate representation. Another advantage is that the quasi-biparabolic representation in each cell may be easily modified to be monotonic and/or positive definite.

The quasi-biparabolic function representing an Eulerian cell can be written in normalized local (x, y) coordinates as

h is the average value in the cell, and (a^x, b^x) and (a^y, b^y) are coefficients of the parabola in the x and the y direction, respectively. Note that the above 2D parabolic function needs only two 1D parabolic fits to determine the coefficients. These coefficients are defined as in the 1D case, and may be further modified for either the monotonic or positive-definite option in the respective two coordinate directions. Moreover, with this definition of h(x, y), it can be easily shown that the surface integral of the density distribution function over the cell is equal to the cell-average value (h):

a. The CISL scheme in 1D

First we consider the 1D CISL advection scheme introduced by Machenhauer and Olk (1996, 1998) to illustrate the remapping algorithm, in a two-time-level context. Figure 1 schematically illustrates the CISL advection in the 1D case. Assume that the arrival points are regularly spaced Eulerian grid points x_j±1/2 at time t + Δt (or time level n + 1), and corresponding departure points are the irregularly spaced Lagrangian points x^*_j±1/2 at time t (or time level n). In Fig. 1, the Eulerian and Lagrangian points are marked by open squares and filled circles, respectively. Let Δx_j = (x_j+1/2 − x_j−1/2) be the grid spacing (cell width) of the Eulerian grid and let the corresponding Lagrangian grid spacing be Δx^*_j = (x^*_j+1/2 − x^*_j−1/2). With this spatial discretization, let hⁿ_j(x) be the known density distribution at the time level n, of the jth Eulerian cell. Then the cell-average density (mass per unit length) hⁿ⁺¹_j of the arrival cell defined by Δx_j at the new time level n + 1 can be determined by using the 1D equivalent to (9):

hⁿ⁺¹_jx_jh^*_jx^*_j(10)

where * denotes the upstream value evaluated at the known time level n and h* is the integral mean value of the density distribution over the Lagrangian cell Δx^*_j; that is,

The density distributions hⁿ_j(x) in Eulerian cells are often represented by piecewise polynomials (Bott 1993; Rasch 1994). Also the density across the cell walls is not necessarily continuous (Laprise and Plante 1995). In the present study, we use the piecewise parabolic function (3) to represent the density distribution for each of the Eulerian cells. This representation is generally continuous across the cell walls. Thus, to determine the average density at the new time level hⁿ⁺¹_j, the rhs of (10) should be evaluated, which implies that the function (3) needs to be integrated (analytically) over the Lagrangian cell [x^*_j−1/2, x^*_j+1/2] as shown in (11).

Geometric interpretation of the remapping is rather easy as illustrated in Fig. 1. The shaded region represents the mass in the Lagrangian cell (shown as area under the parabolic curves A₁B₁). It may be computed as the total accumulated mass toward the line x = x^*_j+1/2 from the common reference line in the left x = x_R minus the accumulated mass toward the line x = x^*_j−1/2 from the reference line x = x_R. If we denote the total mass accumulated toward the line x = x^*_j+1/2from the reference line by A(x^*_j+1/2), then the mass in the Lagrangian cell is

h^*_jx^*_jAx^*_j+1/2Ax^*_j−1/2

Thus, the new cell-average value of density at the future time level, from (10), is

On a uniform grid with a constant wind field, it can be shown that the process of parabolic representation of density distribution in Eulerian cells, followed by upstream estimation of density at Lagrangian cells, is formally equivalent to an SL scheme with cubic interpolation at upstream points (Laprise and Plante 1995).

b. The CISL remapping in 2D

We use a fully 2D remapping rather than using a dimension-splitting approach (Rančić 1995). Figure 2 schematically illustrates our CISL scheme in 2D with (λ, μ) as Cartesian coordinates. The filled circles are the upstream Lagrangian points corresponding to the Eulerian grid (thin lines) intersections. The Lagrangian cell walls are shown as thick lines with midpoints marked by ×s.

The upstream Lagrangian cells are generally quadrilaterals with irregularly oriented sides, and the computation of the exact mass enclosed in such cells is a complicated task, which we want to simplify for the sake of efficiency. Following Machenhauer and Olk (1998), first we approximate the Lagrangian cells by “computational cells,” which are polygons with sides parallel to the coordinate axis λ or μ. Each computational cell is constructed so that it has the same area as the Lagrangian cell it approximates. In Fig. 2 the sides of the computational cells are shown as dashed lines, for convenience we consider a particular computational cell bounded by the polygon BCC₁AA₁D₁DB₁B (the shaded region). The vertical sides (or walls) of the computational cells are constructed with lines through the midpoints (×) of the Lagrangian cell walls and lines along the horizontal sides pass through the Lagrangian points (here, on a map with north upward, “vertical” and “horizontal” lines means lines oriented north–south and east–west, respectively). In general, the computational cell (polygon) has eight sides; however, in certain special cases it can have fewer sides. Thus in Fig. 2, the shaded polygon approximates the Lagrangian cell in question, and mass computed in the polygon approximates that in the Lagrangian cell.

The entire closed domain can be filled with such polygons without any overlaps or cracks (disjoint regions), which ensures exact total mass conservation. Now we discuss mass computation in the shaded polygon BCC₁AA₁D₁DB₁B using the 2D remapping scheme. Let λ = 0 be the common reference line (Fig. 2), from which accumulated mass is computed toward each vertical side of the computational cell. Thus, the mass enclosed (M_C) in the shaded polygon is

where M_S denotes the mass in the rectangular region from the reference line to the specified vertical segment (see Fig. 3 for details).

Generally, the first, second, third, and fourth terms are the accumulated masses toward those sides that end up in the east, west, south, and north sides, respectively. However, to be valid in general the accumulated masses in Eq. (13) must be determined with signs, positive or negative, which depend on the orientation of the Lagrangian cell sides in question. The accumulated masses to be substituted into (13) with the right signs are determined by expressions introduced in the next subsection. Equation (13) may be used as a general formula for determining the mass in a particular cell. As in the case of the Lagrangian cells, the computational cells also share their boundary walls with their neighbors, and therefore it is only necessary to find the accumulated mass corresponding to the east and the south walls of each of the Lagrangian cells, while computing from west to east and south to north.

As seen from Fig. 2, the accumulated mass M_S(AA₁) is equal to the accumulated west-side mass M_S(BB₁) for the neighboring cell located to the west of the current cell, and M_S(CC₁) is equal to the accumulated north-side mass M_S(DD₁) for the neighboring cell south of the current cell. Thus, using Eq. (13) while stepping through the grid, one just adds the accumulated masses M_S(AA₁) and M_S(CC₁) to the current cell, and subtracts the same amount, respectively, from the neighbor to the east and the neighbor to the south.

c. Determination of the accumulated mass

In 2D, (9) can be written as follows:

hⁿ⁺¹AhA(14)

where ΔA is the area of the arrival cell at the future time t + Δt and ΔA* is the area of the upstream Lagrangian cell at time t. In order to predict the value of the average density (mass per unit area) hⁿ⁺¹ at the future time level n + 1, the quantity h*ΔA* in (14) should be known. In other words, we need to evaluate the following integral:

where hⁿ(λ, μ) is the known density distribution at the time level n. First, we represent hⁿ(λ, μ) in each Eulerian cell by the quasi-biparabolic function (4), introduced in the previous section. Trajectory origins corresponding to the Eulerian cell's corner points are then determined, and straight lines joining these points define the Lagrangian cell with area ΔA*. Mass enclosed in the actual Lagrangian cells is finally approximated by that in the computational cells, using the 2D remapping scheme (13). The accumulated masses toward the computational cell walls remain to be estimated.

Computational cells by design are made of rectangular regions. We refer to each of the rectangular regions from the reference line (usually left border of the domain) to a cell wall as a stripe. Thus, accumulated mass toward any computational cell wall, for example, AA₁, (Fig. 2) is the total mass in the stripe PP₁A₁A. Figure 3 shows different formations of stripes (shaded regions). Now, we discuss the computation of accumulated mass toward a cell wall (or the total mass in a stripe). Consider the upper stripe in Fig. 3, bounded by the region [λ_a, λ_b] ⊗ [μ_a, μ_b], then the mass M_S in that stripe is

where h(λ, μ) is the density distribution function in the domain (for ease of notation, the dependence of the function on time index n is suppressed). Let λ_a = λ₁ be the reference line and

λ_bλ_kλ_k+1μ_aμ_bμ_jμ_j+1

as shown in Fig. 3. Using the properties of a definite integral, (16) can now be written as

where h_ij(λ, μ) is the piecewise quasi-biparabolic function representing the density distribution at each Eulerian cell C_ij.

Here an Eulerian grid cell is characterized by two indices increasing in the positive direction of the λ and μ axes, respectively. The cell indices are also used for the southwestern corner point. We use the convention that the indices of a Lagrangian (departure) point are the same as those of the corresponding Eulerian (arrival) point. From Eq. (16) it is seen that the sign of M_S depends on the sign of μ_b − μ_a. In order to get the right signs of the accumulated masses to be used in Eq. (13) for the cell (i, j), we must, according to the above conventions, for a south wall, use μ_a = μ_i,j and μ_b = μ_i+1,j. In the case of an east wall we must use μ_a = μ_i,j+1 and μ_b = μ_i+1,j+1.

For convenience we now introduce the local (x, y) coordinate system for each Eulerian cell C_ij such that

where Δλ_i = λ_i+1 − λ_i and Δμ_j = μ_j+1 − μ_j. For each left and right grid point λ = λ_i and λ = λ_i+1, we have x = −1/2 and x = 1/2, respectively. Let [y_a, y_b] be the values corresponding to [μ_a, μ_b] and x_b be the value corresponds to λ_b in (x, y) coordinates, respectively, so that, x_a, x_b, y_a, and y_b ∈ [−1/2, 1/2]. Now (17) can be modified for the evaluation of accumulated mass toward a cell wall P₁P₂ (Fig. 3) as follows:

The quasi-biparabolic function (4) for any cell C_ij can be expressed as

Substituting (19) into (18) and simplifying, we get

are the “accumulated coefficients.”

Thus, to determine the accumulated mass at the Lagrangian cell walls, at each time step, it is necessary to know three accumulated coefficients for each cell in addition to the five coefficients of the quasi-biparabolic function (4). The integral (20) can be analytically integrated and may further be derived for efficient numerical evaluation. Also, (20) is designed for nonuniform resolution grids. The stripe bounded by the rectangle μ_aP₂P₁μ_b in Fig. 3 may be considered as an elementary stripe. Any other formation of the stripes can be split into such elementary stripes and accumulated mass can be estimated. For example, if the stripe is separated by a grid line (bottom stripe shown in Fig. 3), then the accumulated mass for each stripe above and below the grid line of separation, the sum of which is the total accumulated mass, can be determined individually. The remapping calculation is inherently asymmetrical with respect to the two coordinate directions. However, this has only a negligible impact, at least for remapping in a Cartesian grid. Tests have shown only negligible differences between the results obtained with masses accumulated in either the λ or the μ direction (Machenhauer and Olk 1998).

a. The CISL scheme in the polar regions

In the spherical latitude–longitude coordinate system, the north–south Lagrangian cell walls should be great circle segments to be consistent with the form of the Eulerian cells. Such Lagrangian cell walls would appear as straight-line segments in the (λ, μ) coordinate system over the regions away from pole lines. However, near a pole line in the (λ, μ) plane the north–south cell walls would be curved, and the curvature would increase as the lines move toward the pole (see Fig. 4a). Thus, a straight-line segment (wall) in the polar zones of the (λ, μ) plane will not be a true image of the great circle segment representing north–south wall in the latitude–longitude system. Therefore, the choice of straight-line segments as the north–south walls for the Lagrangian cells in the polar zones is not accurate and this may adversely affect the local mass conservation. In order to circumvent this deficiency, we need to consider a local tangent plane coordinate system, where straight-line segments are good approximations for the north–south walls of the Lagrangian cells. Note that the east–west walls of the Lagrangian cells in the latitude–longitude coordinate system near the polar zones are well represented by straight lines in the corresponding (λ, μ) system. This is mainly because the lengths of these sides become very small near the polar zones in both coordinate systems, and any deviation from the actual east–west sides has only a negligible effect on the cell area.

We introduce two tangent plane coordinate systems, one tangent to the north pole and the other one tangent to the south pole, both with the origin at the pole point. They are both polar systems with the polar angle equal to λ and the polar radius r = 2(1 − μ) in the north pole system and r = 2(1 + μ) in the south pole system. Note that transformations between the spherical coordinate system and both tangent plane systems are area-preserving dλdμ = ∓rdrdλ. For the north pole system the corresponding rectangular coordinates are

and for the south polar region the tangent plane coordinates are given by

Then the corner points of a Lagrangian cell near the poles can be expressed in terms of (X, Y) in the respective tangent planes.

It is possible to compute analytically in these tangent plane coordinate systems the accumulated mass for a strip from the reference line λ = 0 toward a straight line connecting two east-side corner points. This would be the most ideal procedure. However, for simplicity, here we introduce an approximate procedure. In the polar regions where the east cell sides in the (λ, μ) system, which correspond to the straight cell sides in the tangent plane system, are strongly curved lines, they are treated as piecewise straight lines by the introduction of additional points along the curved sides. Thus, all the straight east cell sides in a polar latitude cell row on the tangent plane are divided into a certain number of equal pieces by auxiliary points (X_p, Y_p). These points are then transformed to the (λ, μ) coordinate system using

for the north pole region [for the south pole region λ_p is given as in (26) and the corresponding μ_p = (X²_p + Y²_p)/2 − 1]. Here, they are treated as corner points in auxiliary subcells. The latitude cell row in question is thus divided into a certain number of rows of subcells with straight east cell sides. Therefore, the remapping of these subcell rows can be done exactly as for the nondivided cell rows outside the polar regions, with only small modifications in the code.

b. Estimation of mass in the polar “singular” regions

In general the Lagrangian cells or subcells are quadrilaterals in the (λ, μ) plane with well-defined cell sides as specified in the preceding subsections. Exceptions are the two Lagrangian cells, which include an Eulerian pole, that is, the north pole or the south pole. As an example, Fig. 4a shows schematically such a cell including the north pole, in the upper figure in the tangent plane and in the lower figure in the (λ, μ) plane. The cell in question is denoted by P₀P₁P₂. Also shown is a Lagrangian cell P₀P₃P₄ in the same latitudinal row. Because the cell sides are so close to the Eulerian pole the east and west cell sides are very much curved in the (λ, μ) plane and would require the addition of several auxiliary points in order to be approximated well by piecewise straight-line segments. Furthermore, it is evident that the usual remapping procedure will not work for the cell including the pole point as it is not a closed quadrilateral in the (λ, μ) plane. Therefore, some special treatment would be required for the pole-point cell and in addition all the cells in the same latitude row would have to be divided into several subcells. This could be done, but it would, respectively, complicate the programming and decrease somewhat the efficiency of the whole advection scheme. Instead we test here a simple, approximate method that fits well into the general program structure. We call the degenerated polar cells “singular cells” and call an array of cells along the latitude circles that includes a singular cell a “singular belt.” Our approximate method retains the conservation of mass for the whole singular belt but treats only approximately the conservation for each individual cell in the belt. Figure 4b schematically shows the singular belt (shaded region) on a tangent plane. The dashed circles are regular latitudes, and the longitudes are denoted by dashed straight lines. The corresponding Lagrangian latitudes are shown as ellipses and longitudes as thin straight lines. The Eulerian pole is marked by a filled circle, and the corresponding Lagrangian pole is marked by an open square.

Without loss of generality, we may consider a solid-body rotation case on the sphere (Williamson et al. 1992) to illustrate the computational procedure. Figure 4c schematically shows the distribution of Lagrangian points in the (λ, μ) plane, for such a case when the wind flow is along the pole-to-pole direction. In the figure the Lagrangian pole is denoted as × and located in the second row of Eulerian cells from the pole line (i.e., the directional Courant number along the longitudes, 1 < C_θ ≤ 2). The thick curves are the Lagrangian latitudes on which the cell corner points are marked by open circles. We have seen that for the singular cell (in the shaded region) the mass cannot be constructed using the regular approach. Nevertheless, for the last row of triangular Lagrangian cells with the Lagrangian pole point as the common corner point (Fig. 4b). In Fig 4c this belt is shown as a region bounded by the closed curve Λ_b enclosing the Lagrangian pole point), the computational cells can be constructed and the mass in the individual cells can be estimated in the usual ways.

For computational convenience, we determine the total mass in the singular belt and redistribute it for the constituent cells. We consider the north pole region bounded by the Lagrangian latitude Λ_a (Fig. 4c) and the pole line defined by μ = 1. First, we construct an array of auxiliary cells by drawing lines parallel to the μ axis from the Lagrangian points on the lower boundary line (Λ_a) to the pole line. The computational cells corresponding to the new set of auxiliary cells can now be constructed and mass enclosed in each of such cells can be determined by the regular approach. Total mass bounded by the Lagrangian latitude (Λ_a) is then the sum of the masses in the constituent cells. Also the total mass in the closed region bounded by Λ_b containing the Lagrangian pole can be determined by the regular approach. Thus, the total mass in the singular belt is the difference between the total mass in the closed region bounded by Λ_b and the total mass in the region bounded by Λ_a. This idea can be further generalized for any C_θ > 1, where inside the singular belt we may have concentric closed regions enclosing the Lagrangian pole. For the cases of C_θ ≤ 1, the singular belt does not contain the closed region bounded by Λ_b as shown in Fig. 4c, and in this case the mass in the region bounded by Λ_a gives the total mass in the singular belt.

The next step is to redistribute the total mass (T^M) in the singular belt for each of the constituent cells. We use a conventional SL method for finding the weights of redistribution. First, we estimate the coordinates of the cell centers for each of the cells in the singular belt on the tangent plane, for example, defined by (24) and (25), using the known (λ, μ) coordinates of the corner points. Then, the upstream values of the density, w_i, at these points are determined by employing the computationally efficient quasi-bicubic interpolation (Ritchie et al. 1995).

The mass in the Lagrangian cell C^M_i is then given by

such that Σ_i C^M_i = T^M. The absolute values of the weights are used to avoid negative values.

c. Computational cells separated by the reference line

Accumulated mass is always determined from the common reference line, usually considered to be the west boundary wall of the computational domain. The (λ, μ) plane is east–west periodic and the longitude λ = 0 is a natural choice as a reference grid line. Since the domain is periodic, some of the computational cells may be separated by the vertical reference line. Near the polar regions the computational cells may be advected several grid lengths depending on the zonal wind velocity and may cross the reference line.

The remapping process (13) in such cells needs special attention, since the vertical walls of the computational cells may lie on either side of the reference line. There are many ways to solve this problem. We choose a simple approach described as follows. If a computational cell is separated by the reference line, then the accumulated mass (or the mass in the stripes) toward the vertical walls that appear to the right side of the reference line should be modified. This is done by adding the total mass in the stripes with the same width wrapping around the latitude circles, starting from the reference line and going to the reference line.

a. Solid-body rotation

Solid-body rotation is a commonly used experiment to test an advection scheme over the sphere. The details of the experiment are given in Williamson et al. (1992). We use the parameter values of the experiment setup as given in Williamson and Rasch (1989) and Nair et al. (1999b, hereafter referred to as NCS99).

The velocity components of the advecting wind field are given by

where α is the angle between the axis of solid-body rotation and the polar axis of the spherical coordinate system (Williamson et al. 1992). The flow field is such that when α = 0, the axis of rotation is the polar axis, and when α = π/2, it is in the equatorial plane. The initial scalar distribution is assumed to be a cosine bell. Thus

is the great circle distance between (λ, θ) and the bell center, initially taken as (λ_c, θ_c) = (3π/2, 0). The bell radius R is set to 7π/64 as in Rasch (1994), Li and Chang (1996), LR96, and NCS99. The analytic solution of the advection equation at any time step appears to be identical to the initial condition since the solid-body rotation translates the cosine bell around the globe without incurring any deformation of shape.

The spherical (λ, θ) domain consists of a 128 × 65 uniform-resolution (2.8125°) mesh, where the first and last latitudinal grid lines represent the south and the north poles, respectively. Thus, there are 128 × 64 grid cells spanning the entire spherical domain or the equivalent computational (λ, μ) plane. Note that the scalar field is initially determined as gridpoint values at the cell centers rather than being average values over the cells, as they should have been. The time step and the value of the maximum wind speed u₀ are chosen such that the angular velocity of the rotational flow is either ω = 2π/256 or ω = 2π/72 per time step. In these cases the meridional Courant number C_θ = 0.5 or C_θ = 1.78, and a complete revolution around the globe takes 256 or 72 time steps, respectively.

b. Results of solid-body rotation

Numerical experiments have been performed for α = 0, π/2 − 0.05, and π/2 with exact trajectories as recommended by Williamson et al. (1992). Three different options of the CISL scheme have been used, which are the monotonic option (CISL-M), the positive-definite option (CISL-P), and the CISL scheme without any filtering (CISL-N). In addition to that, a conventional SL scheme with bicubic-Lagrange interpolation (SL-BCL) has been used for comparison.

As discussed in section 4, to improve the local mass conservation, the Lagrangian cells in the polar regions are meridionally subdivided by introducing trajectory points along the east (or west) walls. However, the singular belt is excluded from such division and the total mass in this belt is estimated and redistributed to the constituent cells. For the present study, the three Lagrangian belts closest to the pole in each hemisphere have been “refined” by introducing additional points. We have done two sets of experiments, by introducing three, two, and one additional points (nonuniform refinement) along each east (or west) wall of the Lagrangian cells (starting closest to the pole) and another combination of two, two, and two (uniform refinement) points. The nonuniform refinements (three, two, and one) were found to be slightly more accurate than the uniform refinement. Also we have experimented with different combinations with more Lagrangian belts and additional points. There was no significant improvement in the results with high-resolution refinement of more belts and the above-mentioned combinations were found to be the most cost effective.

Results after one complete revolution around the globe (256 time steps) using the CISL-P and CISL-M schemes are displayed in Fig. 5, together with the analytic solution (dashed contours), where the contour values varies from 0.1 to 0.9 with uniform increments of 0.1. For this experiment we have used three, two, and one additional points on the north–south cell walls in the Lagrangian belts. The top panels in Fig. 5 are for polar flow with α = π/2, while the bottom panels are for equatorial flow with α = 0. Since the flow pattern of CISL-N is very similar to that of CISL-P, the corresponding figures of CISL-N are not shown. For the polar flow, the cosine bell has undergone a small stretching in the flow direction for CISL-P. When the flow is along the equator, the numerical solution obtained by using the CISL-P scheme is very close to the analytic solution. However in both cases the numerical solution with the CISL-M scheme is less accurate than that of CISL-P. Monotonicity constraints resulted in slight degradation of the shape of the cosine bell. Particularly in the central region (Fig. 5; CISL-M), the bell is stretched along the direction of the flow for both cases when α = π/2 and α = 0. Also, for the monotonic case, stretching of the central region of the cosine bell is visible in Fig. 14 of LR96.

Table 1 shows the normalized errors [for definition of error measures, see Williamson et al. (1992) and NCS99] for the solid-body rotation experiment with α = π/2. Also for easy comparison, the results from similar case studies by LR96 and Rasch (1994) are provided in the same table. The error measures show that our results are comparable with the flux-form SL (FFSL) schemes of LR96 with monotonic [FFSL-5(M)] and positive definite constraints [FFSL-3(P)]. The CISL gives better accuracy in l_∞ (normalized maximum absolute error), maximum (normalized overshoot), and minimum (normalized undershoot) errors (monotonic case). However, the FFSL scheme has slightly smaller l₁ (normalized mean absolute error) and l₂ (normalized root mean square error) in both cases shown. Rasch (1994) developed a 2D forward-in-time upwind-biased flux-form scheme and extended it to the “reduced spherical grid” (RG2.8). Table 1 shows that both the CISL and the FFSL schemes are more accurate than Rasch's scheme.

Table 2 shows error measures for solid-body rotation along the equator (α = 0) after one revolution or 256 time steps. Compared with the corresponding results in Table 1 the errors of the CISL-N and CISL-P schemes are consistently smaller for flow along the equator. This is in spite of much higher resolution in the zonal direction in the polar caps, which is in favor of the cross-pole experiments. It indicates that the special procedures introduced in the polar regions might be improved. In the case of the CISL-M scheme the errors are larger in Table 2 than in Table 1. This indicates that the error introduced by monotonization of the quasi-biparabolic function in this case dominates those introduced by the approximation in the polar caps.

Further, we have performed a series of polar flow tests with the CISL scheme to prove its robustness. Figure 6 shows a polar stereographic projection of the cosine bell advection at time steps 32, 64, and 96, as the bell approaches, passes over, and leaves the north pole, respectively. The bottom panel of Fig. 6 shows the cross polar simulation with the shift parameter α = π/2 − 0.05. In both cases there is no visible distortion of the cosine bell as it passes over the pole. Moreover, Fig. 6 (top panel) is very similar to Fig. 13 of LR96. Time traces of l₁, l₂, and l_∞ for the CISL-P scheme with α = π/2 are shown in Fig. 7. Here, error plots are more variable at time steps 64 and 192 (when the bell crosses the poles) with relatively large values for l_∞. However, the curves become again relatively smooth when the bell has crossed the poles. Compared to Figs. 7 and 8 of Rasch (1994), the maximum value of the time traces of error is much smaller with the CISL scheme.

In addition to the regular slow cross-polar advection test, we have done experiments with the strong flow. In this case 72 time steps are used for one revolution (C_θ = 1.78) with α = π/2. For this set of experiments we have used the uniform refinement for the polar Lagrangian belts. Figure 8 shows the position of the cosine bell near the north pole (top-left and -right panels) and over the south pole (bottom-left panel) and at the initial position (bottom-right panel) after 72 time steps (one revolution); the CISL-P scheme is used for the advection. There is no visible distortion for the cosine bell even for the strong polar wind. After one revolution, the numerical solution appears to be very close to the analytic solution (dashed contours). Table 3 shows the standard errors for this particular test. For comparison, errors of regular SL advection with a bicubic-Lagrange interpolation (SL-BCL) are also listed. From this table it is evident that in addition to conserving the mass exactly, which the SL-BCL does not, the accuracy of the CISL scheme is superior to that of the SL-BCL scheme. Note that for this experiment the FFSL schemes of LR96 or Rasch's schemes could not be used since the meridional Courant number C_θ > 1.

c. Deformational flow test

In order to compare the accuracy and robustness of the CISL scheme further with the conventional SL scheme, here we consider a deformation flow test in spherical geometry. NCS99 gives the details of the idealized vortex problem of Doswell (1984) on the surface of the sphere. The flow field is deformational and more challenging than the solid-body rotation. Here we use a different version of the vortex problem than that considered by NCS99. The flow field used here is positive definite, smooth, and the vortex is simulated on the surface of the sphere rather than projected onto a polar tangent plane.

Let (λ′, θ′) be the rotated coordinate system with the north pole at (λ₀, θ₀) with respect to the regular spherical coordinate system (λ, θ). Consider the rotation of the (λ′, θ′) system with angular velocity ω′ such that

In the case of the solid-body rotation test problems, the angular velocity ω′ is a constant, but for the deformational problem we consider an ω′ that varies with respect to the latitude θ′.

A steady circular vortex is defined to have normalized tangential velocity:

where ρ′ = r₀ cosθ′ is the radius of the vortex and r₀ is a constant. The angular velocity is specified as

The analytical solution at time t is

where d is the smoothness parameter for the flow field. The initial condition for the advected scalar is given by ψ(λ′, θ′, 0).

The exact upstream position of a particle at time t that arrives at a point (λ′, θ′) at time t + Δt is given by

How to specify the problem in the unrotated (λ, θ) coordinate system and the derivation of the components of velocity are described in NCS99.

For the present study we have set the parameters (λ₀, θ₀) = (π + 0.025, π/2.2), r₀ = 3, and d = 5. With these conditions two symmetric vortices are created, one near the north pole and the other one near the south pole. The flow field is smooth, and the vortex centers are approximately at 81°N and 81°S. Also, this setup keeps the vortex centers away from the poles to avoid symmetry with the underlying spherical geometry. The point values of the initial field are generated at the cell centers on a 128 × 64 grid of cells, and integrated for three time units with 32 time steps. The corresponding Courant numbers in the λ and θ directions are C_λ = 38.3 and C_θ = 1.65, respectively.

d. Results for the deformational flow test

Figure 9 (top-left panel) shows the initial conditions of the smooth scalar field and (top-right panel) the analytic solution after three time units. The bottom-left panel of Fig. 9 shows the numerical solution after 32 time steps with CISL-N, and the bottom-right panel shows the same with SL-BCL (without any filter). The shape of the vortex center is better captured by the CISL-N scheme as compared to the SL-BCL scheme. Table 4 shows that in addition to exact mass conservation the numerical solution of the CISL scheme is also more accurate. However the very small overshoots and undershoots, that is, the maximum and minimum errors, with the CISL scheme are slightly greater than the corresponding values of the SL-BCL scheme. The main reason for this difference is probably due to the truncation error involved in the difference process of the remapping scheme (13), where the mass in a computational cell is determined as the difference of accumulated masses.