1. Introduction
Land, ocean, and atmosphere components of coupled climate system models are often implemented on different spherical grids, individually designed to enhance the accuracy or capture features unique to their respective settings. Historically, the regular latitude–longitude (RLL; see Table 1 for a complete list of acronyms used in this paper) grid has been the predominant choice for global atmospheric models, but problems associated with the polar singularity persist, and hence this grid is not well suited for highly scalable atmospheric models. Much interest in recent years has been instead directed toward the development of atmospheric solvers defined on more isotropic spherical grids; for example, the so-called cubed-sphere grid, which divides the polar singularities among eight weaker singularities located at the corners of a cube, and is otherwise highly scalable on parallel architectures. The cubed-sphere grid was originally introduced by Sadourny (1972), and more recently reintroduced by Ronchi et al. (1996) and Rančić et al. (1996) with equiangular grid spacing and orthogonality. For the land component, however, the RLL grid does not pose polar singularity problems as is the case for the atmosphere (with the current complexity of land models). Neither does the land model seem to be susceptible to scalability problems since most of the computation is in vertical columns rather than in the horizontal. Hence, for the foreseeable future the RLL grid seems to be a viable and convenient grid for land model components.
An intricate problem introduced by defining the model components on different spherical grids is that the exchange of information between the grids is nontrivial and requires a regridding algorithm. In a coupled climate system model it is paramount that the regridding process is not a spurious source or sink for first-order moment variables such as mass. To prevent the generation of unphysical negative and/or large values, the regridding must also be shape preserving–monotone for mixing-ratio-related variables. Regridding with these constraints, conservation and monotonicity, is a nontrivial problem if higher than first-order accuracy is desired.
The regridding problem is not only limited to a static grid-to-grid information transfer setting. The problem is essentially the same for finite-volume advection schemes where the mass transport into a given cell is given in terms of integrals over overlapping areas. In fact, methods developed for advection schemes can be readily applied in grid-to-grid regridding problems such as articulated by Margolin and Shashkov (2003). A major difference between the advection regridding problem and static grid-to-grid regridding is that the source or target grid is not static for advection problems. Hence, the regridding algorithm must be able to deal with a large class of source or target grids changing dynamically at each time step. For grid-to-grid regridding the problem is static, facilitating certain parts of the algorithm. For example, the regridding problem can be optimized for specific grid pairs. On the other hand, the advection problem is usually constrained by Courant numbers and the number of source and target grid cells is identical, which constrains the overlap regions. On the contrary, grid-to-grid regridding does not have that constraint so many source grid cells can overlap a particular target grid cell and vice versa.
A strategy for doing conservative regridding without ad hoc conservation fixers is to reconstruct a subgrid-cell distribution in each source grid cell with conservation as a constraint and then integrate the subgrid-cell distributions for the respective source grid cells over the overlap areas. This process of conservative transfer of variables between grids is referred to as remapping or rezoning. Depending on the source and target grid cell geometries (i.e., the overlap regions over which one must integrate) can be very complex. Hence, direct integration on the sphere of the overlap areas seems like an almost impossible task in terms of algorithmic complexity. Note, however, that it has been done in Cartesian geometry in the context of advection (see Rančić 1992). The problem can be greatly simplified by making use of the powerful mathematical theorem, Gauss’s divergence theorem, which converts area integrals into line integrals (see Dukowicz and Kodis 1987). This is the approach taken in the most widely used regridding software in the climate community called the Spherical Coordinate Remapping and Interpolation Package (SCRIP; Jones 1999). Also in the algorithm presented in this paper we make use of Gauss’s divergence theorem.
To perform the line integrals on the sphere one usually makes simplifying assumptions about the cell sides. For example, the sides of the grid cells are approximated by straight lines in (λ, θ) space in SCRIP. This obviously leads to exact cell wall representations for the RLL grid but other spherical grids such as the cubed-sphere grids do not share that property (see Fig. 1). The remapping algorithm’s inability to represent the cell sides exactly is here referred to as the geometric error (Lauritzen and Nair 2008, hereafter LN08). In other words, the geometric error is the deviation from exact closed-form integration along the cell walls. Unfortunately, exact closed-form integration is not always achievable, except for a small set of source and target grids. As a consequence, generic conservative remapping schemes, such as SCRIP and the Cascade Remapping between Spherical Grids (CaRS) scheme, must apply potentially crude geometrical approximations in order to yield a conservative and computationally efficient scheme (Fig. 1). Perhaps serendipitously, we find that complete removal of geometric error via exact closed-form integration is possible for the RLL grid and gnomonic cubed-sphere grids, and hence these grids immediately lend to the development of a geometrically exact conservative scheme.
Assuming small geometric error, the order of the remapping algorithm is determined by the accuracy of the subgrid-cell reconstruction. The errors introduced by the reconstruction are referred to as the derivative error. The simplest reconstruction is a piecewise constant (first order) representation in each source grid cell. This method is inherently monotone, but is excessively damping at least in idealized remapping problems. First-order reconstructions are, for example, used in the National Center for Atmospheric Research (NCAR) Coupled Climate System Model, version 3 (CCSM3; Collins et al. 2006), through SCRIP for remapping variables requiring conservation. The effect on climate by using higher-order remapping in a coupled climate system model is unknown (as far as the authors are aware), but it seems instinctive to speculate that it could have a significant effect; especially in areas where remapping is done from a coarse to a fine grid.
Unfortunately, high-order reconstructions are not inherently monotone, making it harder to achieve conservation, monotonicity, and high-order accuracy simultaneously. LN08 choose to apply the cascade remapping approach to achieve monotonocity with high-order reconstructions. This algorithm is referred to as CaRS (i.e., monotone and conservative cascade remapping between spherical grids). In CaRS the remapping problem is split into two one-dimensional problems and hence only one-dimensional limiters are needed to guarantee monotonicity, making it relatively easy to impose shape-preservation simultaneously with higher-order reconstructions. Unfortunately, the CaRS method is inherently prone to geometric errors, although it seems to compensate for these via the higher-order reconstructions that are easily and efficiently applied in one dimension. The geometric errors can be reduced by artificially increasing the resolution in areas of higher geometric error (see LN08 for details). A necessary, but insufficent, condition for completely eliminating the geometric error is to approximate the grid cells in a two-dimensional manner rather than the dimensional split approach such as done in CaRS. However, a fully two-dimensional approach requires fully two-dimensional integrals, reconstructions, and limiters. This greatly increases the complexity of the problem, especially when aiming for higher-order remapping. SCRIP has the option of performing second-order remapping if the user supplies the gradient in latitude–longitude coordinates. In theory SCRIP could be extended to higher-order by including the curvature and high-order derivatives in the line-integral computations but that route was not explored by Jones (1999) and is not a trivial extension.
In this paper we present a new conservative remapping method between gnomonic cubed-sphere grids and the RLL grids hereafter referred to as the Geometrically Exact Conservative Remapping (GECoRe) scheme. GECoRe uses Gauss’s divergence theorem to convert area integrals into line integrals. The line integrals are exact (to machine precision) for given polynomial subgrid-cell reconstructions since the lines along which the line integrals are computed exactly coincide with grid lines on the cubed sphere and RLL grid. Consequently the geometric error is completely eliminated in GECoRe. We also use up to third-order-accurate reconstructions and apply limiters to obtain monotonicity thereby obtaining high-order accuracy and shape preservation simultaneously in two dimensions.
The paper is organized as follows. In section 2 we introduce the mathematical basis for GECoRe, that is, how to compute the potentials needed to convert area integrals into line integrals, how line integrals are computed exactly, and how the fully two-dimensional reconstructions are approximated. Section 3 covers some practical considerations such as finding intersections between line segments of the two grids. The accuracy of GECoRe is assessed in section 4 by comparing standard error measures for remapping analytical functions with GECoRe, SCRIP, and CaRS between equi-angular cubed-sphere and RLL grids with different resolutions. A summary is given in section 5.
2. Geometrically Exact Conservative Remapping
In the context of finite-volume methods, we are given the cell-averaged value of a scalar field for each cell in the source grid, denoted by
a. Source and target grids
Of the numerous choices for the cubed sphere, we will focus on the equiangular cubed sphere, where a sphere is decomposed into six identical regions (panels or faces) and the grid lines on each panel are defined by equispaced central angles (see Fig. 2). Hereafter, we will refer to the equiangular cubed sphere as the alpha, beta, panel (ABP) coordinate system, in reference to coordinates on the sphere being identified by the three-element vector (α, β, np), where (α, β) ∈ [−π/4, π/4] are the coordinates on each panel and np ∈ {1, 2, 3, 4, 5, 6} is the panel index.
Although we focus on the equiangular projection, we should emphasize that the GECoRe method is trivially extended to any cubed-sphere grid that is composed of grid lines parallel to the panel edges (i.e., gnomonic cubed-sphere grids). Notably, conformal or spring-dynamics grids, where grid lines are “warped” near the corners, do not fall into this category [see, e.g., Putman and Lin (2007) for a review of several types of cubed-sphere grids]. Grids with grid lines that are not parallel to panel sides can also be captured via this scheme by replacing the exact line-integral formulas with Gaussian quadrature approximations to the integrals in gnomonic coordinates. Such an approach is taken in the tracer transport scheme described in Lauritzen et al. (2009, manuscript submitted to J. Comput. Phys.). Of course, the geometric error will not be completely eliminated if the cell sides are not great spherical arcs. Note that in such a situation the geometric error could be reduced by approximating the cell sides by several great-spherical arc segments.
b. Overview of the method
c. Summary of the GECoRe algorithm
A GECoRe remapping scheme is initialized as follows:
Perform a search on the source and target grids, classifying line segments by type (i.e., constant α, constant β, constant latitude, or constant longitude) and keeping track of their endpoints and orientation. Each line segment should then be associated with at most one ABP cell and one RLL cell.
Using (9) calculate the weights
Once the GECoRe scheme is initialized for a particular mesh pair, the resulting initialization data can be saved to a file for later use. The actual remapping is then performed as follows:
Calculate the reconstruction coefficients as(p,q) associated with the scalar field, potentially using neighboring cell values.
Use the weights
3. Practical considerations
In this section we present issues relating to the implementation of the GECoRe scheme.
a. Search algorithm
Since we are restricted to RLL and cubed-sphere grids, the search algorithm for finding line segments is dramatically simplified when compared to the SCRIP algorithm. In fact, this knowledge of the coordinate systems allows us to exactly calculate the line segment endpoints up to machine precision. The proposed technique involves first searching along longitude and latitude lines to compute intersection points, binning each line accordingly depending on its corresponding ABP cell. Then a search within each cell can be performed to obtain all lines of constant α and β within an RLL cell. This binning procedure results in memory locality of line segments associated with a given RLL cell, and hence is optimal for remappings from the RLL grid to ABP grid. A similar algorithm can be performed to obtain line segments binned by ABP cell, or the results from the forward algorithm can simply be resorted to obtain the desired result. Note that special attention must be paid to special cases, such as coincident lines, so as to avoid double counting of line segments.
b. Spherical coordinates
Similar to the SCRIP algorithm, certain aspects of the spherical coordinate system introduce additional problems when applied in practice. Unlike SCRIP, the pole points do not pose any particular problem since all calculations are performed along the surface of the cubed sphere, which has no special treatment of the pole points. However, when calculating line segment weights the multiple-valued longitude coordinate in spherical coordinates must be taken into account. Antiderivatives that require the evaluation of the longitude coordinate at each endpoint must ensure that only the “shortest” distance between longitudes is used. Similarly, additional checks must be performed to ensure that antiderivatives that require the evaluation of an arctangent have endpoints evaluated along the same branch of the arctangent function. Failure to take into account either of these factors will result in spurious factors of π being introduced into the calculation over line segments where the longitude coordinate becomes discontinuous.
c. Extensions to higher orders
Although it is not proven here, symbolic computations have shown that the line segment weights
d. Parallelization considerations
The nonlocality of this algorithm required during online calculations is largely constrained to calculating the reconstruction coefficients associated with each cell (which requires a stencil size that increases with the order of accuracy of the method). In this sense we conclude that the GECoRe scheme is potentially highly parallelizable, for example by using parallelization techniques currently employed in an existing finite-volume model.
4. Results
The new remapping algorithm (GECoRe) has been implemented for both RLL to cubed sphere and cubed sphere to RLL remapping schemes and tested on a variety of analytical fields. For comparison, we have provided results from SCRIP and CaRS for the piecewise constant first-order reconstruction (PCoM), piecewise linear second-order reconstruction (PLM), and piecewise parabolic third-order reconstruction (PPM). Particular interest should be paid to comparing the results from SCRIP and GECoRe, since much of the underlying structure of the algorithms in these two cases are directly comparable. We note, however, that the method referred to in this paper as SCRIP PPM is an extension of the SCRIP scheme of Jones (1999) to high-order accuracy. Here we have implemented this scheme by using the PPM method of Colella and Woodward (1984) to compute left and right edge values (ϕL and ϕR, respectively) in each cell, and hence use these values for reconstructing the gradient within each cell. As a consequence, this method is not a true PPM method, in that the subgrid-scale reconstruction is not composed of parabolic terms. As of the current time, no true PPM implementation of SCRIP is believed to exist.
a. Test cases
Our analysis mirrors the approach of LN08, in that we consider three idealized test cases and computed error measures for both equiangular cubed sphere to RLL remapping and vice versa.
b. Error measures
c. Calculation of reconstruction coefficients
Calculation of the reconstruction coefficients a(p,q) that describe the subgrid-cell reconstruction is required for the second- and third-order schemes. Recall that the grid is an equiangular cubed-sphere grid that, when translated to gnomonic coordinates, leads to cell centroids that are far from equidistant. As a consequence, if applied in gnomonic coordinates, an equidistant discrete derivative operator will lead to large derivative errors. We examine two possible solutions to this problem: First, we can compute the derivatives in equiangular coordinates using an equidistant discrete derivative operator and then apply a stretching factor to obtain the derivatives in gnomonic coordinates. Second, we can use a nonequidistant discrete derivative operator in gnomonic coordinates. These methods will be compared, in terms of the resulting error measures, in section 4e.
Although both the parabolic fit (three point) and quartic fit (five point) can be used to derive reconstruction coefficients for the third-order scheme, we have chosen to use the parabolic fit for the second-order scheme and the quartic fit for the third-order scheme. This choice is made since the parabolic fit corresponds most closely to a piecewise-linear reconstruction, whereas the quartic fit corresponds most closely to the piecewise-parabolic reconstruction of Colella and Woodward (1984).
Note that when applying the discretized derivative operator in ABP coordinates, halo regions must be provided for cells along the boundary of each panel. These halo cells should correspond to the cells that would be obtained by extending each panel outward, overlapping cells of the neighboring panels (see Fig. 5). Hence, halo cells do not exactly correspond to boundary cells on the neighboring panels, and their cell-averaged values must be obtained via a 1D remapping. Note that this 1D remapping does not require that the conservation criteria be fulfilled, since the area-averaged property of the interior cells is satisfied for any choice of reconstruction coefficients. For our purposes, we obtained the best accuracy from a straightforward fourth-order nonconservative cubic fit.
d. Discussion
The error measures associated with remapping from the equiangular cubed-sphere grid to RLL grid are given in Figs. 6 and 7, for a high-resolution cubed-sphere grid (Nc = 80 grid lines on each panel) and a medium-resolution cubed-sphere grid (Nc = 40) mapped to a RLL grid with Nλ = 128 longitudes and Nθ = 64 latitudes. The results of remapping from the same RLL grid (Nλ = 128, Nθ = 64) to a high-resolution cubed-sphere grid (Nc = 80) are given in Fig. 8.
It should be emphasized that, in each case, the GECoRe method should perform at least as well as SCRIP, since both methods lead to a derivative error that should be roughly identical for PCoM and PLM. The two methods deviate only in their geometric error, in that the geometrically exact techniques used for GECoRe lead to geometric error that is roughly on the order of machine epsilon. It should be noted that slight deviations from machine epsilon occur in GECoRe because of poorly conditioned function evaluations in some of the antiderivatives (i.e., calculations involving the difference of two nearly equal floating point numbers, or evaluations of arcsine or arccosine near ±1), but for our purposes we can assume these deviations are effectively negligible.
As expected, all error norms for SCRIP, CaRS, and GECoRe tend to decrease when going from PCoM to PLM to PPM. We expect that extending these methods by including the piecewise cubic scheme (PCM, as done for CaRS in LN08) and higher-order reconstructions will lead to smaller, and perhaps worthwhile, improvements in the accuracy of the method (Figs. 6, 7, and 8).
For the first-order piecewise constant method, we observe nearly identical behavior for GECoRe and SCRIP since, in both cases, line segments are effectively integrated along constant fields. Hence, small perturbations in the geometrical orientation of the line segments will not lead to significant differences in the resulting line integral.
However, we find that the error measures in GECoRe and SCRIP deviate significantly for the second- and third-order methods. The effect of geometric error is clearly apparent in our calculations, as GECoRe produces results that are often one or two orders of magnitude better than the associated error from SCRIP. The results are particularly apparent for the smooth field Y22, where each of the error measures shows an improvement of two–four orders of magnitude.
Comparing the spatial distribution of error for the three schemes (see Fig. 9), we observe that the error for SCRIP and CaRS tends to be most significant near the corners, whereas GECoRe has a much more uniform distribution of the resulting error. This result reflects the lack of a contribution from geometric error in the GECoRe scheme, which tends to dominate near the singularities of the cubed-sphere grid. Hence, we can further conclude that, compared to SCRIP and CaRS, the GECoRe scheme tends to be affected less strongly by the shape of the underlying grid.
e. Impact of the reconstruction method
We briefly turn our attention to derivative error in the GECoRe schemes by comparing four methods for computing the reconstruction coefficients in each cell. We focus on the nonequidistant parabolic fit (three point) and quartic fit (five point) methods, both computed in gnomonic coordinates, to the stretched equidistant three- and five-point methods computed in equiangular coordinates. Results from this comparison are given in Fig. 10.
Observe that there is no significant difference between both three-point schemes and both five-point schemes for the nonsmooth Y3216 and VX test cases, since the natural derivative error dwarfs any error that would be present from stretching of the equiangular reconstruction. However, for the smooth test case, there are obvious deviations between the four methods in the third-order PPM scheme. In particular, we clearly observe an immediate benefit to computing the reconstruction coefficients directly in gnomonic coordinates. From the nonsmooth test cases we also observe an obvious benefit to increasing the stencil size.
f. Impact of the monotone filter
To ensure monotonicity in the reconstruction, we employ the monotone filter of Barth and Jespersen (1989). This simple monotone filter simply scales the subgrid-scale reconstruction so that its minimum and maximum values do not exceed the cell averages of the neighboring cells. In the case of the second-order linear reconstruction, the extreme values within a cell will occur at the four corner points. For the third-order reconstruction, the extrema could also possibly occur along the boundary or within the cell. Hence, five additional points must be checked in the third-order scheme.
The effect of applying the monotone filter to the remapping scheme is given in Fig. 11. We observe that this simple monotone limiter tends to reduce the accuracy of the method by an order of magnitude, but, as a consequence, maintains that global extreme points are not enhanced. Tests performed on a simple cosine hill field, which is more susceptible to overshoots and undershoots, actually result in improved accuracy of the remapped field under the monotone limiter (not shown).
As resolution is increased, we expect that the loss of accuracy due to the monotone limiter will be reduced, since higher resolution results in less relative variation in the scalar field. Furthermore, advanced limiters, such as that of Zerroukat et al. (2005), if extended to two dimensions, are certain to result in an improved accuracy of these results.
5. Summary
A general modeling environment consists of different model components, usually implemented on different grid systems. Hence, accurate translation of scalar field data between grid systems is important in order to ensure overall accuracy of the model. For variables such as fluxes conservation is a particularly important property that should also be maintained by the remapping scheme, as it ensures that a given model does not disobey the fundamental laws of nature. The accuracy of existing conservative schemes, such as the Spherical Coordinate Remapping and Interpolation Package and the Cascade Remapping between Spherical Grids scheme is often limited by the capacity of these methods to accurately model the geometry of the problem. Here we have presented a high-order geometrically exact scheme for conservative and monotone remapping of scalar fields between the regular latitude–longitude and gnomonic cubed-sphere geometries. The new remapping scheme is referred to as Geometrically Exact Conservative Remapping.
GECoRe is based on the principle that we can integrate certain fields in exact closed-form along grid lines in the gnomonic cubed-sphere grid and regular latitude–longitude grid. The advantage of this approach is the removal of geometric error associated with low-order approximations to line segments at a similar computational cost to existing techniques. Here we provide mechanisms for constructing schemes up to the third-order, possibly combined with an inexpensive monotone filter.
GECoRe is validated by remapping a standard set of both smooth and rapidly varying test functions. Standard error measures are compared with existing SCRIP and CaRS schemes at low and medium resolutions of the cubed-sphere grid. We observe that the GECoRe scheme excels in both cases when applied at second order or higher, often yielding a one or two order of magnitude improvement over the existing schemes without additional tweaking. However, SCRIP is more general than GECoRe, since it can, in principle, handle any kind of spherical grid. Generalized versions of GECoRe can be obtained by replacing the exact integrals, where the exact formulas are too complicated or do not exist, with Gaussian quadrature of the potentials (Lauritzen et al. 2009, manuscript submitted to J. Comput. Phys.).
Acknowledgments
The authors thank Dr. Ram Nair for his comments and suggestions during the development of this paper and Dr. Rich Loft for his encouragement and administration of the SIParCS program, which allowed this work to be produced. The first author is grateful to NCAR’s SIParCS program and the University of Michigan for providing funding for this project.
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APPENDIX A
Calculation of Antiderivatives
In general, for interpolation between the RLL grid and ABP grid, we must evaluate the contour integral in (17) along four types of line segments:
Lines of constant α. Since dα = 0, any integral along a line of constant α will always evaluate to zero. Hence lines of constant α can be ignored in the computation.
Lines of constant β. In this case the integral can be evaluated directly from (18), on taking β = constant.
Lines of constant longitude (λ). In this case we can rewrite the integrand of (18) in terms of α and λ via the RLL and ABP coordinate relations given in (D1) and (D3) and then integrate to obtain a closed-form expression. Alternatively, transforming these integrals to make use of θ as the dummy variable and then rewriting all α and β components in terms of λ and θ may be helpful when performing the integration. Furthermore, observe that lines of constant λ are lines of constant α on panels 1–4, and hence the resulting integral will evaluate to 0.
Lines of constant latitude (θ). In this case we can rewrite the integrand of (18) in terms of α and θ via the RLL and ABP coordinate relations given in (D1) and (D3) and then integrate to obtain a closed-form expression. Alternatively, it may be helpful to make use of λ as the dummy variable and rewrite all α and β components in terms of λ and θ prior to integrating.
On performing the integration for each of the cases in (2)–(4), we then obtain antiderivatives Is(p,q) that can be evaluated at the endpoints of each line segment in order to give a numerically computed line integral. In Cartesian geometry computing exact line integrals of polynomials is straightforward (see Bockman 1989); however, on the sphere the integration is nontrivial and is hence performed using the computational mathematics software Maple and then simplified by hand. In the following sections we provide Is(p,q) for p + q < 3, as required for constructing remapping schemes up to third-order accuracy as well as the associated subgrid-cell reconstruction functions. Extensions to fourth order and beyond can be obtained via the process of integration also described below.
First-order scheme
- Lines of constant β:
- Lines of constant λ (panels 5 and 6):
- Lines of constant θ (panels 1–4):
- Lines of constant θ (panels 5 and 6):
Second-order scheme
We now turn our attention to the higher-order GECoRe schemes, beginning with the second-order scheme, which is based on a piecewise-linear reconstruction. First, the antiderivatives associated with the background fields used in these high-order schemes are presented without derivation. Second, we provide a derivation of the reconstruction coefficients for this scheme.
Antiderivatives
- Lines of constant β:
- Lines of constant λ (panels 5 and 6):
- Lines of constant θ (panels 1–4):
- Lines of constant θ (panels 5 and 6):
Piecewise linear (second order) reconstruction
Note that the second-order scheme requires knowledge of the area integrals (or line segment weights) for the fields x = tanα and y = tanβ, in addition to the constant field, for a total of three weights per line segment.
Third-order scheme
In this section we provide the antiderivatives and reconstruction coefficients required for the third-order-accurate piecewise-parabolic scheme.
Antiderivatives
- Lines of constant β:
- Lines of constant λ (panels 5 and 6):
- Lines of constant θ (panels 1–4):
- Lines of constant θ (panels 5 and 6):
Piecewise parabolic (third order) reconstruction
The third-order scheme requires knowledge of the area integrals, or line potentials, of the fields x2 = tan2α, xy = tanα tanβ, and y2 = tan2β, in addition to all potentials from the first- and second-order schemes.
Appendix B
High-Order Bisected Element Reconstruction
Observe that for each of the high-order schemes described in appendix A the reconstruction is performed over the gnomonic coordinates, which are inherently discontinuous between panels. Hence, special consideration must be taken when performing remapping over an RLL source volume that covers two or more panels. The simplest solution to this problem is to divide the RLL cell into two or more cells, bisected by the panel edge. All high-order reconstruction coefficients can be maintained in this case, but the first-order cell average must be reevaluated in each subcell.
A second-order-accurate bisected element reconstruction
Since the reconstruction in gnomonic coordinates requires that derivatives are aligned along x and y coordinate axes, additional work must be performed in order to rotate the reconstructed derivative in spherical coordinates to the gnomonic coordinate system. Appendix C gives equations for translating the reconstructed derivatives in gnomonic coordinates from a reconstruction in RLL coordinates for both the second-and third-order schemes.
A third-order-accurate bisected element reconstruction
A third-order-accurate reconstruction for a bisected cell can be obtained as an extension of the method discussed in the previous section, and is necessary for remapping from the RLL grid to CS grid using the PPM scheme. Henceforth, we will use the notation (xi, yi) to denote the centroid of each subcell (i ∈ {1, 2}) in gnomonic coordinates and (λi, θi) to denote the centroid in RLL coordinates.
Appendix C
The Gnomonic Cubed-Sphere Projection
Panels 1–4 (equatorial panels)
Panels 5 and 6 (polar panels)
Change-of-coordinates matrices for high-order schemes
- Panels 1–4 (equatorial panels):
- Panels 5 and 6 (polar panels):
Appendix D
The Equiangular Cubed-Sphere Projection
In this appendix we briefly provide details on coordinate relations and the metric associated with the equiangular cubed-sphere projection. For a more thorough treatment of this material, we refer to Nair et al. (2005).
A point on the cubed sphere in the equiangular projection is normally given in terms of (α, β, np) (ABP) coordinates, where α, β ∈ [− π/4, π/4] and np ∈ {1, 2, 3, 4, 5, 6}. By convention, we choose panels 1–4 to be along the spherical equator, with panels 5 and 6 centered on the southern and northern pole, respectively.
Panels 1–4 (equatorial panels)
Panels 5 and 6 (polar panels)
The equiangular cubed-sphere metric
The cell boundaries of the cubed-sphere south polar panel plotted in Cartesian coordinates. SCRIP approximates cell edges by connecting the cell vertices (filled circles) with straight lines in RLL coordinates (dotted lines). The solid lines are the exact ABP cell walls that are great circle arcs.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
The cell boundaries of the cubed-sphere south polar panel plotted in Cartesian coordinates. SCRIP approximates cell edges by connecting the cell vertices (filled circles) with straight lines in RLL coordinates (dotted lines). The solid lines are the exact ABP cell walls that are great circle arcs.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
The cell boundaries of the cubed-sphere south polar panel plotted in Cartesian coordinates. SCRIP approximates cell edges by connecting the cell vertices (filled circles) with straight lines in RLL coordinates (dotted lines). The solid lines are the exact ABP cell walls that are great circle arcs.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
An illustration of the RLL grid (thin solid lines) and cubed-sphere grid (dotted lines). Thick lines mark the boundaries of each, distinguished by the index given in the upper-right corner. By convention we choose for the southern and northern polar panels to have indices 5 and 6, respectively.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
An illustration of the RLL grid (thin solid lines) and cubed-sphere grid (dotted lines). Thick lines mark the boundaries of each, distinguished by the index given in the upper-right corner. By convention we choose for the southern and northern polar panels to have indices 5 and 6, respectively.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
An illustration of the RLL grid (thin solid lines) and cubed-sphere grid (dotted lines). Thick lines mark the boundaries of each, distinguished by the index given in the upper-right corner. By convention we choose for the southern and northern polar panels to have indices 5 and 6, respectively.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
An example of a quadrilateral target grid cell Ak that overlaps several source grid cells. The region overlapped by both Ak and An is denoted by An,k.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
An example of a quadrilateral target grid cell Ak that overlaps several source grid cells. The region overlapped by both Ak and An is denoted by An,k.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
An example of a quadrilateral target grid cell Ak that overlaps several source grid cells. The region overlapped by both Ak and An is denoted by An,k.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Contours of the analytical function (a) Y22, (b) Y3216, and (c) the vortex fields with one of the vortices centered about (λ0, θ0) = (0, 0, 6). Dotted lines show the regular latitude–longitude grid.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Contours of the analytical function (a) Y22, (b) Y3216, and (c) the vortex fields with one of the vortices centered about (λ0, θ0) = (0, 0, 6). Dotted lines show the regular latitude–longitude grid.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Contours of the analytical function (a) Y22, (b) Y3216, and (c) the vortex fields with one of the vortices centered about (λ0, θ0) = (0, 0, 6). Dotted lines show the regular latitude–longitude grid.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
A depiction of the halo region along the boundary of the top surface (dashed lines), showing the overlap with cells of the neighboring surfaces. Observe that accurate modeling of the halo region only requires 1D interpolation for this choice of grid.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
A depiction of the halo region along the boundary of the top surface (dashed lines), showing the overlap with cells of the neighboring surfaces. Observe that accurate modeling of the halo region only requires 1D interpolation for this choice of grid.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
A depiction of the halo region along the boundary of the top surface (dashed lines), showing the overlap with cells of the neighboring surfaces. Observe that accurate modeling of the halo region only requires 1D interpolation for this choice of grid.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Performance measures for the remapping of Y22, Y3216, and the idealized vortices (VX) from a medium-resolution ABP grid (Nc = 80) to an RLL grid (Nλ = 128, Nθ = 64) using GECoRe, SCRIP, and CaRS with PCoM, PLM, and PPM reconstructions: (left) l1, (middle) l2, and (right) l∞ error.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Performance measures for the remapping of Y22, Y3216, and the idealized vortices (VX) from a medium-resolution ABP grid (Nc = 80) to an RLL grid (Nλ = 128, Nθ = 64) using GECoRe, SCRIP, and CaRS with PCoM, PLM, and PPM reconstructions: (left) l1, (middle) l2, and (right) l∞ error.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Performance measures for the remapping of Y22, Y3216, and the idealized vortices (VX) from a medium-resolution ABP grid (Nc = 80) to an RLL grid (Nλ = 128, Nθ = 64) using GECoRe, SCRIP, and CaRS with PCoM, PLM, and PPM reconstructions: (left) l1, (middle) l2, and (right) l∞ error.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
As in Fig. 6, but with Nc = 40.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
As in Fig. 6, but with Nc = 40.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
As in Fig. 6, but with Nc = 40.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
As in Fig. 6, but remapping from an RLL grid (Nλ = 128, Nθ = 64) to an ABP grid (Nc = 80).
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
As in Fig. 6, but remapping from an RLL grid (Nλ = 128, Nθ = 64) to an ABP grid (Nc = 80).
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
As in Fig. 6, but remapping from an RLL grid (Nλ = 128, Nθ = 64) to an ABP grid (Nc = 80).
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Spatial distribution of the error (×10−2) for the remapping of Y3216 from a coarse-resolution ABP grid (Nc = 40) to an RLL grid (Nλ = 128, Nθ = 64) using (a) GECoRe, (b) SCRIP, and (c) CaRS with piecewise parabolic (third order) reconstructions.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Spatial distribution of the error (×10−2) for the remapping of Y3216 from a coarse-resolution ABP grid (Nc = 40) to an RLL grid (Nλ = 128, Nθ = 64) using (a) GECoRe, (b) SCRIP, and (c) CaRS with piecewise parabolic (third order) reconstructions.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Spatial distribution of the error (×10−2) for the remapping of Y3216 from a coarse-resolution ABP grid (Nc = 40) to an RLL grid (Nλ = 128, Nθ = 64) using (a) GECoRe, (b) SCRIP, and (c) CaRS with piecewise parabolic (third order) reconstructions.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Performance measures for the remapping of Y22, Y3216, and the idealized vortices (VX) from a high-resolution ABP grid (Nc = 80) to an RLL grid (Nλ = 128, Nθ = 64) using GECoRe with four choices of subgrid-scale reconstruction techniques: three-point stretched equiangular (3-St), five-point stretched equiangular (5-St), three-point nonequidistant gnomonic (3-NE), and five-point nonequidistant (5-NE): (left) l1, (middle) l2, and (right) l∞ error.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Performance measures for the remapping of Y22, Y3216, and the idealized vortices (VX) from a high-resolution ABP grid (Nc = 80) to an RLL grid (Nλ = 128, Nθ = 64) using GECoRe with four choices of subgrid-scale reconstruction techniques: three-point stretched equiangular (3-St), five-point stretched equiangular (5-St), three-point nonequidistant gnomonic (3-NE), and five-point nonequidistant (5-NE): (left) l1, (middle) l2, and (right) l∞ error.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Performance measures for the remapping of Y22, Y3216, and the idealized vortices (VX) from a high-resolution ABP grid (Nc = 80) to an RLL grid (Nλ = 128, Nθ = 64) using GECoRe with four choices of subgrid-scale reconstruction techniques: three-point stretched equiangular (3-St), five-point stretched equiangular (5-St), three-point nonequidistant gnomonic (3-NE), and five-point nonequidistant (5-NE): (left) l1, (middle) l2, and (right) l∞ error.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Performance measures for the remapping of Y22, Y3216, and the idealized vortices (VX) from a high-resolution ABP grid (Nc = 80) to an RLL grid (Nλ = 128, Nθ = 64) using GECoRe with (dotted line) and without (solid line) monotone filtering: (left) l1, (middle) l2, and (right) l∞ error.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Performance measures for the remapping of Y22, Y3216, and the idealized vortices (VX) from a high-resolution ABP grid (Nc = 80) to an RLL grid (Nλ = 128, Nθ = 64) using GECoRe with (dotted line) and without (solid line) monotone filtering: (left) l1, (middle) l2, and (right) l∞ error.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
Performance measures for the remapping of Y22, Y3216, and the idealized vortices (VX) from a high-resolution ABP grid (Nc = 80) to an RLL grid (Nλ = 128, Nθ = 64) using GECoRe with (dotted line) and without (solid line) monotone filtering: (left) l1, (middle) l2, and (right) l∞ error.
Citation: Monthly Weather Review 137, 6; 10.1175/2008MWR2817.1
List of acronyms used in this paper.
Observe that (A20) is analogous to Eq. (5) in Jones (1999).
This choice reduces our reconstruction to the well-known PPM devised by Colella and Woodward (1984).
