The bi-Mercator Grid as a Global Framework for Numerical Weather Prediction

1. The bi-Mercator Grid as a Global Framework for Numerical Weather Prediction R. James Purser S.A.I.C. National Centers for Environmental Prediction, Washington D.C.

2. Introduction The popularity of global spectral methods seems destined to decline with the continued increase in resolution and the need to parallelize efficiently. [Caveat: But it seemed that way 20 years ago, too!] Reverting to grid-point methods raises the “polar” problems we are all familiar with.

3. Convex “polyhedral” geometries, such as the cubed sphere, or the icosahedron, have been introduced to avoid the problem of excessive longitudinal resolution but two coordinate singularities are thereby replaced by eight (square grids) or twelve (triangular grids). Of course, these are much weaker singularities, so some low-order numerical schemes are still able to work. Conformal-grid versions of all the polyhedral geometries are possible and, except at the singular points, make the treatment of vector quantities particularly easy and elegant. However, they all suffer from mildly excessive resolution near each of the singularities. This is mitigated through use of a more general variational principle in the grid- generation [conformal grids being solutions to a special form of variational principle – see Purser and Rancic (1998).]

4. Unfortunately, some valuable numerical techniques cannot easily accommodate Singularities ANYWHERE on the computational domain. Examples: High-order numerical schemes, especially if they are “compact” (one-dimensionally implicit in space) do not usually tolerate singularities. Dimensional-splitting “Cascade” methods for interpolating between grids during semi-Lagrangian advection is highly efficient but impossible to implement near a Singularity, since it requires all intermediate interactions to remain “transversal”. The topological complications of smoothed or conformal polyhedral grids can also cause problems, even away from the singularities. Example: Cascade interpolation operates on “X”, then on “Y”; this sequence is disrupted by the ambiguity between “X” and “Y” encountered upon tracing a circuit that contains a singularity.

5. These facts, and the continued desire to keep open the options of employing compact numerics, cascade semi-Lagrangian interpolation, and conformal grids, motivated the examination of composite grid configurations in which the difficulties associated with singularities are replaced with the more manageable inconveniences associated with grid-to-grid interpolations.

6. Bi-Mercator/Yin-Yang Grid A bi-stereographic grid, as used by Dudhia and Bresch (2002) following earlier feasibility studies of Browning et al. (1989) seemed excessively wasteful, given the factor of 2 change in map factor between pole and equator. Overlapping stereographic faces in a cubic arrangement need much interpolation to bring about mutual consistency, including the difficult three-way interpolations and blending at corners. An overlapping pair of transverse Mercator grids seemed an ideal solution but, regrettably, I was unaware of the work being done here in Japan on the new “Yin-Yang” grid when I found essentially the same geometrical solution!

7. The Yin-Yang grid is proposed in the submitted paper: Kageyama, A., and T. Sato, 2003: The “Yin-Yang Grid”: An overset grid in spherical geometry. (submitted to Geochem. Geophys. Geosyst.) Preprint available at: http://arxiv.org/abs/physics/0403123 Also, in this workshop, note the presentations by Peng; Hirai; Komine; Ohdaira.

8. Rectangles, minimal overlap Overlaps trimmed to median

9. Grid construction • The sphere is covered by two rectangular grids, mutually nested • Each is conformal (transverse Mercator) • Each is centered on a geographical pole • “latitude” extends to > 45 degrees and “longitude” extends to >135 degrees both ways

10. Triangular grid is also possible(in two long hexagonal regions)

11. Nonoverlapping conformal analogues of these bi-Mercator square/triangular grids are the conformal cubic and a conformal gridding of an irregular octahedron.

12. Reconciling solutions We may project the two maps “X” and “Y” (named for the Cartesian directions of their principal axes) into the regular “Z” Mercator domain. There are many ways to find a “median” symmetrically lying between the two interlocked boundaries. A third conformal grid may then be constructed as follows: Map the median to segment [0,2*pi] of the real axis of the complex plane; regard the “x” and “y” Z-map coordinates as real and imaginary parts of a function; Fourier transformation allows analytic continuation.

13. The new grid maps conformally to the sphere and bears a symmetric relation to both of the main grids

14. For accurate (high-order) interpolations, we may use the “cascade” technique (Purser and Leslie 1991) (Sides) (Ends)

15. The cascade method allows four (slightly overlapping) sections of the median grid to connect with the two ends and two sides of one main grid. Another partition likewise connect the median grid with the other main grid. If we dispense with the intermediate “median grid”, cascade interpolation requires partitioning of mutual overlap into eight sections (true also of other composite grids). Using “Cascade”, the interpolations are reduced to a (blended) sequence of one-dimensional operations, making accuracy more affordable. A smooth partition of unity (eg. a Beta distribution) provides a suitable weighting to the respective main grids across the width of a suitable “median strip”.

16. Work is beginning at NCEP to use the Yin-Yang/bi-Mercator framework in order to test it as a possible means to unify atmospheric modeling activities for regional and global forecasts. This includes the use of the non-hydrostatic WRF dynamical cores. Accuracy and mass-conservation of this approach will be given special attention. Best width for blending zone needs to be determined. Numerical artifacts at the “seam” need to be avoided at all cost.

17. Summary • Bi-Mercator mapping efficiently grids the sphere conformally, square or triangular. • Grids may be considered analogues of non-overlapping polyhedral conformal grids. • Cascade interpolations may be applied with or without a median grid to reconcile the two solutions accurately. • Unifies regional and global models