R. James Purser *, Zavisa I. Janjic + and Thomas Black NOAA/NWS/NCEP/EMC Camp Springs, MD

1. A bi-cyclindrical “Yin-Yang” global grid geometry applied to the NCEP Nonhydrostatic Mesoscale Model R. James Purser*, Zavisa I. Janjic+ and Thomas Black NOAA/NWS/NCEP/EMC Camp Springs, MD *Science Applications International Corporation +UCAR Visiting Scientist Program 21st Conference on Weather Analysis and Forecasting/17th Conference on Numerical Weather Prediction. 1st-5th August 2005, Washington D.C.

2. At NCEP, we are examining ways by which we can extend existing limited area models, such as the WRF Nonhydrostatic Mesoscale Model (NMM), to the global domain. If we succeed, then NCEP operations might benefit from a forecasting system possessing a common dynamical framework that applies at all scales. The effort of maintaining and improving one unified system should be less than the effort of maintaining and improving separate systems for global and regional domains. We need to choose a gridding compatible with numerical accuracy and computational efficiency in a distributed processor computing environment.

3. The difficulties of gridding the sphere are well known. • No method is completely free of problems. • Latitude-longitude grid:Somewhat inefficient areal coverage, converging grid meridians near poles require speicial additional filtering, two coordinate singularities at the poles themselves need special treatment. • Grids based on polyhedral surfaces mapped to the sphere (e.g., the “cubed sphere”): mulitple coordinate singularities still need special treatment, smooth continuity across facet edges only only achieved at the price of severe grid curvarure near the coordinate singularities. • Overlapping grids:the efficient areal coverage and freedom from coordinate singularities comes at the cost of having to perform frequent two-way interpolations and mergings of the solutions between grids sharing a region of overlap to maintain consistency.

4. Among the class of overlapping grid configurations that involve only two grids, the “Yin-Yang” scheme offers what is probably the most efficient practical way of covering the globe. This configuration was first applied to problems of convection in the Earth’s core and mantle by Kageyama and colleagues at Japan’s Earth Simulator Center[1,2]. It is also being used there for models simulating the oceans[3] and atmosphere[4]. [1] Kageyama, A., and T. Sato, 2004: The “Yin-Yang Grid”: An overset grid in spherical geometry. Geochem. Geophys. Geosyst., Q09005, doi:10.1029/2004GC000734. E-print: asXiv:physics/0403123 v1. [2] Yoshida, M., and A. Kageyama, 2004: Applications of the Yin-Yang grid to a thermal convection of a Boussinesq fluid with infinite Prandtl number in a three-dimensional spherical shell. Geophys. Res. Lett., 31, L12609 doi:10.1029/2004GL019970. [3] Ohdaira, M., K. Takahashi, and K. Watanabe, 2004: Validation for the solution of shallow water equations in spherical geometry with overset grid system. The 2004 Workshop on the Solution of Partial Differential Equations on the Sphere. [4] Komine, K., K. Takahashi, and K Watanabe, 2004: Development of a global non-hydrostatic simulation code using Yin-Yang grid system. The 2004 Workshop on the Solution of Partial Differential Equations on the Sphere.

7. The domain is covered by a pair of transverse cylindrical projection grids. Code under development at NCEP allows either conformal Mercator mapping or simply the rotated spherical “lat-long” coordinates to be used. For minimal overlap each map must extend 270 degrees (3 right angles) in rotated longitude and ±45 degrees from the “equator” in rotated latitude. Their mutual projection axes are at 90 degrees, so they interlock. The median curve of demarcation between the domains of responsibility of the two grids is not unique – any suitable closed curve within the overlap invariant to the “dicyclic group of order 8” symmetry of the Yin-Yang configuration will serve this purpose. In the geographical Mercator representation this curve has a wave-2 form; it can be adequately shaped using Fourier sine coefficients in the Mercator domain with only waves 2, 4 and 6. Coefficients are optimized under conditions that cause the curve to fill out, with relatively straight sides, the smallest sized rectangle in the transverse-cylindrical domain that contains the curve, while preserving a very uniform parameterization along the curve.

8. In practice, we need a finite overlap all round in order to give all the interpolation operators a sufficient stencil and to allow some progressive blending of the two solutions according to an appropriate measure of effective distance across the chosen median. Here, we show the median sitting between the two limits of the blending zone. The projection used here is the map’s own transverse Mercator (the broadening of the zone close to both sides of the rectangle is an artifact of this projection).

9. With the median analytically defined as a simple Fourier sum in Mercator projection, a conformal “analytic continuation” supplies a natural way of defining the transverse distance across the overlap zone with the median itself at transverse coordinate zero:

10. Interpolations are by adaptations of methods already in use at NCEP for interpolation to data in the assimilation schemes. We call these methods “diamond” interpolation schemes informally.

11. A family of smooth blending functions is obtained by iterating the function, f(x) = (3x – x3)/2, between x = -1 and x = +1. y=f[f(x)] y=f(x) y=x y = +1 x = +1 y = -1 x = -1

12. Summary • We are in the process of applying “Yin-Yang” grid geometry to a B-grid version of the NMM. • Experiments will determine the optimum combinations of overlap zone width, order of accuracy of interpolations, order of blending. • Evaluation of cost relative to the lat-long grid will need to be carried out.