Journal of the Earth Simulator, Volume 3, September 2005, 20 – 28 20 Dissection of a Sphere and Yin-Yang Grids Akira Kageyama The Earth Simulator Center, Japan Agency for Marine-Earth Science and Technology, 3173–25 Showa-machi, Kanazawa-ku, Yokohama, 236-0001 JAPAN Email: kage@jamstec.go.jp (Received February 28, 2005; Revised manuscript accepted March 24, 2005) Abstract A geometrical dissection that divides a spherical surface into two identical pieces is considered. When the piece is symmetric in two perpendicular directions, the two pieces are called yin and yang and the dissection is yin-yang dissection of a sphere. The yin and yang are mapped each other by a rotation M on the sphere where M 2 = 1. Therefore, the yin’s landscape viewed from yang is exactly the same as the yang’s land- scape viewed from yin, and vice versa. This complemental nature of the yin-yang dissection leads to the idea of new spherical overset grid named Yin-Yang grid. The flexibility of the yin-yang dissection of a sphere enables one to patch the piece with an orthogonal, quasi-uniform grid mesh. Since the two pieces are identi- cal, one computational routine that involves individual calculation in each grid is used for two times, one for yin grid and another for yang. Other routines that involve data transformation between yin and yang are also recycled for two times because of the complemental nature of the grids. Due to the simplicity of the underly- ing grid geometry, the Yin-Yang grid suits to massively parallel computers. Keywords: dissection, Yin-Yang grid, spherical grid 1. Introduction Recently, we proposed a new spherical grid system named “Yin-Yang grid” [1], which is a kind of overset or Chimera grids applied to a spherical surface or spherical shell. The Yin-Yang grid is already used with good suc- cess in simulations of geodynamo [2, 3], mantle convec- tion [4], and coupled simulation of atmosphere and ocean [5, 6]. In this paper, we review the Yin-Yang grid with a special emphasise on the motivation and basic ideas behind it. In these ten years, we have been performing simulation research of magnetohydrodynamic (MHD) dynamo in spherical shells. For the spatial discretization of the MHD equations, we used a finite difference method with a prospect of its increasing importance in the era of mas- sively parallel computers. The base grid system adopted in our previous spherical shell MHD code was the lati- tude-longitude (Lat-Lon) grid, which is defined on the spherical polar coordinates (r, , ) with radius r, colati- tude , and longitude . For the spherical surface S, the grid mesh of the Lat-Lon grid is uniform when it is seen in the computational space (1) but it is far from uniform when it is seen in the real space. A numerical problem in the Lat-Lon grid is the exis- S : = , , - /2 /2 , , ≤ ≤ tence of the coordinate singularity on the north pole (2) and the south pole (3) In the spherical polar coordinates, all the differential operators should be represented in three different forms for three local regions of S. Take the gradient operator ∇ =(∇ r , ∇ , ∇ ) as an example. Define a local region S' by (4) which is the spherical surface without the poles. The gra- dient operator is represented by (5) which is a familiar form. On the other hand, the gradient operator should be represented in other, unfamiliar, forms for the north and south poles as (6) = r , r 1 , + r 1 2 for Pn , ∇ ∂ ∂ ∂ ∂ ∂ ∂ ∂ z = r , r 1 , r sin 1 for , S' ∇ ∂ ∂ ∂ ∂ ∂ ∂ S' : = S - P n - P s , P s : = = , . P n : = = 0, ,