Mobius Net Cubed-Sphere Gnomonic Grids
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Mobius Net Cubed-Sphere Gnomonic Grids

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    A ‘gnomomic grid’ on the spherical surface is one whose grid lines are great circles and therefore centrally project as straight lines in a plane tangent to the sphere. For a cubed-sphere, a gnomonic grid on each face projects onto the plane tangent to the sphere at the center of the face to form a net of orthogonal lines on this plane. Each one of the two families of lines associated with a given face corresponds to the family of planes they belong to that share an axis of mutual intersection through the center of the sphere. The angular distribution of these planes at this shared axis can be chosen to define a characteristic type of gnomonic cubed grid. For the purposes of atmospheric numerical modeling and data assimilation, the most natural choice is to have the planes defining the grid lines to be equally-spaced in angle (the ‘equiangular’ gnomonic cube grid). With this convenient choice, the problem of interpolating from the grid to targets located close to a cube-edge is a relatively simple one because the family of lines of which the edge is a member is smoothly continued in passing from one face to its neighbor. Thus, extrapolating the grid of one face needed to supply a sufficient margin for subsequent centeredstencil interpolations to arbitrary target points in that cube’s face involves only one-dimensional interpolations to generate the extra needed grid points. However, the equiangular gnomonic grid is not the only one possessing this property. By a minor adjustment to the angular spacing of the grid lines of each family, it is possible to make all three families of grid lines shared by the three faces that meet at a cube-corner engage in perfect three-way intersections in the vicinity of that corner. This is analogous to the construction of a ‘M¨obius net’ configuration of lines in classical projective geometry. When a form of grid has this property one obtains further simplification of the subsequent interpolations, especially when broad stencils are desired. A proposed method of constructing such a M¨obius-net gnomonic cubed sphere grid is described.
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