Regional
refinement and grid hierarchiesThe horizontal grid is derived from the regular icosahedron. The geometric properties of this Platonic solid body are explained for example on:
The projection of the regular icosahedron on the unit sphere provides a regular grid on the sphere with 20 equilateral spherical triangles, 30 great circle edges, and 12 vertices. Its dual grid is the projection of the regular dodecahedron on the sphere.
The Delaunay triangulation then allows to refine each triangle into n2 smaller tringles by dividing each edge into n sections. This procedure may be repeated as often as necessary. In our implementation we allow for small prime numbers (n1 = 2, 3, 5) in the first triangulation step, but restrict ourselve to ni=2,...,k = 2 in the following refinement steps. If the refinement procedure is applied equally to all base triangles, then after k refinement steps the resulting grid has N = 20 × (n1 × 2k-1)2 triangular cells.
An important point is to note that the regularity of the spherical triangles of the base grid is lost, though the differences in areas between triangles or lengths between edges remains small. This break in symmetry is obvious in the dual grid that consists now of pentagons and hexagons.
There are some possibilities to cure the effects of the break in symmetry, at least for some properties: The use of a method called equal area subdivision, which uses small circles for the construction of triangle edges (Kimerling et al.), may warrant equally sized triangles at each refinement level at the expense of biasing the distances from an edge midpoint to the two adjacent triangle centers. The use of small circles may alternatively be used to solve this problem, but again by slightly disturbing the size of the triangles. A further means to improve the grid is the spring dynamics as in the NICAM model.
The geometric informations as for example positions of vertices, cell areas and edge lengths, and the topology describing the neighborhood relations between vertices, cells and edges are generated by a "grid generator" that is a separate software independent of the ICON circulation models.
The globally refined spherical grid can be truncated to specific regions, for example in regional atmospheric models, or in ocean models. The selection methods implemented currently are:
a subset of the base triangles of the regular spherical icosahedral grid
cells with their centers within a selected longitude and latitude range
cells with their centers within a given distance to a given longitude and latitude
cells selected by a mask defined in longitude and latitude
Regional
refinement and grid hierarchiesIn some applications the domain will be refined stronger in one or
more regions. The further refinement may consist of one or more
refinement steps by bi-section of the spherical edges. The original
uniformly refined grid of the entire domain, the parent grid, and the
whole cascades of regional higher refined sub-grids, or patches,
build the grid hierarchy that is used for the solution of the
equations.
The example to the right shows a parent grid
resulting from 3 consecutive refinement steps, where the first
refinement step is a tri-section (n1 = 3, n2
= n3 = 2). A first regional refinement step is
applied to the area between 10N and the North Pole, and between about
50E and 40W. A second embedded region is refined once more.
Within the ICON model, the data are structered in patches, which
are connected pieces of cells. Technically, patches are lists of
cells building the patch area with the information on the neighbor
patches and possible parent or children patches. During the
integration, information has to be exchanged between neighboring
patches for those cells which are part of the inner and outer halo of
a patch.
The
second example shows a parent grid resulting from 3 consecutive
dyadic refinement steps (ni = 2, i
= 1, 2, 3). On this global grid a patch is defined as the set of
cells within the area between about the Equator and 50N, and between
55W and 110W. Cells belonging to the interior are marked in with a
red circle. Cells along the boundary, which belong to the patch are
marked in light blue. These cells form the internal halo of the
patch. The neighboring cells outside of the patch, building the
external halo, are marked in dark blue. The edges delimiting the
patch are marked with green circles.
More information on the grid are given in description of the ICON
prototype SWM in section 3: The
discretization grid and the discrete operators.
(ags, 27.10.2008)