Carving up the Globe

This week, That’s Maths (TM007) describes various ways of dividing up the sphere. This is an important problem in geometry, biology, chemistry, astronomy, meteorology and climate modelling.

The problem of defining a uniform distribution of points on the sphere has challenged mathematicians for centuries. The vertices of the five Platonic solids achieve this but, in practical applications, millions of points are required. Many approximations to uniform distributions have been developed.

New ideas are emerging all the time. A variety of grids were discussed at a recent workshop, Solution of Partial Differential Equations on the Sphere, at the Isaac Newton Institute in Cambridge.

Latitude-longitude grid.

The regular latitude-longitude (or lat-long) grid has a serious problem: the meridians of longitude all converge at the poles. This leads to difficulties with the numerical schemes used to solve partial differential equations on the sphere: the solution may be inaccurate, or even unstable. Alternatives are needed. Several of the grids under current investigation are shown below.

Cubed-sphere grid

Cubed Sphere grid. Image from                           http://dx.doi.org/10.1016/j.jcp.2005.08.027

The cubed-sphere grid, which maps points on a sphere onto the six faces of a cube, gives a conformal grid, with quasi-uniform resolution. Instead of the two poles of a regular lat-long grid, there are now eight poles. However, they are much easier to handle than those of the lat-long grid.

Geodesic grid (Icosahedral-hexagonal grid)

Geodesic grid (Icosahedral-hexagons). Image from http://kiwi.atmos.colostate.edu/BUGS/geodesic/

Geodesic grids are used in many models, for example the atmospheric general circulation models at Colorado State University. They are comprised mostly of hexagons but, as a result of Euler’s polyhedron theorem (V − E + F = 2), there must be precisely twelve pentagons.

Variable resolution grid

Adaptive unstructured grid used in the MPAS model.
http://mpas.sourceforge.net/

Above is an adaptive grid, with variable resolution. Called a Spherical Centroidal Voronoi Tessellation, it is used in the MPAS (Model for Prediction Across Scales) that allows variable resolution on the sphere, on limited areas of the sphere. Note the smaller cells over North America, giving enhanced resolution there.

The Yin–Yang grid

Yin-Yang grid. Image from                                   http://dx.doi.org/10.1016/j.pepi.2008.06.025

The Yin–Yang grid is composed of two identical grids with the same shape and size, one a part of the normal latitude–longitude grid, the other identical but rotated. The Yin and Yang grids combine to cover the entire globe, overlapping each other at their borders. The Yin–Yang grid has been applied to simulations of various fields, including global circulation models of the atmosphere and ocean.

 


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