Geodesic grid

A geodesic grid is a technique used to model the surface of a sphere (the Earth) with a subdivided polyhedron, usually an icosahedron.

Introduction

When modeling the weather, ocean circulation, or the climate, partial differential equations are used to describe the evolution of these systems over time. Because computer programs are used to build and work with these complex models, approximations need to be formulated into easily computable forms. Some of these numerical analysis techniques (such as finite differences) require the area of interest to be subdivided into a grid — in this case, over the shape of the Earth.

A common approach is to use a longitude/latitude grid (a Cartesian grid), where each cell is defined as the intersection of a longitude and latitude line. This approach can be easily represented on a computer as a rectangular grid, accessible using the longitude and latitude as an ordered pair. It has the advantage of being simple and easy-to-understand. There are two downsides, however. First, there are many more cells near the poles than the equator (oversampling of poles), meaning a larger grid is needed to represent the equator properly. Second, there are two singularities at the poles where the lines of latitude converge and the longitude term loses significance. This requires smaller time steps to assure stability.

Another approach gaining favor uses grids generated by the subdivision of an icosahedron, generated by iteratively bisecting the edges of an icosahedron and projecting the new vertices onto a sphere. In this "geodesic grid", each of the vertices in the resulting geodesic sphere corresponds to a cell. The resulting structure can be visualized as cutting twenty triangles from a hexagonal grid and arranging them over the surface of a sphere. [For this reason, geodesic grids are also known as icosahedral-hexagonal grids.] Alternatively, using the vertices of the bisected icosahedron directly as vertices in the mesh provides the dual polyhedron of the mesh described above. This triangular mesh is useful for representing the grid graphically.

The geodesic grid inherits many of the virtues of 2D hexagonal grids, and specifically avoids problems with singularities and oversampling near the poles. Along the same line, different Platonic solids could also be used as a starting point instead of an icosahedron — cubes are common in other applications, such as video games.

Positive traits

* Largely isotropic.
* Resolution can be easily increased by binary division.
* Does not suffer from over sampling near the poles like more traditional rectangular longitude/latitude square grids.
* Does not result in dense linear systems like spectral methods do (see also Gaussian grid).
* No single points of contact between neighboring grid cells. Square grids and isometric grids suffer from the ambiguous problem of how to handle neighbors that only touch at a single point.
* Cells can be both minimally distorted and near-equal-area. In contrast, square grids are not equal area, while equal-area rectangular grids vary in shape from equator to poles.

Negative traits

* More complicated to implement than rectangular longitude/latitude grids in computers

History

The earliest use of the (icosahedral) geodesic grid in geophysical modeling dates back to 1968 and the work by Sadourny, Arakawa, and Mintz [cite journal | last=Sadourny | first=R. | coauthors=A. Arakawa; and Y. Mintz | title=Integration of the non-divergent barotropic vorticity equation with an icosahedral-hexagonal grid for the sphere | journal=Monthly weather review | volume=96 | pages=351–356 | year=1968 | doi=10.1175/1520-0493(1968)096<0351:IOTNBV>2.0.CO;2 | doilabel=10.1175/1520-0493(1968)0960351:IOTNBV2.0.CO;2] and Williamson. [cite journal | last=Williamson | first=D. L. | title=Integration of the barotropic vorticity equation on a spherical geodesic grid | journal=Tellus | volume=20 | pages=642–653 | year=1968] [Williamson, 1969] Later work expanded on this base. [cite journal | last=Cullen | first=M. J. P. | title=Integrations of the primitive equations on a sphere using the finite-element method | journal=Quarterly Journal of the Royal Meteorological Society | volume=100 | pages=555–562 | year=1974 | doi=10.1002/qj.49710042605] [Cullen and Hall, 1979.] [cite conference | last=Masuda | first=Y. | coauthors=H. Ohnishi | title=An integration scheme of the primitive equation model with an icosahedral-hexagonal grid system and its application to the shallow-water equations | booktitle=Short- and Medium-Range Numerical Weather Prediction | publisher=Japan Meteorological Society | pages=317–326 | year=1987] [cite journal | last=Heikes | first=Ross | coauthors=David A. Randall | title=Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part I: Basic design and results of tests | journal=Monthly Weather Review | volume=123 | pages=1862–1880 | year=1995 | doi=10.1175/1520-0493(1995)123<1862:NIOTSW>2.0.CO;2 | doilabel=10.1175/1520-0493(1995)1231862:NIOTSW2.0.CO;2

cite journal | last=Heikes | first=Ross | coauthors=David A. Randall | title=Numerical integration of the shallow-water equations on a twisted icosahedral grid. Part II: A detailed description of the grid and an analysis of numerical accuracy | journal=Monthly Weather Review | volume=123 | pages=1881–1887 | year=1995 | doi=10.1175/1520-0493(1995)123<1881:NIOTSW>2.0.CO;2 | doilabel=10.1175/1520-0493(1995)1231881:NIOTSW2.0.CO;2] [Randall "et al.", 2000; Randall "et al.", 2002.]

References

External links

* [http://kiwi.atmos.colostate.edu/BUGS/geodesic/ BUGS climate model] page on geodesic grids
* [http://www.sou.edu/cs/sahr/dgg/ Discrete Global Grids] page at the Computer Science department at Southern Oregon University