Shaping the weather sphere
THAT’S MATHS:FOR MILLENNIA, mathematics has attracted the attention of leading philosophers, scholars and scientists. The problems studied have ranged from the practical to the abstract.
The application of maths to the physical world expanded dramatically following the development of calculus by Isaac Newton and Gottfried Wilhelm Leibniz. Applications of mathematics now extend far beyond the traditional field of theoretical physics to areas as diverse as biology, chemistry, ecology, finance, genetics and volcanology.
Newton’s achievements established Cambridge as a world-leading mathematical centre, and its reputation for excellence has continued to the present day. In 1992, the 350th anniversary of Newton’s birth, the Isaac Newton Institute was established. The institute supports research in all areas of pure and applied mathematics, hosting conferences and workshops where mathematical scientists come together for lectures and for research collaboration.
I recently participated in a workshop there titled “Solving Partial Differential Equations on the Sphere”. These equations are used to model numerous physical systems. In particular, they are the basis of weather prediction and climate modelling. The workshop was about novel ways of solving these equations using computers with highly parallel architectures, which can perform thousands of calculations simultaneously.
On the Earth’s surface, latitude and longitude are normally used to specify positions. But since the meridians converge towards the poles, a regular grid with points spaced evenly at, say, one degree apart has points that cluster together at high latitudes. This causes errors and instabilities for the numerical schemes used to make weather forecasts.
For a circle, it is simple to pick a set of uniformly distributed points: just divide the circle into equal segments. For a sphere, it is not so easy, and the question arises: can we define a uniform distribution of points on the spherical Earth? There are some possibilities. For example, if we embed a cube in the globe, we get a uniform grid of eight points, represented by the corners of the cube. But we don’t need eight points; we need something like eight million. Can the globe be divided into millions of parts in a completely regular way?
The Greeks showed, thousands of years ago, that there are only five solutions to this problem, corresponding to the five so-called Platonic solids. The cube, with six square faces meeting at right angles, is well known. The tetrahedron, made from four equilateral triangles, is also perfectly regular, as are the octahedron and the icosahedron, with eight and 20 such faces respectively. Finally, there comes the dodecahedron, with 12 regular pentagonal faces.
Plato identified the four elements, fire, earth, air and water, with four of the solids and regarded the fifth, the dodecahedron, as representing the entire cosmos. This idea was developed by Kepler in a magnificent, if ultimately invalid, model of the solar system published in 1596 in his Mysterium Cosmographicum (Mystery of the Cosmos).
For each of these “regular polyhedra”, all faces are the same, all edges equal and each vertex or corner identical. Any of them can be embedded in a sphere to generate a perfectly regular grid. Alas, Plato presented a proof that these are the only possibilities: no sixth solution exists. Thus, we are limited to, at most, 20 regularly spaced points, not nearly enough for practial applications.
Several ingenious and exotic approximations to a perfectly regular arrangement have been devised. One, called the “yin-yang grid”, splits the sphere into two pieces along a curve like the seam of a tennis ball, with a regular grid applied to each piece. Numerous other solutions are under study. Any one of these may prove to be the best candidate for generating more accurate weather forecasts in the future.