The Coordinate System

Given that the surface of the eath is curved one might think that it would be difficult to construct a coordinate system that behaves like a rectangular coordinate system on a small scale but curves around the earth on a large scale.

To approach this problem we ask. How is a sphere like a cube? To a topologist they are nearly identical as it doesn’t take much effort to deform one into the other.

Imagine we had a cubic balloon. We blow it up a little and it is a cube. We draw a separate grid on each face of the cube shaped balloon. Then we blow it up more until it becomes a sphere. If our grid was fine enough then even with the balloon inflated to a spherical shape it still might look like a grid of squares is covering the balloon.

This should give confidence that we can divide the surface of a sphere up into sections that look like squares even though they cover a round object. So to do this we start thinking about what a square balloon might look like if it was inflated into a sphere.  The center of the top face will be mapped to the north pole. The center of the bottom face will be mapped to the south pole, the four faces on the side of the cube will divide the earth into four more sections. And the center point of each of these sections will be mapped to somewhere on the equator.

There are 360 degrees of longitude in the earth. Thus each section covers 90 degrees of longitude. If  we take the prime meridian as one edge then each face adjacent to the prime merridian will be centered at +/-45 degrees longitude.  Additionally the center points of the remain faces will be at +/-135 degrees of longitude.

These results are summarized in the following table. The remaining columns will be discussed in another post:

gridPoints
id	parent	level	left	right	top	bottom	up	down	lat	long	time	P_Theory
1	0	1	5	4	6	3	0	0	90	0	12	0.5
2	0	1	5	4	3	6	0	0	-90	0	12	0.5
3	0	1	5	4	1	2	0	0	0	45	12	0.5
4	0	1	3	6	1	2	0	0	0	135	12	0.5
5	0	1	6	3	1	2	0	0	0	-45	12	0.5
6	0	1	4	5	2	1	0	0	0	-135	12	0.5

The Concept

Should a climate model attempt to predict the weather or is weather far too complex a process to include in a climate model. If we neglect the weather will we miss important cycles such as El Niño and La Niña? There is a temptation in modeling to try to include as many factors as possible to hopefully simulate reality.

However, increasing the complexity of the model increases the potential for error and ads degrees of freedom which are often tuned to fit data rather then taken from fundamental measurement of physics. It is my hope that this blog and the corresponding project

https://sourceforge.net/projects/earthcubed/ (not approved yet):

will succeed by attempting the avoid the complexities inherent in small time step transient models. Instead of focusing on short term transients (i.e. weather) the focus will be more on the global energy balance. This is not to say the model will be simple. The earth will be represented with three special dimensions and two temporal dimensions.

The first temporal dimensions will be the time of day, the seconded temporal dimension will be the day. The first temporal dimension is used to express the steady state solutions.  The second temporal dimension will only be used if dynamic simulations are to be preformed.

The special coordinate system will be built based upon a cube. That is the earths surface will initially be divided up into six curved faces and each division will intersect at right angles.

Further refinement will be done by subdividing each face into grids both vertically and horizontally. The vertical grid will be scaled based on the air density and the horizontal grid will be chosen to make the three dimensional partitions roughly cubical.

With this method the space of the atmosphere will be recursively divided up into a uniformly spaced coordinate grid which will locally behave like a Cartesian coordinate system but globally it will be curved around a cube.