• Introduction
  
• Data Assimilation
  
• SSI
  
• Harmonics
• Variables
  
• Example
• Sources

Where on Earth are you?

Most people use latitude and longitude to represent where they are on the planet.  However, that's not the only method!  For example, you could use Euclidean space, but that is often inconvenient on a curved surface like a sphere.  Some forecast models use spherical harmonic functions as coordinates.

What are spherical harmonics?

If you had a glass sphere, and you banged on it, you would hear a certain sound*; this sound is due to a vibration on the surface of the sphere, which began when you banged on it.  If you banged on it even harder, you might hear two or  three different sounds at the same time.  If you banged on it INFINITELY harder, you would hear infinitely many sounds, because infinitely many waves would occur all over the sphere. This infinite spectrum of different sounds can be represented visually as waves on a sphere, (as in the image below) or mathematically by eigenfunctions of the Laplace spectra:

http://lmb.informatik.uni-freiburg.de/masters/malingyu/01/index.en.html This is what the vibrations on the sphere look like

where P is the Legendre polynomial, φ is the latitude, θ is the longitude, l is the degree of the polynomial, m is number of wave peaks (wavenumber) along longitude lines, and m-l is the number of wave peaks (wavenumber) along latitude lines.

In the figure above, the top left sphere is where l=m=0, down is increasing l, and right is increasing m.  The furthest left represents only waves along the longitudinal axis, and the furthest right represents only waves along latitudes.  Note: there is a left side of the pyramid in Figure 1, but it is the same as the right side but rotated.

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Triangular Truncation

Since there are infinitely many of these harmonic functions, the series has to be cut off, or truncated, somewhere; only some of the harmonic functions can be used.  Often models that use spherical harmonics as coordinates are called Spectral Triangular, which refers  to which harmonics (of the infinite spectrum) were chosen.  For a spectral triangular truncation (which is used in SSI) the harmonics are chosen by choosing three numbers to be equal:
1. K: the largest value of the longitudinal wavenumber (m) + half the latitudinal wavenumber (m-l) 
2. M: the largest longitudinal wavenumber (m)
3. L: the largest degree of the polynomial (l)
This means not every harmonic of each wavenumber will be chosen: in Figure 1 there will be more of the furthest right and furthest left harmonics on the pyramid than center harmonics.
In SSI, K, M, and L are all equal to 62.

* Spheres do not sound nice: that's probably why musical instruments are not spherical!  For better listening, I recommend banging on something tubular, though this would create a vibration of a different shape.