# Earth Cubed

## Spectral Solutions to the prognaustic equations

In some GCMs spectral solutions (see [1] For instance) to the prognaustic equations are sought out.  This is either done for long term predictions or for reasons of stability testing. Another reason to use spectral solutions, is to relate points, in space and time, so that we can extrapolate both spacialy and temporarily given that we are given limited data.

Spectral solutions also seem to be useful for separating fast gravity waves from slower nonlinear dynamics as is done in:

http://www.mi.uni-hamburg.de/216.0.html?&L=1

I don’t really have much to say about this at the moment. I am wonder what is the basis for using spherical harmonics, for the basis expansion. It is true that spherical harmonics form a linear independent basis but they also arise as the solution to Laplace’s equation.  I’m wondering if there is any connection.

I was thinking a bit about what predictive properties these basises must have. If we are trying to use information at Y(P2) to estimate Y(P1), then the amount the point P2 contributes to the basis expansion is:

Y(P1)=sum_i f_i(P1)fi(P2)Y(P(2))

where:

Y(P1) is the value of the property at P1
Y(P2) is the value of the property at P2
f_i(P1) is the ith basis funciton evaluated at P1
f_i(P2) is the ith basis function evaluated at P2

Notice that the quantity:

sum_i f_i(P1)fi(P2)

acts like a correlation coefficent between Y(P1) and Y(P2)

it is not clear in general if this quantity will converge but if we instead of choosing a single point Y(P2) integrate about some region around Y(P2) then the higher frequency basis functions will become less significant to this correlation.

Therefore it might be usefull to develop some expressions for the integral of this over a sufrace area corresponding to the smallest grid resolution. It is worth considering basis functions that may be easier to integrate then, integrating spherical harmonics.

Another question, is with regards to the choice of extrapolation funciton what will give good extrapolation along the z component. There are solid spherical harmonics, but the verticle components of these are not orthogonal.

[1] A multi-layer spectral model and the semi-implicit method by B J. Hoskins and A.J. Simmons