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The IFS uses a spectral transform method to solve numerically the equations governing the spatial and temporal evolution of the atmosphere.  The idea is to fit a discrete representation of a field on a grid by a continuous function.  This is achieved by expressing the function as a truncated series of spherical harmonics:

\begin{eqnarray*}
A(\lambda,\mu,\eta,t) & = & \sum_{l=0}^{\mbox{T}} \sum_{m = -l}^{l}  \psi_{lm}(\eta,t) Y_{lm}(\lambda,\mu)
 & = &  \sum_{l=0}^{\mbox{T}} \sum_{m=-l}^{l}}  \psi_{lm}(\eta,t) P_{l}^{m}(\mu) e^{im\lambda}
 \end{eqnarray*}

where μ = sinθ with λ  the longitude and θ the latitude of the grid point, T is the spectral truncation number and Y _lm are the spherical harmonic functions which are products of the associated Legendre polynomials,

P_{l}^{m}(\mu) 

and the Fourier functions,

e^{im\lambda}.

The spectral coefficients ψ_lm are computed from the discrete values known at each point of a Gaussian grid on the sphere by

• a Fast Fourier Transform in the zonal direction followed by
• a slow/fast Legendre transform in the meridional direction.

At each time step in the IFS:

• derivatives, semi-implicit correction and horizontal diffusion are computed in spectral space;
• explicit dynamics, semi-Lagrangian advection and physical parametrizations are computed in grid point space.

The representation in grid point space is on the Gaussian grid.  The grid point resolution is determined by the spectral truncation number, T.