Some background

Spectral representation of the IFS

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{equation*}
A(\lambda_i,\mu_i,\eta,t) = \sum_{l=0}^{\mbox{T}} \sum_{|m|\leq l}^{\mbox{T}}  \psi_{lm}(\eta,t) Y_{lm}(\lambda_i,\mu_i)
\end{equation*}

where λ_i  the longitude and μ_i = sin(θ_i) with θ_i the latitude of grid point i, T is the spectral truncation number and Y_lm are the spherical harmonic functions.

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.

Gaussian grids

Naming convention

The Gaussian grids are defined by the quadrature points used to facilitate the accurate numerical computation of the integrals involved in the Fourier and Legendre transforms. The grids are labelled by N where N is the number of latitude line between the pole and the equator.  For example, for the N640 Gaussian grid, there are 640 lines of latitude between the pole and the equator giving 1280 latitude lines in total.

The grid points in latitude, θi, are given by the zeros of the Legendre polynomial or order N:  P_N⁰(μ_i = sin(θi))=0.  A consequence of this is that a Gaussian grid has:

• no latitude points at the poles;
• no line of latitude at the equator.

Regular (or full) Gaussian grid

A regular Gaussian grid has the following characteristics:

• there are 4N longitude point along each latitude circle;
• the longitudinal resolution in latitude-longitude is given by 90^o/N;
• the points get closer together (i.e. more crowded) as the latitude increases towards the poles;
• the total number of grid points is 8N².

Reduced (or quasi-regular) Gaussian grid

A reduced Gaussian grid:

• has the same number of latitude lines (4N) as the corresponding regular Gaussian grid;
• the number of longitude points decreases towards the poles;
• has a quasi-regular grid spacing in distance at each latitude;

• provides a uniform CFL (Courant–Friedrichs–Lewy) condition.

Up to and including IFS cycle 41r1, ECMWF has used a standard reduced Gaussian grid.  This has 4N longitude points at the latitude nearest to the equator, with the number of longitude points reducing in blocks as the latitudes approach the poles.

With the horizontal resolution increase at IFS cycle 42r1, ECMWF introduces a slightly different form of the reduced Gaussian grid which is referred to as the octahedral grid.