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  is the longitude and μ _i = sin(θ_i) with λ  the longitude and θ_i the latitude of the 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.

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.

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 θ_k, are given by the zeros of the Legendre polynomial or order N:  P_N⁰(μ_{i k}=sin(θi)θ_k) = 0.  A consequence of this is that a Gaussian grid has:

• latitude lines which are not equally spaced;
• 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 reduced Gaussian grid or, more simply, the octahedral grid.

Relationship between spectral truncation and grid point resolution - linear, quadratic and cubic grids

The relationship between the spectral resolution, governed by the truncation number T, and the grid resolution depends on the number of points on the grid at the frequency equator at which the shortest wavelength field is sampled on the grid at the equator:

linear:  each wavelength is sampled by 2 grid points → 4N_L = 2 (T_L + 1)

quadratic: each wavelength is sampled by 3 grid points → 4N_Q = 3 (T_Q + 1)

cubic: each wavelength is sampled by 4 grid points → 4N_C = 4 (T_C + 1)

By changing the frequency number of points at which the shortest wavelength is sampled on the grid, the effective grid point resolution can be increased while keeping the spectral truncation number constant.

Grid point computations, which are often non-linear, benefit from the higher resolution of the grid produced with cubic sampling.  Such a grid has no aliasing, less numerical diffusion and more realistic surface fields.