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Some backgroundSpectral representation of the IFSThe 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:
where λi is the longitude and μi = sin(θi) with θi the latitude of grid point i, T is the spectral truncation number and Ylm are the spherical harmonic functions. The spectral coefficients ψlm are computed from the discrete values known at each point of a Gaussian grid on the sphere by
At each time step in the IFS:
The representation is in grid point space is on the Gaussian grid. The grid point resolution is determined by the spectral truncation number, T. |
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Gaussian gridsNaming conventionThe 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: PN0(μi = sin(θi))=0. A consequence of this is that a Gaussian grid has:
Regular (or full) Gaussian gridA regular Gaussian grid has the following characteristics:
Reduced (or quasi-regular) Gaussian gridA reduced Gaussian grid:
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. |
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Relationship between spectral truncation and grid point resolution - linear, quadratic and cubic gridsThe relationship between the spectral resolution, governed by the truncation number T, and the grid resolution depends on the frequency at which the shortest wavelength field is sampled on the grid at the equator: linear: each wavelength is sampled by 2 grid points → 4NL = 2 (TL + 1) quadratic: each wavelength is sampled by 3 grid points → 4NQ = 3 (TQ + 1) cubic: each wavelength is sampled by 4 grid points → 4NC = 4 (TC + 1) |
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By changing the frequency 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. |
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Increasing the horizontal resolutionThe relationship between the spectral truncation, the sampling (linear, quadratic or cubic) and the grid point resolution allows for three possible approaches for increasing the horizontal resolution:
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The horizontal resolution upgrade at IFS cycle 42r1 will be achieved by:
This new grid will be referred to simply as a cubic octahedral grid. |
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