Versions Compared

Old Version 24

changes.mady.by.user Paul Dando

Saved on

compared with

New Version 25

changes.mady.by.user Paul Dando

Saved on

Key

  • This line was added.
  • This line was removed.
  • Formatting was changed.

...

Section
Column

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 lines 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, θk , are given by the zeros of the Legendre polynomial or order N:  P N 0 (μk=sinθk) = 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;
  • latitude lines which are symmetric about the equator.

Regular (or full) Gaussian grid

A regular Gaussian grid has the following characteristics:

  • there are 4N longitude points along each latitude circle;
  • the longitudinal resolution in latitude-longitude is given by 90o/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;
  • has a decreasing number of longitude points 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.

Column
N48

Example regular Gaussian grid

N48

Corresponding standard reduced Gaussian grid

Section

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 equator, 4N, at which the shortest wavelength field is sampled:

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

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

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

Until the implementation of IFS cycle 18r5 on 1 April 1998, the IFS used a quadratic grid. The introduction of the two-time level semi-Lagrangian numerical scheme at IFS cycle 18r5 made possible the use of a Linear Gaussian Grid reflected by the TL notation.  The linear grid has been used since then, up to and including IFS cycle 41r1. At IFS cycle 42r1, the cubic grid will be used and will be indicated by the TC notation.

...