# 4.1 OpenIFS: Gaussian grids

OpenIFS uses the so-called 'Gaussian grid' to represent variables in physical space. A Gaussian grid does not have regular latitude spacing but nearly so.

A Gaussian grid does not have a latitude at the equator, nor exactly at the poles. The latitudes are symmetric about the equator.

The model uses several representations of a Gaussian grid as explained below.

# 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 N=640 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 of order 2xN (i.e., the total number of latitude lines from pole to pole):  P 2N 0 k=sinθk) = 0A 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;
• each latitude circle has a grid point at 0o longitude;
• the longitudinal resolution in degrees is 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 (2N) as the corresponding regular Gaussian grid;
• has a grid point at 0o longitude on each latitude circle;
• 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 IFS cycle 41r1, ECMWF has used a original 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 planned horizontal resolution increase, 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.

## Notation

The following notation is used when referring to the full (regular), original reduced and octahedral reduced Gaussian grids:

• FXXX - full (regular) Gaussian grid with XXX latitude lines between the pole and equator
• NXXX - original ECMWF reduced Gaussian grid with XXX latitude lines between the pole and equator
• OXXX - octahedral ECMWF reduced Gaussian grid with XXX latitude lines between the pole and equator

# Examples

## Gaussian grid with 320 latitudes between pole and equator

latitude
number
reduced points
(original - N320)
reduced points
(octahedral - O320)
regular points
(full - F320)
latitude
11820128089.784877
22524128089.506203
33628128089.225883
44032128088.945191
54536128088.664358
65040128088.383457
76044128088.102518
86448128087.821556
97252128087.540577
107256128087.259589
117560128086.978592
128164128086.697590
139068128086.416583
149672128086.135573
1510076128085.854560
1610880128085.573545
1712084128085.292528
1812088128085.011510
1912592128084.730490
2013596128084.449469
21144100128084.168447
22144104128083.887425
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25180116128083.044353
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31216140128081.358199
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64018201280-89.784877

## Gaussian grid with 640 latitudes between pole and equator

latitude
number
reduced points
(original - N640)
reduced points
(octahedral - O640)

regular points

(full - F640)

latitude
11820256089.892396
22524256089.753005
33228256089.612790
44032256089.472390
54536256089.331918
65040256089.191413
76044256089.050889
86048256088.910352
97252256088.769808
107256256088.629259
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128164256088.348150
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## Gaussian grid with 1280 latitudes between pole and equator

latitude
number
reduced points
(original - N1280)
reduced points
(octahedral - O1280)
regular points
(full - F1280)
latitude
11820512089.946188
22524512089.876478
33228512089.806357
44032512089.736143
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