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Comment: corrections from Sylvie

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Section
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What is the octahedral grid ?

The octahedral grid has been inspired by the Collignon Projection of the sphere onto an octahedron. It is a form of reduced Gaussian grid with the same number of latitude lines located at the same latitude values as those of a standard Gaussian grid but with the number of longitude points at each latitude circle computed according to the formula:

Mathdisplay
\begin{eqnarray*}
\mbox{N}_{lat}(lat_N) & = & 20 \\
\mbox{N}_{lat}(lat_i) & = & \mbox{N}_{lat}(lat_{i+1}) + 4,  \mbox{ for } i=\mbox{N} - 1,\ldots,1
\end{eqnarray*}

In other words, there are 20 longitude points at the latitude circle closest to the poles with the number of points increasing continuously by 4 at each latitude towards the equator. This is in contrast to the standard reduced Gaussian grid where there are 'jumps' between blocks of latitudes with a constant number of longitude points (a restriction imposed by the Fast Fourier Transform routines being used prior to IFS cycle 41r2).

As a consequence, the zonal resolution of the octahedral grid varies more with latitude than for the standard reduced Gaussian grid. This can be seen in the figures to the right.

Note in particular that the octahedral grid has 4N + 16 longitude points at the latitude circle closest to the equator whereas the standard reduced Gaussian grid has 4N longitude points at the latitude circle closest to the equator.

There are also fewer total grid points in the octahedral grid compared to the standard reduced Gaussian grid.For example, the N1280 octahedral grid has about 20% fewer grid points than the equivalent N1280 linear grid. Generally, there are 4xNx(N+8) grid points in the octahedral grid or of resolution N.

The octahedral grid has been shown to improve the calculation of local derivatives in grid point space.

Column

Comparison of the resolution variation with latitude for the reduced Gaussian grids

Comparison of the variation of resolution with latitude for the reduced Gaussian grids.  The full red and yellow curves show the resolution for the standard reduced Gaussian grids at N1024 and N1280 (TC1023 and TC1279, respectively).  Note that the resolution remains more or less constant at 10km and 8km, respectively, as the latitude varies.  The corresponding curves for the N1024 (TCO1023) and N1280 (TCO1279) octahedral grids are shown by the dashed and full blue lines, respectively.  Note that the resolution for the N1280 octahedral grid varies from about 8 km at the equator, increasing to about 10 km at 70oN and 70oS before decreasing again towards the poles. The resolution of the N640 standard reduced Gaussian grid used fro for HRES at IFS cycle 41r1 is at about 16 km. Also shown are the regular Gaussian grids at N1024 (black dashed line) and N1280 (black full line).

Comparison of the zonal variation in resolution between standard reduced and octahedral grids

Comparison of the zonal variation in resolution for the N1280 standard reduced Gaussian grid (left) with the corresponding octahedral grid (right) on the surface of the sphere

 

 

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Section

Gaussian grid descriptions

Descriptions of the Gaussian grids introduced for the horizontal resolution upgrade at IFS cy42r1 and used for HRES and ENS are available:

The descriptions provide the latitude values and the number of longitude points at each latitude for both the standard and octahedral reduced Gaussian grids.

The number of points at each latitude is encoded as the PL array in the Grid description section of the GRIB header of a grid point field.  This is accessible with grib_api using the pl key.

 At grib_api 1.14.0, a new computed key isOctahedral is introduced which allows users to query the grid type. For the cubic octahedral grid, isOctahedral=1; otherwise, isOctahedral=0.

 

 

 

Section

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:

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

where μ = sinθ with λ  the longitude and θ the latitude of the grid point, T is the spectral truncation number and Y lm 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

  • 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.

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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 represented:

linear grid:  each wavelength is represented by 2 grid points → 4N

Mathinline
\simeq
 2(TL + 1)

quadratic grid: each wavelength is represented by 3 grid points → 4N = 4N 

Mathinline
\simeq
3(TQ + 1)

cubic grid: each wavelength is represented by 4 grid points → 4N = 4N 

Mathinline
\simeq
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 is used and is indicated by the TC notation.

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