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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_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 λi  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

  • a Fast Fourier Transform in the zonal direction followed by
  • a slow/fast Legendre transform in the meridional direction.

 

 

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 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)

By changing the way the shortest wavelength is sampled on the grid, the effective grid point resolution can be increased without increasing the spectral resolution.

The table shows some examples of the correspondence between linear, quadratic and cubic sampling:

Spectral truncationlinearquadraticcubic
T1279NL640NQ960NC1280
T1023NL512NQ768NC1024
T639NL320NQ480NC640

 

 

 

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 is grid point space is on the Gaussian grid.  The grid point resolution is governed by the spectral truncation number, T.

Section

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, 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:

  • 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 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;
  • 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 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 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)

By changing the way the shortest wavelength is sampled on the grid, the effective grid point resolution can be increased without increasing the spectral resolution.

 

 

 

Section

 

Section
Column

What is the octahedral grid ?

The octahedral grid has been inspired by the Collignon Projection of the sphere onto a 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*}
N_{lat}(lat_N) & = & 20 \\
N_{lat}(lat_i) & = & N_{lat}(lat_{i+1}) + 4,  \mbox{ for } i=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 which was 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.

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.

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

Column

Figure 1

Comparison of the resolution for the reduced Gaussian grids with latitude.  The full red and yellow curves show the resolution for the standard reduced Gaussian grids at N1024 and N1280.  Note that the resolution remains more or less constant at 10km and 8km, respectively, as the latitude varies.  The corresponding curves for the N1024 and N1280 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 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).

Figure 2

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