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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.

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 determined by the spectral truncation number, T.

...

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)

Note

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

 

 

 

Section

Increasing the horizontal resolution

  1. increase the number of wave numbers in the spectral representation but keep the same grid
  2. increase the number of wave numbers in the spectral representation but keep the same sampling frequency (linear, quadratic or cubic)
  3. keep the number of wave numbers in the spectral representation constant
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 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.

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