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Sectioninfo

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 determined 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)

Note

By changing the frequency at which the shortest wavelength is sampled on the grid, the effective grid point resolution can be increased while keeping the spectral truncation number constant.

Section

Increasing the horizontal resolution

The relationship between the spectral truncation, the sampling (linear, quadratic or cubic) and the grid point resolution allows for three possible approaches for increasing the horizontal resolution:

  1. increase the number of wave numbers in the spectral representation but keep the same Gaussian grid by reducing the sampling frequency (e.g, quadratic → linear);
  2. increase the number of wave numbers in the spectral representation and the Gaussian grid resolution but keep the sampling frequency constant;
  3. keep the number of wave numbers in the spectral representation constant and increase the Gaussian grid resolution by increasing the sampling frequency.

...

The horizontal resolution upgrade at IFS cycle 42r1 will be achieved by:

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The planned horizontal resolution upgrade employs the so-called cubic reduced Gaussian grid (with spectral truncation denoted by TC) instead of the current linear reduced Gaussian grid (denoted by TL) where the shortest wave is represented by four rather than two grid points. By increasing the number of grid points used to represent the shortest wave, more resolution is added in grid-point space while keeping the spectral truncation constant,

To reduce further the computational cost, the new IFS cycle implements a modification to the grid, the octahedral reduced Gaussian grid (with spectral truncation denoted by TCO).  The octahedral grid is a form of the reduced Gaussian grid but applying a new rule for computing the number of points per latitude circle.

In this page, we refer to the reduced Gaussian grid as described by Hortal and Simmons (Use of Reduced Gaussian Grids in Spectral Models; ECMWF Tech. Memo. 168, 1990 - see also Hortal M., and A.J. Simmons, 1991, Mon. Wea. Rev. 119 1057-1074) and used by the IFS up to cycle 41r1 as the original reduced Gaussian grid. The new octahedral form of the reduced Gaussian grid is described in this page.

Table of Contents

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Section
Column

What is the octahedral grid ?

The octahedral grid has been inspired by the Collignon Projection of the sphere onto a an octahedron.

 

Projection of the sphere onto an octahedronImage Added

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 original Gaussian grid but with the number of longitude points at each latitude circle between the pole and equator computed according to the formula:

Mathdisplay
\begin{eqnarrayequation*}
\mbox{N}_{lat}(lat_N) &_i} = & 204 \\
N_{lat}(lat_i) & = & N_{lat}(lat_{i+1}) + 4,  \mbox{ fortimes i  + 16 \mbox{    for  } i =N- 1, \ldots,1 \mbox{N}
\end{eqnarrayequation*}

In other words, :

  • there are 20 longitude points at the latitude circle closest to the poles
with
  • ;
  • the number of points
increasing continuously
  • increases by 4 at each latitude line from the pole towards the equator.

This is in contrast to the standard original 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 up to IFS cycle 41r241r1).

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

Note in particular that the octahedral grid has 4N 4N + 16 longitude points at the latitude circle closest to the equator whereas the standard original reduced Gaussian grid has 4N 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 original reduced Gaussian grid. For example, the N1280 O1280 octahedral grid has about 20% fewer grid points than the equivalent N1280 linear gridoriginal reduced grid. Generally, an octahedral grid with N latitude lines between the pole and equator has 4xNx(N+9) grid points.

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

Tip
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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

Note that the first character of the grid name is an upper case letter.

Figure 1

Image Removed
Column
Column

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

Comparison of Gaussian gridsImage Added

Variation of the resolution (characteristic length scale) with latitude for the reduced Gaussian grids.  The full red and yellow curves show red curve shows the resolution for the standard original reduced Gaussian grids grid at N1024 and N1280 (TC1279).  Note that the resolution remains more or less constant at 10km and 8km , respectively, as the latitude varies.  The corresponding curves curve for the N1024 and N1280 octahedral grids are O1280 (TCO1279) octahedral grid is shown by the dashed and full blue lines, respectivelyblue curve.  Note that the The resolution for the N1280 O1280 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 original reduced Gaussian grid used for HRES at IFS cycle 41r1 is at about 16 km. Also shown are is the regular Gaussian grids grid at N1024 F1280 (black dashed line) and N1280 (black full linecurve).

Figure 2

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

Comparison of the zonal variation in resolution for the N1280 standard original reduced Gaussian grid (left) with the corresponding O1280 octahedral grid (right) on the surface of the sphere

 

 

Section

Gaussian grid descriptions

Descriptions of the Gaussian grids introduced for the planned horizontal resolution upgrade 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 original 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 as the pl key.

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

From grib_api 1.14.4 onwards, the computed key gridName can be used to query the grid name.  For an octahedral with 1280 latitudes between pole and equator gridName=O1280; for an original reduced Gaussian grid with 640 latitudes between pole and equator, gridName=N640 while the corresponding regular Gaussian grid has gridName=F640.

Section

Further information

For further background information see:

Children Display

See also:

  • Malardel S., et al. 2016:  "A new grid for the IFS",  ECMWF Newsletter No.146 - Winter 2015/16 (pages 23-28).
  • Wedi N.P., 2014: Increasing the horizontal resolution resolution in numerical weather prediction and climate simulations:  illusion or panacea ?" Phil. Trans. R.Soc.A, 372,  doi: 10.1098/rsta.2013.0289.
Section

Frequently asked questions

Will the change to the octahedral grid affect me if I use regular lat-lon data ?

No, users of regular lat-lon data will be unaffected by the change of model grid.  They will, however, benefit from the increase of the model horizontal resolution.

Will ECMWF make data available on the original regular and reduced Gaussian grids ?

Yes, users will still be able to request data, both in dissemination and MARS, on the original regular and reduced Gaussian grids. Note, however, that this will be interpolated from the values on the octahedral reduced Gaussian model grid. See also I make computations involving flux parameters or accumulated fields (for example, to de-accumulate precipitation) and am advised to work on the model grid: which grid should I use ?

I use point data (e.g., for meteograms, vertical profiles, etc) - what do I need to do ?

Users of point data should note that the coordinates of the nearest grid point will have changed.  Users should take particular care for coastal points for which the nearest grid point may have changed from a land point to a sea point or vice versa.

Are the new land-sea mask and orography for the octahedral grid available ?

Yes, the new land-sea masks and orography fields for HRES at TCO1279 (O1280), ENS Leg A at TCO639 (O640) and ENS Leg B TCO319 (O320) are available for download:

I use the orography in spectral representation - will I be affected ?

Although the spectral resolution for HRES remains constant, the spectral orography has changed.  If you have this as a static file then this should be updated with the new version.

Do I need to upgrade the version of GRIB API I use in order to decode data on the octahedral grid ?

Version 1.12.3 of grib_api can decode fields on the octahedral grid correctly.  As of grib_api 1.14.0, a new computed key isOctahedral is introduced which allows the grid type to be queried. For the octahedral grid, isOctahedral=1; otherwise, isOctahedral=0.

Can GRIBEX decode data on the octahedral grid ?

GRIBEX is no longer supported by ECMWF and will therefore not be upgraded to support the octahedral grid.

What should I check in my program to make sure it will work with the octahedral grid ?

Any GRIB decoding program that makes use of the PL array to obtain the number of longitude points on each latitude circle should be able to handle the octahedral grid.  Programmers should note that the largest number of points at any one latitude circle increases from 4N in the original Gaussian grid to 4N+16 in the octahedral grid and adjust any data structures accordingly to cater for this.

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I make computations involving flux parameters or accumulated fields (for example, to de-accumulate precipitation) and am advised to work on the model grid:  which grid should I use ?

For performing computations with accumulated fields, users are advised to request data on the octahedral grid.

Is there any change to the vertical resolution as part of the planned resolution upgrade ?

No, only the horizontal resolution is increased.  The vertical resolution remains at L137 for HRES and L91 for ENS.

What will happen if I retrieve data from MARS using grid=av ?

Users retrieving data from MARS with the keyword, grid=av ("archived value") will retrieve data on the model grid.  For data from the upgraded model this will be the octahedral grid.

What will happen if I retrieve data from MARS using grid=1280 ?

This behaviour is unchanged. By default, users retrieving data from MARS with the keyword, grid=1280 will retrieve data on the regular Gaussian grid with 1280 latitude lines between the pole and equator (equivalent to grid=F1280).

Will ERA-Interim fields also use the cubic octahedral grid ?

No, the horizontal resolution upgrade applies only to ECMWF HRES and ENS operational forecasts, including the monthly extension. It will affect the additional runs in support of the Optional Programme for Boundary Conditions (BC).

Will the ECMWF System 4 Seasonal Forecasts (SEAS) also use the octahedral grid ?

No, the horizontal resolution upgrade applies only to ECMWF HRES and ENS operational forecasts, including the monthly extension.