**Overview****Introduction to Spherical Harmonics and Spectral Transformations****ERA-Interim reduced N128 Gaussian grid****References**

The first section of this documnet provides in-depth background to spherical harmonics and the spectral transform technique by discussing technical aspects and terminology essential to a clear understanding and informed usage of fields archived as spectral coefficients by ECMWF. In spectral models, the spectral space representation of fields is tightly coupled to the corresponding physical latitude-longitude transform grid (a Gaussian grid), and vice versa. The second section discusses the ERA-Interim transform grid, which in this case is a reduced N128 Gaussian grid.

over a truncated wavenumber space where is the zonal (east-west) wavenumber,
and can be regarded as a "total" wavenumber whereby
gives the number of zeroes between the poles (*not including the poles*)
of the spherical harmonic function . Hence,
can be interpreted as a type of meridional (north-south) wavenumber.
In the spherical harmonic expansion, the coordinate variables are represented by
where is longitude,
( being latitude)
and time. (We have omitted a vertical coordinate for simplicity
of presentation.) In addition, the polynomials defined by

are associated Legendre polynomials of order and degree , and

are complex-valued spectral coefficients where is the
complex conjugate operator. Finally, for simplicity of presentation we set ,
resulting in a single constant spectral truncation parameter which
corresponds to what is commonly referred to as *triangular truncation*. (The usual convention is
to use rather than
to designate triangular truncation.) In the
() wavenumber space
prescribes a triangular region of spherical harmonic modes indicated by the filled circles in the diagram below.
Modes outside of this triangle are set to (open circles).

Spherical Harmonic Wavenumber Space |
---|

Other types of truncation may also be used, such as *rhomboidal*, but triangular truncation is the most common
and is used in ERA-Interim (ECMWF, 2002), and for example the NCAR CCM (Kiehl *et al.*, 1998) and
CSM (Boville and Gent, 1998). Triangular truncation is frequently referred to as *isotropic* in the sense
that every position and direction on the sphere is treated identically, that is, spectral solutions obtained
using triangular truncation are invariant with respect to a coordinate rotation.
Washington and Parkinson (1986), and Hack (1992), discuss many aspects of spectral truncation in more detail.

Equation (1) represents the *spectral synthesis* of a scalar field
from a truncated series of spectral coefficients and spherical harmonic functions
. Conversely, in a *spectral analysis* stage, the spectral coefficients are
obtained by a discretized version of

where the inner integral (highlighted in light blue)

is a forward Fourier transform applied in the zonal (east-west) direction. The forward Fourier transform is computed at each circle of latitude using a discrete fast Fourier transform (FFT). The outer integral

is evaluated in the meridional (north-south) direction using Gaussian quadrature

where denotes Gaussian grid points in the meridional direction, the corresponding Gaussian weight at point , and the number of Gaussian grid points in the meridional direction. The are given by the roots of the Legendre polynomial and the by

The Gaussian grid points are synonymous with Gaussian
latitudes , the relation being ,
and the *number* of Gaussian grid points
is synonymous with the *number* of Gaussian latitudes.

The *spectral analysis* stage represented by Equation (2) can for illustrative purposes be constructed sequentially from two
arrays as shown in the following diagram. If we choose equally
spaced longitudes and Gaussian latitudes, then to the columns of the array in the left panel
we assign the FFTs, one for each circle of latitude. The length
of the FFTs is (due to the *Nyquist frequency* limit, only
half of the possible Fourier modes are retained,
,
and the Fourier "mode" is the mean value of
at and time
). In the array in the right panel, we store the spherical harmonic spectral coefficients
obtained by Gaussian quadrature of the Fourier coefficients
, associated Legendre polynomials ,
and Gaussian weights . As an example, we show the how the
spectral coefficient is computed from the sum of the products of
(the in the left
panel), , and .
(In passing we note that represents the average value of
at time and fixed level.)

Array of Complex Fourier Coefficients |
Array of Complex Spherical Harmonic Spectral Coefficients |
---|---|

Observe that the dimensions of the spectral coefficient array in the right panel are , and
that a triangular truncation has been applied. Also, keep in mind that there
are no modes to compute in the region, as
for the associated Legendre polynomials . Hence, values of the spectral
coefficient array at () indices outside of the light-gray region need not be computed
and are set to . In general, the triangular truncation
is chosen such that ,
or where is the
number of east-west grid points, to avoid *aliasing* potentially extraneous small spatial scales onto large spatial
scales. (However, this is not to say that modes beyond cannot be
or are not computed — indeed, modes beyond may be retained
depending on the post-processing conventions of the operational, reanalysis, or modeling center, with the
implicit understanding that the appropriate truncation was applied during model integrations.)

Once spectral coefficients are obtained during a *spectral analysis* stage, the equivalent physical
representation can be obtained by a *spectral synthesis* via Equation (1). To follow up on our example,
we may choose to focus on only a single mode such that .
The real part of the
spherical harmonic function is shown in the following figure. Noting that
we observe a pattern of alternating negative and positive regions with zonal wavenumber 2 (with nodes at ±45°E and
±135°E) and 3 meridional nodes at 0°N and ±35.3°N. Of course, in a complete
*spectral synthesis* we are summing over many different spherical harmonic modes (multiplied by the corresponding
spectral coefficient).

Real part of Spherical Harmonic Function |
---|

blue represents negative values, red positive values |

Equations (1) and (2) (*spectral synthesis* and *spectral analysis*) constitute a transformation pair in which a
scalar variable at time
may be represented in *physical space*, Equation (1), or *spectral space*, Equation (2).
Both are equivalent representations of and each possess
context-dependent computational advantages and disadvantages. The transformation pair is bound together by
the *transform grid*, constituting the *physical*, or *grid-point space*, defined by the
Gaussian latitudes and equally spaced longitudes.
Throughout the preceding discussion, we have chosen to emphasize the time-dependent nature of the scalar
variable by adhering to notation that retains explicit reference
to time, especially in the spectral coefficients . This allows
us to conclude this section by discussing the broad outline of a typical forecast (integration) cycle in
modern spectral GCMs.

GCMs have as their basis the *primitive equations* which describe atmospheric dynamics
and thermodynamics employing six equations in six unknowns — three momentum equations relating the east, north, and vertical
components of velocity () to the gradient of pressure
(Newton's second law of motion in a noninertial reference frame), a mass continuity equation
relating local changes in density to divergence
or convergence of atmospheric mass, a thermodynamic equation
relating temperature to the storage and conversion of thermal and other
forms of energy into work (first law of thermodynamics), and an equation of state relating
, ,
and (the ideal gas law).
Through various approximations and combinations, the primitive equations are reduced to a set of *model
equations* that are of two types, these being *prognostic equations * of scalar
variables (these predictive variables are hereafter referred to as
*prognostic variables*), and *diagnostic equations * of variables that are most
conveniently computed from the prognostic variables. In a typical GCM, the resulting prognostic variables
are vorticity , divergence ,
temperature , and surface pressure ,
with specific humidity added for the inclusion of atmospheric moisture.
The horizontal components of both the prognostic and diagnostic sets of model equations are then subjected to the *spectral transform method*,
converting these equations to an equivalent representation in spectral space, i.e. in terms of spectral coefficients.

To examine the outlines of a model forecast cycle let us focus on a single prognostic variable, in this case vorticity . The prognostic equation for vorticity contains linear terms , horizontal derivatives , nonlinear terms (usually products of individual space- and time-dependent terms), and parameterizations of forcing terms . In the spectral method, the forecast cycle begins in grid-point space at time and involves the point-by-point multiplication of the individual parts of nonlinear terms, as well as calculation of physical parameterizations (see lower left panel in diagram below). (Linear operations and horizontal derivatives are to be carried out exclusively in spectral space.) In the second phase of a forecast cycle (upper left panel), nonlinear terms and physical parameterizations are transformed to spectral space. The spectral form of nonlinear terms and physical parameterizations , as well as the spectral form of linear operations and horizontal derivatives , are then used to compute the right-hand side of the prognostic equation for vorticity, i.e. . In the third phase of the forecast cycle (upper right panel), the spectral coefficients are integrated forward in time one time step to time . In addition, from the diagnostic set of equations, the spectral coefficients of the individual parts of nonlinear terms are computed as functions of ().

Finally, the forecast cycle is completed by transforming the individual
parts of nonlinear terms from spectral space to grid-point space (lower right panel). In ERA-Interim,
at time the spectral form of the prognostic variables
and the grid-point form of parameterized and derived fields are archived together which results in
some variables being represented by spherical harmonics, and some variables represented on the physical
transform grid — the ERA-Interim physical transform grid is the topic of the next section^{2}.

^{2}Our discussion of an integration cycle in a spectral model
has purposely omitted reference to *data assimilation* which combines *objective analysis*
and *initialization*, key elements in ECMWF operational and reanalysis programs. For a
brief overview of *data assimilation* at ECMWF, see pages 462-468 in Holton (1992).

For Gaussian grids in general,
one may intuit that as the pole is approached, the curvilinear distance between meridians of longitude decreases
markedly (e.g. from 78.170 *km* just north and south of the Equator to 0.733 *km* near the poles for
*regular* ), introducing a distortion in the
area bounded by the four sides of a grid cell. Because of the isotropic nature of triangular truncation,
it has been posited that a prescribed and sufficient grid resolution at the Equator should also serve as a sufficient
grid resolution over the entire sphere, alleviating the poleward distortion of grid cells. Beginning with Machenhauer
(1979), this premise led to the investigation
of *reduced Gaussian grids* in which the number of longitudes per circle of latitude decreases as one approaches the poles.
The operational aspects of using *reduced Gaussian grids* was more fully developed at ECMWF by Hortal and Simmons
(1991), providing the basis for using a *reduced Gaussian grid*
for ERA-40 and a *reduced Gaussian grid* for ERA-Interim. (Hortal and Simmons, 1991, found no significant loss of accuracy in using the reduced grid for short- and medium-range
forecasts of vorticity and geopotential height compared to the full grid. In addition to no significant loss of accuracy,
there are real reductions in computational time (20-25%) and storage requirements (30%) for
*reduced * compared to *regular *
at T106 spectral truncation. The reader is referred to Hortal and Simmons, 1991, for further details.)

We provide graphical and numerical representations of *reduced and regular Gaussian grids*
via the following two links. The *reduced Gaussian
grid* is used in ERA-Interim.

*reduced and regular Gaussian grids:*Figures*reduced and regular Gaussian grids:*Numerical values

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