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Table of contents
Chapter 1. Introduction
Chapter 2. Basic equations and discretization
Chapter 3. Semi-Lagrangian formulation
Chapter 4. Computational details
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4.1 Scanning structure
Each timestep of the model integration procedure consists of three scans.
At the beginning of the timestep, the model fields at time
are specified in grid-point form (as described in Subsection
2.2.6 of Chapter 2 `Basic
equations and discretization' , these fields are `partially time-filtered').
The grid-point values of the model variables are contained in a `grid-point
work-file', held on a secondary storage device and organized as a random-access
file with one record for each latitude row. Meanwhile, the model fields
at time 
are specified in spectral form, all the spectral coefficients being
held in central memory. The first scan consists of Legendre transforms to
compute the Fourier coefficients of the model variables at time 
on each latitude row, the results being written out to a `Fourier
work-file', again organized with one record for each latitude row. During
the first scan, latitude rows are processed in north/south pairs with the
members of each pair being equidistant from the equator, in order to make
use of the symmetries of the Legendre polynomials (see for example Temperton,1991). Once this first scan
has been completed, the spectral coefficients are no longer required and
the central memory arrays can be released for use during the next scan.The second scan steps through the latitude rows, starting at the row nearest the North Pole and proceeding southwards. At each row, the corresponding records of the grid-point values at time
and the Fourier coefficients at time 
are read in. Fourier transforms then provide grid-point values of
the fields (together with any required horizontal derivatives) at time 
. At this juncture, the time-filtering of the fields at time 
is completed, while `partially time-filtered' fields
at time 
are also computed and written out to the grid-point work-file ready
for the next timestep.The grid-point calculations for the present timestep continue using the time-filtered values at
and the unfiltered values at time 
. The right-hand sides of the equations, discretized in semi-Lagrangian
form as described in Chapter 3, are computed with
terms being grouped separately depending on whether they will be evaluated
at the departure point, the midpoint or the arrival point of the trajectory.
The results of these calculations, together with the horizontal wind components
and the vertical velocity 
, are then stored in a `rotating buffer' which contains
values for a number of consecutive latitude rows. The grid-point calculations
described so far correspond to the southernmost row contained in this buffer.
Next, the focus of the computation returns to the central row of the buffer.
Values of the wind fields and the right-hand sides of the equations are
now available at a sufficient distance to the north and south of the central
row for the trajectory calculations to be performed and for the semi-Lagrangian
timestep to be implemented, thus furnishing provisional values at 
. As described in Section 3.4, the contributions
from the physical parametrization schemes can then be incorporated to complete
the calculation of the right-hand sides 
of Eqs. (3.26)-(3.30).These right-hand sides are now Fourier transformed and the coefficients are written out to another Fourier work-file, again organized with one record for each latitude row but this time with a special structure which will be exploited in the third scan. The computation then proceeds southwards to the next pair of `southernmost' and `central' rows, the values computed for the new southern row overwriting those in the buffer for the previous northernmost row, which are no longer required.
At the start of the second scan, there is clearly an initialization phase during the first few rows when only the first part of the above calculations can be done. Similarly, at the end of the scan there is a `winding-down' phase during which the first part of the calculations has already been done, and only the second part is required. The same logical structure is also used to run the Eulerian version of the model, but in this case the width of the `rotating buffer' can be reduced to that for a single latitude row.
The third scan performs direct Legendre transforms to obtain the provisional spectral coefficients at time
from the Fourier coefficients computed in the second scan, using
Gaussian quadrature. The calculation proceeds one zonal wavenumber at a
time. Here we make use of the special structure of the Fourier work-file;
although the file was written row by row, it can be read in `transposed'
fashion, wavenumber by wavenumber. The direct Legendre transforms first
exploit the symmetries of the Legendre polynomials, and then complete the
calculations using highly efficient matrix multiplication routines. To see
how this is achieved, notice that since a single Legendre transform can
be written as a matrix/vector multiplication of the form 
, a set of simultaneous transforms for the same zonal wavenumber
but for different variables and model levels can be written as
which is indeed in the form of a matrix multiplication
. A similar technique could have been used in the first scan, and
this has been incorporated in the latest version of the model.After the transformation to spectral space, the semi-implicit equations are solved and the horizontal diffusion is implemented as described in Section 3.4, thus completing the calculation of the spectral coefficients at time
. At the end of the third scan, the whole model
has been advanced by one timestep.Next Section
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08.04.2002 |
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