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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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2.2 Discretization
2.2.1 Vertical discretization
To represent the vertical variation of the dependent variables
, 
, 
and 
, the atmosphere is divided into 
layers. These layers are defined by the pressures at the interfaces
between them (the `half-levels'), and these pressures are given by
|
(2.11) |
for
. The 
and 
are constants whose values effectively define the vertical coordinate
and 
is the surface pressure field.The values of the
and 
for all 
are stored in the GRIB header of all fields
archived on model levels to allow the reconstruction of the `full-level'
pressure 
associated with each model level (middle
of layer) from 
(
)by using (2.11) and the surface pressure field.The prognostic variables are represented by their values at `full-level' pressures
. Values for 
are not explicitly required by the model's vertical finite-difference
scheme, which is described below.The discrete analogue of the surface pressure tendency equation (2.10) is
|
(2.12) |
 .
|
(2.13) |
From (2.11) we obtain
|
(2.14) |
is the divergence at level 
,
|
(2.15) |
 .
|
(2.16) |
The discrete analogue of (2.9) is
|
(2.17) |
and from (2.11) we obtain
|
(2.18) |
is given by (2.14).Vertical advection of a variable X is now given by
|
(2.19) |
The discrete analogue of the hydrostatic equation (2.6) is
|
(2.20) |
|
(2.21) |
is the geopotential at the surface.
Full-level values of the geopotential, as required in the momentum equations
(2.1) and (2.2), are given by
|
(2.22) |
where
and, for 
,
 ,
|
(2.23) |
The remaining part of the pressure gradient terms in (2.1) and (2.2) is given by
|
(2.24) |
given by (2.23) for all .Finally, the energy conversion term in the thermodynamic equation (2.3) is discretized as
|
(2.25) |
, 
, is defined by (2.23) for 
, and
|
(2.26) |
The reasons behind the various choices made in this vertical discretization scheme are discussed by Simmons and Burridge (1981); basically the scheme is designed to conserve angular momentum and energy, for frictionless adiabatic flow.
2.2.2 Time discretization
To introduce a discretization in time, together with a semi-implicit correction, we define the operators
,
represents the value of a variable at
time 
, 
the value at time 
, and 
the value at 
. In preparation for the semi-Lagrangian treatment to be developed
in section 3, we also introduce the three-dimensional advection operator
|
(2.27) |
Introducing the semi-implicit correction terms, Eqs. (2.1)-(2.4) become:
|
(2.28) |
|
(2.29) |
|
(2.30) |
|
(2.31) |
where
is a parameter of the semi-implicit scheme; the classical scheme
(Robert 1969) is recovered with 
. The semi-implicit correction terms are linearized
versions of the pressure gradient terms in (2.1)-(2.2) and the energy conversion term in
(2.3). Thus 
is a reference temperature (here chosen to be independent
of vertical level), while 
and 
are matrices such that
 ,
|
(2.32) |
 .
|
(2.33) |
where the half-level pressures appearing in (2.32) and (2.32) are reference values obtained from (2.11) by choosing a reference value
of 
, and the coefficients 
are based on these reference values. The reference values adopted
for the semi-implicit scheme are 
and 
.The integrated surface pressure tendency equation (2.14) becomes
|
(2.34) |
|
(2.35) |
2.2.3 Horizontal grid
A novel feature of the model is the optional use of a reduced Gaussian grid, as described by Hortal and Simmons (1991). Thus, the number of points on each latitude row is chosen so that the local east-west grid length remains approximately constant, with the restriction that the number should be suitable for the FFT (
). After some experimentation, the `fully
reduced grid' option of Hortal and Simmons was implemented;
all possible wavenumbers (up to the model's truncation limit) are used in
the Legendre transforms. A small amount of noise in the immediate vicinity
of the poles was removed by increasing the number of grid points in the
three most northerly and southerly rows of the grid (from 6, 12 and 18 points
in the original design of the T213 grid to 12, 16 and 20 points respectively).
Courtier and Naughton
(1994) have very recently reconsidered the design of reduced Gaussian grids.2.2.4 Time-stepping procedure
The time-stepping procedure for the Eulerian
-
version of the model follows closely that outlined
by Temperton (1991)
for the shallow-water equations. At the start of a time-step, the model
state at time 
is defined by the values of 
, 
, 
, 
and 
on the Gaussian grid. To compute the semi-implicit corrections, the
values of divergence 
, 
and 
are also held on the grid, where 
and
 .
|
(2.36) |
The model state at time
is defined by the spectral coefficients
of 
, 
, 
, 
and 
. Legendre transforms followed by Fourier transforms
are then used to compute 
, 
, 
, 
, 
, 
, 
and 
at time 
on the model grid. Additional Fourier transforms are used to compute
the corresponding values of 
, 
. 
, 
and 
. The meridional gradients of 
and 
are obtained using the relationships
.
All the information is then available to evaluate the terms at time
on the left-hand sides of (2.28)-(2.31) and (2.34), and thus to compute `provisional'
tendencies of the model variables. These tendencies (together with values
of the variables at 
are supplied to the physical parametrization routines, which increment
the tendencies with their respective contributions. The semi-implicit correction
terms evaluated at time-levels (
) and 
are then added to the tendencies. Ignoring
the horizontal diffusion terms (which are handled later in spectral space),
and grouping together the terms which have been computed on the grid, (2.28)-(2.31) and (2.34) can be written in the form
|
(2.37) |
|
(2.38) |
|
(2.39) |
|
(2.40) |
|
(2.41) |
The right-hand sides
-
are transformed to spectral space via Fourier transforms followed
by Gaussian integration. The curl and divergence of
(2.37) and (2.38) are
then computed in spectral space, leading to
|
(2.42) |
 .
|
(2.43) |
Eqs. (2.39), (2.41) and (2.43) can then be combined with the aid of (2.36) to obtain an equation of the form
|
(2.44) |
and total wavenumber 
, where the matrix
|
(2.45) |
values of 
in a vertical column. Once 
has been found, the calculation of 
and 
can be completed, while 
and 
have already been obtained from (2.40) and (2.42).Finally, a `fractional step' approach is used to implement the horizontal diffusion of vorticity, divergence, temperature and specific humidity. A simple linear diffusion of order
is applied along the hybrid coordinate surfaces:
|
(2.46) |
, 
or 
. It is applied in spectral space to the 
values such that if 
is the spectral coefficient of 
prior to diffusion, then the diffused value 
is given by
|
(2.47) |
A modified form of (2.47) is also used for the temperature
, to approximate diffusion on surfaces of constant pressure rather
than on the sloping hybrid coordinate surfaces (Simmons,
1987). The operational version of the model uses fourth-order horizontal
diffusion 
2.2.5 Time filtering
To avoid decoupling of the solutions at odd and even time steps, a Robert filter (Asselin 1972) is applied at each timestep. The time-filtering is defined by
|
(2.48) |
where the subscript
denotes a filtered value, and 
, 
and 
represent values at 
, 
and 
, respectively.Because of the scanning structure of the model (see Chapter 4 `Computational details' ), it is convenient to apply the time-filtering in grid-point space, and to split (2.48) into two parts:
|
(2.49) |
|
(2.50) |
The `partially filtered' values computed by (2.49) are stored on a grid-point work file and passed from one time-step to the next. Thus, the information available after the transforms to grid-point space consists of partially filtered values at time
together with unfiltered values at time 
. The filtering of the
fields can then be completed via (2.50), which after shifting by one timestep
becomes:
 .
|
(2.51) |
The computations described in Section 2.2.4 are performed using these fully filtered values at time
and the unfiltered values at time 
. Once (2.51) has
been implemented, values of 
are also available to implement (2.49) for the partially filtered values
to be passed on to the next timestep.2.2.6 Remarks
Ritchie (1988) noted that for a spectral model of the shallow-water equations, the
-
form and the 
-
form gave identical results (apart from round-off error). In extending
this work to a multi-level model, Ritchie (1991) found that this equivalence
was not maintained. This was in fact a result of some analytic manipulations
in the vertical, used to eliminate between the variables in solving the
equations of the semi-implicit scheme, which were not exactly matched by
the finite-element vertical discretization of Ritchie's model.In the case of the model described here, the corresponding elimination between the variables is purely algebraic, and the equivalence between the
-
form and the 
-
form is maintained apart from one small exception due to the use
of the hybrid vertical coordinate. In the 
-
model, the gradients of the geopotential 
are computed in grid-point space (from the spectrally
computed gradients of 
, 
and 
), while in the 
-
model 
itself is computed and transformed separately
into spectral space, where its Laplacian is added into the divergence equation.
Since 
is not a quadratic function of the model variables there is some
aliasing, which is different for the two versions of the model. In practice
the differences between the 
-
model and the 
-
model were found to be very small, and in the case
of a pure sigma-coordinate the two models would be algebraically equivalent.The
-
model is nevertheless considerably more economical
than its 
-
counterpart in terms of the number of Legendre transforms
required. In addition to the transform of 
referred to above, four Legendre transforms are saved in the treatment
of the wind fields using the procedures described by Temperton
(1991) for the shallow-water equations. The number of multi-level Legendre
transforms is thereby reduced from 17 to 12 per time-step.2.2.7 
as spectral variable
In preparation for a further reduction in the number of Legendre transforms required by the semi-Lagrangian version of the model, the modified Eulerian version includes an option to keep the virtual temperature
, rather than the temperature 
, as the spectral variable. In the time-stepping procedure,
Legendre transforms followed by Fourier transforms are used to compute 
, 
and 
at time 
on the model grid; the corresponding values of 
, 
and 
are then computed using the corresponding values of 
, 
and 
. The thermodynamic equation (2.3) is then stepped forward in time
exactly as before. After the physical parametrization routines, the `provisional'
value of 
is combined with 
to compute a provisional value of 
. The semi-implicit correction terms evaluated at time-levels
and 
are then added to the provisional value of 
, just before the transform back to spectral space.There are corresponding slight changes in the semi-implicit correction terms. The linearized hydrostatic matrix
in (2.28)-(2.29)
and (2.36) now operates
on 
rather than on 
. From the point of view of the semi-implicit scheme, (2.30)
has implicitly been replaced by an equation of the form
|
(2.52) |
although as explained above it is not necessary to formulate or compute the missing terms explicitly. Hence, (2.39) is replaced by
|
(2.53) |
and the solution of the semi-implicit equations in spectral space proceeds just as before.
This change of spectral variable results in only insignificant changes to a 10-day model forecast, but permits useful economies in the semi-Lagrangian version to be described in the next chapter.
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08.04.2002 |
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