Geography Reference
In-Depth Information
case. In (13.19) there are terms involving the s
1 time level on both sides of the
equal sign, and these involve the values at a total of three grid points. Thus, the
system (13.19) must be solved simultaneously for all points on the grid. If the grid
is large, this may involve inverting a very large matrix, and so is computationally
intensive. Furthermore, it is usually not feasible to utilize the implicit approach for
nonlinear terms, such as the advection terms in the momentum equations. Semi-
implicit schemes in which the linear terms are treated implicitly and the nonlinear
terms explicitly have, however, become popular in modern forecasting models.
These are discussed briefly in Section 13.6.
+
13.3.5
The Semi-Lagrangian Integration Method
The differencing schemes discussed above are Eulerian schemes in which the time
integration is carried out by computing the tendencies of the predicted fields at a set
of grid points fixed in space. Although it would be possible in theory to carry out
predictions in a Lagrangian framework by following a set of marked fluid parcels,
in practice this is not a viable alternative, as shear and stretching deformations tend
to concentrate marked parcels in a few regions so that it is difficult to maintain
uniform resolution over the forecast region. It is possible to take advantage of
the conservative properties of Lagrangian schemes, while maintaining uniform
resolution, by employing a semi-Lagrangian technique. This approach permits
relatively long time steps while retaining numerical stability and high accuracy.
The semi-Lagrangian method can be illustrated in a very simple fashion with the
one-dimensional advection equation (13.6). According to this equation the field q is
conserved following the zonal flow at speed c. Thus, for any grid point, x
m
=
mδx,
and time t
s
=
sδt:
q
˜
x
s
m
,t
s
q (x
m
,t
s
+
δt)
=
(13.24)
x
s
m
Here,
˜
is the location at time t
s
for the air parcel that is located at point x
m
at
time t
s
+
δt. This position in general does not lie on a grid point (see cross marked
on Fig. 13.1), so that evaluation of the right-hand side of (13.24) requires interpo-
lating from the grid point values at time t .Forc>0 the position
x
s
m
lies between
the grid points x
m
−
p
and x
m
−
p
−
1
where p is the integer part of the expression
cδt/δx (a measure of the number of grid points traversed in a timestep). If linear
interpolation is used
˜
q
˜
x
s
m
,t
s
=
αq
x
m
−
p
−
1
,t
s
+
α) q
x
m
−
p
,t
s
(1
−
=
x
m
−
p
−˜
x
s
m
/δx. Thus, in Fig. 13.1 p
where α
=
1, and to predict q at point
A data are interpolated between the points m
=
1 and m
=
2 to the point shown
by the cross.
