Geography Reference
In-Depth Information
The result is
if
l
c
2
ν
G
+
v
0
+
f
2
c
2
ˆ
ˆ
R
=
u
0
−
ˆ
k
ˆ
(13.66)
0
(kν
G
+
−
k
2
l
2
0
(13.67)
c
2
2ν
G
+
ˆ
ˆ
G
+
=
if l )
u
0
+
ˆ
(lν
G
+
+
if k )
v
0
+
ˆ
+
(kν
G
−
−
k
2
l
2
0
(13.68)
c
2
2ν
G
−
ˆ
ˆ
G
−
=
if l )
u
0
+
ˆ
(lν
G
−
+
if k )
v
0
+
ˆ
+
It is readily verified that if
u
0
and
ˆ
v
0
are in exact geostrophic balance with
ˆ
ˆ
ˆ
=
ˆ
ˆ
G
+
=
ˆ
0. When the observed velocities are not
exactly in geostrophic balance, (13.66)-(13.68) give the relative weightings for
the projections onto the Rossby mode and the two gravity modes. For the linear
system it is possible to set the gravity modes to zero and to initialize with the
projection of the observed fields onto the Rossby mode. Note from (13.66) that
the relative weighting of the velocity observations (which appear in the form of
relative vorticity) increases as the scale of the disturbance decreases.
As explained in the previous section, merely setting the gravity modes to zero
does not work for the nonlinear system. In that case when the normal mode projec-
tion defined by (13.62) and (13.66) is used to determine the amplitude coefficients
in (13.61) and the result is substituted into (13.58)-(13.60), the unbalanced advec-
tive terms will lead to large initial accelerations. This problem can be solved by
including the gravity modes, but with their amplitudes adjusted so that their initial
tendencies vanish. Because the equations are nonlinear it is necessary to carry out
the solution iteratively, using the linear solution as a first guess.
In practice the nonlinear normal mode method works very well when applied
to primitive equation forecast models. The method can be modified to incorporate
not only nonlinear advection, but also the effects of diabatic heating in the tropics
in order to preserve the Hadley circulation in initial data.
0
, then
0
and
G
−
=
R
13.7.3
Four-Dimensional Data Assimilation
The traditional approach to objective analysis works well as long as the predic-
tion model is limited to a region of adequate data coverage (such as the North
American continent) and all observations are made at the standard 00UTC and
12UTC times so that they can be directly incorporated into synoptic analyses.
For global analysis the traditional approach is less successful, as in many oceanic
regions there are very few conventional data sources available, and it is necessary
to rely on asynoptic data (i.e., observations at nonstandard times, such as those
from ships, aircraft, and satellites) and on estimates from earlier forecasts. Asyn-
optic data have proved invaluable, especially over the oceans and in the Southern
