Geography Reference
In-Depth Information
Fig. 13.9
A schematic diagram of the domain of initial dependence for a prediction point in midtropo-
sphere at midlatitudes (denoted by a star) as a function of forecast time span. Atmospheric
and ocean elevations are given on a logarithmic scale, increasing upward and downward,
respectively. The stripped area is the interface zone. (After Smagorinsky, 1967. Reproduced
with permission of the American Meteorological Society.)
forecast would not be possible. The atmosphere is a continuum with a continuous
spectrum of scales of motion. No matter how fine the grid resolution is made,
there will always be motions whose scales are too small to be properly observed
and represented in the model. Thus, there is an unavoidable level of error in the
determination of the initial state. The nonlinearity and instability of atmospheric
flow will inevitably cause the small inherent errors in the initial data to grow and
gradually affect the larger scales of motion so that the forecast flow field will
eventually evolve differently from the actual flow.
A very simple example of this process of error growth was presented by Lorenz
(1984). Lorenz illustrated the general problem of predictability by considering the
first-order quadratic difference equation
Y
s
Y
s
+
1
=
aY
s
−
(13.71)
which may be solved for Y
s
+
1
by iteration if the constant a and the initial condition
Y
0
are specified. The solid line in Fig. 13.10 shows a portion of the solution
sequence with a
1.5. This line represents a control run that may
be regarded as defining the “observations.” The dashed line is the solution for a
case in which the initial value Y
0
is perturbed by 0.001, whereas the dash-dot line
=
3.75 and Y
0
=
