Geography Reference
In-Depth Information
In general, the ensemble mean provides a better forecast than individual mem-
bers of the ensemble, as the ensemble mean tends to filter out the most uncertain
aspects of the forecast. The spread of ensemble members provides an important
indication of the likely accuracy of the ensemble mean forecast. If there is a large
spread among members of the ensemble, not all members can be correct, and the
reliability of the forecast should be judged as low. However, when most mem-
bers of the ensemble agree, the reliability of the forecast should be high (provided
that the forecast model is unbiased). In summary, the ensemble prediction system
not only can improve forecast reliability, but also can provide information on the
degree of reliability of a specific forecast.
PROBLEMS
13.1.
Show that for the barotropic vorticity equation on the Cartesian β plane
(13.26) enstrophy and kinetic energy are conserved when averaged over the
whole domain, that is, that the following integral constraints are satisfied:
ζ
2
2
∇
d
dt
d
dt
ψ
·∇
ψ
dxdy
=
0
dxdy
=
0
2
Hint
: To prove energy conservation, multiply (13.26) through by
−
ψ and
use the chain rule of differentiation.
13.2.
Verify the expression (13.31) in the text. Use periodic boundary conditions
in both x and y.
13.3.
The Euler backward method of finite differencing the advection equation
is a two-step method consisting of a forward prediction step, followed by
a backward corrector step. In the notation of Section 13.3.2 the complete
cycle is thus defined by
2
ˆ
q
m
−
1,s
σ
q
m
−ˆ
ˆ
q
m,s
=−
q
m
+
1,s
−ˆ
2
ˆ
q
m
−
1
σ
q
m
+
1
−ˆ
q
m,s
+
1
−ˆ
ˆ
q
m,s
=−
q
m
is the first guess for time step s
ˆ
+
where
1. Use the method of Section
13.3.3 to determine the necessary condition for stability of this method.
13.4.
Carry out truncation error analyses analogous to that of Table 13.1 for the
centered difference approximation to the advection equation but for the
cases σ
=
0.95 and σ
=
0.25.
