Geography Reference
In-Depth Information
From inspection of (13.14) it is clear that the solution will remain finite for
s
provided that θ
p
is real. If θ
p
is imaginary, then one term in (13.14) will
grow exponentially and the solution will be unbounded for s
→∞
→∞
. This sort of
behavior is referred to as
computational instability
. Now because
sin
−
1
(σ sin p)
θ
p
=
(13.17)
θ
p
will be real only for
|
σ sin p
|≤
1, which can only be valid for all waves (i.e.,
for all p)ifσ
≤
1. Thus, computational stability of the difference equation (13.8),
requires that
σ
=
cδt/δx
≤
1
(13.18)
which is referred to as the Courant-Friedrichs-Levy (CFL) stability criterion.
The CFL criterion for this example states that for a given space increment δx the
time step δt must be chosen so that the dependent field will be advected a distance
less than one grid length per time step. The restriction on σ given by (13.18)
can be understood physically by considering the character of the solution in the
x, t plane as shown in Fig. 13.1. In the situation shown in Fig. 13.1, σ
1.5.
Examination of the centered difference system (13.8) shows that the numerical
solution at point A in Fig. 13.1 depends only on grid points within the shaded
region of the figure. However, because A lies on the
characteristic
line x
=
−
ct
=
0,
the true solution at point A depends only on the initial condition at x
0 (i. e.,
a parcel which is initially at the origin will be advected to the point 3δx in time
2δt). The point x
=
0 is outside the
domain of influence
of the numerical solution.
Hence, the numerical solution cannot possibly faithfully reproduce the solution to
the original differential equation since, as shown in Fig. 13.1, the value at point
A in the numerical solution has no dependence on the conditions at x
=
0. Only
when the CFL criterion is satisfied will the domain of influence of the numerical
solution include the characteristic lines of the analytic solution.
Although the CFL condition (13.18) guarantees stability of the centered differ-
ence approximation to the one-dimensional advection equation, in general the CFL
criterion is only a necessary and not sufficient condition for computational stabil-
ity. Other forms of finite differencing the one-dimensional advection equation may
lead to more stringent limits on σ than given in (13.18).
The existence of computational instability is one of the prime motivations for
using filtered equations. In the quasi-geostrophic system, no gravity or sound waves
occur. Thus, the speed c in (13.18) is just the maximum wind speed. Typically,
c<100 m s
−
1
so that for a grid interval of 200 km a time increment of over
30 min is permissible. However, in the nonhydrostatic equations commonly used
in cloud-resolving models, the solution would have characteristics corresponding
to acoustic modes, and to assure that the domain of influence included such char-
acteristics c would need to be set equal to the speed of sound, which is the fastest
=
