Geography Reference
In-Depth Information
where r is the spin-down rate due to boundary layer dissipation, ε is as defined
below (7.100), and K
s
is the resonant stationary Rossby wave number defined
in (7.93).
It is clear from (10.73) that the form drag will have a strong maximum when
β
K
2
, as shown schematically in Fi
g.
10.17. The last term on the right in
(10.71), however, will decrease li
ne
arly as u increases. Thus, for suitable param-
eters there will be three values of u (labeled A, B, and C in Fig. 10.17) for which
the wave drag just balances the external forcing so that steady-state solutions may
exist. By perturbing the solution about the points A, B, and C, it is shown easily
(Problem 10.12) that point B corresponds to an unstable equilibrium, whereas the
equilibria at points A and C are stable. Solution A corresponds to a low-index
equilibrium, with high-amplitude waves analogous to a blocking regime. Solution
C corresponds to a high-index equilibrium with strong zonal flow and weak waves.
Thus, for this very simple model there are two possible “climates” associated with
the same forcing.
The Charney-DeVore model is a highly oversimplified model of the atmosphere.
Models that contain more degrees of freedom do not generally have multiple steady
solutions. Rather, there is a tendency for the (unsteady) solutions to cluster about
certain climate regimes and to shift between regimes in an unpredictable fashion.
Such behavior is characteristic of a wide range of nonlinear dynamical systems
and is popularly known as
chaos
(see Section 13.8).
u
=
Fig. 10.17
Schematic graphical solution for steady-state solutions of the Charney-DeVore model.
(Adapted from Held, 1983.)
