432
by first unstable steady state to begin a new cycle. This transition phase
is the longest lasting stage of the whole cycle; its timescale is determined
by the baroclinic spin-down time (J" J.l) -1.
u
,..
H
to...4 ~ u
~
,
-,
3.'
U
3~
~
V
U
0.5 0
,. ,
50
'00
""
:zoo
,.. 300 0
50
'00
'50
:zoo
,.. 300
-(0)
.... ~I
Figure 9: Time series of two of the variables for the same parameter values as for Figure 8.
The variables shown are the dimensionless values of the barotropic and baroc1inic mean
zonal velocities U a nd V.
The existence of this complex attractor is linked to a homo clinic bifurcation of a type described first by L.P. Shilnikov. For certain parameter
values a homo clinic orbit exists which connects the stable and the unstable
manifold of the unstable steady state in the center of the oscillatory phase.
This homo clinic orbit is the limiting case of an unstable periodic orbit that
bifurcates in a Hopf-bifurcation from the low-index or from the high-index
state. It can be shown (Guckenheimer and Holmes 1993) that depending
on the ratio of the eigenvalues of the Jacobi-matrix of the steady state this
homo clinic orbit can be accompanied by quite complex dynamical behavior, including the existence of horseshoe-maps in the flow and of infinitely
many periodic orbits of arbitrary high period.
4 Comparison with high-resolution numerical models
of zonal channel ftow
Finally we want a comparison of the low-order model with a system which
is closer to reality than this model with extremely reduced physics. Since
there are no adequate observational data to determine possible multiple
equilibria or temporal variability of the ACC we have to build a less realistic 'strawman' to compare with, i. e. we will use another model. Of
by first unstable steady state to begin a new cycle. This transition phase
is the longest lasting stage of the whole cycle; its timescale is determined
by the baroclinic spin-down time (J" J.l) -1.
u
,..
H
to...4 ~ u
~
,
-,
3.'
U
3~
~
V
U
0.5 0
,. ,
50
'00
""
:zoo
,.. 300 0
50
'00
'50
:zoo
,.. 300
-(0)
.... ~I
Figure 9: Time series of two of the variables for the same parameter values as for Figure 8.
The variables shown are the dimensionless values of the barotropic and baroc1inic mean
zonal velocities U a nd V.
The existence of this complex attractor is linked to a homo clinic bifurcation of a type described first by L.P. Shilnikov. For certain parameter
values a homo clinic orbit exists which connects the stable and the unstable
manifold of the unstable steady state in the center of the oscillatory phase.
This homo clinic orbit is the limiting case of an unstable periodic orbit that
bifurcates in a Hopf-bifurcation from the low-index or from the high-index
state. It can be shown (Guckenheimer and Holmes 1993) that depending
on the ratio of the eigenvalues of the Jacobi-matrix of the steady state this
homo clinic orbit can be accompanied by quite complex dynamical behavior, including the existence of horseshoe-maps in the flow and of infinitely
many periodic orbits of arbitrary high period.
4 Comparison with high-resolution numerical models
of zonal channel ftow
Finally we want a comparison of the low-order model with a system which
is closer to reality than this model with extremely reduced physics. Since
there are no adequate observational data to determine possible multiple
equilibria or temporal variability of the ACC we have to build a less realistic 'strawman' to compare with, i. e. we will use another model. Of