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and for the baroclinic component
∂
Further studies have successively reduced the dimen-
sion of the system to the absolute minimum required
for vacillation to isolate a simple sufficient mechanism
for vacillation. For example,
Pedlosky and Frenzen
[1980]
derived from the quasi-geostrophic two-layer equations a
set of ordinary differential equations of the form
∂t
2
F
ψ
d
+
β
∂ψ
d
∂x
2
∇
−
+
u
s
∂
2
ψ
d
+
u
d
∂
2
ψ
s
+2
Fu
d
∂ψ
s
∂x
∂x
∇
∂x
∇
+
J
ψ
s
,
2
ψ
d
+
J
ψ
d
,
2
ψ
s
∇
∇
−
2
FJ (ψ
s
,
ψ
d
)
dA
dt
=
B
2
ψ
d
+
Re
−
1
2
F
−
γ A
,
(3.9)
2
2
ψ
d
=
−
r
∇
∇
−
∇
(3.6)
c
A
2
+
V
k
,
dB
dt
=
γ
2
(B
−
−
γ A)
+
A
−
(3.10)
followed by a suitable spectral expansion and truncation,
for example, Fourier modes for a straight channel [
Früh
,
1996],
=
γ
A
2
αV
k
,
dV
k
dt
−
(3.11)
N
φ(t)
s
,
d
cos
nπy
ψ
s
,
d
=
where
A
is the amplitude of the represented baroclinic
wave,
B
=
dA/dt
+
γ A
,and
V
k
is a set of
k
=1,
...
,
M
cross-channel modes of the mean-flow correction to cap-
ture the mean-flow correction adequately; cf. equation
(3.8). In this model, only the wave amplitude is repre-
sented explicitly but not the phase speed.
Pedlosky and Frenzen
[1980] showed that this system
can be reduced to a form equivalent to the classic Lorenz
equations, originally derived as the simplest model for
Rayleigh-Bénard convection [
Lorenz
, 1963a]. The rela-
tionship between the Lorenz equations and the two-layer
model equations was subsequently analyzed and discussed
by
Lovegrove et al.
[2001].
n
=1
χ(t)
mn
x
sin
nπy
.
M
,
N
s
,
d
cos
2
mπ
s
,
d
sin
2
mπ
x
+
σ(t)
mn
+
α
α
m
,
n
=1
(3.7)
To satisfy the lateral boundary conditions, only some of
the cross-channel Fourier modes are possible, but the non-
linear interactions in the equations result in terms of the
other modes, and this energy must be projected onto those
that do satisfy the boundary conditions. If the product of
two wave terms has a zero zonal wave number (
m
=0),it
has a mean-flow structure of the form sin
nπy
while the
modes satisfying the boundary conditions are of the form
cos
πy
. This means that each term of radial mode
n
has
to be expressed by a series of zonal flow correction terms
with
3.4.WAVE INTERACTIONS
3.4.1. Wave Triad Interactions
1
1
)
n
+
c
n
=
2
π
n
A common description of the underlying processes is
to identify the transfer to and from the eddy kinetic and
potential energies through nonlinear wave interactions
which arise explicitly in the advection term
(
u
−
(
−
2
.
(3.8)
n
2
−
The Reynolds number term was originally omitted
in the low-order models as it was assumed that the
Stewartson layers at the side boundaries were “passive”
while the relevant dissipation occurred through Ekman
suction from the Ekman layers as the horizontal bound-
aries. However,
Smith
[1974] showed that the side bound-
aries are involved in the energy balance for the fluid inte-
rior and, in particular, that their absence resulted in a
nonphysical energy source of mean-flow kinetic energy.
One of the earliest applications by
Lorenz
[1963b] of a
truncation of a two-layer model to investigate specifically
amplitude vacillation arrived at a 14-dimensional system,
capturing a barotropic zonal flow, a baroclinic zonal flow,
and two different radial modes of a wave with a common
zonal wave number, each represented by a cosine and a
sine component of the stream function as well as a tem-
perature component. Depending on the parameter values,
this system produced steady wave solutions, periodic vac-
illations, as well as aperiodic flow that appeared to arise
from a homoclinic bifurcation.
)
u
of the
Navier-Stokes or “primitive” equations or the Jacobian
J(
·∇
2
ψ
,
ψ)
in the vorticity-stream function form of the
momentum equations when the flow field is expanded into
Fourier modes. The product of sine and cosine terms leads
to contributions to the equations for the modes with mode
numbers of the sum and difference of the two terms in the
product. This leads to the notion of wave triads: two waves
with zonal and radial wave numbers
(m
,
n)
and
(m
,
n
)
combine in the multiplication to structures with wave
numbers
(m
,
n
)
=
(m
∇
n
)
, which then appear in
the evolution equation for those respective modes. These
possibilities are constrained for two-dimensional and non-
divergent flow such that energy has to flow to both larger
and smaller scales in such a way that both kinetic energy
and entrophy are conserved [
Fjørtoft
, 1953]. In addition,
Hasselmann
[1967] pointed out that within a triad only
the wave with the highest frequency can support energy
transfer to the other two members of this triad, which
m
,
n
±
±
