Geoscience Reference
In-Depth Information
3.3.3. Low-Order Models
of the same zonal wave, amplitude vacillation is due to
the interference of two vertical modes of the same zonal
wave. This is in accord with the CFD results from
Lu
and Miller
[1997], who identified the vertical transport of
energy associated with periodic changes in the vertical tilt
of the wave structure during the vacillation cycle.
Weng
and Barcilon
[1988] added more zonal wave modes to the
model, but in a way that did not allow for wave-wave
interactions. With this they demonstrated that wave-mean
flow interactions alone are sufficient to give rise to wave
number vacillation as well as amplitude vacillation and
structural vacillation.
The picture of a low with a regular spatial structure has
led to a number of low-order dynamical systems models
of amplitude vacillation in which the components are as
follows:
1. A constant forcing, often represented as a constant
vertical shear velocity, applied positively to a pair of wave
mode amplitude equations
2. A mean-flow correction equation, coupled to the
wave amplitude (the larger the wave, the stronger the
correction), which counteracts the forcing (applied nega-
tively to the wave amplitude equations)
3. Dissipation applied negatively to both the wave
amplitude and the mean-flow correction equations
This suggests that the system requires at least three
dimensions, but the energy transfer routes indicated in
Figure 3.4 suggests that four components are needed.
Translating the amount of necessary information to a nor-
mal mode decomposition, this would suggest a traveling
barotropic wave (consisting of two modes or amplitude
and phase), a traveling baroclinic wave (also two degrees
of freedom), and a mean-flow correction. While this adds
up to five degrees of freedom, it is recognized that one
of the phases can be eliminated by a suitable coordinate
transformation, leaving four degrees of freedom. Reduc-
ing this to only three degrees of freedom would only
be possible if either the relative phase or the relative
amplitude between the barotropic and baroclinic mode is
constant.
3.3.5. Two-Layer Models
Two-layer quasi-geostrophic models are a standard tool
in geophysical fluid dynamics [e.g.,
Pedlosky
, 1987], and
can be formulated for the stream function in each layer or
for the barotropic and baroclinic components. For exam-
ple,ona
β
plane in the layer formulation with
ψ
i
, with
i
= 1,2 for the upper and lower layers, respectively, the
equations can be written as
∂
∂t
+
J (ψ
i
,
)
q
i
=
r
2
∇
1
Re
∇
2
ψ
i
+
2
q
i
·
−
(3.4)
with
2
ψ
i
+
βy
+
(
1
)
−
i
F
i
(ψ
1
−
q
i
=
∇
−
ψ
2
)
,
i
= 1 for upper layer, 2 for lower layer,
∂x
2
+
∂
2
∂
2
3.3.4. Eady-Type Models
∇
=
∂y
2
,
The basic instability as developed by
Eady
[1949] led
to the formulation of the nonlinear dynamics of finite-
amplitude waves driven by a linear vertical shear.
Weng
et al.
[1986] expressed the flow through a (nondimen-
sional) stream function
φ
=
J(ψ
,
q)
=
∂ψ
∂x
∂q
∂y
−
∂ψ
∂y
∂q
∂x
,
r
=
√
E
Ro
=
νf
0
D
2
L
U
, the dissipation parameter,
yz
+
ϕ
+
φ
,where
yz
is
the basic Eady profile,
ϕ
thewaveield,and
φ
the mean-
flow correction, which they then expressed in modes using
e
ikx
sin
lπy
for the horizontal component of the fields and
sinh
μz
and cosh
μz
for the vertical structure. Using this,
Wengetal.
[1986] followed a bifurcation scenario from the
initial instability to a steady wave with the lowest radial
(sin
πy
) wave number, which then underwent a period
doubling bifurcation, followed eventually by the growth
of the second radial mode, sin 2
πy
, which is referred to
as structural vacillation. However, while period doubling
has been observed in the two-layer experiment, the only
well-documented period doubling in the thermally driven
annulus was associated with strong stationary forcing due
to an imperfection in the apparatus [
FrühandRead
, 1997].
Weng and Barcilon
[1987] suggested that, while structural
vacillation is due to the interference of two radial modes
−
−
ρ
0
f
0
L
2
g(ρ
2
−
F
i
=
, the Froude number,
ρ
1
)D
i
Re =
UL
ν
=
1
r
2
Ro
, the Reynolds number.
The equivalent form for the barotropic component,
ψ
s
=
(ψ
1
+
ψ
2
)
/
2, and the baroclinic component,
ψ
d
=
(ψ
2
−
ψ
1
)
/
2, is for the barotropic component (
s
= sum)
∂
∂t
∇
2
ψ
s
+
β
∂ψ
s
∂x
+
u
s
∂
2
ψ
s
+
u
d
∂
2
ψ
d
∂x
∇
∂x
∇
+
J
ψ
s
,
2
ψ
s
+
J
ψ
d
,
2
ψ
d
∇
∇
2
ψ
s
+
Re
−
1
4
ψ
s
=
−
r
∇
∇
(3.5)
