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even reversed radial eddy heat flux in the lee side of the
barrier.
The experiment described has not been designed to
resemble some particular large-scale flow that can be
found in nature. However, it is tempting to point to simi-
larities between experimental and real flows. The ACC is
an annulus-like large-scale flow that is partially blocked
at the Drake Passage, connecting the Pacific and the
Atlantic oceans. For the ACC, eddy fluxes play a more
central role in the dynamical and thermodynamical bal-
ances than in other oceans [
Rintouletal.
, 2001]. Of course,
many processes important for the ACC dynamics are not
captured by the experiment, e.g., wind stress, coastal as
well as bottom topography, and the high-latitude
β
-effect
[
Harlander
, 2005;
Afanasyev et al.
, 2009, chapter 5 this
issue]. Nevertheless, a meridional overturning circulation
is present in the ACC and the experiment and baroclinic
instability plays an important role for the ACC dynamics
due to the meridional transport of heat toward the pole.
where
x
(t
+
)
is the state vector at time
t
+
and
G
()
the
propagator matrix that maps the state at time
t
to time
t
+
.
G
()
bears the argument
since it can be estimated by
G
()
=
C
()
C
(
0
)
−
1
,
(17.7)
where
C
()
=
(
F
1
)
T
F
0
,
(17.8)
C
(
0
)
=
(
F
0
)
T
F
0
(17.9)
are the covariance matrices with lag
and lag zero.
The eigenvectors of
G
()
are the POPs, whereas the
eigenvectors of
C
(
0
)
are the EOFs. Usually, the POPs are
sorted (in decreasing order) with respect to the e-folding
times
τ
i
=
)
, where the
λ
i
are the eigenvalues of
G
()
. The period of the POP is given by
T
=2
π/
arg
λ
.
We used surface temperature data to compute EOFs
and POPs. The data have been recorded by an infrared
camera that has a noncooled microbolometer detector
with a spectral range of 7.5
−
1
/
ln
(
|
λ
i
|
−
14.0
μ
m and a tempera-
ture resolution smaller than 0.08K with an accuracy of
±
17.3.3. Principal Oscillation Patterns and Singular
Vectors
1.5K at 30
◦
C. The spatial resolution of the infrared sen-
sor is 640
480 pixels. To reduce the size of the covariance
matrices we smoothed the data by using a running aver-
age over areas of 2
×
In many cases, EOFs can be interpreted as the pat-
terns of natural oscillations, i.e., as the eigenmodes of the
system under consideration. From a mathematical point
of view such an interpretation is not justified. However,
empirically estimating the eigenmodes and then compar-
ing them with the EOFs can reveal the connection between
modes and patterns of variability [
Harlanderetal.
, 2009b].
Empirically estimated normal modes are called principal
oscillation patterns (POPs) in the meteorological litera-
ture [
Hasselmann
, 1988]. In the following we will compare
EOFs and POPs computed from our annulus data. In a
subsequent step we will then estimate optimal growing ini-
tial perturbations, called singular vectors from the POPs.
Theoretically, SVs converge to POPs in the limit
t
2 pixels. In Figure 17.12a we display
the EOF1 that explains 27% of the total variance. Further,
in Figure 17.12b, the real part of the least damped POP1
with a damping time of 1827.6 s and a period of 62.5 s for
an experiment with
=6rpmand
T
= 8K is shown.
It should be noted that EOF2 is a phase shifted version
of EOF1 with nearly the same explained variance. Obvi-
ously, EOF1 and POP1 agree very well. Taking EOF1 and
EOF2, as well as the real and imaginary parts of POP1
together, both patterns propagate with the same phase
speed. This suggests that for the data considered the EOFs
represent the eigenmodes of the system.
Let us next use the POPs to estimate the dominant
patterns of nonmodal instability. This procedure needs an
appropriate filtering and the EOF analysis is a suitable
method for this purpose.
×
→∞
.
As we will see, this holds for the empirically estimated SVs
as well. SV growth might explain the increased irregularity
that occurs for annulus flows at Taylor numbers larger that
10
8
. We will briefly discuss this by employing laboratory
data and model results.
17.3.3.2. Empirical Singular Vectors.
Instability is
related to exponentially growing eigenmodes and thus to
POPs with
>
1. Interestingly, when finite time
intervals are considered, growth rates of certain initial per-
turbations can exceed the growth rates of the most unsta-
ble modes. Moreover, even when all modes are damped
(
|
λ
i
|
17.3.3.1. Principal Oscillations Patterns.
Generally
speaking, the POP method is promising when the dynam-
ical process under consideration is linear to first approx-
imation. This sounds very restrictive in particular when
geophysical applications are the target. However, the
method has successfully been applied to a wide class of
geophysical data [
von Storch and Zwiers
, 1999].
The linear system considered reads
<
1), such particular initial perturbations can
grow dramatically during finite time intervals. The pertur-
bations with the largest growth rates are called singular
vectors or optimal perturbations. They play an impor-
tant role not only in atmospheric ensemble predictions
[
Kalnay
, 2002] but also for the theory of instability and
turbulence [
Trefethen et al.
, 1993;
DelSole
, 2007].
|
λ
i
|
x
(t
+
)
=
G
()
x
(t)
,
(17.6)
