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transform method to make the (horizontal) velocity com-
ponents (
u
,
v
) complex and then formed extended time
series by combining the two complex time series. With
this new time series we built the covariance matrix and
computed its eigenvectors (i.e., the CEOFs) and the cor-
responding time-dependent coefficients.
approximately 0.009rad
/
s), according to the linear Eady
model.
To summarize the main results of the CEOF analy-
sis, we find that in the rather stable
m
= 3 regime the
presence of higher modes and their linear interaction
with the leading wave mode can give rise to slow mod-
ulations. However, the existence of the
m
= 4 mode in
the
m
= 3 wave regime cannot be explained by linear
theory.
17.3.1.1. Analysis of PIV Measurements.
As des-
cribed, PIV measurements were performed to detect
complex flows during the traverse between two reg-
ular wave regimes. The measurements presented here
have been conducted at Ta = 1.74
17.3.1.2. Analysis of LDV Measurements.
Next we dis-
cuss the results from the MSSA of the LDV data. As
mentioned above, a strong feature of the MSSA is its
ability to detect oscillating/propagating features in noisy
data. Thus, the MSSA seems to be more suitable than
EOF analysis to find excited propagating modes in flow
with structural vacillations and “irregular wave regimes.”
Note that we call a wave irregular when it shows signif-
icant transient features. In contrast, for a turbulent flow
regime a dominant wave can no longer be observed. Here
we apply the MSSA to a parameter point in the irreg-
ular flow regime, i.e., at Ta = 3.76
10
7
,Ro = 1.30 (i.e.,
=0.50rad
/
s,
T
= 6.7K). That parameter point is close
to the transition from the steady wave regime of wave
number
m
= 3 to the structural vacillation (SV) regime,
i.e.,
m
= 3 (SV) (cf. Figure 17.2, but note that here
T
= 7.5 K and that for larger
T
transitions occur at
larger Ta).
The eigenvalue spectrum (not shown) is dominated by
the first eigenvalue that contains more than 40% of the
total variance of the flow, and the second eigenvalue
includes about 10% of the total variance. It should be
noted that the variance distribution depends on the data
quality and on Ro and Ta. The lower row of Figure 17.4
shows the corresponding real part of the CEOFs and
the upper row their time evolution. Note that, in gen-
eral, the real and imaginary parts of the CEOFs and their
time series show a 90
◦
phase difference [
von Storch and
Zwiers
, 1999].
The velocity field can be reconstructed via (17.5).
CEOF1 together with the time-dependent coefficient
determines a prograde propagating wave (i.e., a wave prop-
agating in the direction of the annulus revolution) with
wave number
m
= 3; CEOF2, in contrast, determines a
rather regular and slowly retrograde wave (propagating
in the opposite direction of the annulus revolution) with
wave number
m
= 4. By combining the first two patterns
(which then contain about 50% of the total variance), a
wavy jet flow with dominant wave number
m
= 3 is found
which shows slow vacillations due to an interference with
the (weak) mode pattern with
m
= 4. Further details
and also patterns for different Ta and Ro are discussed in
Harlander et al.
[2011].
From the time-dependent coefficients (Figure 17.4,
top panel), the drift rates of the dominant mode
m
=3
and of the weak mode
m
= 4 are found to be 0.021 and
−
×
10
8
,Ro = 0.14
×
(
=2.32rad
/
s,
T
= 6.9K).
Figure 17.5 (top panel) shows the preprocessed LDV
data. The data are presented in the form of a space-time
diagram, where the abscissa runs from 0 to 2
π
, covering
the spatial structure of the radial velocity at midradius of
the annulus. Although the flow is much more noisy than
for the PIV experiment, we clearly can identify a wave pat-
tern with
m
= 4 that propagates prograde with a phase
speed of 0.011rad
/
s.
The eigenvalue spectrum is broad (Figure 17.5, upper
right part). Noise is usually part of the flat tail in the eigen-
value spectrum. Here, we define the noise level to be at 1%
of the total variance. Note that this is a qualitative mea-
sure, and it is not the exact signal-to-noise level of our
experiments. The first two eigenvalues explain 36% of the
total variance and the next two eigenvalues are also clearly
above the defined noise level.
Similar to the description of flow field reconstruction
using the CEOF method, reconstructing the data by using
just the first two space-time EOFs (ST-EOFs) gives a
filtered version of the original data (Figure 17.5, bot-
tom left). Instructive is the reconstruction by ST-EOF
3 and 4, explaining at least 5% of the total variance
(Figure 17.5, bottom right). The reconstruction reveals
a wave pattern with wave number
m
= 5 that propa-
gates essentially with the same phase speed as the dom-
inant
m
= 4 wave. Finally, note that the
m
=5 wave pat-
tern shows slight amplitude vacillations. Moreover, its
phase speed is less constant than the phase speed of the
dominant wave.
Roughly speaking, for the irregular flow regime, the
first wave modes seem to be less dispersive than for the
0.007rad
/
s, respectively. Slow retrograde propagating
modes are rather exceptional but have been reported
earlier [
Früh and Read
, 1997]. It appears that the prop-
agation of the weak mode is strongly affected by the
dominant mode of the system and that linear wave theory
fails to describe its anomalous retrograde propagation, as
all unstable baroclinic modes should propagate with the
volume-averaged mean flow (which was estimated to be
