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time. Note that
t
j
+1
=
t
j
+
j
,where
is the sampling
interval, and that the first measurement was done at time
t
1
and the last at time
t
P
=
t
N
+
n
,where
n
is an inte-
ger number. Subtracting the mean from each time series
x
i
=
X
i
−
X
, we can put the data into a generalized
N
periods sufficiently long to allow for the application of
multivariate statistical techniques. While the PIV data
consist of the horizontal flow field, the LDV data con-
sist of 20 time series regularly distributed along a circle in
the annulus at mid-radius, i.e.
(a
+
b)/
2 (see Figure 17.1).
Note that the sampling rate of the LDV measurement is 10
times larger than the one of the PIV measurements. How-
ever, with respect to a fixed spatial point, the LDV samples
the data just once per revolution. Still, short-lived struc-
tures typical for more transient flows are better resolved
in the LDV data.
The PIV system is used to measure the horizontal veloc-
ity components 15 mm below the fluid surface. Each
experiment lasts typically 50
τ
,where
τ
is the revolution
period of the annulus. We sample the PIV data with
τ/
2,
i.e. two observations per revolution. The PIV camera is
mounted in an inertial frame above the cylinder, that is,
the camera does not corotate with the cylinder. To obtain
the velocity components in the corotating frame, we sub-
tract the solid-body velocity
×
M
data matrix
⎛
⎞
x
1
(t
1
+
n) x
2
(t
1
+
n)
···
x
M
(t
1
+
n)
⎝
⎠
x
1
(t
2
+
n) x
2
(t
2
+
n)
···
x
M
(t
2
+
n)
F
n
=
,
.
.
.
.
.
.
···
x
1
(t
N
+
n) x
2
(t
N
+
n)
x
M
(t
N
+
n)
where
n
,
n
= 0,1,2,
...
, defines a time delay. An
extended matrix
F
m
can be defined as
F
m
=
F
,
F
2
,
F
3
,
...
,
F
m
.
(17.5)
Note that
F
m
is an
N
mM
matrix formed by the
m
submatrices
F
,
...
,
F
m
. EOFs are the eigenvectors of
(
F
0
)
T
F
0
, space-time EOFs (MSSA modes) with time
window
m
are the eigenvectors of
F
m
F
m
,POPsarethe
eigenvectors of
P
=
((
F
1
)
T
F
0
)((
F
0
)
T
F
0
)
−
1
,and
singular vectors with optimization time
and based on
the Euclidean norm are the eigenvectors of
P
T
P
. All these
eigenvectors form an orthogonal basis. Therefore, the data
vector at time
i
can be written as
F
=
j
×
v
=
r
from each observed
PIV velocity rigid-field. A preprocessing of the data is
needed to eliminate erroneous vectors and to homogenize
the data.
The radial velocity component is measured with the
LDV that was fixed in the inertial frame, too. The mea-
surements take place 2 mm below the fluid surface at
midradius of the annulus. The large data set is reduced
by an appropriate averaging. Furthermore, linear interpo-
lation is applied to obtain a homogeneous data set with
regular grid distance
=18
◦
and
t
=
τ/
20. These pre-
processed LDV data are then analyzed using the MSSA
software toolkit by
Dettinger et al.
[1995].
The MSSA software package is particularly suited to
detect (intermittent) oscillations in noisy time series as
well as in multivariate data. With regard to the aims of
our study, this makes the method particularly suitable
to find structures in the transition region to the quasi-
chaotic regime where the waves become more and more
irregular. In the specific experimental setup used here, this
regime occurs when the Taylor number is larger than 10
8
and the thermal Rossby number is smaller than 0.5 (cf.
Figure 17.2).
Owing to the fixed PIV camera, the errors of PIV obser-
vations grow with growing angular velocity of the cylin-
der. In contrast, using LDV, the radial velocity component
can be observed even for large angular velocities of the
annulus with high accuracy. Thus, LDV data from the
irregular wave regime will be analyzed by using the MSSA
method (question (ii)). Instead, the preprocessed PIV data
were analyzed by using the CEOF method with the focus
on question (i).
A detailed description of our approach to find cou-
pled propagating patterns with the CEOF is given by
Harlanderetal.
[2011]. Briefly speaking, we use the Hilbert
×
a(t
i
)
j
j
,
where
j
is the
j
th eigenvector from an EOF, MSSA, POP,
or SV analysis, and the coefficients
a(t
i
)
j
are found by a
suitable projection of the eigenvectors on the data vector.
In Section 17.3.4 we discuss a novel orthogonal decom-
position that is not premised on a statistical basis. Here we
use a mesh-free data reconstruction method that is based
on RBFs. Using these basis functions, we can decompose
the horizontal velocity data into a sum of divergence-free
and curl-free parts. Such a decomposition can be very
useful in discriminating different waveields in the annulus.
17.3.1. EOF and MSSA Analysis of Wave Interactions
With PIV and LDV Measurements
In this section, we present the analysis of velocity data
from classical
f
-plane thermally driven rotating annulus
experiments with
η
= 0.38,
= 1.8, and Pr = 7.16
(see (17.2) and (17.4)), recovered by PIV and LDV (see
Harlander et al.
[2011] for details). The following ques-
tions will be addressed: (i) Can the statistical analysis
detect coexisting wave modes during a traverse between
two regular wave regimes? (ii) Can we find coexisting
modes in the transition region to the quasi-chaotic regime
or is the flow dominated by random fluctuations?
The data we used are sampled twice (PIV) or 20 times
(LDV) per revolution of the annulus and cover time
