Geoscience Reference
In-Depth Information
(T
−
2
)
) includes all flow regimes. Such regime
diagrams are the basis of many studies and they have been
experimentally derived already by
Fowlis and Hide
[1965]
and have been refined later by other authors e.g. [
Früh and
Read
, 1997;
von Larcher and Egbers
, 2005a]. The range of
azimuthal wave numbers
m
is restricted by the dimensions
of the gap.
Hide and Mason
[1970] found an empirical law
for the minimum and maximum wave number,
m
min
have been acquired by the optical laser Doppler velocime-
try (LDV).
On the other hand, also linear, multivariate statistical
techniques have been successfully applied to highlight
certain aspects of the motion in the rotating annu-
lus. [
Read
, 1993], e.g., used multivariate singular system
analysis (MSSA) for phase portrait reconstructions of
the annulus flow. Complex empirical orthogonal function
(EOF) analysis have been applied to data from a rotating
annulus with bottom topography [
Pfefferetal.
, 1990]. The
focusof thisworkwastoidentifyfeaturesof thewaveprop-
agationasafunctionof theTaylornumber.
MundtandHart
[1994] constructed a reduced low-dimensional model of
two-layerbaroclinicinstabilitybyprojectingthegoverning
equations onto the EOFs of numerical flow simulations.
The same should be possible by using EOFs from annulus
laboratory data [
Stephen et al.
, 1997, 1999]. Finally,
Read
et al.
[2008] used EOFs deduced from numerical simula-
tionstoidentifystructuralchangesof thedominantmodes
in the annulus when the Taylor number is increased.
The differentially heated rotating annulus has also been
used as a test bed for studies on weather predictabil-
ity [
Young and Read
, 2008;
Ravela et al.
, 2010]. Young
and Read studied the breakdown of predictability for
numerically deduced irregular flow regimes. In this con-
text breeding vectors (close relatives to singular vectors)
play an important role. Such vectors are an orthogonal
decomposition for flows with
nonorthogonal
eigenmodes.
The present chapter is organized as follows. In Section
17.2 we will give details on the experimental appara-
tus we use and the governing nondimensional parame-
ters. Then, in section 17.3 we will present a summary
of laboratory studies on annulus flows we performed
over the previous few years. In particular, we describe
the multivariate orthogonal decomposition techniques we
applied to the laboratory data. In Section 17.3.1 we ana-
lyze PIV and LDV data at the transition between two
different wave regimes by applying the complex EOF
analysis and MSSA. Subsequently, in Section 17.3.2 we
analyze data from an annulus with a broken azimuthal
symmetry. Similar to
Pfeffer et al.
[1990], we are inter-
ested in the wave propagation characteristics in a rotating
annulus with “topography”. The data have been retrieved
simultaneously by thermography and PIV measurements.
Complex EOF analysis is able to decompose the flow
into features typical for the flow up- and downstream
of the annulus constriction. This study was motivated
by specific large-scale ocean currents like the Antarctic
Circumpolar Current where the “gap width” of the flow
depends on longitude. In Section 17.3.3 we decompose
surface temperature data of the annulus flow in principal
oscillation patterns (POPs), that is, the linear eigenmodes,
and in modes of maximal growth, called singular vectors
(SVs). In contrast to the traditional approach, we deduce
(
∝
≤
m
≤
m
max
, known as the Hide criterion reading,
π
4
b
+
a
b
3
π
4
b
+
a
b
a
≤
m
≤
a
,
(17.1)
−
−
with
a
(
b
) as the inner (outer) radius of the gap.
One of the most fascinating aspects of the differentially
heated rotating annulus is its rich time-dependent flow
behavior. It is therefore not surprising that many stud-
ies have focused on this aspect. A phenomenon that has
attracted much attention over many years is the so-called
amplitude and structural vacillation, which is a modu-
lation of the amplitude and the wave shape in distinct
subregions of the regime diagram mentioned above. Wave
dispersion and structural vacillation have been observed
by e.g.
Pfeffer
and
Fowlis
[1968] using streak photographs
and by
Harlander et al.
[2011] by particle image velocime-
try (PIV). They showed the simultaneous presence of two
subsequent wave modes and argued that some part of the
vacillation might result from the different phase speeds of
the two modes (see also
Yang
[1990]).
However, wave dispersion cannot explain the existence
of multiple wave modes during a traverse of the regular
wave regime. Therefore,
Lindzen et al.
[1982] numerically
investigated a nonlinear version of Eady's baroclinic insta-
bility problem for the annulus, and
Barcilon and Drazin
[1984] investigated the problem by asymptotic techniques.
In both studies, regions in the regime diagram could be
identified where two modes with the same wave num-
ber may grow. Later,
Früh
[1996] and
Früh and Read
[1997] suggested that resonant wave triads are responsi-
ble for certain amplitude vacillations (see also the review
on amplitude vacillations in this topic in chapter 3). Such
triads, besides the dominant mode, involve two other,
weaker modes. Energy is redistributed between the mem-
bers of a triad, and the dominant pattern vacillates with a
characteristic time. Geostrophic turbulence, i.e., the irreg-
ular flow regime, is generally found at high rotation rates
[
Morita and Uryu
, 1989;
Read et al.
, 1992;
Pfeffer et al.
,
1997].
The nonlinear behavior of the annulus flow motivated
a number of contributions using nonlinear time series
analysis to better understand the physical mechanisms.
Read et al.
[1992] and
Früh and Read
[1997], for exam-
ple, used time series of temperature from probes in the
fluid interior. In contrast,
Sitte and Egbers
[2000] and
von
Larcher and Egbers
[2005a] used velocity time series that
