Geoscience Reference
In-Depth Information
(a)
(b)
Wave amplitudes
Wave amplitudes
12
0.3
10
0.25
m
= 1
m
= 2
m
8
= 3
m
= 4
0.2
m
= 1
m
= 2
m
6
= 3
m
= 4
0.15
4
0.1
2
0.05
0
0
0
600
800
1000
1200
1400
1600
1800
2000
2200
1000
2000
3000
4000
5000
6000
7000
8000
9000
10000
#
Time
Figure 3.2.
Typical time series of wave amplitudes for (a) amplitude vacillation and (b) a modulated amplitude vacillation. Quan-
tities shown are Fourier amplitudes from the temperature fields calculated by the model described by
Randriamampianina and
Crespo Del Arco
[2014].
[
Christiansen
, 2000;
Pogoreltsev et al.
, 2009;
Studer
et al.
, 2012], or seasurface temperature (SST) fluctuations
[
Watterson
, 2001], and for climate fluctuations [
Son and
Lee
, 2006].
A widely accepted definition of amplitude vacillation
(AV) is now that it is a (fairly) regular oscillation of the
magnitude of a well-defined wave mode while the spa-
tial structure remains (essentially) unchanged. A more
complex form of vacillation is modulated amplitude vac-
illation (MAV) [
Read et al.
, 1992;
Früh and Read
, 1997],
which frequently displays chaotic oscillations and usually
involves fluctuations of several wave modes of different
spatial structure. A periodic amplitude vacillation of a
wave number 2 and a chaotic modulated amplitude vac-
illation are illustrated in Figure 3.2, both taken from
the direct numerical simulations of an air-filled annulus
described by
Randriamampianina et al.
[2006] and
Ran-
driamampianina and Crespo Del Arco
[2014] in this topic.
A special case of a flow that appears like a modulated
amplitude vacillation is the superposition of two steady
wave modes; this has been termed
interference vacillation
[
Ohlsen and Hart
, 1989;
Lindzen et al.
, 1982].
For laboratory experiments of baroclinic flows, these
types of amplitude vacillations are contrasted to “struc-
tural vacillation”, “tilted-trough vacillation” or “shape
vacillation” [
Hide and Mason
, 1975;
Tamaki and Ukaji
,
1993;
Pfeffer et al.
, 1997;
Früh et al.
, 2007;
Hide
, 2011;
Read et al.
, 1992;
Früh and Read
, 1997]. These flows are
mainly characterized by distinct changes in the shape but
little changes in the power of the waves. While these vac-
illations can occur at a distinct time scale, they tend to
be much less regular than amplitude vacillations, and the
time scale is shorter than that of the typical amplitude
vacillation.
Hart
[1972] also reported the phenomenon
of “frequency vacillation” in the mechanically driven
two-layer experiment that appeared to involve an oscilla-
tion of the wave speed independently of the wave ampli-
tude though a similar time series was later interpreted by
Hart
[1979] as “wave number vacillation” where the flow
structure vacillated between mode 1 and mode 2.
3.1.1. Nondimensional Parameters
In the thermally driven baroclinic annulus, the two prin-
cipal non-dimensional parameters are usually the Taylor
number and the thermal Rossby number. The Taylor
number,
Ta =
fL
2
ν
2
L
d
=
4
2
(b
a)
5
−
,
(3.1)
ν
2
d
is essentially the ratio of the Coriolis term to viscous dis-
sipation, where
=
f /
2 is the angular velocity of the
annulus,
L
and
d
the horizontal and vertical length scales
(
L
=
b
a
with
a
the radius of the inner cylinder and
b
that of the outer), and
ν
the kinematic viscosity. Using an
aspect ratio
γ
=
L/d
, the Taylor number can be equated
to the Ekman number as Ta =
γ E
−
2
.
The thermal Rossby number,
−
gα d T
2
(b
=
(3.2)
a)
2
−
