18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
Coriolis effects on the elliptical instability in cylindrical and spherical
rotating containers
Michael Le Bars, St?phane Le Diz?s & Patrice Le Gal
Institut de Recherche sur les Ph?nom?nes Hors ?quilibre - CNRS UMR 6594
49 rue F. Joliot Curie, B.P. 146, 13384 Marseille Cedex 13, France
lebars@irphe.univ-mrs.fr
Abstract :
The effects of Coriolis force on the elliptical instability are studied experimentally in cylindrical and spherical
rotating containers embarked on a table rotating at a fixed rate. For a given set-up, changing the ratio a0a2a1 of global
rotation to flow rotation leads to the selection of various unstable modes due to the presence of resonance bands,
in close agreement with the normal mode theory. No instability takes place when a0 a1 ranges between -3/2 and -1/2
typically. When decreasing a0a3a1 toward -1/2, resonance bands are first discretized for a0a4a1a6a5a6a7 and progressively
overlap for a8a10a9a12a11a14a13a16a15
a0a3a1
a15
a7 . Simultaneously, the growth rates and wavenumbers of the prevalent stationary
unstable mode significantly increase, in quantitative agreement with the viscous short-wavelength analysis. New
complex resonances have been observed for the first time in the sphere, in addition to the standard spin-over. We
argue that these results have significant implications in geo- and astrophysical contexts.
1 Introduction
The elliptical instability corresponds to the three-dimensional destabilisation of two-dimensional
rotating flows with elliptical streamlines (see the review by Kerswell, 2002, and references
therein). It has first been discovered in the context of strained vortices, but it generally appears
in any turbulent flow exhibiting some coherent structures with elliptical motion as well as in
a large range of industrial and natural systems (e.g. in the wake vortices behind aircrafts, in
planetary liquid cores, in binary stars and accretion disks), where the ellipticity is generated
either by vortex interactions or by tidal effects.
In most practical cases, the strain field responsible for the elliptical pattern rotates around
the same axis as the flow, but with a different rate and possibly in an opposite direction. In the
present paper, we thus systematically study the effects of Coriolis force on the elliptical insta-
bility, both in a rotating cylinder and in a rotating spheroid. Our experimental set-up is inspired
from Malkus (1989): it is similar to the one already used in Eloy et al. (2003) and Lacaze et al.
(2004) respectively. Contrary to former devices, it permits to analyse the growth and the satu-
ration of the elliptical instability. A deformable and transparent container - either a cylinder of
radius a17a18a20a19a22a21a24a23a26a25a28a27 cm and height a17a29a30a19a31a21a33a32a34a23a36a35 cm or a hollow sphere of radius a17a18a20a19a22a21a24a23a37a32a38a25a34a27 cm - is set
in rotation about its axis a39a41a40a43a42a45a44 with an angular velocity a17a46a48a47 up to a49a34a50a34a50 rpm and is simultaneously
deformed elliptically by two fixed rollers parallel to a39a51a40a43a42a34a44 . The container is filled with water
seeded with anisotropic particles (Kalliroscope). A light sheet is formed in a plane containing
the rotation axis for visualisation, allowing the measurement of wavelengths and frequencies of
excited modes. Besides, the whole set-up (with also the camera and light projector) is placed on
a 0.5m-diameter rotating table, which allows rotation with angular velocity a17a46a53a52 up to a54a56a55a34a50 rpm.
Our protocol is the same all along the experiments presented here. First, we set the global rota-
tion to its assigned value and wait for solid body rotation to take place in the container. Then we
start the rotation of the container: a spin?up phase first takes place, before the possible develop-
ment of an instability. All presented experiments are carried out near the instability threshold:
the characteristic growth time is then much larger than the spin?up time and decorrelation of
both phenomena is expected.
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
2 Theoretical and experimental study in the cylinder
2.1 Theoretical approaches
The elliptical instability mechanism has been reviewed in Kerswell (2002). It is associated
with the parametric resonance of two inertial waves of the undistorted circular flow induced by
the underlying strain field (e.g. Waleffe, 1990; Kerswell, 2002). For small deformations, the
global (or normal mode) theory permits to calculate explicitly the conditions of resonance for
a given geometry and provides information on the structure of the eigenmodes. Results for the
elliptical instability in a cylinder with Coriolis effects have been obtained by Kerswell (1994).
Numerous resonances with various structures can be excited by changing the global rotation
rate a46a53a52 a19 a17a46a53a52a1a0 a17a46a48a47 only, except in a forbidden band for a46 a52 between a2 a49 a0 a21 and a2 a32 a0 a21 where the
elliptical instability cannot develop.
In addition to the conditions for resonance given by the global approach, the local approach
allows the analytical determination of the growth rate of the instability. It is based on the inviscid
short?wavelength Lagrangian theory developed by Bayly (1986) and Craik & Criminale (1986).
In this approach, perturbations are assumed to be sufficiently localised in order to be advected
along flow trajectories and are searched in the form of local plane waves. This method has been
applied to the elliptical instability with global rotation by Le Diz?s (2000). He determined the
exponential growth rate at order 1 in eccentricity a3
a4
a19 a5a6
a6
a7 a8
a49a10a9
a21 a46 a52
a35
a39
a32
a9
a46 a52
a44a12a11a1a13
a3a15a14a16a2
a8
a32
a2
a21a18a17a32
a9
a46
a52 a17a20a19a22a21a24a23
a39a26a25a33a44
a11
a14a12a27 (1)
where a25 is the angle between the flow rotation axis and the wavevector.
Assuming that the viscous dissipation is of order a3 , viscous effects on the localised pertur-
bations can be easily taken into account by adding the viscous damping rate a2a29a28 a14 Rea30a32a31 (Craik &
Criminale, 1986). Here Re is the Reynolds number defined by Re a19 a17a46a48a47 a17a18 a14 a0a34a33 , a33 the kinematic
viscosity of the fluid and a28 the wavevector of the perturbation. Viscous effects on the surface of
the container for plane wave perturbations can be estimated using the work of Kudlick (1966)
and introduce corrections of order Rea30a32a31a26a35 a14 . For given values of a39 a17a18 a27 a17a29 a27 a17a46 a47 a27a36a33a37a27 a3a28a44 , bands of insta-
bility then take place depending on the global rotation rate a46 a52 , each band corresponding to a
given axial structure determined by the number a38 of axial half-periods. Two examples of these
theoretical predictions are shown in figure 1, together with our experimental data.
2.2 Experimental study
A series of experiments was performed using a cylinder of height a17a29 a19 a21a24a32a34a23a35 cm and eccen-
tricity a3 a19 a50 a23a50a34a39 a27 , systematically changing a17a46 a47 and a17a46 a52 . Good agreement is found with the
linear inviscid global approach: stationary mode with a sinusoidal rotation axis and various
wavelengths (figure 2) as well as other more exotic modes recognised by their complex radial
structure and/or by their periodic behaviour can be selected by changing the dimensionless ratio
a46 a52 only, providing the Reynolds number is large enough.
The growth rate of the stationary mode can be determined experimentally: from sequences
of images, we measure the maximum amplitude of the sinusoidally deformed rotation axis; its
temporal evolution is then fitted with an exponential growth, which can be compared to the
exponential growth rate determined by the local theory (see figure 1). First, one can notice that
the threshold for instability agrees with the theory, with for instance the sharp disappearance of
resonant modes at a46 a52a40 a19 a2 a50 a23a26a27a28a21 a50 a54 a50 a23a50a34a50 a35 for a17a46a48a47 a19 a50 a23a26a27 a50 a27 a54 a50 a23a50a34a50 a27 Hz. Besides, measurements
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
0
0.1
0.2
0.3
0.4
0.5
global rotation rate WG
g
r
o
w
t
h
r
a
te
s/e
-0.6 -0.4 -0.2 0 0.2 0.4
n=4
n=3
n=5
n=6
n=7
n=12n=15
(b)(a)
-0.6 -0.4 0 0.2 0.4 0.6 0.8 10
0.1
0.2
0.3
0.4
0.5
0.6
0.7
global rotation rate WG
g
r
o
w
t
h
r
a
te
s/e
-0.2
n=3
n=2
n=14
n=11
n=8
n=6
n=5
n=4
other instable modes
with n=20-25
Figure 1: Viscous growth rate of the elliptical instability determined by the local analysis as a function of
the global rotation rate a0a2a1 for a given cylinder of radius
a3
a4a6a5a8a7a10a9a12a11a14a13 cm, height
a3
a15a16a5a8a7a18a17a19a9a21a20 cm, eccentricity
a22
a5a24a23a18a9a25a23a27a26a28a13 , filled with water (
a29
a5a30a17a31a23a33a32a35a34a31a36a38a37a31a39a27a32a41a40 ): (a)
a3
a0a43a42
a5a44a23a18a9a45a13a19a23a27a13a47a46a48a23a18a9a25a23a28a23a27a13 Hz (Re a5a44a7a10a9a21a20a27a23a50a49a51a17a31a23a28a52 ) and
(b)
a3
a0 a42
a5a53a23a18a9a45a7a28a13a28a13a54a46a55a23a18a9a25a23a28a23a27a7 Hz (Re a5a56a17a19a9a45a7a18a17a57a49a58a17a31a23 a52 ). Triangles stand for experimental measurements and
solid lines for theoretical predictions. The predicted number n of axial half-wavelengths increases by a17
from the right to the left on each resonant band, starting from a59 a5a60a7 in (a) and a59 a5a60a61 in (b); measured
values are indicated above each experimental point. Note that in (a), additional resonances were observed
for a0 a1 in the range a62a64a63 a23a18a9a45a13a19a23a65a11a67a66 a63 a23a18a9a21a20a27a23a27a61a69a68 ; nevertheless, because of their small wavelength and their rapid
growth rate, quantitative measurements were not accurate.
WG= 0 Hz
n=4
~
WG=+0.499 Hz
n=2
~
WG=+0.145 Hz
n=3
~
WG=-0.069 Hz
n=5
~
WG=-0.134 Hz
n=6
~
WG=-0.163 Hz
n=9
~
WG=-0.186 Hz
n=11
~
WG=-0.203 Hz
n=14
~
WG=-0.231 Hz
n=20
~
WG=-0.257 Hz
n=25
~
Figure 2: Variation of the wavelength of the elliptical instability versus the global rotation
a3
a0 a1 for a
given cylinder of radius
a3
a4a6a5a8a7a10a9a12a11a14a13 cm and height
a3
a15a16a5a8a7a18a17a19a9a21a20 cm with an eccentricity
a22
a5a55a23a18a9a25a23a27a26a28a13 rotating at
a3
a0 a42
a5a8a23a18a9a45a13a19a23a27a13a70a46a71a23a18a9a25a23a28a23a27a13 Hz (Re a5a8a7a10a9a21a20a27a23a72a49a73a17a31a23 a52 ). In these pictures, the rotation axis is horizontal.
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
of the growth rate qualitatively agree with the theory, regarding the general increasing trend
when a46 a52 decreases toward a2 a32 a0 a21 , and also regarding the specific shape of one resonance band
(see for instance in figure 1a the band around a46 a52 a19 a50 a23a26a21 a39 a27 that we have explored in detail).
Quantitatively, orders of magnitude also agree, but theoretical values always overestimate ex-
perimental values. Three main explanations can be provided. First, non-linear effects were not
taken into account in the theory, but are expected to be stabilising (Eloy et al., 2003). Then,
it is worth recalling that the theoretical estimate is based on a short?wavelength (i.e. large k)
asymptotic analysis: the discrepancy could therefore be associated with finite k effects. The last
source of discrepancy is experimental, since in our set-up, rollers only deform the central part
of the cylinder.
3 Theoretical and experimental study in the sphere
The eigenmodes of the sphere have been studied in the non-rotating case by Greenspan (1968).
His study can be modified to take into account an additional Coriolis force, similarly to what
has been done for the cylindrical case. The global rotation leads to exactly the same changes as
in the cylinder. Hence, in contrast with the non-rotating case where the only exact resonance in
the sphere leads to the spin?over mode (i.e. a solid body rotation around the axis of maximum
strain, see Lacaze et al., 2004), the analytical study suggests that more complex instabilities can
be triggered by the global rotation.
A series of experiments was performed in the sphere of radius a17a18 a19 a21a24a23 a32 a25a34a27 cm with a fixed
eccentricity a3 a19 a50 a23a21 a50 , systematically changing a17a46a53a52 and a17a46a48a47 to excite various resonances. In
the explored range a2 a50 a23a55a1a0 a46 a52 a0 a50 , we observed the same behaviour as in the cylinder:
when a46a53a52 decreases towards a2 a32 a0 a21 , the number of axial structures as well as the growth rate
of the instability rapidly increase (see figure 3), until the instability suddenly disappears in the
vicinity of a46 a52a3a2 a2 a32 a0 a21 . Excited modes are in good agreement with analytical predictions for
a46a53a52 ranging in a resonance band of
a54a56a50
a23
a50a28a49 typically around the theoretical perfect?resonance
value. With our experimental device, the visualisation in the sphere was not precise enough to
allow a systematic measurement of the growth rate of the elliptical instability, but we determined
experimentally the viscous threshold of instability for two given values of the flow rotation
rate: a46 a52a40 a19 a2 a50 a23a27a34a27a34a25 a54 a50 a23a50a34a50 a35 for a17a46a48a47 a19 a50 a23a26a27 a50 a32 a54 a50 a23a50a34a50 a27 Hz and a46 a52a40 a19 a2 a50 a23a27a34a27a24a32 a54 a50 a23a50a34a50 a35 for
a17
a46 a47 a19
a50
a23a25 a35 a25
a54 a50
a23
a50a34a50
a27 Hz. We recall that in the absence of global rotation, the only perfect
resonance and the only observed mode in the vicinity of threshold (i.e. at low Reynolds number)
is the spin-over, corresponding to a single additional rotation around the axis of maximum strain
(see Lacaze et al., 2004).
4 Conclusion
In this paper, we have presented the analytical and experimental study of the influence of Cori-
olis force on the elliptical instability. For a given container - either cylindrical with a fixed
aspect ratio a17
a29 a0
a17
a18 or spherical -, the global rotation rate allows to select various resonances, in
good agreement with the global theory. In particular, we have observed in the sphere numer-
ous complex stationary modes at relatively low values of the Reynolds number, in addition to
the simple spin-over that takes place in the non?rotating case. For both the cylinder and the
sphere, when decreasing progressively the global rotation rate, we have observed that various
bands of resonance coexist for a46 a52a5a4 a46a53a52a40 a2 a2 a32 a0 a21 , first separated by large regions of stabil-
ity (especially for cyclones), then progressively overlapping (especially for anticyclones). All
resonances sharply disappear once the global rotation rate reaches a critical value a46 a52a40 a2 a2 a32 a0 a21 .
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
WG=0 n=1~ WG=-0.170 Hz n=2~ WG=-0.196 Hz n=3~
WG=-0.264 Hz n=6~WG=-0.242 Hz n=5~WG=-0.226 Hz n=4~
WG=-0.279 Hz n=7~
Figure 3: Pictures of the flow structure associated with the elliptical instability for different global
rotation rates a0a1 a52 in the deformed sphere with an eccentricity a2a4a3a6a5a8a7a10a9a11a5 and a fixed fluid rotation
a0
a1 a47
a3a12a5a8a7a10a13a11a5a14a5a16a15a4a5a8a7a17a5a14a5a18a13 Hz (Re a3a20a19a11a7a22a21a24a23a26a25a27a19a28a5a18a29 ). The measured number n of axial half-wavelengths is
also indicated. In these pictures, the rotation axis is horizontal.
Focusing on the stationary modes, we have shown that the instability wavenumber as well as its
growth rate significantly increase and reach a maximum just before a46 a52a40 . In the cylindrical ge-
ometry, all these results agree quantitatively with the theoretical estimations taking into account
the viscous corrections. Our conclusions in the cylinder and in the sphere also agree qualita-
tively with the general trend observed by Afanasyev (2002) in vortex pairs and by Stegner et al.
(2005) in Karman vortex streets, even if our experimental set-up is totally different (i.e. their
vortices are not confined and are subjected to rather large elliptical deformations). Indeed, both
studies report the systematic destruction of elliptical anticyclones by a sinusoidal mode with a
decreasing wavelength when a46 a52 decreases up to a certain critical value, corresponding to the
overlapping resonances mentioned here. We thus argue that this behaviour is universal, except
for the explicit value of a46 a52a40 that will depend both on the considered vortical structure and on the
value of the eccentricity (see also Sipp et al., 1999; Le Diz?s, 2000).
Conclusions in the spherical geometry are especially interesting in the geophysical and as-
trophysical contexts. For instance, complex motions can be expected in the Earth?s core in
addition to the simple spin-over excited by both precession and elliptical instability. More
generally, one can imagine that binary stars and moon?planet systems where the elliptical in-
stability is expected to take place, encounter various bands of instability during their evolution:
depending on the relative changes in their rotation and revolution rates, different and complex
histories regarding energy dissipation and flow motions can thus be expected. Clearly, the role
of the elliptical instability in natural flows, as suggested for instance by Kerswell & Malkus
(1998), still demands more works, in order to fully understand the implications of all natural
complexities on the standard and well-known hydrodynamical model (see also Lacaze et al.,
2006; Le Bars & Le Diz?s, 2006).
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18 ?me Congr?s Fran?ais de M?canique Grenoble, 27-31 ao?t 2007
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