The combined effects of Coriolis force and buoyancy effects on the dynamics of a weakly elliptical bounded vortex are treated theoretically as well as experimentally. As predicted theoretically, stratification and rotation have antagonist contributions to the stability of an elliptical vortex. Thus if the stratification is strong enough (Nb>Omega_c, Nb and Omega_c being respectively the Brunt-Väisälä frequency and the rotation rate of the flow in a frame rotating with the elliptical deformation at angular velocity Omega_t), we have observed that only anticyclones (such that |Wa|<Omega_c with Wa=2(Omega_c+Omega_t)) are unstable, whereas the cyclones are always stable. In addition if the stratification is weak, instability areas over change. These instability thresholds found theoretically have been observed experimentally with a good accuracy and the measured growth rate are in a good agreement with those predicted by a linear stability analysis in the limit of small deformation

LE BARS M.

LE BLANC S.

LE DIZÈS, Stéphane

LE GAL P.

LE GUIMBARD, David

Colloque avec actes et comité de lecture. Internationale.International audienceThe combined effects of Coriolis force and buoyancy effects on the dynamics of a weakly elliptical bounded vortex are treated theoretically as well as experimentally. As predicted theoretically, stratification and rotation have antagonist contributions to the stability of an elliptical vortex. Thus if the stratification is strong enough (Nb>Omega_c, Nb and Omega_c being respectively the Brunt-Väisälä frequency and the rotation rate of the flow in a frame rotating with the elliptical deformation at angular velocity Omega_t), we have observed that only anticyclones (such that |Wa| Ωc, Nb et Ωc étant respectivement la fréquence de Brunt-Väisälä et la vitesse de rotation de l’écoulement dans un repère tournant avec la déformation elliptique à la vitesse angulaire Ωt), nous observons que seuls les anticyclones (tels que |Wa| < Ωc avec Wa = 2(Ωc + Ωt)) sont instables. Les cyclones étant toujours stables. Par ailleurs si la stratification est faible (Nb < Ωc), les zones d’instabilité s’inversent. Ces différents seuils d’instabilité prévus théoriquement ont été observés expérimentalement avec une grande précision et les taux de croissance mesurés sont en bon accord avec ceux calculés à l’aide d’une étude de stabilité linéaire effectuée dans la limite d’une faible déformation elliptique

Le Guimbard, David

Le Bars, M.

Le Blanc, S.

Le Gal, P.

Le Dizès, Stéphane

HAL-INSU

Experimental and theoretical study of the elliptic instability in a rotating stratified flow

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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
Experimental and theoretical study of the elliptic instability in a rotating
stratified flow
D. Le Guimbard1, M. Le Bars2, S. Le Blanc1, P. Le Gal2 & S. Le Dizès2
1. LSEET
Avenue de l’Université, BP 20132, 83957 La Garde Cedex - FRANCE
david.guimbard@lseet.univ-tln.fr, sl@univ-tln.fr
2. IRPHE
Technopôle de Château-Gombert 49, rue Joliot Curie, BP 146, 13384 Marseille Cedex - FRANCE
lebars@irphe.univ-mrs.fr, legal@irphe.univ-mrs.fr, ledizes@irphe.univ-mrs.fr
Abstract :
The combined effects of Coriolis force and buoyancy effects on the dynamics of a weakly elliptical bounded vortex
are treated theoretically as well as experimentally. As predicted theoretically, stratification and rotation have
antagonist contributions to the stability of an elliptical vortex. Thus if the stratification is strong enough (Nb > Ωc,
Nb and Ωc being respectively the Brunt-Väisälä frequency and the rotation rate of the flow in a frame rotating with
the elliptical deformation at angular velocity Ωt), we have observed that only anticyclones (such that |Wa| < Ωc
with Wa = 2(Ωc + Ωt)) are unstable, whereas the cyclones are always stable. In addition if the stratification
is weak (Nb < Ωc), instability areas over change. These instability thresholds found theoretically have been
observed experimentally with a good accuracy and the measured growth rate are in a good agreement with those
predicted by a linear stability analysis in the limit of small deformation.
Résumé :
Les effets combinés de la force de Coriolis et des effets de flottabilité sur la dynamique d’un tourbillon ellip-
tique confiné sont étudiés expérimentalement et théoriquement. Comme prévu théoriquement, la stratification et
la rotation ont une contribution antagoniste sur la stabilité d’un tourbillon elliptique. Ainsi si la stratification est
suffisamment forte (Nb > Ωc,Nb et Ωc étant respectivement la fréquence de Brunt-Väisälä et la vitesse de rotation
de l’écoulement dans un repère tournant avec la déformation elliptique à la vitesse angulaire Ωt), nous observons
que seuls les anticyclones (tels que |Wa| < Ωc avecWa = 2(Ωc+Ωt)) sont instables. Les cyclones étant toujours
stables. Par ailleurs si la stratification est faible (Nb < Ωc), les zones d’instabilité s’inversent. Ces différents seuils
d’instabilité prévus théoriquement ont été observés expérimentalement avec une grande précision et les taux de
croissance mesurés sont en bon accord avec ceux calculés à l’aide d’une étude de stabilité linéaire effectuée dans
la limite d’une faible déformation elliptique.
Key-words :
Elliptic Instability; rotation; stratification
1 Introduction
The stability of a weakly elliptical confined vortex in a rotating stratified fluid has been per-
formed with local techniques such asWKBmethods by several authors such as Kerswell (2002)
or Miyazaki et al. (1992), but to the best of our knowledge this has not yet been performed with
a global theory which is the purpose of the present work. This paper is organized as follows. In
section 2, we focus on the theoretical study based on a global analysis in the general context.
In section 3, we show some new observed resonances and we compare them with the normal
mode theory. In the last section, we summarize the main results.
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
2 Theoretical study
In the present paper,we consider a confined vortex in solid body rotation. Variables are nondi-
mensionalized with its radius R as a characteristic length scale and its velocity Ω−1c as a char-
acteristic time scale. To perform our theoretical study of the linear stability of a confined weak
elliptical vortex in a rotating stratified flow, we assume our vortex has a stationary strain field in
the frame rotating at the global rotation rate Ωt. So that at leading order in ε the stream function
is:
Ψ = −r
2
2
(1− ε sin 2θ) (1)
Following previous works on the global linear theory of the elliptical instability, we can
show by taking wave solutions of the form (neutral modes):
(u, b, p) = (ur(r) cos kz, uθ(r) cos kz, uz(r) sin kz, b(r) sin kz, p(r) cos kz)ei(mθ−ωt) (2)
that the pressure is solution of the following Bessel equation of the first kind:
r
d
dr
(r
dp
dr
) + (α2r2 −m2)p = 0, (3)
where the axial wave number α is given by:
α = k
√
W 2a − λ2
λ2 − n2 (4)
withWa = 2(1 + Ω˜t), n = N˜b and λ = m− ω (where the tildes represent nondimensionalized
variables). Note that the vertical boundary conditions uz|z=0,H = 0 leads to the discretization of
the axial wave-number k = mpiR/H ,m being an integer corresponding to the number of axial
half-periods. By just looking at the expression of α and the formal expression of the neutral
modes which are just combinations of Bessel functions of the first kind, we can easily verify
that waves only exist if:
|Wa| < 1 and n > 1 or |Wa| > 1 and n < 1. (5)
Moreover the dispersion relation of this problem is given by the radial boundary condition
ur|r=1 = 0 and reads simply in the general case as:
(λ+Wa)Jm−1(α) = (λ−Wa)Jm+1(α). (6)
Waleffe (1990) and many other authors found that the mechanism of instability was a triadic
resonance between two neutral modes (m1, k1, ω1), (m2, k2, ω2) of the undeformed flow and the
underlying strain field. In the context of the study the conditions of resonance simply are:
m2 = m1 + 2, k1 = k2, ω2 = ω1. (7)
To clarify our notations the (−1, 1, i) resonance corresponds to the resonance between the two
modes m1 = −1 and m2 = 1 while the i refers to ith root of the dispersion relation (6) giving
the radial structure of those modes. As we will see in the experimental study we have only
observed the stationary mode (−1, 1, 1) resonance for large stratification and weak rotation
(n > 1 and |Wa| < 1), so that in this region, natural tendency for the simplest mode regarding
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
its structure to take place alone has been verified. So we focused our analysis on this resonance
and derived an explicit expression for the growth rate:
σ =
√
ε2σ2in − (Im(sv) + A∆k)2 −
(α2 + k2)B
Re
− Re(sv)√
Re
(8)
(see Figure 1), where σin is the inviscid growth rate, Re = ΩcR2/ν is the Reynolds number,
Im(sv) and Re(sv) represent the imaginary and real part of the viscous frequency shift due to
boundaries and calculated for m = −1, ∆k represents the detuning of the axial wave number
and every terms (also given in appendix) have explicit formulations depending on (α, n,Wa, k).
We can remark that sv represent the viscous damping due to the presence of boundaries. Note
finally that α is a function ofWa from (6) because ω = 0 for the (−1, 1, 1) resonance.
Figure 1: Contours of the viscous growth rate given by (8) of the elliptical instability mode (−1, 1, 1)
determined by global analysis as a function of the adimensionalized Brunt-Väisälä frequency n and the
absolute vorticity Wa for a given radius R = 2.75cm, height H = 19cm, Brunt-Väisälä frequency
Nb = 2.70rad/s and eccentricity ε = 0.085. Note that the Reynolds number increases as n decreases.
Moreover the inviscid growth rate is explicitly given by:
σin =
∣∣∣∣∣ (n2 − 1)(Wa + 1)2(α2 + (Wa − 1)2)4(Wa − 1)2(Wa + 1)(n2 +Wa) + 4(W 2a − n2)α2
∣∣∣∣∣ (9)
which generalizes previous ones. The WKB result given by Kerswell (2002) is recovered when
k tends to infinity, whereas for finite k Waleffe (1990) result in the absence of stratification
(n = 0) and global rotation (Wa = 2) is obtained.
3 Experimental study
In the experiments we used a rotating deformable and transparent cylinder (with angular fre-
quency Ωc) filled with a solution of linear stratified salted water (with Brunt-Väisälä frequency
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
Nb and experimentally obtained with the classical two buckets method) which is elliptically
deformed by two rollers. Moreover the all set-up (see Figure 2) is fixed on a rotating table of
angular frequency Ωt. The experimental control parameters are the Reynolds number based
on the angular frequency of the cylinder Ωc, the Brunt-Väisälä frequency Nb and the absolute
vorticity of the flowWa = 2(1 + Ω˜t).
Figure 2: Experimental set-up with the deformed cylinder placed on the rotating table.
We performed a series of experiments with a constant height H = 19cm and Brunt-Väisälä
frequency Nb around 2.70rad/s systematically changing Wa. We also changed the eccentricity
from 0.085 to 0.179. New observations were made. First we saw the (−1, 1, 1) resonance in
the high stratified region (n > 1 and |Wa| < 1). It is in a good agreement with theory which
predicts that this mode is always dominant except in a few narrow bands. The threshold in this
region has been well observed and for |Wa| ∼ 1 or n ∼ 1 the instability disappears. We also
observed the (−1, 1, 1) resonance in the region of weak stratification (n < 1 and |Wa| > 1) and
we also observed other resonance such as the (0, 2, 1) one whereas in the high stratified region
the only (−1, 1, 1) resonance has been observed. The different thresholds were observed too
in this region. Secondly the number of wave-length predicted by the theory in this region is in
a very good agreement with the observations presented in Figure 3. However, we never saw
instabilities in the region nearWa = 0 in contradiction with the theoretical predictions.
4 Conclusions
In this paper we presented briefly the global theory and shown some new analytical results about
the growth rate of the (−1, 1, i) resonance of the elliptical instability. These results recover
some results already found with other techniques such as the WKB method or finite axial wave-
number in the unstratified case. Moreover the shift in frequency of the growth rate due to
boundaries has been calculated in the general case of a rotating stratified fluid for a cylinder.
One can recover Kudlick (1966) or Kerswell et al. (1995) formulas for the special case n = 0.
Furthermore the theoretically predicted regions where the elliptical instability occurs and its
thresholds (n < 1 and |Wa| > 1 or n > 1 and |Wa| < 1) have been observed experimentally
and are in good agreement with the theory. We can notice that in the region of high stratification
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
(n > 1 and |Wa| < 1) the main resonance and the only resonance observed was the (−1, 1, 1)
one. This behavior is similar to the one noticed by Le Bars et al. (2007) for the destruction of
elliptical anticyclones in the non stratified case.
Figure 3: Variation of the wave-length of the (−1, 1, 1)mode for a given cylinder of radiusR = 2.75cm,
heightH = 19cm and eccentricity ε = 0.085 filled with stratified salty water of Brunt-Väisälä frequency
Nb = 2.68rad/s for a fixed absolute vorticityWa = 0.7. From the top to the bottom the value of n and
m which is the number of axial half wave length increase.
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18 ème Congrès Français de Mécanique Grenoble, 27-31 août 2007
5 Appendix. Mathematical expressions
A =
(1− n2)(α2 +W 2a − 1)
k(α2 + k2 + (Wa − 1)(Wa + n2))
B = 1− α
2n2(W 2a + α
2 − 1)
2(α2 + k2 + (Wa − 1)(Wa + n2))
The viscous correction due to the presence of boundaries has two contributions. We write
sv = Ir + Iz where Ir is the contribution of the surface r = 1 of the cylinder and Iz is the
contribution of the surfaces z = 0 and z = H . The two expression of Ir and Iz has been
calculated using a classical approach based on the hypothesis that the effect of viscosity is
taken in account in boundaries layers of magnitude Re−
1
2 . So we obtained:
I1r = C
[
(1− i) + (1 + isgn(n2 − 1)) k
2
|n2 − 1| 32
]
I1z = C
[
(α2 + (Wa − 1)2)(1 + isgn(Wa − 1))
|Wa − 1| 32
+
(α2 + (Wa + 1)(Wa − 3))(1− isgn(1 +Wa))
|Wa + 1| 32
]
R
H
with
C =
(W 2a − 1)(1− n2)√
2(α2 + k2 + (Wa − 1)(Wa − n2))
.
References
Kerswell, R.R. 2002 Elliptical Instability. Ann. Rev. Fluid Mech. 34 83-113
Kerswell, R.R., Barenghi, C.F. 1995 On the viscous decay rates of inertial waves in a rotating
circular cylinder. J. Fluid Mech. 285 203-214
Kudlick, M. 1966 On the transient motions in a contained rotating fluid. PhD thesis, MIT.
Miyazaki, T., Fukumoto, Y. 1992 Three-dimensional instability of strained vortices in a stably
stratified fluid. Phys. Fluids A 4 2515-2522
Waleffe, F. A. 1990 On the three-dimensional instability of strained vortices. Phys. Fluids 2
76-80
Le Bars, M., Le Dizès, S., Le Gal, P. 2007 Coriolis effects on the elliptical instability in cylindri-
cal and spherical rotating containers. Under consideration for publication in J. Fluid Mech.
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Experimental and theoretical study of the elliptic instability in a rotating stratified flow

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