On-off intermittency in spherical Couette flow

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On-off intermittency in spherical Couette flow

Raphaël Raynaud, Stephan Fauve, Emmanuel Dormy

To cite this version:

Raphaël Raynaud, Stephan Fauve, Emmanuel Dormy. On-off intermittency in spherical Couette flow. European GDR Dynamo, Oct 2012, Nice, France. �hal-02428471�

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On-off intermittency in spherical Couette flow

Raphaël Raynaud

1 , Stéphan Fauve

2 , Emmanuel Dormy

1

raphael.raynaud@ens.fr

1_LRA, 2_{LPS, Département de Physique, École normale supérieure}

A

BSTRACT

∗_{figure from DPL Nonlinear Dynamics Lab}

We perform 3D numerical simulations of dynamos in a Couette flow generated by two contra-rotating spheres. We show that the magnetic field may display on-off inter-mittency at relatively low magnetic Prandtl number, close to the onset of dynamo

ac-tion. The basic signature of this

phe-nomenon lies in series of short bursts of the magnetic energy (“on” phases) separated by low energy phases (“off” phases). The length of the “off” phases increases as the system gets closer to the threshold. We successfully compare our observations to predictions based on a canonical model and find it provides an accurate estimate of the threshold.

O

N

-

OFF INTERMITTENCY

:

A SIMPLE MODEL

˙

X = [µ + ζ(t)] X

− X

3

with

µ

distance to the threshold,

µ

→ 0

ζ

Gaussian white noise,

(

hζ(t)i = 0

hζ(t)ζ(t

0

)

i = 2Dδ(t − t

0

)

Properties











“off” phases

=

⇒

ln(E

_b

)

follows a random walk

PDF

=

⇒

P (E

_b

)

∼ E

_b(2µ/D)−1 moments

=

⇒

hE

_bn

i ∝ µ , ∀n

E

QUATIONS OF MAGNETOHYDRODYNAMICS

•

Navier-Stokes equation

∂ u

∂t

+ (u

· ∇) u +

2 ERe

(e

z

× u) = −

1 Re

∇Π +

1 Re

∇

2

_u+

1 Re

1 E

+

Re

χ

(

∇ × B) × B

•

Induction equation

∂ B

∂t

=

∇ × (u × B) +

1 Rm

∇

2

_B

•

Boundary conditions

– no-slip conditions for the velocity field

– insulating or ferromagnetic (

B

× n = 0

) outer sphere

– insulating, conducting, or ferromagnetic inner sphere

R

ESULTS

Time series of the magnetic energy in linear & log scale

1 2 3 4 5 6 7 8 time 1e3 0.0 0.5 1.0 1.5 2.0 2.5 energy

1e 4Re=1460; Pm=0.2; Ek=1e-03; theta=180; BC:condIC

Eb 1 2 3 4 5 6 7 8 time 1e3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 energy

Eb 0.0 0.2 0.4 0.6 0.8 1.0 1.2 time 1e4 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 energy

Eb 0 1 2 3 4 5 6 7 8 time 1e3 0.0 0.5 1.0 1.5 2.0 2.5 3.0 3.5 4.0 energy

Eb 1 2 3 4 5 6 7 8 time 1e3 10-16 10-15 10-14 10-13 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 energy

Re=1460; Pm=0.2; Ek=1e-03; theta=180; BC:condIC Eb 1 2 3 4 5 6 7 8 time 1e3 10-9 10-8 10-7 10-6 10-5 10-4 10-3 energy

Re=1470; Pm=0.2; Ek=1e-03; theta=180; BC:condIC Eb 0.0 0.2 0.4 0.6 0.8 1.0 1.2 time 1e4 10-12 10-11 10-10 10-9 10-8 10-7 10-6 10-5 10-4 10-3 energy

Re=1475; Pm=0.2; Ek=1e-03; theta=180; BC:condIC

Eb 0 1 2 3 4 5 6 7 8 time 1e3 10-8 10-7 10-6 10-5 10-4 10-3 energy

Re=1500; Pm=0.2; Ek=1e-03; theta=180; BC:condIC Eb

S

TATISTICS OF THE MAGNETIC ENERGY

PDF (with ferromagnetic boundary conditions)

12 10 8 6 4 log Eb 100 101 log p.d.f. Eb Rm=300.0 Rm=305.0 Rm=310.0 Rm=320.0 Rm=600.0 2.98 3.00 3.02 3.04 3.06 3.08 3.10 3.12 Rm 1e2 1.2 1.0 0.8 0.6 0.4 0.2 0.0 slope

Linear scaling of the moments

2.98 3.00 3.02 3.04 3.06 3.08 3.10 3.12 Rm 1e2 0.5 0.0 0.5 1.0 1.5 2.0 2.5 Eb moments 1e 4 2e moment mean

C

ONCLUSIONS

On-off intermittency has so far never been observed experimentally in dynamo experiments. This phenomenon has already been reported in a few numerical simulations using an analytic ABC forcing and periodic boundary conditions (Sweet 2001, Alexakis and Ponty 2008). Here we show that on-off intermittency can also be observed with more realistic forcing and boundary conditions.

C

HANGING THE BOUNDARY CONDITIONS

With insulating boundary conditions

5.25 5.30 5.35 5.40 5.45 5.50 5.55 5.60 5.65 Rm ×102 −0.5 0.0 0.5 1.0 1.5 2.0 2.5 3.0 Eb moments

×10−4 insulating inner core

2e moment mean 5.25 5.30 5.35 5.40 5.45 5.50 5.55 5.60 5.65 Rm ×102 −1.2 −1.0 −0.8 −0.6 −0.4 −0.2 0.0 0.2 p.d.f . slope

insulating inner core

With a conducting inner core and an insulating outer sphere

2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 Rm ×102 −0.5 0.0 0.5 1.0 1.5 2.0 Eb moments

×10−4 conducting inner core

2e moment mean 2.90 2.91 2.92 2.93 2.94 2.95 2.96 2.97 2.98 2.99 Rm ×102 −1.1 −1.0 −0.9 −0.8 −0.7 −0.6 −0.5 p.d.f . slope

On-off intermittency in spherical Couette flow