ITP 2008

cattaneo@flash.uchicago.edu

MRI Driven turbulenceand dynamo action

Fausto Cattaneo

University of ChicagoArgonne National Laboratory

ITP 2008

MRI in cylindrical annulus

velocity

inn

er c

ylin

der

Simulations by Obabko, Fischer, FC

• Incompressible fluid

• Finite viscosity and magnetic diffusivity

• Periodic in the vertical

• Pm = 0.5

• Not a dynamo at Re=6,000

• Probably a dynamo at Re=60,000

ITP 2008

What happens if Pm 0?

Two issues:

• Dynamo action becomes impossible. Turbulence decays. No enhanced transport

• Dynamo action remain possible but amplitude of fluctuations decreases with decreasing Pm. Asymptotically enhanced transport becomes ineffective

ITP 2008

Dynamo action at small Pm

mP

Re

Turbulence

mcrit P Re

1.0

Two possibilities:

• There exist a critical value of Pm below which dynamo action is impossible for any Re

• Dynamo action is always possible:

– Asymptotically Rmcrit independent of Pm (σ=1)

– Asymptotically Rmcrit increases with decreasing Pm

ITP 2008

Small Pm regime

1

1

01

1

0 Rm Re

)()()( 222 lexulxuu

Assume Re>>1. • Dynamo action driven by strongly fluctuating velocity. • Introduce roughness exponent

<1 rough velocity ( =1/3 corresponds to K41)

• Related to energy spectrum: E(k)k-p p=1+2

• At small scales ( O() ) action of viscosity is to smooth out velocity

1

Pm

ITP 2008

Structure functions

x5

inertial

x50

002.0x

)(2 u

512ny

Simulations by Bodo, Mignone, FC

• Shearing box simulation

• Isothermal EOS

• Finite vertical flux

• Simulations using PLUTO

• No explicit dissipation

ITP 2008

Kazantsev model

increasing roughness

1

Res

olu

tion

par

amet

er

dynamo

No dynamo

From Boldyrev & FC

ITP 2008

What do we need?

x5 x50

002.0x

)(2 u

512ny

• Asymptotic regime is reached when η 50 x

• If =1/3 dynamo action requires 30 η 1500 x ( 2 for periodic systems)

• Requirement to reach asymptotic regime with =1/3 ν /η 0.1 Pm 0.14/3 0.046

ITP 2008

Amplitude effects

Effective transport ( - BBr ) depends on:

• Correlations

• Amplitude of fluctuating quantities (mostly B)

)/()ln(

22*

Rm

uB

For a flux rope in equilibrium with an axi-symmetric stagnation point flow,

peak field satisfies (Galloway, Proctor & Weiss)

Valid for Pm >>1

ITP 2008

B-field (vertical) vorticity (vertical)

Pm = 8.0

Pm = 0.125

What happens if Pm < 1?S

imu

lati

ons

by

Em

onet

& F

C

Magneto-convection

g

hot

cold

ITP 2008

From magneto-convection

ITP 2008

Conclusion

• Present simulations not suitable to inform us on small Pm regime

• At present there is no result, theoretical or numerical indicating that dynamo action becomes impossible at small Pm

• Results for Pm > 1 not representative of Pm <<1

• Informative simulations are possible, but very demanding