MRI Driven turbulence
and dynamo action
Fausto Cattaneo
University of Chicago
Argonne National Laboratory
cattaneo@flash.uchicago.edu
ITP 2008
MRI in cylindrical annulus
•
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
inner cylinder
•
•
velocity
Simulations by Obabko, Fischer, FC
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
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
Pm
1.0
Turbulence
Re crit  Pm 
Re
ITP 2008
Small Pm regime
Assume Re>>1.
• Dynamo action driven by strongly fluctuating velocity.
• Introduce roughness exponent 
u 2 ()  u(x  l)  u(x) el
2
  2
•
•
 <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
0

 Re
1
1
0

 Rm
1
1
1
  

Pm  
 
 
ITP 2008
Structure functions
•
•
•
Shearing box simulation
Isothermal EOS
Finite vertical flux
•
•
Simulations using PLUTO
No explicit dissipation
x  0.002
ny  512
inertial
u 2 ()
5x
50x

Simulations by Bodo, Mignone, FC
ITP 2008
Kazantsev model
Resolution parameter
From Boldyrev & FC
dynamo
No dynamo
1 
increasing roughness
ITP 2008
What do we need?
• 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
x  0.002
ny  512

u ()
2
5x
50x

ITP 2008
Amplitude effects
Effective transport ( - BBr ) depends on:
• Correlations
• Amplitude of fluctuating quantities (mostly B)
For a flux rope in equilibrium with an axi-symmetric stagnation point flow,
peak field satisfies (Galloway, Proctor & Weiss)
2
u
B*2 
( /  )
ln( Rm )
Valid for Pm >>1
ITP 2008
What happens if Pm < 1?
B-field (vertical)
vorticity (vertical)
Magneto-convection
Pm = 8.0
cold
g
Simulations by Emonet & FC
hot
Pm = 0.125
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
ITP 2008