itp 2008 [email protected] mri driven turbulence and dynamo action fausto cattaneo...
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ITP 2008
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