Large Scale Dynamo
Action in MRI Disks
Role of stratification
Dynamo cycles
Mean-field interpretation
Incoherent alpha-shear dynamo
Axel Brandenburg (Nordita, Stockholm)
Unstratified MRI turbulence
5123
w/o hypervisc.
Dt = 60 = 2 orbits
No large scale field
(i) Too short?
(ii) No stratification?
2
Vorticity and Density
See poster by Tobi Heinemann on density wave excitation!
3
Small scales dominate
4
Animated spectra
Red
EM
Black
EK
5
<By >
Old stratified runs
Brandenburg et al. (1995)
6
Different boundary conditions
Bx  B y  Bz , z  0
Bx , z  B y , z  Bz  0
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Tall disk with
potential field
b.c.
2p x 2p x 8p
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z-t diagram
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Longer and bigger
Parity not always well defined
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Mean-field interpretation
A
 U  B  ε  J
t
 effect and turbulent
aagnetic diffusivity
turbulent emf

j
ε  ub
 ij B j ij* J j
• Correlation method
– MRI accretion discs (Brandenburg & Sokoloff 2002)
– Galactic turbulence (Kowal et al. 2005, 2006)
• Test field method
– Stationary geodynamo (Schrinner et al. 2005, 2007)
– Shear flow turbulence (Brandenburg 2005)
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Calculate full ij and ij tensors
A
 U  B  J
t
Original equation (uncurled)
A
 U  B  u  b  J
t
Mean-field equation
a
 U  b  u  B  u  b  u  b  j
t
fluctuations
Response to arbitrary mean fields
a pq
 U  b pq  u  B pq  u  b pq  u  b pq  j pq
t
12
Test fields
 cos kz 


11
B  0 ,
 0 


 sin kz 


21
B  0 
 0 


 0 


12
B   cos kz  ,
 0 


 0 


  sin kz 
 0 


B 22

pq
j
 ij B
pq
j
ijk B
pq
j ,k
Example:


11
1
 11 cos kz 113k sin kz
21
1
 11 sin kz  113k cos kz
 11   cos kz sin kz  111 

  
 21 
113k    sin kz cos kz  1 
*
11
12*  123 113 
 *


  *   


213 
 223
22 
 21
13
Validation: Roberts flow
  cos k x x sin k y y 


U  u rms   sin k x x cos k y y 
 2 cos k x cos k y 
x
y 

a pq
 U  b pq  u  B pq  u  b pq  u  b pq  j pq
t
SOCA
k x  k y  kf / 2
SOCA result
   13 Rmurms
 t  13 Rmurms kf-1
normalize
 0   13 urms
 t0  13 urms kf-1
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(i) Turbulence: kinematic 
and t independent of Rm
 0   13 urms
0  13 urms kf1
Galloway-Proctor:
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Full alpha tensor for MRI
yy negative, as before (Brandenburg et al. 1995)
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Full eta tensor
xx and xx the same and positive (new)
yx always positive (new)
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(ii) Shear turbulence
ε  δ  J  μJ  t J
0
μ  



0
12*    
*
 21
  
Growth rate
*
 S
21
12* 


 1 

2
2
 t k1
 t   t k1  t 

Use S<0, so need negative *21 for dynamo
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Case with just shear
Similar to G. Lesur’s plot of yesterday!
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Dependence on Sh and Rm
Again, yx always positive
20
Gaussian fluctuations
21
Fluctuations of ij and ij
Incoherent  effect
(Vishniac & Brandenburg 1997,
Sokoloff 1997, Silantev 2000,
Proctor 2007)
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Onset and saturation of
incoherent alpha-shear
dynamos
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Conclusions
• Cycles w/ stratification
• Test field method robust
– Even when small scale
dynamo
–   0 and   t0
• Rotation and shear: *ij
– WxJ not (yet?) excited
– Incoherent  works
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