Turbulent Dynamo

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Turbulent Dynamo
Stanislav Boldyrev (Wisconsin-Madison) Fausto
Cattaneo (Chicago)
Center for Magnetic Self-Organization in
Laboratory and Astrophysical Plasmas
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Questions
1. Can turbulence amplify a weak magnetic field?

• What are the scales and growth rates of
• magnetic fields?

Kinematic Theory
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MHD Equations
?tv (vr)v -rp (rB)B ??v f
Kinematic dynamo
?tB r(vB) ??B
ReVL/?
- Reynolds number
RmVL/?
- magnetic Reynolds number
Pm?/?
- magnetic Prandtl number
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Kinematic Turbulent Dynamo Phenomenology
V(x, t) is given.
?tB r(vB) ??B
Consider turbulent velocity field V(x,t) with the
spectrum
?V? / ?1/3
? 1/K
?? ?/?V?/ ?-2/3
smaller eddies rotate faster
Magnetic field is most efficiently amplified by
the smallest eddies in which it is frozen. The
size of such eddies is defined by resistivity.
B
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Kinematic Turbulent Dynamo Phenomenology
Role of resistivity ?
Small Prandtl number,
Large Prandtl number,
PM ?/? 1
PMÀ 1
Dynamo growth rate
Dynamo growth rate
? 1/??
? 1/??
smooth velocity ? V?/ ?
rough velocity ? V? / ?? ?1/3
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Phenomenology Large Prandtl Number Dynamo
Magnetic lines are folded
Cattaneo (1996)
Schekochihin et al (2004)
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Phenomenology Small Prandtl Number Dynamo
Small Prandtl number PM?/? 1 Stars,
planets, accretion disks, liquid metal
experiments
?V? ?1/3
?? ?/?V? / ?2/3
?
?? / (RM)-3/4
Dynamo growth rate
? 1/???/ (RM)1/2
RM
Numerics Haugen et al (2004)
S.B. F. Cattaneo (2004)
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Kinematic Turbulent Dynamo Theory
V(x, t) is a given turbulent field
?tB r(vB) ??B
Two Major Questions
1. What is the dynamo threshold, i.e., the
critical magnetic Reynolds number RM, crit ?
2. What is the spatial structure of the growing
magnetic field (characteristic scale,
spectrum)?
These questions cannot be answered
from dimensional estimates!
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Kinematic Turbulent Dynamo Theory
Dynamo is a net effect of magnetic line
stretching and resistive reconnection.
RM gt RM, crit stretching wins, dynamo
RM lt RM, crit reconnection wins, no dynamo
RMRM, crit stretching balances reconnection
??
B
When RM exceeds RM, crit only slightly, it takes
many turnover times to amplify the field
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Kinematic Turbulent Dynamo Kazantsev Model
homogeneity and isotropy
incompressibility
Dynamo
No Dynamo
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Kazantsev Model Large Prandtl Number
If we know ?(r, t), we know growth rate and
spectrum of magnetic filed
Large Prandtl number PM?/? À 1

• Kazantsev model predicts
• 1. Dynamo is possible
• 2. EM(K)/ K3/2

Numerics agree with both results
Schekochihin et al (2004)
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Kazantsev Model Small Prandtl Number
PM?/? 1
Is turbulent dynamo possible?
Batchelor (1950) analogy of magnetic field and
vorticity. No
Kraichnan Nagarajan (1967) analogy with
vorticity does not work.
?
Vainshtein Kichatinov (1986)
Yes
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Small Prandtl Number Dynamo Is Possible
PM?/? 1
PM?/? À 1
Keep RM constant. Add small-scale eddies
(increase Re).
Kazantsev model dynamo action is always
possible, but for rough velocity (PM1) the
critical magnetic Reynolds number (RMLV/?) is
very large.
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Kazantsev Model Small Prandtl Number
L/??
Theory S. B. F. Cattaneo (2004)
smooth velocity
Kolmogorov (rough) velocity
PMÀ 1
PM 1
Simulations P. Mininni et al (2004) A. Iskakov et
al (2007)
RM, crit
May be crucial for laboratory dynamo, PM 1
Re
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Kinematic Dynamo with Helicity
It is natural to expect that turbulence can
amplify magnetic
field at K K0
Can turbulence amplify
magnetic field at K K0 ?
large-scale dynamo
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Dynamo with Helicity Kazantsev Model
hs v (r v)d3 x ? 0
given
energy
helicity
magnetic energy
magnetic helicity
need to find
Equations for M(r, t) and F(r, t) were derived by
Vainshtein and Kichatinov (1986)
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Dynamo with Helicity Kazantsev Model
Two equations for magnetic energy and magnetic
helicity can be written in the
quantum-mechanical spinor form
hs v (r v)d3 x ? 0
hs v (r v)d3 x 0
S. B. , F. Cattaneo R. Rosner (2004)
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Kazantsev model and ?-model
? - model
assumes scale separation
r
Kazantsev model, NO scale separation
The ? - model approaches the exact solution only
at r? 1
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Conclusions

• Main aspects of kinematic turbulent dynamo is
  relatively
• well understood both phenomenologically and
  analytically.

2. Dynamo always exists, but
3. Separation of small- and large-scale may be
not a correct procedure.