Turbulent Dynamo
Stanislav Boldyrev (Wisconsin-Madison)
Fausto Cattaneo (Chicago)
Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas
Questions
1. Can turbulence amplify a weak magnetic field?
2. What are the scales and growth rates of
magnetic fields?
Kinematic Theory
MHD Equations
tv + (v¢r)v = -rp + (r£B)£B + v + f
Kinematic dynamo
tB = r£(v£B) + B
Re=VL/
Rm=VL/
- Reynolds number
Pm=/
- magnetic Prandtl number
- magnetic Reynolds number
B
V1
B
V2
Kinematic Turbulent Dynamo: Phenomenology
V(x, t) is given.
tB = r£(v£B) + B
Consider turbulent velocity field V(x,t) with the spectrum:
EK
» 1/K
K-5/3
V / 1/3
t» /V/ -2/3
smaller eddies rotate faster
K0
B
K
K
Magnetic field is most efficiently
amplified by the smallest eddies
in which it is frozen.
The size of such eddies is defined
by resistivity.
Kinematic Turbulent Dynamo: Phenomenology
EK
Role of resistivity 
K-5/3
EM(K)
K0
K
Small Prandtl number,
K
K
Large Prandtl number,
PM´ /¿ 1
PMÀ 1
Dynamo growth rate:
Dynamo growth rate:
 » 1/t
“rough” velocity  V /  =1/3
 » 1/t
“smooth” velocity  V/ 
K
Phenomenology: Large Prandtl Number Dynamo
EK
K-5/3
Large Prandtl number
PM=/ À 1
Interstellar and
intergalactic media
EM(K)
K0
K
K K
Magnetic lines are folded
B

Cattaneo (1996)
Schekochihin et al (2004)
Phenomenology: Small Prandtl Number Dynamo
EK
K-5/3
EM(K)
K0
K
K
K
Small Prandtl number
PM=/ ¿ 1
Stars, planets,
accretion disks,
liquid metal
experiments
V » 1/3
t » /V / 2/3
 /
(RM)-3/4

Dynamo growth rate:
 » 1/t/ (RM)1/2
[S.B. & F. Cattaneo (2004)]
RM
Numerics: Haugen et al (2004)
Kinematic Turbulent Dynamo: Theory
tB = r£(v£B) + B
V(x, t) is a given turbulent field
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!
Kinematic Turbulent Dynamo: Theory
Dynamo is a net effect of magnetic line stretching and resistive reconnection.
RM > RM, crit : stretching wins,
RM < RM, crit : reconnection wins,
dynamo
no dynamo
RM=RM, crit: stretching balances reconnection:

B
When
RM exceeds RM, crit only slightly, it takes
many turnover times to amplify the field
Kinematic Turbulent Dynamo: Kazantsev Model
homogeneity and isotropy
incompressibility
No Dynamo
Dynamo
Kazantsev Model: Large Prandtl Number
EK
K-5/3
EM(K)
K0
Large Prandtl number:
K
K K
PM=/ À 1
Kazantsev model predicts:
1. Dynamo is possible;
2. EM(K)/ K3/2
Numerics agree with both results:
Schekochihin et al (2004)
If we know (r, t), we know growth rate
and spectrum of magnetic filed
Kazantsev Model: Small Prandtl Number
EK
K-5/3
PM=/ ¿ 1
EM(K)
K0
K
K
K
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
Direct numerical simulations:
(2004): No  (2007): Yes
resolution
Small Prandtl Number: Dynamo Is Possible
EK
EK
K-5/3
EM(K)
K0
K KK
PM=/ À 1
K-5/3
EM(K)
K0
K
K
K
PM=/ ¿ 1
Keep RM constant. Add small-scale eddies (increase Re).
Kazantsev model: dynamo action is always possible, but for rough velocity (PM¿1)
the critical magnetic Reynolds number (RM=LV/) is very large.
Theory
S. B. & F. Cattaneo (2004)
L/
Kazantsev Model: Small Prandtl Number
smooth velocity
Kolmogorov
(rough) velocity
PMÀ 1
May be crucial for
laboratory dynamo, PM¿ 1
RM, crit
Simulations
P. Mininni et al (2004)
A. Iskakov et al (2007)
PM¿ 1
Re
Kinematic Dynamo with Helicity
It is natural to expect that
turbulence can amplify magnetic
field at K ¸ K0
K-5/3
EK
K0
K
K
K
Can turbulence amplify
magnetic field at K ¿ K0 ?
“large-scale dynamo”
Yes, if velocity field
has nonzero helicity
Dynamo with Helicity: Kazantsev Model
h=s 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)
Dynamo with Helicity: Kazantsev Model
Two equations for magnetic energy and magnetic helicity
can be written in the quantum-mechanical “spinor” form:
r
r
h=s v¢ (r £ v)d3 x = 0
h=s v¢ (r £ v)d3 x  0
S. B. , F. Cattaneo & R. Rosner (2004)
Kazantsev model and -model
 - model
assumes scale
separation
r
Kazantsev model,
NO scale separation
The  - model approaches the exact solution only at r
1
Conclusions
1. Main aspects of kinematic turbulent dynamo is relatively
well understood both phenomenologically and analytically.
RM, crit
2. Dynamo always exists, but
1
PM
3. Separation of small- and large-scale may be not a correct
procedure.