Modulation of Solar and Stellar Activity Cycles

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Small-scale dynamos
Fausto Cattaneo Department of Mathematics Univers
ity of Chicago
cattaneo_at_flash.uchicago.edu
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What is a small-scale dynamo?
A hydromagnetic dynamo is a sustained mechanism
to convert kinetic energy into magnetic energy
within the bulk of an electrically conducting
fluid
Concentrate on magnetic field generation on
scales comparable with or smaller than the
velocity correlation length (cf. mean field
theory).

• 50s - 60s Is MHD turbulence self excited?
  (Batchelor 1950 Schlüter Biermann 1950
  Saffman 1963 Kraichnan Nagarajan 1967)
• 80s -90s Fast dynamo problem (Childress
  Gilbert 1995)
• Now Large scale numerical simulations

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Astrophysically speaking
In a rotating body, the Rossby radius, Ro,
defines the spatial scale above which rotation
becomes important

• L gt Ro
• Differential rotation
• Helical turbulence
• Large-scale dynamo action (mean field
  electrodynamics)
• Llt Ro
• Non helical flows
• Small-scale dynamo action

Earth Ro ? 50 km Sun Ro ? 50,000 km
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Stellar examples

• Small scale photospheric fields
• Solar granulation
• Size 1,000 km
• Lifetime 5-10 mins
• Basal flux
• Chromospheric emission increases with rotation
• Residual emission at zero rotation
• Possibly acoustic

G. Scharmer SVST, 07/10/97
From Schrijver et al. 1989
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Physical parameters
Rm
Pm 1
103
simulations
102
Liquid metal experiments
Re
107
103

With the exception of numerical simulations,
everything has either Pmgtgt1 or Pm ltlt1 (typical
!!)
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Rough vs smooth velocities
Introduce longitudinal (velocity) structure
functions
(assuming stationarity, homogeneity, isotropy)
Where ? is the roughness exponent. ?lt1 ? rough
velocity
with
plt3 ? rough velocity
Equivalently
Turnover time shorter at small scales
Reynolds number deceases with scale
Define scale d for which Rm(d)1
diffusion dominates
advection dominates
Similarly for Re
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Magnetic Prandtl number dependence
Smooth velocity case
Rough velocity case
Both cases have comparable values of Rm
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Rough vs smooth

• The difference between the smooth and rough cases
  (for dynamo action) is related to the behaviour
  of the velocity at the reconnection scale d?
• Slowly varying
• Strongly fluctuating
• Smooth and rough velocities have very different
  dynamo properties
• As Pm decreases through unity dynamo velocity
  changes from smooth to rough

Begin with smooth case (easier)
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Dynamo action
Magnetic field grows if on average rate of field
generation exceeds rate of field destruction

• Magnetic field generation due to line stretching
  by fluid motions
• Magnetic field destruction due to enhanced
  diffusion

Dynamo growth rate depends on competition between
these two effects. In a chaotic flow both effects
proceed at an exponential rate.
Why cant we have a gradient dynamo for finite ??
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Chaotic flows
Fluid trajectory given by
Follow deformation of cube of fluid of initial
size ?x over a short time ?t. New size
If deformation proceeds at exponential rate on
average, flow is chaotic. ?1, ?2, ?3 are the
Lyapunov exponents . (incompressibility ?1 ?2
?3 0 )
?1 Rate of stretching ?3 Rate of
squeezing
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Example of 2-D chaotic flow
Simple example of smooth solenoidal flow with
chaotic streamlines.
Streamlines
Red and yellow correspond to trajectories with
positive (finite time) Lyapunov exponents
Cattaneo Hughes
Planar flow ? ?2 0, and ?1 -?3
Finite time Lyapunov exponents
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Line stretching in a chaotic flow
In a chaotic flow the length of lines increases
exponentially (on average)
is the topological entropy. It satisfies
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What about dissipation?
Do (fast) dynamos operate at the rate ?T?
?3 local rate of growth of gradients ?
dissipation also increases exponentially. In two
dimensions ?1 - ?3. Magnetic field is
destroyed as rapidly as it is generated.
In 2D magnetic flux behaves like a scalar. Thus
Two-dimensional dynamo action is impossible
(Zeldovich 1957)
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Introduce third dimension
Simple modification leads to dynamo action
(Galloway Proctor 1992)
Three-dimensional but still y-independent.
However still have ?1 - ?3
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Enhanced diffusion
Diffusion of magnetic field is determined by two
processes

• Growth in the magnitude of the gradients
• Geometry of the sign reversals of the field lines

Effectiveness of diffusion of a vector field
depends also on how the field lines are arranged
Effective
Ineffective
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Enhanced diffusion
Effective diffusion of vector quantities depends
on magnitude of gradients and orientation.
Packing becomes important.
from Cattaneo
k is the cancellation exponent. Measures the
singular nature of sign reversals (Du Ott 1994 )
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(Fast) dynamo growth rate
Enhanced diffusion depends on the (exponential)
growth of gradients and on field alignment
Local stretching Local contraction
Conjecture by Du Ott (1995) for foliated fields
as Rm ? ?
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The flux problem
Fractal dimensions can be constructed similarly
to ? (Hentschel Procaccia 1983)
Define averages (Cattaneo, Kim, Proctor Tao
1995)
For smooth velocities m is the cancellation
exponent (Bertozzi et al 1994)
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Convectively driven dynamos

• Plane-parallel layer of fluid
• Boussinesq approximation

Simulations by Emonet Cattaneo
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Convectively driven dynamos field structure

• Ra500,00 Pm5
• kinematically efficient (i.e. growth rate)
• nonlinearly efficient (i.e. total magnetic
  energy)

detail
top
middle
volume
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Convectively driven dynamos field intensity
P(56) 0.5 P(U/2) 0.2 P(U) 0.03

• PDF is exponential in the interior. Super
  equipartition fields possible.
• PDF is stretched exponential near the surface
  ?extreme intermittency.
• Relatively strong fields can be found everywhere
  strongest near the surface.

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Extrapolation to Pm ltlt 1

• Dynamo action may disappear as Pm becomes small

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Kinematic and dynamical issues

• Kinematic issue Does the dynamo still operate?
• (Batchelor 1950 Nordlund et al. 1992
  Brandenburg et al. 1996 Nore et al. 1997
  Christensen et al 1999 Schekochihin et al. 2004
  Yousef 2004)
• Dynamical issue Dynamo may operate but become
  extremely inefficient

yes
Re1100, Rm550, S0.5
Simulations by Cattaneo Emonet
no
Re1900, Rm950, S1
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Dynamo action in rough velocity fields (schematic)

• Is dynamo action always possible provided L?, and
  hence Rm, are sufficiently large?
• Does the dynamo growth rate become independent of
  k?, and hence Pm, as k? ? ? (Pm? 0)?

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Kazantsev-Kraichnan models

• Previous questions can be answered within the
  framework of an analytical model of dynamo action
  by Kazantsev (1968)

• Velocity incompressible, homogeneous, isotropic,
  stationary, Gaussian random process with zero
  correlation time.
• Longitudinal velocity correlator
• Closed form equation for magnetic field
  correlator
• Equation similar to Schrodinger equation with
  1/r2 potential in inertial range
• Existence of bound states (dynamo instability)
  depends on ?

Several extensions Vainshtein Kichatinov
(1986), Kim Hughes (1997), Kleorin
Rogachevskii (1997) Also Zeldovich, Ruzmaikin
Sokoloff (1990)
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Properties of Kazantsev dynamos

• Magnetic field correlator H is localized near
  1/k?
• Correlator decreases exponentially with r for r gt
  1/k?
• Localization decreases with decreasing ß (i.e.
  correlators for rough velocities are more spread
  out)

Dynamo action is always possible. Numerical
resolution required to describe dynamo solution
increases sharply as velocity becomes rougher
(Boldyrev Cattaneo 2004)
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Dynamical effects convectively driven dynamos
Nonlinear dynamo efficiency depends on generation
of moderately strong fluctuations

• Peak amplification for steady axi-symmetric flux
  rope maintained by convection (Galloway et al.)

• Expression derived assuming a1 (laminar
  velocity)
• Assumes inertia terms are negligible (not true if
  Regtgt1)

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Easier case magneto-convection

• Relax requirement that magnetic field be self
  sustaining (i.e. impose a uniform vertical field)
• Construct sequence of simulations with externally
  imposed field, 8 Pm 1/8, and S ??? 0.25
• Adjust Ra so that Rm remains constant

Simulations by Emonet Cattaneo
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Magneto-convection results
B-field (vertical)
vorticity (vertical)
Pm 8.0
Pm 0.125
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Magneto-convection results

• Energy ratio flattens out for Pm lt 1
• PDFs possibly accumulate for Pm lt 1
• Evidence of regime change in cumulative PDF
  across Pm1
• Possible emergence of Pm independent regime

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Conclusion

• Huge difference between dynamo action driven by
  smooth and rough velocities
• Varying the magnetic Prandtl number across unity
  in a turbulent environment changes the dynamo
  from smooth to rough
• In all cases dynamo action is very likely if Rm
  is large enough
• Some evidence of a Pm independent regime at small
  (but not miniscule) values of Pm ?

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The end
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Vectors and gradients
Induction equation
Scalar advection-diffusion
Scalar variance equation
Scalar gradient equation
Dynamo action as a way to reduce diffusion
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Kazantsev model
Velocity correlation function
Isotropy reflectional symmetry
Solenoidality
Magnetic correlation function
Kazantsev equation
Renormalized correlator
Funny transformation
Funny potential
Sort of Schrödinger equation