Coherent Structures and spectra in turbulent dynamos

1
Coherent Structures and spectra in turbulent
dynamos

• Steve Tobias (Applied Maths,
• University of Leeds)
• Fausto Cattaneo (University of Chicago)

2
Talk Outline

• Motivation
• The Dynamo Problem
• The equations and parameters
• Outstanding problems in dynamo theory
• Coherent Structures in dynamos
• Theory
• Scalings. Competition
• Identification of a efficient dynamo scale
• A numerical example
• How good is the theory?
• Filtered Dynamos
• Other applications turbulent dynamos with shear
• Why are coherent structures so important?

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The Dynamo Problem

• Magnetic Fields are ubiquitous in astrophysics
• appear on a large range of spatial and temporal
  scales
• Planets, Stars, Galaxies, Accretion Discs
• These magnetic fields play a dynamic role in the
  evolution of these astrophysical objects
• Accretion
• Star formation
• Stellar spin-down
• Solar sunspots/flares/coronal mass
  ejections/solar wind
• Fundamental question Where do these fields come
  from?

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The Suns Magnetic Field
Galileo
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The Dynamo Problem Equations

• In many astrophysical/geophysical/laboratory
  situations magnetic fields are generated by
  turbulent motions
• High Reynolds number leads to turbulence.

Lorentz force
advection/stretching
diffusion
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Example parameter setthe Sun
BASE OF CZ
PHOTOSPHERE
(Ossendrijver 2003)
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Some outstanding problems in dynamo theory I

• The large-scale magnetic field problem
  (astrophysical)
• Given a turbulent flow which has energy on a wide
  range of scales how is it possible to generate
  systematic coherent magnetic fields.
• Interaction of turbulent fields and flows in the
  presence of rotation could lead to correlations
  that give systematic behaviour (kinematic)
• Mean field electrodynamics
• Nonlinear driving of flows by magnetic fields may
  lead naturally to correlations.
• High Rm leads naturally to small-scale field
  generation.
• Can shear help?

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Some outstanding problems in dynamo theory II

• The saturation mechanism for dynamos
• If velocity fields can be found that lead to
  growth of the magnetic field.
• How does the dynamo saturate?
• What is the nature of the magnetic force in the
  momentum equation that leads to the saturation of
  the dynamo instability
• Various mechanisms proposed
• Reduction to marginality
• Increase in dissipation
• Modification of Lagrangian properties of the
  flow.

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Some outstanding problems in dynamo theory III

• The low Pm problem
• Given a turbulent velocity field at high Re
• Can this support growing magnetic fields if Rm ltlt
  Re

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Some outstanding problems in dynamo theory III

• The low Pm problem
• Given a turbulent velocity field at high Re
• Can this support growing magnetic fields if Rm ltlt
  Re

Eu
EB
ku
kB
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Outstanding problems in dynamo theory

• Are there generic answers to the problems posed
  above?
• Does the answer depend on the nature of the
  turbulence?
• To what extent can these be answered using the
  statistical techniques of turbulence?
• Do coherent structures play a role
• Are they beneficial or detrimental to dynamo
  action? (Tobias Cattaneo 2008 PRL)
• Can a theory be constructed for dynamos in
  turbulence with coherent structures? (Tobias
  Cattaneo 2008 JFM)
• Can this theory tell us anything about the
  outstanding problems in dynamo theory?

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Coherent structures- vortices

• Turbulence is a hard problem the superposition
  of random elements and coherent structures
• In geophysical and astrophysical fluid mechanics
  coherent structures often take the form of vortex
  tubes/patches (stratification rotation).
• Structures that contain the local
  helicity/enstrophy of the turbulence.
• Found on all scales.
• Vortices are formed as coherent structures in
  turbulence, convection, shear and differential
  rotation instabilities.
• Vortices play a large role in determining the
  dynamics of the turbulence (e.g. Vincent
  Meneguzzi 1991 Frisch 1995).
• Vortices contain non-trivial phase information
  about the turbulence.

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An aside Coherent Structures - Vortices.
Vortices arise due to the interaction of shear
and rotation due to the nonlinear saturation of
Rossby waves. May be important for planet
formation?
Bracco et al (1999)
In compressible convection there is a strong
asymmetry between upflows and downflows. Dynamics
dominated by strong downward sinking plumes
from vertices of downflow network.
VORTEX TUBES
Brummell et al (2002)
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The Dynamo Problem

• Rm is the non-dimensional measure of
  advection/stretching to diffusion
• Rm is typically very large! This is good for
  dynamo action
• The induction equation here is non-dimensionalised
  in the units of turnover time. Shorter turnover
  times mean better dynamos
• Fix ideas by looking at flow on one scale

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The induction equation Big and Slow versus small
and quick
BIG EDDY Rm large (good) tc
large (bad)
SMALL EDDY Rm small (bad)
tc small (good)
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1-scale dynamos

• Flow is generally assumed to have one (large)
  scale.
• Field is generated on scales smaller than flow
• Central Question Can a dynamo be maintained as
  Rm grows large?
• i.e. is the dynamo fast
• Plot growth-rate as a function of Rm.
• Note growth-rate is measured in units of the
  inverse turnover time of the eddy.
• So fast eddies will have large growth-rates, slow
  eddies small growth-rates.

Can only be fast if Flow exhibits
Lagrangian Chaos (Klapper Young, Vishik)
This formalism only makes sense if RmgtgtRe i.e.
Pmgtgt1
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Dynamo properties of turbulent cascades of
interacting coherent structures

• Investigate dynamo properties of turbulent
  cascades of eddies at high Rm, Re
• Analytically
• Use superposition of coherent fast dynamos
• Numerically
• Examine kinematic dynamo properties of sample
  velocity fields
• Easily computable (relatively)
• Contain coherent structures
• Have a parameter that changes the spectral slope
  of the turbulence

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Theory for turbulent Spectra

• Consider a turbulent cascade of eddies E(k)k-p
• If we consider each eddy (localised on a scale
  k-1 in isolation) what can we say about their
  dynamo properties
• The large-scale eddies are large and slow, so
  they have a
• Large Rm
• Long turnover time
• The small-scale eddies are small and quick, so
  they have
• Small Rm
• Short turnover time
• For example if v k-a then
• Rmk-(a1)
• T k-(1-a)

Small Rm Fast
Large Rm Slow
For Kolmogorov turbulence a1/3
So assuming a growth-curve shape for each
individual eddy We can see which are the active
dynamo modes
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a0.2
max(s) can be as k?0 (usually at k10-20)
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Dynamo properties of turbulent cascades Numerics

• How well does this theory do?
• Illustrate via an example. How to get a velocity
  field?
• Magnetic fields generated by dynamo action have
  to be fully three dimensional (Cowlings Theorem)
• Could try to do fully 3D simulations of
  Navier-Stokes. Would need 40963 calculation.
• Instead use a 2 ½ D velocity field
• i.e.
• Then can use a 2D calculation.
• But how do we evolve u to get a realistic
  spectrum?

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Pseudo 3-d dynamos Active Scalar Equations

• l 2 ? 2d Navier Stokes l lt 2 ? pseudo 3d
  dynamics LOCAL
• 1 surface QG (see e.g. Pedlosky)

Bx
q

• High Re, Rm calculations
• 0.00001, h 0.001, l 1
• (20482, 40962 calcs)

Steepness of spectrum depends on l (e.g.
Pierrehumbert et al 1992, Constantin 1999)
(Tobias Cattaneo, JFM 2008)
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Dynamics varies depending on l
k4 l2 2D NS
k4 l1
k100 l1
Surface QG
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Spectrum varies depending on l
k4 l2 2D NS Complete inverse cascade
Enstrophy Spectrum
k100 l1
k4 l1
Surface QG
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Bx
q/w
Field generation by small-scale vortices Stays
coherent for many turnover times
NOTE it is vortices of a certain size that are
doing all the work
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Close-ups
q
q
Bx
Bx
Vortices, vortices all the way down
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Numerical Growth Rates
0.0001
0.001
0.01
Growth-rate versus kz for l1. Fast dynamo
characteristics Preferred kz is quite large How
can we understand what velocity scale is doing
the work as far as the dynamo is concerned?
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Filtered Dynamos

• Solve pseudo-3d NS equations
• Get coherent structures with eddies on various
  scales and appropriate phase relations
• Filter velocity field
• Cut off large scales
• Cut off small scales
• Solve induction equation with filtered velocity
  field
• Growth rates
• Form of generated field.

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Filtered Dynamos
k1
k0
k200
k10
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Filtered Dynamos
Keeping scales gt kfil
Keeping scales lt kfil
Growth rate begins to fall significantly when
filter reaches 10-50. Both ways so the
velocity scales 10-50 are most significant for
Dynamo action (pretty crude).
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What does the theory say
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Why should we worry so much about coherent
structures?
Randomised Phase
Coherent Structures

• These two flows have the same spectrum
• They have very different dynamo properties

Morphology of resulting field for coherent case
very different from random case (more and better
folding)
Tobias Cattaneo PRL (2008)
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The role of a large-scale shear

• What happens if a large-scale shear is added to a
  turbulent cascade with coherent structures (e.g.
  rotating convection)?
• The theory enables the computation of the local
  Strouhal number at the dynamo eddy scale.(NB
  shear may change energy spectrum)
• For low Rm (or very strong shear) the dynamo
  scale is large enough to be affected by the shear
  (St(kd)1) or even given by the shear.
• If Rm is high enough St(kd) is small and the
  small-scale dynamo can proceed largely unaffected
  (directly) by shear

a0.2
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Conclusions

• Turbulence is a lot more difficult than just the
  superposition of random waves (not homogeneous or
  isotropic)
• For dynamos in astrophysical and geophysical
  systems the coherent bit might be more important
  than the random bit.
• For flows at low Pm, (Re gtgt Rm) the important
  distinction is not between fast and slow but
  between quick and pedestrian (even at very high
  Rm)
• Competition between small/fast eddies and
  large/slow ones. In general the ones that are
  pretty fast and pretty large win!
• Coherent structures within the turbulence may
  play an important role.
• There is no problem having a dynamo at low Pm
• Can get some idea of the behaviour of coherent
  structures from
• Quick dynamo theory work out local growth
  rates
• Filtered spectra bear out quick dynamo
  analysis
• How much do different scales really interact?
  (CT 2006 POF)