A dynamo without many of the usual ingredients

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A dynamo without many of the usual
ingredients! A public service announcement
Nic Brummell Kelly Cline Fausto Cattaneo Nic
Brummell [phone] JILA, University of
Colorado brummell_at_solarz.colorado.edu
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Large-scale dynamo theory

• We seem to strive very hard to build a
  large-scale dymamo out of the usual suspects
  ingredients
• a-effects, rotation, differential rotation,
  turbulent diffusivities, turbulent transport
• Everytime we look nonlinearly, our intuitive
  ideas come up against obstacles
• Turbulent Re stresses are complex
• a-effects and turbulent diffusivities are
  quenched
• etc
• Does a dynamo NEED turbulence to work, or can it
  work IN SPITE of turbulence?

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A dynamo! The movie
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A dynamo!

• Strong magnetic field maintained!
• Strong toroidal field is generated in a cyclic
  manner
• Polarity of the strong field reverses

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A dynamo! Longer time

• Diffusion time 300 time units
• gt even more convincing is a dynamo
• Remarkably, also shows periods of reduced
  activity!

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WTF!

• WHAT THE HECK IS THIS THING?!
• Answer
• A dynamo driven entirely by magnetic buoyancy
• That does requires NEITHER rotation NOR
  turbulence!
• But is intrinsically nonlinear and non-kinematic

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Large-scale dynamo intuitive picture
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Model configuration Localised velocity shear
Configurations used Build one strong
structure Have field that will diffuse Velocity
shear Early work U(y,z) f(z) cos(2 p
y/ym) Dynamo work U(y,z) f(z)
sawtooth(y) Magnetic field
B0(0,By,0) Early work By 1
Dynamo work
1 (zgt0.5)
- 1 (zlt0.5)
Sawtooth profile

By
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Weak initial field Non-static quasi-equilibrium

• Velocity ramps up
• Magnetic field By stretched into Bx by
  velocity shear.
• Strong tube-like magnetic structure forms in
  region of strongest shear.

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Weak initial field Non-static quasi-equilibrium

• Magnetic structures created by shear.
• Density drops due to contribution from magnetic
  pressure.
• Density drives a roll-like flow up through the
  centre of the structure.
• Balance achieved between creation of magnetic
  field by induction due to the shear and resistive
  diffusion and advection by magnetic buoyancy
  driven flow.
• Flows gt non-static

Bx (shaded, ve dark) (By,Bz) arrows
Density perturbation (shaded, ve dark) (v,w)
arrows
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Weak initial field Non-static quasi-equilibrium

• System eventually decays due to diffusion
  between the By /- parts (hence
  quasi-equilibrium)

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Stronger initial field K-H instability

• Increasing initial field strength increases the
  poloidal flow strength induced by the toroidal
  magnetic structure.
• At t40, an instability occurs
• Instability is of Kelvin-Helmholtz type
  sinusoidal variations in velocity components
  associated with shears in vertical and
  horizontal.
• Instability mechanism
• Initial field purely poloidal
• Poloidal field sheared -gt toroidal
• Toroidal field creates magnetic buoyancy
• Magnetic buoyancy induces roll-like poloidal
  flows
• These steepen the shear
• Shear then becomes K-H unstable

K-H
Hydrodynamic instability but magnetically induced
(gt non-kinematic)
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A linear magnetically-induced K-H instability

• Background state
• u0 (y dyU, 0, 0) (linear)
• B0 (t BydyU , By, 0) (Bx growing in time)
• Perturbation
• u1 (ect ei(ky ykz z), 0, 0)
• Perturbation generates a perturbation magnetic
  field, Bx1
• Creates a density perturbation
• Creates a buoyant flow, with horizontal y
  component v1
• This interacts with the original perturbation
  steepening the shear.
• Can show that can find c with a real positive
  part gt linear instability.

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Stronger initial field K-H instability

• Effect of instability is to kink geometry of
  structures.
• Note this is NOT a magnetic kink instability!
• K-H modes advect/wrap magnetic field into
  helical shape

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Stronger initial field poloidal field generation

• K-H flows create two poloidal loops -- CCW
  above, CW below out of STRONG toroidal field
• gt STRONG poloidal field created
• Stronger poloidal field gt stronger toroidal
  field

B components
Feedback loop created for dynamo! So let it
run
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A dynamo! The movie
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A dynamo!

• Strong magnetic field maintained!
• Strong toroidal field is generated in a cyclic
  manner
• Polarity of the strong field reverses

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A dynamo! Longer time

• Diffusion time 300 time units
• gt even more convincing is a dynamo
• Remarkably, also shows periods of reduced
  activity!

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A dynamo!

• Mechanisms (complicated!)
• Dynamo
• Two poloidal loops created, upper one opposing
  original field
• Sign of By reversed between loops
• Weaker toroidal field created which rises
• K-H acts on this to create poloidal loop in
  upper region with original direction
• Combines with lower loop (diffusion) to start
  process again.
• Reversal
• Strongest structure created
• Dredges in toroidal field from sides to switch
  polarity
• Inactivity periods
• Failed polarity reversal

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An even wackier dynamo!

• Things to note
• There is a minimum initial magnetic field
  required to trigger the K-H instability and
  therefore the dynamo i.e. the mechanism is NOT
  KINEMATIC.
• The dynamo saturates in equipartition with the
  shear energy source
• Higher Rm (e.g. Rm 2000 cf earlier Rm
  1000, varying the magnetic Prandtl number)
• dynamo behaves irregularly irregular
  production of structures, polarity no longer so
  obvious
• Work in progress to determine large Rm
  behaviour does it turn off (no more
  reconnection)?
• Lower Rm (e.g. Rm 500 cf earlier Rm 1000,
  varying the magnetic Prandtl number)
• Diffuses away UNLESS raise initial field
  strength significantly
• Then can trigger K-H gt dynamo, but different
• NO RISE! Statistically steady travelling wave
  K-H rather than intermittent K-H

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An even wackier dynamo!
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The role of magnetic buoyancy

• Dual roles of magnetic buoyancy in the
  large-scale dynamo
• Limiter
• Magnetic buoyancy limits the growth of the
  magnetic field by removing flux from the region
  of dynamo amplification
• Magnetic buoyancy instabilities then control
  the dynamo amplitude
• BUT magnetic buoyancy does not actively
  contribute to the amplification process
• Driver
• If the poloidal field regeneration is
  associated with rising and twisting structures,
  then magnetic buoyancy is the very mechanism that
  drives the dynamo.
• First case dynamo operates IN SPITE of magnetic
  buoyancy
• Second case dynamo operates BECAUSE of magnetic
  buoyancy

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Dynamo conclusions

• A new class of dynamo mechanisms (as far as we
  know)
• A dynamo driven solely by the action of shear
  and magnetic buoyancy
• Fully self-consistent
• No Coriolis forces required to twist toroidal
  into poloidal
• Intrinsically nonlinear cannot quantify in
  terms of an a-effect (and if you do attempt to,
  get meaningless result).
• What is the role of turbulence? This is VERY
  LAMINAR! Does hydrodynamic turbulence enhance or
  decrease the dynamo effeciency? Enhanced
  diffusion helps reconnection processes? OR loss
  of coherence kills dynamo? Add noise to the
  dynamo simulations ( work in progress ? )

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Obvious questions

• What determines the strength of the emerging
  structures?
• tshear-buoyant gtgt tequilib gt structures have
  characteristics of equilibrium
• tshear-buoyant tequilib gt structures have
  characteristics set by instability
• tshear-buoyant depends on stratification
  (poloidal flows)
• tequilib does not (depends on
  balance of stretching and tension)
• Buoyancy forces set upper limit on strength of
  structures by setting maximum time for shear
  amplification mechanism to act

See Geoffs talk next!

• What are the writhe and twist of 3D structures
  (observational signatures)?
• components of the magnetic helicity, invariant
  in the limit of zero resistive diffusion when
  integrated over a volume (flux tube) surrounded
  by unmagnetized material.
• Writhe and twist could be defined by
  thresholding, but would be ambiguous.
• Integral would not be invariant, due to
  fieldlines entering and exiting volume.
• Leads to question are our structures isolated or
  encased in flux surfaces?

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Effects of parameter variation

• Ingredients
• Basic equilibrium
  stratification, buoyancy, magnetic tension vs.
  diffusion
• Instability
  interplay of basic shear, buoyancy and tension
• KH secondary instability local Re, shear
  profile
• complicated parameter space! BUT can extract
  general trends
• Increasing Re increases growth rate of
  shear-buoyant instability
• facilitates the
  development of secondary KH instability
• Increasing Rm also increases the growth rates
  of the shear-buoyant instability
• Changing background field strength depends on
  regime
• -- very weak initial field gt remains in
  equilibrium (no feedback on shear)
• -- stronger initial field gt tension
  resistance overcome by increased magnetic
  buoyancy gt shear-buoyant instability
• -- even stronger initial field gt equilibrium
  again, tension wins

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Question What is a flux tube?
The observation of intense, intermittent
,isolated, frozen-in elements of magnetic
field on the sun has led to the notion of a
MAGNETIC FLUX TUBE Do such flux surfaces
really exist? Important question because can lead
to very different models of evolution. e.g. Do
not need non-axisymmetric rise of annulus or
drainage down tubes to remove mass if the tube is
not defined by a closed surface. Usefulness of
the flux tube concept hinges on the existence of
flux surfaces (although may hold up even if
magnetic field lies close to surfaces). So

• compact, typically cylindrical, region of
  magnetic field
• really isolated magnetic field inside, none
  outside
• divided by magnetic flux surface
• flux surfaces are material surfaces (in an
  ideal fluid)
• fluid inside stays inside, fluid outside stays
  outside, unless leaves through ends

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Examine magnetic fieldlines

• We will examine the nature of magnetic fieldlines
  in the three general states found
• equilibrium
• primary instability
• secondary instability
• We take a 3-D snapshot of the magnetic fields,
  pick a starting point and integrate along the
  magnetic field lines.

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Fieldlines in equilibrium state
Recurrence maps of 15 fieldlines stacked
vertically in XY- and YZ-planes. Points of return
are commensurate hits same points over and over
again periodicity of lines is same as
box. Fieldlines map out only a line
Projection of 1 fieldline onto XY-plane (i.e.
viewed from above)
Projection of 15 fieldlines stacked vertically
onto YZ-plane (i.e. viewed from the end)
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Fieldlines primary instability
Recurrence maps of 15 fieldlines stacked
vertically in XY- and YZ-planes. Points of return
migrate in X and Y but not Z Fieldlines map out
a PLANE, i.e. FLUX SURFACES.
Projection of 1 fieldline onto XY-plane (i.e.
viewed from above)
Projection of 15 fieldlines stacked vertically
onto YZ-plane (i.e. viewed from the end)
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Fieldlines primary instability
Time sequence
Recurrence maps of 15 fieldlines (stacked
vertically) in YZ-planes. Planes remain as planes
throughout.
Contours in YZ-plane
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Fieldlines secondary instability
Recurrence maps of 15 fieldlines (stacked
vertically) in YZ-planes. Fieldlines fill volume
during the 3D stages.
Time sequence
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Fieldlines secondary instability

• Recurrence map (YZ-plane)
• single instance in time
• 3D KH kinked structure
• 5 returns
• initial positions inside structure
• Fieldlines do NOT remain within structure.
• Neighbouring fieldlines diverge rapidly (chaotic?)

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Fieldlines secondary instability

• Lyapunov map (YZ-plane)
• single instance in time
• 3D KH kinked structure
• Points within 3D structure show large lyapunov
  exponents
• Trajectories diverge rapidly
• Chaotic!

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Comments, thoughts, conclusions(?)

• Three types of fieldline topology found,
  depending on degree of symmetry present
• Fieldlines lie on surfaces but individual lines
  do not cover the surface
• Fieldlines lie on surfaces and individual lines
  do cover the surface
• Fieldlines are volume filling (chaotic)
• Structures are not necessarily encased in flux
  surfaces
• There is no easily defined inside and outside
• Fluid is free to flow in and out (leak out) of
  the structure
• Despite the fact that this is not our idealised
  picture, this may actually HELP in many
  problematic circumstances, e.g. axisymmetric rise
  of a flux tube.
• What are the dynamics of leaky structures?

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