A topological view of 3D global magnetic field reversal in the solar corona

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A topological view of3D global magnetic field
reversalin the solar corona

• Rhona Maclean
• Armagh Observatory
• 5th December 2006

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Solar minimum and maximum

• minimum large-scale field is bipolar, from north
  and south poles relatively simple topology
• maximum most of flux is in active regions,
  large-scale field topology much more complex

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The sunspot cycle

• sunspot numbers follow regular pattern shown in
  butterfly diagram
• flux from sunspot following polarities
  preferentially transported towards pole by
  meridional circulation
• this cancels with polar flux then builds up again
  with opposite sign ? solar cycle

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What is magnetic topology?

• topological analysis of magnetic fields
• tool for understanding fields
• 3D structure
• connectivity
• evolution
• can be applied to theoretical models or
  numerical/observational datasets
• topological features are prime sites for magnetic
  reconnection ? coronal heating

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Point source approximation
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Magnetic null points

• structure of magnetic field near generic
  potential null point with B0
• fan/separatrix surfaces divide space into regions
  of different magnetic connectivity flux domains
• separatrices can intersect in separator
  fieldlines that join two null points

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Example topology

• 2 positive and 2 negative sources (spheres)
• 1 positive and 1 negative null point (tetrahedra)
• 2 spine fieldlines
• 2 separatrix domes
• 1 separator
• called intersecting state

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Towards global topologies

• wanted to extend concept of magnetic topology to
  global field of Sun, with spherical photosphere
• physical idea for field
• effective source strength of 1 outside sphere
• fieldlines everywhere normal to photosphere and
  evenly spaced over surface
• balancing source elsewhere will make photosphere
  a flux surface
• any number of balanced sources can now be used

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Four-source topologies (22 case, part I)
detached state
nested state
intersecting state
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Four-source topologies (22 case, part II)
dual intersecting state new state!
coronal null state
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Four-source topologies (31 case)
separate state
enclosed state
upright null state
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Bifurcations changes in topology

• local bifurcations
• create or destroy null points
• do not change connectivity
• global bifurcations
• create or destroy flux domains
• leave nulls unchanged
• quasi-bifurcations
• change dominant flux domain
• no effect on nulls or connectivity

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22 bifurcation diagram
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Model setup for field reversal

• six balanced point sources of magnetic flux
• initially dominant bipole
• large active region in each hemisphere
• modelling large-scale global magnetic field
• mimic sunspot cycle by changing source strengths
  from solar min to max and back to min
• sequence of 17 topological changes takes place

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Initial state solar minimum

• polar flux is dominant
• active regions are separate, just starting to
  emerge

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Poles 1, active regions 0.1

• active regions magnetically connect over equator
• formation of transequatorial loops
• separator B2-A2 created

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Poles 0.25, active regions 1

• increasing flux in active regions means first
  leading then following active region sources
  dominate topology
• P3-N1 dominant flux domain here
• active region bipoles now completely connected
• separator B1-A1 created

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Poles 0.001, active regions 1

• all polar flux topologically isolated inside
  simple flux domains
• following polarities dominate during reversal due
  to Joys Law
• only B1-A1 separator remains
• essentially intersecting topology

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Poles 0.001, active regions 1

• polar sources have reversed sign at solar maximum
  
• new polar sources topologically isolated
• still intersecting topology with B1-A1 separator

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Poles 0.01, active regions 1

• flux of P1 (polar flux) connects back in to
  topology as it gains strength
• all separators present again

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Poles 1, active regions 1

• polar flux continues to strengthen and regains
  dominance of coronal magnetic field

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Poles 1, active regions 0.05

• active regions lose their influence and
  disconnect
• transequatorial loops severed
• back to solar minimum field reverts to bipolar
  nature but with reversed direction compared with
  initial state

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Example of description in terms of four-source
states

• just after reversal poles isolated, following
  polarity flux dominates topology
• B1-B2 hybrid separate
• A1-A2 hybrid separate
• B1-A1 pure intersecting
• B1-A2 hybrid nested
• B2-A1 hybrid nested
• B2-A2 compound detached

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Conclusions

• modelled topological nature of global magnetic
  field reversal
• simple model of large-scale field captures many
  features that could be compared with observations
• sequence of 17 topological changes
• each state can be described in terms of
  combination of four-source states

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Greens function method

• find potential field due to point source on
  sphere, with Bn specified on surface
• Greens function is solution of differential
  equation due to point source
• exterior Neumann problem find F outside sphere,
  given grad(F) on boundary
• use modified version of usual Greens function
• then find F by integrating over surface