A topological view of3D global magnetic field reversal

in the solar corona

Rhona Maclean

Armagh Observatory

5th December 2006

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

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

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

Point source approximation

Magnetic null points

• structure of magnetic field near generic potential null point with B=0

• fan/separatrix surfaces divide space into regions of different magnetic connectivity: flux domains

• separatrices can intersect in separator fieldlines that join two null points

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”

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

Four-source topologies (2+2 case, part I)

detached state nested state intersecting state

Four-source topologies (2+2 case, part II)

coronal null state dual intersecting state:new state!

Four-source topologies (3+1 case)

separate state enclosed state upright null state

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

2+2 bifurcation diagram

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

Initial state: solar minimum

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

Poles ±1, active regions ±0.1

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

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

Poles ±0.001, active regions ±1

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

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

Poles ±0.01, active regions ±1

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

• all separators present again

Poles ±1, active regions ±1

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

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

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

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

Green’s function method• find potential field due to point source on sphere, with

Bn specified on surface

• Green’s function is solution of differential equation due to point source

• exterior Neumann problem: find Φ outside sphere, given grad(Φ) on boundary

• use modified version of usual Green’s function:

• then find Φ by integrating over surface:

rr

rrrr

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ar

aaG

2

ln2

4

1,

m

iiiar n GSdGB

1

,, rrrrrr