A topological view of 3D global magnetic field reversal in the solar corona Rhona Maclean Armagh Observatory 5 th 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 statenested stateintersecting state

Four-source topologies (2+2 case, part II) coronal null state dual intersecting state: new state!

Four-source topologies (3+1 case) separate stateenclosed stateupright 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 B n 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: