1. A topological view of 3D global magnetic field reversal in the solar corona Rhona Maclean Armagh Observatory 5 th December 2006

2. 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

3. 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

4. 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

5. Point source approximation

6. 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

7. 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”

8. 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

9. Four-source topologies (2+2 case, part I) detached statenested stateintersecting state

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

11. Four-source topologies (3+1 case) separate stateenclosed stateupright null state

12. 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

13. 2+2 bifurcation diagram

14. 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

15. Initial state: solar minimum polar flux is dominant active regions are separate, just starting to emerge

16. Poles ±1, active regions ±0.1 active regions magnetically connect over equator formation of transequatorial loops separator B2-A2 created

17. 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

18. 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

19. 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

20. Poles ±0.01, active regions ±1 flux of P1 (polar flux) connects back in to topology as it gains strength all separators present again

21. Poles ±1, active regions ±1 polar flux continues to strengthen and regains dominance of coronal magnetic field

22. 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

23. 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

24. 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

26. 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: