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

1. A topological view of3D global magnetic field reversalin the solar corona Rhona Maclean Armagh Observatory 5th 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 state nested state intersecting state

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

11. Four-source topologies (3+1 case) separate state enclosed state upright 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 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: