1. SHINE 20061 Formation and Eruption of Filament Flux Ropes A. A. van Ballegooijen 1 & D. H. Mackay 2 1 Smithsonian Astrophysical Observatory, Cambridge, MA 2 University of St. Andrews, Scotland, UK

2. SHINE 20062 Filament Flux Ropes Filaments are cool plasma (~10 4 K) embedded in hot corona and supported by a magnetic flux rope:

3. SHINE 20063 Formation of Flux Ropes by Reconnection Pneuman (Sol. Phys. 88, 219, 1983) proposed that helical flux ropes are formed by reconnection in the low corona:

4. SHINE 20064 Formation of Flux Ropes by Reconnection Van Ballegooijen & Martens (ApJ, 343, 971, 1989) proposed reconnection is associated with magnetic flux cancellation in a sheared arcade:

5. SHINE 20065 Modeling Coronal Magnetic Fields Mean-field approach to modeling the evolution of coronal magnetic fields (van Ballegooijen et al., ApJ, 539, 983, 2000): The coronal magnetic field is sum of mean and fluctuating components. The mean field B(r,t) describes the large-scale structure of the corona. The fluctuating field δB(r,t) describes small-scale twists of the coronal field lines, produced by footpoint motions on the scale of the solar granulation ( ~ 1 Mm). Small-scale reconnection events in the corona cause diffusion of the mean magnetic field. Such events are responsible for coronal heating (e.g., Parker 1972). Simplest possible model: isotropic diffusion.

6. SHINE 20066 Modeling Coronal Magnetic Fields At the photosphere, the mean field is subject to large-scale surface flows (v θ and v φ ) and diffusion (D): Surface diffusion allows for the observed flux cancellation at polarity inversion lines. Above model assumes there is only horizontal diffusion of the radial field, no radial diffusion of horizontal field. Therefore, once formed, coronal flux ropes cannot “submerge” below the photosphere. Observations indicate D ≈ 600 km 2 s -1 (Wang et al. 1989). For coronal diffusion driven by photospheric footpoint motions, η ~ D.

7. SHINE 20067 Evolution of Two Sheared Bipoles Mackay & van Ballegooijen (ApJ, 641, 577, 2006) simulated the interaction of two bipolar active regions using a mean-field approach. Both bipoles have sheared magnetic field along the polarity inversion line (PIL):

8. SHINE 20068 Evolution of Two Sheared Bipoles Strong magnetic shear develops on the internal PIL of the trailing bipole on day 18 (left), and on the external PIL on day 40 (right):

9. SHINE 20069 Evolution of Two Sheared Bipoles Lift-off of the flux rope lying above the internal PIL of the trailing bipole: day 17day 22day 25

10. SHINE 200610 Evolution of Two Sheared Bipoles Reconnection of field lines below the erupting flux rope:

11. SHINE 200611 Evolution of Two Sheared Bipoles Heights of flux ropes (solid lines) and magnetic nulls (dashed) vs. day of simulation (C=initial null, T=trailing bipole, L=lead bipole, E=external): DAY OF SIMULATION RADIUS (solar radii)

12. SHINE 200612 Evolution of Two Sheared Bipoles Mackay & van Ballegooijen (ApJ, 642, 1193, 2006) constructed connectivity maps (blue=bipolar, green=cross-bipolar, red=open, yellow=periodic):

13. SHINE 200613 Improved Models of Diffusion What is the effect of coronal reconnection on magnetic helicity? where η 2 describes perpendicular diffusion, and η 4 describes hyper- diffusion (Boozer 1986; Strauss 1988). If reconnection (and heating) events are highly localized, H ≈ constant. To satisfy this helicity constraint, the mean-field induction equation can be modified as follows:

14. SHINE 200614 Evolution of an Ω-loop Simulate coupled evolution of an Ω-loop with perpendicular diffusion in the convection zone (η 2 = 600 km 2 s -1 ) and hyper-diffusion in the corona (η 4 = 3×10 11 km 4 s -1 ). Initial configuration produced by pushing a twisted flux rope up from the base of the convection zone:

15. SHINE 200615 Evolution of an Ω-loop Strong magnetic shear builds up along the polarity inversion line: t = 0 dayst = 4 days

16. SHINE 200616 Evolution of an Ω-loop After 10 days, sufficient flux has cancelled at photosphere for the coronal field to erupt: Contours of B z in a horizontal slice (z=0.1 R): Contours of λ in a vertical slice:

17. SHINE 200617 Evolution of an Ω-loop Global configuration of the magnetic field after 10 days:

18. SHINE 200618 Conclusions Filaments/prominences are embedded in highly sheared, weakly twisted fields (“coronal flux ropes”). These coronal flux ropes are not identical to the Ω-loops that emerge through the photosphere. The emerged field is reconfigured by reconnection processes that enhance the magnetic shear. Two types of reconnection are important: a) reconnection associated with photospheric flux cancellation; b) small-scale reconnection which produces “diffusion” of the mean coronal magnetic field and reduces the degree of twist of flux ropes. The mean field approach is an effective tool for modeling the evolution of coronal and sub-surface magnetic fields over long periods (many days). Magnetic helicity constraints have been included into such models.