Magnetic Hyperdiffusion in the Solar Corona

1. Magnetic Hyperdiffusionin the Solar Corona A. A. van Ballegooijen and S. R. Cranmer Harvard-Smithsonian Center for Astrophysics • Role of hyperdiffusion in • Evolution of coronal magnetic field • Heating of coronal plasma

2. Introduction Coronal heating is closely tied to the presence of magnetic fields: SOHO/MDI, Apr 8, 2000 Soft X-rays (T ~ 3 MK), Yohkoh/SXT, 1991

3. Magnetic Helicity The corona contains S-shaped and inverse-S-shaped structures (“sigmoids”), indicating magnetic fields are sheared and/or twisted: Two sigmoids, May 15, 1998 Sigmoid before and after eruption, June 8-9, 1998.

4. Coronal Magnetic Flux Ropes Magnetic energy can be stored in the solar corona in the form of flux ropes, highly sheared, weakly twisted fields overlying polarity inversion lines (PIL) on the photosphere. In equilibrium, the flux rope is held down by an overlying arcade of coronal loops. filament embedded inside flux rope PIL

5. Emergence of Twisted Flux Ropesinto the Corona From simulations by Magara (2006): Comparison with Yohkoh SXT and TRACE images:

6. Formation of Flux Ropes by Reconnection Pneuman (1983) shows how helical flux ropes can be formed by reconnection in a sheared coronal arcade:

7. Flares and Coronal Mass Ejections Coronal flux ropes are key to understanding solar flares and coronal mass ejections (CME). When a flux rope becomes too strong compared to the overlying arcade, it may lose its equilibrium and be ejected from the corona, producing a CME: A current sheet forms in the wake of the erupting flux rope (Lin & Forbes 2000) flare loops

8. Role of Flux Ropes in Coronal Heating? • There are two possible sources of energy for heating the corona: • Energy may propagate into the corona from the convection zone. • For example, Parker (1972) proposed that the corona is heated by • twisting/braiding of magnetic field lines due to small-scale, random • footpoint motions. • Energy may already be stored in the corona. Coronal flux ropes • contain large amounts of magnetic free energy. Some of this energy • may be converted into heat.

9. Heating of Coronal Flux Ropes Resistive MHD turbulence within the flux rope can convert mean-field energy into heat via reconnection. The process can be described in terms of hyperdiffusion.

10. Theory of Coronal Magnetic Fields In ideal MHD, the evolution of the magnetic field is described by the magnetic induction equation: Corresponding equations for magnetic energy and helicity: helicity flux

11. Concept of Helicity Flux Alfvén waves transport magnetic helicity. Consider a wave propagating along a cylindrical flux tube: Helicity flux:

12. Mean-Field Theory of the Corona The Sun likely has stochastic magnetic fields (Lazarian & Vishniac 1999). Electric- and magnetic fields have mean and fluctuating components (subscripts 0 and 1), and the mean-field induction equation reads Corresponding magnetic energy equation: where we assume that the mean field is force free:

13. Constraints from Helicity Conservation There are two ways of deriving the equation for the mean magnetic helicity: where H0 is the helicity flux due to the fluctuations: Expression for mean electromotive force (e.g., Bhattacharjee & Hameiri 1986):

14. Hyperdiffusion and Coronal Heating Boozer (1986) and Bhattacharjee & Hameiri (1986) argue that the helicity flux is proportional to the gradient of α0: where η4 is the hyperdiffusivity. Then the induction equation reads and the magnetic energy equation has a dissipative term:

15. Estimating the Hyperdiffusivity Consider a small volume in the stochastic field, so that α0(z) can be approximated as a linear function of position. Small-scale reconnection creates magnetic connections between different planes α0 = constant, launching Alfvén waves that carry helicity from high to low α0: Component of helicity flux perpendicular to a plane α0 = constant:

16. Formation of Coronal Flux Ropes What is the effect of diffusion on the evolution of solar magnetic fields? Modeling the evolution of an Ω-loop with second-order diffusion in the convection zone and hyperdiffusion in the corona (van Ballegooijen & Mackay 2007, ApJ, 659, 1713): Modeling assumes η4 = 1011 km4 s-1 in corona. Uses magnetofriction to keep the magnetic field close to a force free field:

17. Formation of Coronal Flux Ropes On day 10 magnetic flux is submerging below the photosphere:

18. Formation of Coronal Flux Ropes The coronal field initially has weak left-handed twist (left), but on day 15 of the simulation the field has become strongly sheared (right) because the photosphere presents a barrier for the submergence of axial flux: The coronal field is highly sheared but weakly twisted, consistent with observations (without diffusion the field would be strongly twisted).

19. Formation of Coronal Flux Ropes As more and more flux cancels, axial field builds up in the corona, until the field loses equilibrium and erupts on day 20 of the simulation:

20. Coronal Heating Model Total heating rate is a sum of a direct contribution from footpoint motions (Parker 1972) and an indirect contribution from hyperdiffusion: • where s is position along coronal loop, L is the loop length. • Parameters: λphot = 200 km, τphot = 600 s, Bphot = 1500 G. • Modeling approach: • construct 3D magnetic model of an active region (Bobra et al. 2007), • including α0(r) distribution; • compute heating rate ε(s) along selected field lines; • compute temperature and density along field lines.

21. Coronal Heating Model • 3D magnetic model of AR 9997 on 2002/06/18 with highly sheared fieldlines (blue/magenta) along the polarity inversion line. • Model is based on: • MDI magnetogram • (red/green contours). • BBSO Hα image with • filament (not shown). • Coronal loops seen • in TRACE 171 Å.

22. Coronal Heating Model Select a magnetic field line (blue) that matches an observed coronal loop (red): Total and direct heating rates as functions of position along field line:

23. Summary • Magnetic diffusion plays an important role in the evolution of coronal fields and the formation of coronal flux ropes. • There are two sources of energy for heating the corona: • a) energy that propagates into the corona directly from the • photospheric footpoints (energy provided by footpoint motions); • b) energy that is already stored in the corona in the form of • magnetic shear. • The latter can be described in terms of hyperdiffusion, and may dominate the heating in coronal flux ropes.