1. Jan 13, 2009ISSI1 Modeling Coronal Flux Ropes A. A. van Ballegooijen Harvard-Smithsonian Center for Astrophysics, Cambridge, Massachusetts, U.S.A Collaborators: M. Bobra, S. Cranmer, Y. Su

2. Jan 13, 2009ISSI2 Flux Ropes in Active Regions Bobra et al. (2008) constructed non-linear force-free field (NLFFF) models of an active region (NOAA 9997/10000), based on an observed TRACE loop (panel b). Models are constructed by inserting a flux rope into a potential field, then applying magneto-frictional relaxation. Model #8 (best fit to observations) Left panels show the flux rope (overlying arcade not shown). Model parameters: Axial flux: Φ axi = 4×10 20 Mx (W section) Φ axi = 14×10 20 Mx (E section) Poloidal flux: F pol = -10 10 Mx/cm.

3. Jan 13, 2009ISSI3 Flux Ropes on the Quiet Sun Cavities observed in X-rays are thought to be due to coronal magnetic flux ropes: Cavity

4. Jan 13, 2009ISSI4 Flux Ropes on the Quiet Sun Goal is to apply flux-rope insertion method to the quiet Sun (coronal cavities, filament channels). Here I present initial results for 2007 April 12-19: Polar CH Low-Latitude Coronal Hole

5. Jan 13, 2009ISSI5 Part I: Modeling Coronal Flux Ropes Starting from full-disk magnetogram (SOLIS), a latitude-longitude map of the radial field is constructed. This provides the lower boundary condition for the 3D magnetic model. Note: both coronal holes have positive polarity (white).

6. Jan 13, 2009ISSI6 Modeling Coronal Flux Ropes There are two polarity inversion lines. The H  image (Kanzelhöhe) from Apr 19 shows only a small filament fragment: Red=positive, Green=negative

7. Jan 13, 2009ISSI7 Modeling Coronal Flux Ropes This filament fragment is observed as a prominence with Hinode/SOT (left) and XRT (right) on April 25, 2007 (Heinzel et al. 2008):

8. Jan 13, 2009ISSI8 Modeling Coronal Flux Ropes Despite the absence of long H  filaments, I manually select two flux rope paths. Two artificial sources (± 10 20 Mx) have been added at the left ends. Red=positive, Green=negative

9. Jan 13, 2009ISSI9 Modeling Coronal Flux Ropes Compute potential field (partial sphere geometry, R max = 2.4 R s ), create “cavities” above selected paths, and insert sinistral flux ropes (axial flux 10 20 Mx; poloidal flux 5x10 9 Mx/cm): Selected field lines after only 1000 iterations. Overlying arcades not shown. Then apply magneto-frictional relaxation for 20,000 iterations.

10. Jan 13, 2009ISSI10 Modeling Coronal Flux Ropes Spatial distribution of current-helicity  (x,y,z) in the NLFFF after 20,000 iterations:

11. Jan 13, 2009ISSI11 Modeling Coronal Flux Ropes Side view of northern flux rope (  in vertical plane along PIL). There is a current layer between the flux rope and its surroundings. Note the “waviness” of this current layer.

12. Jan 13, 2009ISSI12 Modeling Coronal Flux Ropes The same waviness is seen in a second model with reduced axial- and poloidal fluxes: It is due to the break-up of the current layer into channels with twisted field lines:

13. Jan 13, 2009ISSI13 Modeling Coronal Flux Ropes Field-line dips (blue) occur in disconnected patches everywhere along PIL, not just at observed filament fragment (prominence):

14. Jan 13, 2009ISSI14 Modeling Coronal Flux Ropes Selected field lines in the flux ropes (left) and overlying arcades (right) in the first model, rotated 60  : In both models the eastern part of the northern flux rope is unstable, probably due to the lack of an overlying arcade.

15. Jan 13, 2009ISSI15 Modeling Coronal Flux Ropes Reconnection between northern flux rope (left) and polar field (far right) in eastern part of first model:

16. Jan 13, 2009ISSI16 Part II: Coronal Heating There are two sources of energy for coronal heating: Energy may propagate into the corona from the convection zone. 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 (van Ballegooijen & Cranmer 2008).

17. Jan 13, 2009ISSI17 Coronal Heating The idea is that the magnetic fields in a coronal flux rope are to some degree stochastic (Lazarian & Vishniac 1999). Resistive MHD turbulence within the flux rope converts mean magnetic energy into heat via reconnection. The process can be described in terms of hyperdiffusion, a type of magnetic diffusion in which the magnetic helicity is conserved (Boozer 1986; Bhattacharjee & Hameiri 1986) :

18. Jan 13, 2009ISSI18 Coronal Heating In the model by van Ballegooijen & Cranmer (2008), the total heating rate ε is a sum of a direct contribution from footpoint motions and a contribution from hyperdiffusion: where L is the loop length. Parameters: u ph = 0.35 km/s, τ ph = 600 s, B ph = 1500 G, λ turb = 10 3 km. Modeling approach: construct NLFFF model, α 0 (r) compute heating rate ε compute temperature and density

19. Jan 13, 2009ISSI19 Coronal Heating Temperature (left) and density (right) for model with hyperdiffusion, in vertical cross-section of the flux rope (y=0):

20. Jan 13, 2009ISSI20 Coronal Heating Temperature (left) and density (right) for model without hyperdiffusion:

21. Jan 13, 2009ISSI21 Summary Developed preliminary models for flux ropes on polar crown and sub-polar crown channels based on observed photospheric fields (SOLIS). Remains to be seen whether surrounding arcades can hold down the flux rope(s). Modeled the heating in coronal flux ropes, including the effect of hyperdiffusion. Depending on the spatial distribution of α, such models may explain the coronal cavities associated with flux ropes.