Current sheets and the Tectonics Model for Solar Coronal Heating

1. Current sheets and the Tectonics Model for Solar Coronal Heating Bhattacharjee and C.-S. Ng Center for Magnetic Self-Organization Space Science Center University of New Hampshire

2. Introduction • The Magnetic Carpet, observed by Yohkoh, SOHO, Trace, and now Hinode, has important implications for the coronal heating problem. • This Magnetic Carpet covers the Sun and is constituted of magnetic fragments that are in a continual dynamical state of emergence, break-up, merging, and cancellation. • Approximately 95% of the photospheric flux closes within the magnetic carpet in low-lying loops, leaving only 5% to form large-scale connections [Schriver and Zwaan 2000]. • Photospheric flux in the quiet Sun is replaced approximately 10-40 hrs [Hagenaar 2001]. Recycling time of magnetic flux in the corona, in which reconnection plays a strong role, is of the order of 1.4 hrs [Close et al. 2004].

5. Parker’s Nanoflare Model (1972, 1994) • Current density in the corona is in the form of thin and intense current sheets (tangential discontinuities). • “…the X-ray luminosity of the Sun… is a consequence of a sea of reconnection events---nanoflares---in the local surfaces of tangential discontinuity…”

6. Parker's Model (1972) Straighten a curved magnetic loop Photosphere

7. Reduced MHD equations low  limit of MHD

8. Magnetostatic equilibrium Key Question: What is the nature of the solutions, given a sufficiently complicated footpoint mapping? Questions regarding Parker’s claim: van Ballegooijen [1985], Longcope and Strauss [1994]. Cowley et al. [1997] and others.

9. A theorem onParker's model For any given footpoint mapping connected with the identity mapping, there is at most one smooth equilibrium. Caveat: A proof based on reduced MHD equations, periodic boundary condition in x [Related results by Aly 2005 and Low 2006]

10. Implication An unstable but smooth equilibrium cannot relax to a second smooth equilibrium, hence must have current sheets.

11. Possible current sheet topology in line-tied geometry

12. c. f. recent results in QSLs • Quasi-separatrix layers (QSLs) [Titov et al. 2004]: • Strong current layers can form in high Q region.

13. c. f. recent results in QSLs • A typical constant Q surface[Titov et al. 2004] where strong current layers tend to form. • Note that the QSL rotates 90 degrees from top to bottom. • Qualitatively consistent with the geometry of our predicted current sheet. From [Titov et al. 2004]

15. Tectonics model of coronal heating • Proposed byPriest, Heyvaerts and Title [2002]---by obvious analogy with geophysical plate tectonics. • Heating is provided by dissipation and reconnection via current sheets at separatrix (or quasi-separtrix) surfaces between neighboring cells due to footpoint motion. [Priest, Heyvaerts and Title, 2002]

16. Tectonics model of coronal heating • In arguing that heating by reconnection is more important than Ohmic dissipation, Priest, Heyvaerts and Title [2002] invoke a calculation of the average Ohmic heating rate per unit solar surface (for short times): • It is obviously too small for coronal heating due to the 1/2 dependence on resistivity. (note that w2/ is of the order of the resistive time scaler , which is usually much larger than the coherence time coh) • Goal: to study the dependence on resistivity by simulation for long times, varying the coherence time, which will show that  is roughly independent on , when coh is much smaller than r.

17. Reduced MHD equations low  limit of MHD

18. Reduced MHD equations -- 2D • Only one transverse coordinate (x) so that nonlinear terms are identically zero. This geometric constraint deliberately excludes for the moment nonlinear dynamics such as instability and reconnection.

19. Boundary conditions • Periodic boundary condition in x: • Line-tied boundary condition in z= 0andz = L : • For a random footpoint driving, has a coherence time .

20. Constant drive --- simulation

21. Random drive • Random of the form: where is small enough such that .

22. Random drive --- typical fields

23. Random drive --- heating rate • Average heating rate decreases when  increases. Dependence less than 1/, but very different from 1/2.

24. Random drive --- heating rate/small coh • Average heating rate almost independent of  .

25. Random drive --- transverse B/small coh • hasalmosta1/2 dependence.

26. Random drive --- very small  • can still get unphysically large fora small. • The time it takes for to settle to a average level is longer than we can run it for, forsmall. However, this figure shows that it is already over an order of magnitude larger than Bz • Physically, the growth of By will be limited by processes such as instabilities and reconnection.

27. w lc B L Bz B B from random footpoint motion

28. B from random footpoint motion If dissipation is due to Ohmic heating with resistivity  where is the statistically expected distance moved by a footpoint with velocity v0 in a random walk motionin a resistive time . • Heating rate is independent of . • If w  v0coh, , which is roughly of the same order of magnitude required for coronal heating. • However, is unphysically large fora small.

29. B from random footpoint motion If dissipation is due instability/reconnection when By ~ f Bz • Heating rate is independent of f(and dissipation mechanism). • If w  v0coh, , which is roughly of the same order of magnitude required for coronal heating. • Now there is no unphysically large , if instability and reconnection dissipates energy fast enough when f ~ O(1).

30. Conclusions • Parker’s model of nanoflares: some rigorous results on current sheets at quasi-separatrix layers. A tectonics model of coronal heating driven by the magnetic carpet [Priest, Heyvaerts and Title 2002] is considered using 2D RMHD simulations. It is shown numerically and by scaling analysis that for a random footpoint driving, the heating rate <H> is independent of , and By is proportional to 1/1/2 when coh is much less than r . In realty, the growth of By would be limited by instabilities or reconnection. We conjecture that the heating power delivered will be independent of the threshold at which instability occurs. To be tested in 3D studies.