Energy & Helicity Flux into the Corona

1. Energy & Helicity Flux into the Corona • I’ll discuss secular trends, ‘the relatively consistent movement of a variable over a long period’ – ‘DC’ as opposed to ‘AC’ (wave) flux • I’ll assume ideal MHD applies, E = -(v x B)/c • First, express fluxes in equations… • …then in terms of boundary motions that • we all know and love!

2. Energy Flux • Poynting Flux, with E = -(v x B)/c: • Rewrite using BAC – CAB rule, take z-comp: (*) (*)

3. Emergence of Energy • Archetype is vertical, perpendicular flow – e.g., vz advecting a flux tube’s apex (*) • NB: does not include vertical, parallel flow – e.g., continued emergence of a “U-bolt” (*)

4. Emergence of Free Energy (EF) • EF from emerging non-potential fields (*)  related to helicity emergence • EF from shearing or twisting (*)  propagation of twist (*) (cf., Nightingale’s rotating sunspots) • possibly from emerging potential fields into potential background fields!  Depends on global topology! (*)

5. Relative Helicity, Defined • Refers to two B-fields and their vector potentials: • B is the actual magnetic field, which may or may not vanish (Bn  0) on the boundary V of V. • BP is the current-free magnetic field that matches the actual field Bn on the boundary. • A is a vector pot’l for B, B = ( x A), and AP is vector pot’l for BP (“potential field’s potential”). • In the general case (barring symmetry), B BP H  0. • This helicity is gauge invariant (Berger & Field, 1984).

6. Vector Potential, AP • By construction, n· ( x A) = Bz , (·AP)= 0, and (AP· n) = 0. • Regions of Bz 0 on the photosphere are sources of AP, which is a 2-D vector field. • AP is circumferential (*), and completely specified by Bz – knowing Bh unnecessary!

7. Helicity Flux – Poynting-like Eqn. • From Faraday’s Law for B/t , and assuming A/t = -cE = (v x B), Berger & Field computed dH/dt across a surface (z = normal direction) • Rz is a helicity flux density (Mx2/cm2/s).

8. Emergence of Helicity • Emergence of helicity-carrying (“helical”— basically, twisted!) fields. (*) • e.g., current-carrying fields observed by Leka et al. (1996) • Emergence of non-helical (potential) fields, resulting in non-potential topology!  Depends on global topology! (*)

9. Helicity Flux by Shearing/Twisting • Motion of magnetic flux along contours of AP leads to a flux of helicity. (*) • Contours are circular, so relative circular motion, a.k.a., braiding, leads to helicity injection. • Propagation of twist along flux tube (*) should lead to observable foot point braiding.

10. Q. How can we determine the photospheric velocity? • (Not just any photospheric velocity – we want the velocity that affects the photospheric magnetic field.) • Doppler info can give LOS velocity, e.g., Chae (2004) • Local Correlation Tracking (LCT), e.g., Chae (2001) • Newer methods: Kusano et al. (2002), ILCT, MEF, Minimum Structure, ??? • More at Velocity Shootout in WG1 splinter session on Fri!

11. Démoulin & Berger’s (2003) Hypothesis • LCT applied to Bz or BLOS from photospheric magnetograms gives the pattern velocity. • Pattern velocity uLCT is related to plasma velocity v by: dE/dt and dH/dt can be written in terms of (uLCT Bz)!

12. dE/dt and dH/dt in terms of uLCT Bz • From LCT on vector magnetograms, can compute fluxes of magnetic energy and helicity. • With LOS magnetograms, and an assumed Bh (e.g., FFF that matches coronal observations), can estimate energy & helicity fluxes.

13. Helicity & Energy Fluxes • Barring symmetry, if B  BP, then H  0. |dH/dt| > 0  B  BP • Since BP is unique, and minimum energy given Bz(x,y), B  BPalso means free energy is present, EF 0. • Knowledge of Bz specifies AP. • Knowledge of v allows computation of dE/dt & dH/dt.

14. Emerging Omega Loop (BACK)

15. “Trombone Slide” Emergence (BACK)

16. Free Energy from Emergence of Twisted Flux (BACK)

17. Free Energy from Shearing Flows (BACK)

18. Free Energy Propagating Twist (BACK)

19. Free Energy from Non-potential Topology (BACK)

20. Vector Potential Pictorial (BACK)

21. Braiding Motions Inject Helicity (BACK)

22. Vector Potential Pictorial (BACK)

23. Vector Potential Pictorial (BACK)