1. Jan. 27, 2005HMI team meeting The Minimum Energy Fit The Slowest Motion Required by Induction Dana Longcope Montana State University Work supported by DoD MURI grant

2. Jan. 27, 2005HMI team meeting The data slices of MHD simulation (Tetsuya Magara) Time-resolved vector magnetograms 00:03:16 00:04:05

3. Jan. 27, 2005HMI team meeting The Induction equation Vertical induction Eq. Known from data

4. Jan. 27, 2005HMI team meeting The Induction equation Vertical induction Eq. To be found 1 Equation 3 unknowns

5. Jan. 27, 2005HMI team meeting The Induction equation Vertical induction Eq. 5 more unkowns Horizontal induction Eq. 1 Equation 3 unknowns

6. Jan. 27, 2005HMI team meeting Solving Induction for v Introduce unknown scalar potentials Induction Eq.  Possion Eq. for 

7. Jan. 27, 2005HMI team meeting Processing data Region w/ info for velocity + t2t2 t1t1 t 3/2

8. Jan. 27, 2005HMI team meeting Processing data - t2t2 t1t1 t 3/2

9. Jan. 27, 2005HMI team meeting Solve Poisson Eq.  (x,y) inside Region  0 on bndry  Induct’n eq. is exactly satisfied

10. Jan. 27, 2005HMI team meeting Finding other components free fields Define function to optimize Solution will have smallest v consistent w/ data Bonus: v.B=0

11. Jan. 27, 2005HMI team meeting The Minimization dvz:dvz:dvz:dvz: dy : known held fixed Elliptic operator

12. Jan. 27, 2005HMI team meeting Magnetogram grid BzBz BhBh BzBz BhBh BzBz BhBh BzBz BhBh BzBz BhBh BzBz BhBh BzBz BhBh BzBz BhBh BzBz BhBh

13. Jan. 27, 2005HMI team meeting Solving for y y yy y f vyvy vyvy vxvx vxvx yyy y vzvz f vzvz f vzvz f vzvz f vzvz f vzvz f vzvz f vzvz f vzvz yy y y yy y y Finite Difference Elliptic operator Variables on staggered mesh

14. Jan. 27, 2005HMI team meeting Solving for y Solve elliptic equation Within strong-field boundary Current implementation: Relaxation method

15. Jan. 27, 2005HMI team meeting Minimizing the Energy Relaxation steps Actual flow Alternate dy & d v z minimization

16. Jan. 27, 2005HMI team meeting Comparison of Results MHDMEF

17. Jan. 27, 2005HMI team meeting Comparison of Results MHDMEF F up = 3.3 X 10 22 cm 3 /s F up = 1.5 X 10 22 cm 3 /s

18. Jan. 27, 2005HMI team meeting Real data: AR8210 IVM (U.Hawaii) IVM (U.Hawaii) 3 min cadence 3 min cadence 1.1” resolution 1.1” resolution Ambig’ty res. Ambig’ty res. Canfield et al. 1993 Canfield et al. 1993 avg. 5 ‘grams avg. 5 ‘grams D t = 30 min. D t = 30 min. (Courtesy KD Leka) Boundary |B z |=60 G

19. Jan. 27, 2005HMI team meeting Real data: AR8210 dB z /dt (grey) f (x,y) (contours)

20. Jan. 27, 2005HMI team meeting Real data: AR8210

21. Jan. 27, 2005HMI team meeting Doppler Flows Find u z (x,y) by other means Find u z (x,y) by other means (e.g. Doppler measurments) (e.g. Doppler measurments) Incorporate using new functional Incorporate using new functional i.e. find consistent flow with small horizontal velocities which best matches observations Mismatch w/ Doppler

22. Jan. 27, 2005HMI team meeting Doppler Flows Use v z (x,y) from solution as Doppler signal

23. Jan. 27, 2005HMI team meeting Data Considerations Cadence:Cadence: –Large D t  problems ( D t ~ 15-30 min) Co-alignmentCo-alignment Seeing produces spurious v(x)Seeing produces spurious v(x) Ambiguity resolutionAmbiguity resolution – t n  t n+1 consistency - important Doppler measurements (easily used)Doppler measurements (easily used) –Best: v for magnetized plasma

24. Jan. 27, 2005HMI team meeting Implementing the Algorithm Define outer boundary (necessary?)Define outer boundary (necessary?) (trickiest step at present) (trickiest step at present) Solve elliptic equation(s) – relaxation?Solve elliptic equation(s) – relaxation? (could be made much faster) (could be made much faster) Artifacts near PILArtifacts near PIL (smoothing, lotsa relaxin’) (smoothing, lotsa relaxin’) Use v(x) from previous step(s) as additional constraintUse v(x) from previous step(s) as additional constraint