1. Dec. 10, 2004RHESSI/SOHO/TRACE The Minimum Energy Fit Consistent with Induction at Minimum Possible Cost Dana Longcope Montana State University Work supported by DoD MURI grant

2. Dec. 10, 2004RHESSI/SOHO/TRACE The data slices of MHD simulation (Tetsuya Magara) Time-resolved vector magnetograms 00:03:16 00:04:05

3. Dec. 10, 2004RHESSI/SOHO/TRACE The Induction equation Vertical induction Eq. Known from data

4. Dec. 10, 2004RHESSI/SOHO/TRACE The Induction equation Vertical induction Eq. To be found 1 Equation 3 unknowns

5. Dec. 10, 2004RHESSI/SOHO/TRACE The Induction equation Vertical induction Eq. 5 more unkowns Horizontal induction Eq. 1 Equation 3 unknowns

6. Dec. 10, 2004RHESSI/SOHO/TRACE Solving Induction for v Introduce unknown scalar potentials Induction Eq.  Possion Eq. for 

7. Dec. 10, 2004RHESSI/SOHO/TRACE Processing data Region w/ info for velocity + t2t2 t1t1 t 3/2

8. Dec. 10, 2004RHESSI/SOHO/TRACE Processing data - t2t2 t1t1 t 3/2

9. Dec. 10, 2004RHESSI/SOHO/TRACE Solve Poisson Eq.  (x,y) inside Region  0 on bndry  Induct’n eq. is exactly satisfied

10. Dec. 10, 2004RHESSI/SOHO/TRACE Finding other components free fields Define function to optimize Solution will have smallest v consistent w/ data

11. Dec. 10, 2004RHESSI/SOHO/TRACE Minimizing the Energy Relaxation steps Actual flow

12. Dec. 10, 2004RHESSI/SOHO/TRACE Comparison of Results MHDMEF

13. Dec. 10, 2004RHESSI/SOHO/TRACE Comparison of Results MHDMEF F up = 3.3 X 10 22 cm 3 /s F up = 1.5 X 10 22 cm 3 /s

14. Dec. 10, 2004RHESSI/SOHO/TRACE Comparison of Results Point-to-point comparison (is this fair?) vzvz vyvy up r=69%r=79% up

15. Dec. 10, 2004RHESSI/SOHO/TRACE 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

16. Dec. 10, 2004RHESSI/SOHO/TRACE Doppler Flows Use v z (x,y) from solution as Doppler signal

17. Dec. 10, 2004RHESSI/SOHO/TRACE The Abbett Challenge (ANMHD) - 03:12:00 06:24:00 Assume small changes!

18. Dec. 10, 2004RHESSI/SOHO/TRACE Find f and then relax Relaxation steps Actual flow Why is minimum above W of actual flow !?

19. Dec. 10, 2004RHESSI/SOHO/TRACE Comparison of Results ANMHD MEF

20. Dec. 10, 2004RHESSI/SOHO/TRACE SummarySummary MEF: 2 vec. m-grams  p-spheric flow vMEF: 2 vec. m-grams  p-spheric flow v Flow is consistent w/ Induction Eq.Flow is consistent w/ Induction Eq. Has smallest possible |v| overallHas smallest possible |v| overall Can incorporate Doppler signalCan incorporate Doppler signal Test problems:Test problems: –Magara: good/excellent –Abbett: fair/poor

21. Dec. 10, 2004RHESSI/SOHO/TRACE MEF: 2 vec. m-grams  p-spheric flow vMEF: 2 vec. m-grams  p-spheric flow v Flow is consistent w/ Induction Eq.Flow is consistent w/ Induction Eq. Has smallest possible |v| overallHas smallest possible |v| overall Can incorporate Doppler signalCan incorporate Doppler signal Test problems:Test problems: –Magara: good/excellent –Abbett: fair/poor time between m-grams may be too great approximate w/ induction @ constant v approximate w/ induction @ constant v SummarySummary