1. HMI & Photospheric Flows 1.Review of methods to determine surface plasma flow; 2.Comparisons between methods; 3.Data requirements; 4.Necessary computational resources; 5.Possible improvements to methods.

2. General Approach From 2D data arrays, f 1 (x 1,x 2 ) & f 2 (x 1,x 2 ), find vector flow v(x 1,x 2 ) consistent with: 1.Observed evolution,  f(x 1,x 2 ) = f 2 (x 1,x 2 ) – f 1 (x 1,x 2 ) 2.Other possible assumptions: –Magnetic induction eqn.,  B n /  t =  t  (v n B t -v t B n ) –Continuity equation,  f/  t +  t  (v t f) = 0 –Doppler velocities – more later v(x 1,x 2 ) might have 2 or 3 components

3. General Approach, cont’d: Ideally, with finite difference equations, cadence should beat “Courant cadence”   t C =  x/v max analog of numerical Courant condition: –time step limited by propagation speed of information  x  pixel size; v max  expected max. flow speed “low cadence” is  t >  t C  t   t C very rare in solar physics! (usually,  t >>  t C )

4. Pixels =.5” ~ 363 km, resolution ~ 1.5” ~ 1100 km Photospheric c sound  (  kT/m) 1/2  9 km/s Courant Cadence:  t HMI  (363 km)/(9 km/s)  40 sec. LOS Mag. Field Cadence,  t LOS ~ 60 sec. Vector Mag. Field:  t VEC ~ 600 sec. Typical v ~ 2 km/s, and resolution ~ 1100 km, so  t PRACTICAL ~ 550 sec. HMI Capabilities

5. Current Methods 1.Local Correlation Tracking (LCT) 2.“Inductive” Methods (ILCT, MEF, …) 3.Feature Tracking (FT)

6. 1. Local Correlation Tracking (LCT) Take subregions,  pixels wide, of f 1 & f 2, find, e.g., –shift  x that minimizes difference  f ; or –shift  x of peak in (Fourier) correlation func’n Sub-pixel shifts found by interpolation – SLOW! Most algorithms solve advection equation,  f/  t + (v t   t ) f = 0 Can be used on intensity images, LOS, & vector magnetograms from HMI. Cadence must be slow enough that  f noise <  f advection Workable with very low cadence data:  t  100  t C

7. 27 Jan 2005HMI Planning Meeting7 LCT applied to magnetograms: Démoulin & Berger’s (2003) analysis of flux transport velocity Motion of flux across photosphere, u f, is a combination of horizontal & vertical flows acting on non- vertical fields.

8. LCT, cont’d Hence, flows u LCT from LCT on magnetograms: 1.are not generally identical to plasma velocity v 2.solve advection equation, not continuity equation 1.Given vector B, can assume u f = u LCT, and thereby find v from u LCT algebraically (ADC). 2. Q: How good does LCT do? A: Pretty good!

9. 27 Jan 2005HMI Planning Meeting9

10. 27 Jan 2005HMI Planning Meeting10

11. 27 Jan 2005HMI Planning Meeting11 A Comparable Data Set: Flare Genesis Experiment Balloon-borne (Antarctic) observations of NOAA 8844, 25 Jan 2000 54 vector magnetograms, ~2.3/5.3 min. per hi-res:.18” pixels (130.5 km), ~520 x 520 pix LCT differenced over t i +/-10, for  t ~85 min. Doppler maps, too! (No info. on method.) Tracking of white light images underway

12. 27 Jan 2005HMI Planning Meeting12 FGE Movie

13. 27 Jan 2005HMI Planning Meeting13 FGE: White Light vs. Mag

14. 27 Jan 2005HMI Planning Meeting14 FGE: Larger 

15. 27 Jan 2005HMI Planning Meeting15 Near future: Improvement in sub-pixel interpolation – added speed. Future: Convert to FORTRAN; parallelize. Compute on tiles, not on each pixel. Modifications to LCT

16. 2. Inductive Methods Use finite diff. approx. to magnetic induction equation’s normal comp. as add’l constraint. Purely inductive methods need  t   t C Methods currently available: ILCT, MEF, Kusano et al. (2002), MSR (Georgoulis et al., 2005, in prep.) All methods return (v x, v y, v z ) at photosphere, where (v  B) = 0; parallel flow unconstrained by ind’n eqn. Post-processing with Doppler data can give v || B –NB: NOT Doppler from Stokes I (Chae et al., 2004)

17. Inductively Derived Flows are Consistent with Induction Eqn’s Normal Component! Directly measured Derived Inductively

18. Directly measured Derived by new method? What about other components?DerivedInductively From NLFFF Extrapolation? at photosphere, z = 0above photosphere, z > 0

19. 27 Jan 2005HMI Planning Meeting19 A) ILCT: Modify LCT solution to match induction equation Solve for ,  with 2D divergence and 2D curl (n- comp), and the approximation that u f =u LCT : Let NB: if only B LOS is known, we can still solve for ,  !

20. B) Minimum Energy Fit (MEF) Also uses induction equation’s normal component to derive flow, with additional assumption that integral of squared velocity is minimized. Applicable to vector magnetograms. More from D. Longcope, shortly!

21. Other Inductive Methods Kusano et al. (2002): get v from LCT flow, derive additional flow for consistency with induction equation. Georgoulis (2005, in prep): Use (i) “minium structure” & (ii) “coplanarity” assumptions, with (iii) induction equation to derive (iv) velocity perpendicular to magnetic field. (System overconstrained.)

22. 27 Jan 2005HMI Planning Meeting22 Prelim. Comparison of Inductive Methods Used MHD simulations of Magara (2001) Given B(x,y,z=0,t), “practioners” computed v(x,y,z=0,t), and were then told actual v.

23. 27 Jan 2005HMI Planning Meeting23 Some Prelim Comparisons

24. 27 Jan 2005HMI Planning Meeting24 Some Prelim Comparisons

25. 27 Jan 2005HMI Planning Meeting25 Some Prelim Comparisons

26. 27 Jan 2005HMI Planning Meeting26 Some Prelim Comparisons

27. 3. Feature Tracking Useful with WL images & magnetograms. Algorithms: –White Light: L. Strous –Active region fields: B. Welsch, G. Barnes –Quiet Sun fields: C. DeForest, M. Hagenaar, C. Parnell, B. Welsch Does not return v(x,y); rather, gives velocity of “patches” of photosphere. Easily incorporated in pipeline.

28. 27 Jan 2005HMI Planning Meeting28 Feature Tracking in AR 8038

29. 27 Jan 2005HMI Planning Meeting29 Conclusions Planned data cadences are compatible with existing velocity inversion algorithms. LCT can be used to derive flows in HMI’s intensity, LOS, and vector field maps. ILCT, MEF suitable for determining three- component photospheric magnetic flows. Doppler data from Stokes’ profiles (zero crossing of V, or central minima of Q,U) desirable. Significant improvement in computational performance of LCT algorithms is needed for real- time analysis.