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

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.

General Approach• From 2D data arrays, f1(x1,x2) & f2(x1,x2), find vector

flow v(x1,x2) consistent with:

1. Observed evolution, f(x1,x2) = f2(x1,x2) – f1(x1,x2)

2. Other possible assumptions: – Magnetic induction eqn., Bn/t = t(vnBt-vtBn)– Continuity equation, f/t + t(vtf) = 0 – Doppler velocities – more later

• v(x1,x2) might have 2 or 3 components

General Approach, cont’d:

• Ideally, with finite difference equations, cadence should beat “Courant cadence” tC = x/vmax

• analog of numerical Courant condition: – time step limited by propagation speed of information

x pixel size; vmax expected max. flow speed

• “low cadence” is t > tC

t tC very rare in solar physics!

(usually, t >> tC)

• Pixels = .5” ~ 363 km, resolution ~ 1.5” ~ 1100 km• Photospheric csound (kT/m)1/2 9 km/s • Courant Cadence:

tHMI (363 km)/(9 km/s) 40 sec.

• LOS Mag. Field Cadence, tLOS ~ 60 sec. • Vector Mag. Field: tVEC ~ 600 sec.

• Typical v ~ 2 km/s, and resolution ~ 1100 km, so tPRACTICAL ~ 550 sec.

HMI Capabilities

Current Methods

1. Local Correlation Tracking (LCT)

2. “Inductive” Methods (ILCT, MEF, …)

3. Feature Tracking (FT)

1. Local Correlation Tracking (LCT)

• Take subregions, pixels wide, of f1 & f2, 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 + (vtt) f = 0• Can be used on intensity images, LOS, & vector

magnetograms from HMI. • Cadence must be slow enough that fnoise < fadvection

• Workable with very low cadence data: t 100tC

27 Jan 2005 HMI Planning Meeting 7

LCT applied to magnetograms: Démoulin & Berger’s (2003) analysis of flux transport velocity

Motion of flux across photosphere, uf, is a combination of horizontal & vertical flows acting on non-vertical fields.

nn

ttf v

B

Bvu 0)B(

B

nftn

tu

LCT, cont’d

Hence, flows uLCT 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 uf = uLCT, and thereby find v from uLCT algebraically (ADC).

2. Q: How good does LCT do? A: Pretty good!




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 ti +/-10, for t ~85 min.

• Doppler maps, too! (No info. on method.)

• Tracking of white light images underway


FGE Movie


FGE: White Light vs. Mag


FGE: Larger


• Near future: Improvement in sub-pixel interpolation – added speed.

• Future: Convert to FORTRAN; parallelize.

• Compute on tiles, not on each pixel.

Modifications to LCT

2. Inductive Methods

• Use finite diff. approx. to magnetic induction equation’s normal comp. as add’l constraint.

• Purely inductive methods need t tC

• Methods currently available: ILCT, MEF, Kusano et al. (2002), MSR (Georgoulis et al., 2005, in prep.)

• All methods return (vx, vy, vz) at photosphere, where (vB) = 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)

Inductively Derived Flows are Consistent with Induction Eqn’s Normal Component!

zyxzyxz B

t

B)( v)( vvBB

Directly measuredDirectly measured Derived InductivelyDerived Inductively

)v()v(

)v()v(

yzyzyxyxy

xzxzxyxyx

Bz

Bxt

B

Bz

Byt

B

vBvB

vBvB

Directly measuredDirectly measured Derived by Derived by new method?new method?

What about other components?

Derived Derived InductivelyInductively

From NLFFFFrom NLFFFExtrapolation?Extrapolation?

at photosphere, z = 0 above photosphere, z > 0


A) ILCT: Modify LCT solution to match induction equation

• Solve for , with 2D divergence and 2D curl (n-comp), and the approximation that uf=uLCT:

.ˆBvB n tttntnfn vBuLet

2tn

t

B

2B tLCTn u

NB: if only BLOS is known, we can still solve for , !

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!

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.)


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.


Some Prelim Comparisons







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.


Feature Tracking in AR 8038


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.

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

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