Azimuth disambiguation of solar vector magnetograms

M. K. GeorgoulisM. K. Georgoulis

JHU/APL11100 Johns Hopkins Rd., Laurel, MD 20723, USA

Ambiguity Workshop

Boulder, Co, 26 Sep 2005

Outline

Boulder, 09/26/05

Azimuth disambiguation: a brief introduction

Techniques:

Examples and comparison

Conclusions

Structure minimization

Nonpotential magnetic field calculation

Introduction

BBlonglong

BBtrantranss

BBtrantranss

SunSun

Therefore:

is unambiguously measured

is unambiguously measured

The orientation of is ambiguous

with two equally likely -differing solutions

longB

transB

transB

Vector magnetic field measurements are performed across and perpendicular to the observer’s line of sight

The deduced quantities are ,

and the azimuth of

longB transB

transB

The properties of the transverse Zeeman effect remain invariant under the transformation πφφ

φ

Boulder, 09/26/05

The problem at hand

Any physically meaningful disambiguation technique should rely on the local, rather than the line-of-sight, magnetic field components

Therefore, the location of the target magnetic configuration is crucial

BB & BB transhlongz

If the target AR is close to disk center

everywhere πΔφ

If the target AR is far from disk center

BB & BB transhlongz variableand π Δφ

Boulder, 09/26/05

The structure minimization technique

Technique developed by Georgoulis, LaBonte, & Metcalf (2004)

Semi-analytical, relying on physical and geometrical arguments

Assumptions and analysis:

Consider Ampere’s law where :bBB

bB4π

cJ and b

4π

cBJ where; JJJ 2121

BJ 2 Notice that

zz 21 J and J Both can be readily calculated

The current density becomes fully known if is found zB/2J

The current density maximizes on the interface between flux tubes

2J

2J Minimizing the magnitude leads to the minimum possible interfaces (structure) between flux tubes, i.e. to space-filling magnetic fields This is the minimum structure approximation

Boulder, 09/26/05

The structure minimization technique (cont’d)

The magnitude becomes minimum when 2J

y

Bb

x

Bb

bb

b

z

Byx2

y2x

z

2J zB/Both and do not show inter-pixel dependences since the only differentiated quantity is the ambiguity-free B

There are only two possible values for and at each location

2J zB/

To perform azimuth disambiguation, we assume that

in sunspots (physical argument)

has to be minimum in plages, because there and

0 zB/

z2J zBB z BJ 2(geometrical argument)

Boulder, 09/26/05

Implementation - Analysis

Vertical magnetic field Normalized wl continuum We construct the following quantity:

1ω0 , ω1ω re wheJωz

BωF pps2ps z

pω Typical - profile pω

The function F is free of inter-pixel dependences and has only two values at any given location. We choose the azimuth solution that minimizes FThis azimuth solution is treated as a good initial guess

Boulder, 09/26/05

Implementation - Numerics

The final solution is reached numerically, assuming smoothness of the azimuth solution and eliminating artificial gradients in the vertical fieldClose to disk center

The two solutions for Bz are very similar; the two azimuth solutions are very different

Far from disk centerAzimuth solutions may be similar; the two solutions for Bz are very different

The initial azimuth solution is smoothed via an iterative Jacobi relaxation process

The initial solution of Bz is filtered by means of the Lee filtering technique

Boulder, 09/26/05

Synopsis of the structure minimization technique

Strengths:

Fast (average running time ~ 5 – 10 min) and fully automatic

Effective - Propagation of a local erroneous solution is precluded because there are no inter-pixel dependences

Applicable to ARs both close and far from disk center

Weaknesses:

Ambiguous - Different numerical treatment when the AR is “ far ” or “ close ” to disk center (How “ far ” is far and how “ close ” is close?)

Restrictive - Initial solution reached from assumptions regarding sunspots and plages – what about other types of magnetic structure (canopies, EFRs, etc.) ?

Too much power on smoothing – limited control during the numerical phase

Boulder, 09/26/05

Nonpotential magnetic field calculation

Technique developed by Georgoulis (2005)

Semi-analytical, self-consistent solution, no final smoothing

Analysis:

Any closed magnetic structure has a potential and a nonpotential component:

npp BBB Gauge conditions:

All three components above are divergence-free:

0B and , 0B , 0B npp

The total and potential component share the same boundary condition on Bz, so the nonpotential component is purely horizontal:

0nB so nBnBSnpSpS

The nonpotential component is responsible for any electric currents

J /c π4BB np Boulder, 09/26/05

Nonpotential magnetic field calculation (cont’d)

Assumption: Assume that the vertical electric current density is knownThen the definition of the nonpotential field is

zznpSnpnp Jc

4πB and , 0nB , 0B

From this definition, is fully-determined (Chae 2001): npB

y )(Jkk

ik x )(J

kk

ikB z2

y2x

x1z2

y2x

y1np

Therefore, if we know we can find the distribution of Bz whose potential extrapolation + best match the observed horizontal magnetic field

npB

npB

This can be done iteratively, starting from any random, but relatively smooth, initial configuration of Bz

The problem is how to find Jz, or some proxy of it, prior to the disambiguation

Boulder, 09/26/05

Finding a proxy vertical electric current density

Objective: Extract as much information on Jz as possible from the ambiguity-free longitudinal magnetic fieldBRelation between the heliographic and the line-of-sight magnetic field components:

zy,x,j ; 1,2i ; BγBβBαB z ji ηy ji ξ xjj i

Because of the azimuthal ambiguity we have

2 η1 η2 ξ1 ξ BB and BB

Therefore, the average of the two ambiguity solutions is fully known and ambiguity-free :

21 BB 1/2

B z γy γx γBB 1/2B zzzy zx 21av

From this we can find a proxy vertical current density

z avp B π4

cJ

z

Boulder, 09/26/05

Examples of the proxy vertical current

Longitudinal magnetic field Proxy vertical current densityThe used proxy generally underestimates the actual vertical current

density with the bias depending on the target’s position on the solar disk

1B

2B

avB

1B

2B

avB

Close to disk center

Far from disk centerTherefore, we are pursuing a minimum-current azimuth solution Boulder,

09/26/05

Examples and comparison

Structure minimization Nonpotential magnetic field calculation

AR 9026

IVM, NOAA AR 9026

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Examples and comparison (cont’d)

Structure minimization Nonpotential magnetic field calculation

ASP, NOAA AR 7205, 06/24/92

AR 7205

Boulder, 09/26/05

The nonpotential field calculation in action

Nonpotential magnetic field vector

onB

Proxy vertical electric current density

zz 21 BB

2

1

z

21p hhzBB

2

1J

Boulder, 09/26/05

Limitations of the nonpotential field calculation

The nonpotential field calculation performs a potential extrapolation in each iteration. Therefore, it generally requires flux-balanced magnetic structures on the boundary The technique relies on Bz. Therefore, it may be compromised where magnetic fields are strongly horizontal and / or show only a weak vertical component

Simulation fan_simu_ts56

h

z

B

B

0.01B

B ,G 500 B

h

zh

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Other examples of nonpotential field disambiguation

Simulation cl1

ASP, 03/11/03

Boulder, 09/26/05

Synopsis of the nonpotential field calculation technique

Boulder, 09/26/05

Strengths:

Very Fast (average running time ~ 1 – 5 min) and fully automatic

Effective to ARs both close and far from disk center

Physically sound and eliminates the need of smoothing

Weaknesses:

Biased toward Bz – it may be compromised in case of strong horizontal fields and very weak vertical fields

Can the technique be improved further?

Yes, if one uses a more reliable estimate of the proxy vertical current density The “off – center” case of the minimum vertical current (Semel &

Skumanich 1998)

Modeling of the vertical current density (Gary & Demoulin 1995) However, any refinement should not

compromise the computational speed of the technique !

Conclusions

Two reliable techniques for a routine azimuth disambiguation of solar vector magnetograms

Both are fast and automatic

Near real-time disambiguation of SOLIS, Solar-B, SDO/HMI data

Structure minimizatio

n

Non-potential field

calculation

Boulder, 09/26/05

http://sd-www.jhuapl.edu/FlareGenesis/Team/Manolis/codes/ambiguity_resolution/

Want to try yourself ? Check out the nonpotential field method @