jan. 27, 2005hmi team meeting the minimum energy fit the slowest motion required by induction dana...

Jan. 27, 2005 HMI team meeting

The Minimum Energy FitThe Minimum Energy FitThe Minimum Energy FitThe Minimum Energy Fit

The Slowest Motion Required The Slowest Motion Required by Inductionby Induction

Dana LongcopeDana LongcopeMontana State UniversityMontana State University

Work supported by DoD MURI grantWork supported by DoD MURI grant


The dataThe dataThe dataThe data

slices of MHD simulation slices of MHD simulation (Tetsuya Magara)(Tetsuya Magara)

Time-resolved vector magnetogramsTime-resolved vector magnetograms

00:03:16 00:04:05


The Induction equationThe Induction equationThe Induction equationThe Induction equation

hzhz Bt

BvB

zvVertical induction Eq.Vertical induction Eq.

Known from dataKnown from data



hzhz Bt

BvB

zvVertical induction Eq.Vertical induction Eq.

To be foundTo be found

1 Equation 3 unknowns1 Equation 3 unknowns



hzhz Bt

BvB

zv

zzzB

t

zh

hz

hz

hhhhh

vv B

Bv

BvBvB

Vertical induction Eq.Vertical induction Eq.

5 more unkowns5 more unkowns

Horizontal induction Eq.Horizontal induction Eq.

1 Equation 3 unknowns1 Equation 3 unknowns


Solving Induction for Solving Induction for vvSolving Induction for Solving Induction for vv

2zv hhzhz Bt

B

vB

zvB ˆvz hzh B

Introduce unknown scalar Introduce unknown scalar potentialspotentials

Induction Eq. Induction Eq. Possion Eq. for Possion Eq. for


Processing dataProcessing dataProcessing dataProcessing data

Region w/ info for velocityRegion w/ info for velocity

+

t2

t1

t3/2


Processing dataProcessing dataProcessing dataProcessing data

-

12

1,2,

tt

BB

t

B zzz

t2

t1

t3/2


Solve Poisson Eq.Solve Poisson Eq.Solve Poisson Eq.Solve Poisson Eq.2hz

t

B

(x,y) inside(x,y) insideRegionRegion

00 on bndryon bndry

Induct’n eq.Induct’n eq.is is exactlyexactlysatisfiedsatisfied


Finding other componentsFinding other componentsFinding other componentsFinding other components

z

hh B

zBv

ˆv)v,( zz

free fieldsfree fields

dydxW zh 22

zz v)v,()v,( v

Define function to optimizeDefine function to optimize

Solution will have smallest Solution will have smallest vv consistent w/ data consistent w/ data

Bonus:Bonus: v .B=0v .B=0


The MinimizationThe MinimizationThe MinimizationThe Minimization

2||

)ˆ(v

B

zB

hz

2z

2z B

)v(ˆ

B

hzBz

vvzz::

::

known

known

held fixed

held fixed

Elliptic operator


Magnetogram gridMagnetogram gridMagnetogram gridMagnetogram grid

BzBh BzBh BzBh

BzBh BzBh BzBh

BzBh BzBh BzBh


Solving for Solving for Solving for Solving for

vy

vy

vx

vx

vz

vz

vz

vz

vz

vz

vz

vz

vz

2z

2z B

)v(ˆ

B

hzBz

Finite DifferenceFinite DifferenceElliptic operatorElliptic operator

Variables on Variables on staggered meshstaggered mesh


Solving for Solving for Solving for Solving for

2z

2z B

)v(ˆ

B

hzBz

Solve elliptic equationSolve elliptic equationWithin strong-fieldWithin strong-field

boundaryboundary

Current implementation:Current implementation:Relaxation methodRelaxation method


Minimizing the EnergyMinimizing the EnergyMinimizing the EnergyMinimizing the Energy

Relaxation steps

Actual flow

dydxW zh 22

zz v)v,()v,( v

Alternate & vz minimization


Comparison of ResultsComparison of ResultsComparison of ResultsComparison of Results

MHD MEF


Comparison of ResultsComparison of ResultsComparison of ResultsComparison of Results

MHD MEF

Fup = 3.3 X 1022 cm3/s Fup = 1.5 X 1022 cm3/s


Real data: AR8210Real data: AR8210Real data: AR8210Real data: AR8210

• IVM (U.Hawaii)IVM (U.Hawaii) 3 min cadence3 min cadence 1.1” resolution1.1” resolution• Ambig’ty res.Ambig’ty res. Canfield et al. 1993Canfield et al. 1993• avg. 5 ‘gramsavg. 5 ‘grams• t = 30 min.t = 30 min.

(Courtesy KD Leka)(Courtesy KD Leka)

Boundary |Bz|=60 G



dBdBzz/dt /dt (grey)(grey)

(x,y) (x,y) (contours)(contours)


Doppler FlowsDoppler FlowsDoppler FlowsDoppler Flows

dydxuW zzh 22

zz v)v,()v,( v

• Find Find uuzz(x,y)(x,y) by other means by other means (e.g. Doppler measurments)(e.g. Doppler measurments)• Incorporate using new functionalIncorporate using new functional

i.e. find consistent flow with small i.e. find consistent flow with small horizontal velocities which best horizontal velocities which best matchesmatches observations observations

Mismatch w/ DopplerMismatch w/ Doppler


Doppler FlowsDoppler FlowsDoppler FlowsDoppler Flows

Use vUse vzz(x,y) from solution as Doppler signal(x,y) from solution as Doppler signal


Data ConsiderationsData ConsiderationsData ConsiderationsData Considerations• Cadence:Cadence:

– Large Large t t problems problems ((t ~ 15-30 min)t ~ 15-30 min)

• Co-alignmentCo-alignment• Seeing produces spurious Seeing produces spurious vv((xx))• Ambiguity resolutionAmbiguity resolution

– ttn n t tn+1n+1 consistency - consistency - importantimportant

• Doppler measurements Doppler measurements (easily used)(easily used)– Best: Best: vv for magnetized plasma for magnetized plasma


Implementing the Implementing the AlgorithmAlgorithm

Implementing the Implementing the AlgorithmAlgorithm

• Define outer boundary Define outer boundary (necessary?)(necessary?)

(trickiest step at present)(trickiest step at present)

• Solve elliptic equation(s) Solve elliptic equation(s) – relaxation?– relaxation?

(could be made much faster)(could be made much faster)

• Artifacts near PILArtifacts near PIL

(smoothing, lotsa relaxin’)(smoothing, lotsa relaxin’)

• Use Use vv((xx)) from previous step(s) as from previous step(s) as additional constraintadditional constraint

jan. 27, 2005hmi team meeting the minimum energy fit the slowest motion required by induction dana...

Documents

data slide

ar8210 slide

s slide

g slide

satisfied slide

doppler signal slide

staggered mesh slide

y contours