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

Dec. 10, 2004 RHESSI/SOHO/TRACE

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

Consistent with InductionConsistent with Induction

at Minimum Possible Costat Minimum Possible Cost

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 bndry on 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


Minimizing the EnergyMinimizing the EnergyMinimizing the EnergyMinimizing the Energy

Relaxation steps

Actual flow

dydxW zh 22

zz v)v,()v,( v


Comparison of ResultsComparison of ResultsComparison of ResultsComparison of Results

MHD MEF



MHD MEF

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



Point-to-point comparison (is this fair?)Point-to-point comparison (is this fair?)

vz vy

up

r=69% r=79%

up


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


The Abbett Challenge The Abbett Challenge (ANMHD)(ANMHD)

The Abbett Challenge The Abbett Challenge (ANMHD)(ANMHD)

-

12

1,2,

tt

BB

t

B zzz

03:12:00

06:24:00

Assume small changes!


Find Find and then relaxand then relaxFind Find and then relaxand then relax

Relaxation steps

Actual flow

Why is minimum Why is minimum above W of actual above W of actual flow !?flow !?



ANMHD MEF


SummarySummarySummarySummary• MEFMEF: 2 vec. m-grams : 2 vec. m-grams p-spheric flow p-spheric flow vv

• Flow is consistent w/ Induction Eq.Flow is consistent w/ Induction Eq.• Has smallest possible Has smallest possible ||vv|| overall overall• Can incorporate Doppler signalCan incorporate Doppler signal• Test problems: Test problems:

– Magara:Magara: good/excellent good/excellent– Abbett: Abbett: fair/poorfair/poor


• MEFMEF: 2 vec. m-grams : 2 vec. m-grams p-spheric flow p-spheric flow vv

• Flow is consistent w/ Induction Eq.Flow is consistent w/ Induction Eq.• Has smallest possible Has smallest possible ||vv|| overall overall• Can incorporate Doppler signalCan incorporate Doppler signal• Test problems: Test problems:

– Magara:Magara: good/excellent good/excellent– Abbett: Abbett: fair/poorfair/poortime betweentime between m-grams may be too great m-grams may be too great

approximate w/ induction @ constant approximate w/ induction @ constant vv

SummarySummarySummarySummary

Documents

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