preprocessing of vector magnetograms for extrapolation of coronal magnetic fields
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Preprocessing of Vector Magnetograms for Extrapolation of Coronal Magnetic Fields. General Discussion. Assumption: force-free: with Force-free condition holds above 400km - PowerPoint PPT PresentationTRANSCRIPT
Preprocessing of Vector Magnetograms for Extrapolation of
Coronal Magnetic Fields
General Discussion
• Assumption: force-free:
with
• Force-free condition holds above 400km– Measurements might be affected by non-magnetic
effects– Extrapolation should take into account of plasma
conditions
0BJ
)()()( xBxxJ
BJ
Test case & Reality Low&Lou Bastille Event
Preprocessing
• Problems on real magnetograms– Force-free ?– Noise (lack of smootheness)– Hidden problems?
• Philosophy: – Different mgms, close to the original one– Staying within given errorbars
Preprocessing: Minimization
• Force-freeness [Molodenskii (1969)]
– Magnetic-field Stress Tensor T with ext. forces F
– Force-free condition
0
dsTx ik
s i
ijjiij BBBT
2
2
1
4
1
BB
0 B
FT
ˆ
0F
Preprocessing: Minimization
• Force-freeness– Transforming surface integral
into an integral along contour L
– Since the magnetic field weakens as the diameter of contour L increases, surface integral will approach to zero
– Magnetic field decreases with height
0321
S
k
L
kk dsTz
dxTdyT
S
k constdsT3
S
kdsT 03
Preprocessing: Minimization
• Torque-Freeness– Same procedure, but
multiply vectorially with r
– Then integrate over surface S, the z-component of the resulting equation
– Seperating out the divergence part
– Similar as before
rFT
ˆ
0
S
yzyyyxxzxyxx
z
T
y
T
x
Tx
z
T
y
T
x
Tyds
0
yzxzyyxyyxxx xTyTz
xTyTy
xTyTx
ds
0yzxy xTyTds
How to Preprocess: Minimization
• Force-free
• Torque-free
0 dxdyBBdxdyBB z
surface
yz
surface
x
surface
yxz dxdyBBB 0222
surface
yxz dxdyBBBx 0222
surface
yxz dxdyBBBy 0222
0 dxdyByBBxBsurface
zxzy
Preprocessing: Minimization
• Smootheness
– Current Method: Laplacian 4th order
– Pro: smoothes the Magnetogram, because of averaging the discontinuities
– Contra: flattens the whole Magnetogram
Numerical Expressions
– L* : treshold to control smoothing» L4=0 no curvature
2
222
22
1
mgmyxz
mgmzy
mgmzx BBBBBBBL
24 *),( LyxBL
22
222
2
2222
mgmzxzy
mgmyxz
mgmyxz ByBBxBBBByBBBxL
2
2
/)2,(121)2,(12
1),(25),(3
4),(34
/),2(121),2(12
1),(25),(3
4),(34
),(
yyyyy
xxxxx
hhyxBhyxByxBhyxBhyxB
hyhxByhxByxByhxByhxB
yxB
Normalizations
2
222222
2
2221
)(
)(
mgmzyx
mgmzyx
BBByxLN
BBBLN
2
2
2
4
/)2,(121)2,(12
1),(25),(3
4),(34
/),2(121),2(12
1),(25),(3
4),(34
)(
yyyyy
xxxxx
hhyxBhyxByxBhyxBhyxB
hyhxByhxByxByhxByhxB
LN
Minimizing functional
)()()( 4
44
2
22
1
11 LN
L
LN
L
LN
LL
]1.0...005.0[
1
4
21
)()( 2
22
1
111212 LN
L
LN
LL
Method of Minimization: Simulated Annealing
• Usual Minimizing(Newton-Raphson, etc.) of a function leads to nearest local minimum, depends on starting point
• Current Problem: – functional with several constraints– Several local minima
• Simulated annealing finds also local minima, but leaves a chance to get out of it, in order to find better local minima
How does Simulated Annealing Work?
• Choose randomly new value of Bx,y,z – But within a small interval (~± 0.5%)
• Calculate L, check if LNEW ≤ LOLD– TRUE: Take Bnew
x,y,z & forget Boldx,y,z,
– FALSE: Take with probability ~ exp{-(LNEW - LOLD)}
• Repeat – To scan the whole domain– Domain within given errorbars of the measurement (depends if
transversal or normal component of Magnetogram)
Results: preprocessing just force- & torque-freeness
Quality of force- & torque-freeness
Results: preprocessing just force- & torque-freeness
Magnetogram
Results: preprocessing both constraintsMagnetogram
Results: preprocessing both constraints Quality of force-freeness
Compared Results
σJ
Low & Lou semi-analytical solution 0
extrapolation of Low & Lou Magnetogram 0.018
Noise introduced 0.214
preprocessed: force- & torquefreeness 0.196
preprocessed: smootheness 0.108
preprocessed: both constraints 0.094
Conclusions
• Extrapolation successful, but on real cases lack of force-freeness (σJ ~ 10-1)
• Preprocessing: – successful to make magnetogram force-free
better agreement with initial assumption
slightly better results, but not enough– Difficulties to smooth without flattening
have to find a better smoothing term!