Practical Calculation of Magnetic Energy and Relative Magnetic Helicity Budgets in Solar Active...
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Practical Calculation of Magnetic Energy Practical Calculation of Magnetic Energy and Relative Magnetic Helicity and Relative Magnetic Helicity BudgetsBudgets in in
Solar Active RegionsSolar Active Regions
Manolis K. Georgoulis Research Center for Astronomy and Applied Mathematics
Academy of Athens, Athens, Greece
Helicity Thinkshop on Solar PhysicsHelicity Thinkshop on Solar Physics Beijing, 12-17 Oct. 2009
D. Rust, B. LaBonte, A. Pevtsov, A. Nindos, M. Berger, T. Wiegelmann, and a number of NASA research grants
Thanks to: Prof. H. Zhang & the Organizers of this Meeting for kind support
Constraints to coronal evolution placed by magnetic helicity
Helicity rates vs. helicity budgets
Calculation of magnetic energy and relative magnetic helicity budgets
2 / 17Beijing, 12 – 17 Oct. 2009
OutlineOutline
Via the linear force-free (LFF) field approximation
Via the nonlinear force-free (NLFF) field approximation
Volume-integral evaluation using extrapolation results
Surface-summation evaluation using photospheric magnetic connectivity
Preliminary results
Correlations between LFF and NLFF energy and helicity budgets
NLFF field energy and helicity budgets
An energy-helicity criterion for eruptive solar active regions
Conclusions – future prospects
Why should magnetic helicity be important for solar Why should magnetic helicity be important for solar coronal activity?coronal activity?
3 / 17Beijing, 12 – 17 Oct. 2009
Theoretical reasons : Observational reasons :
We can see it (!) and there is increasing evidence of its presence in eruptive active regions and CMEs
from Rust & LaBonte (2005)
Magnetic helicity cannot be dissipated effectively by magnetic reconnection
2M
m
m1/2M
m
m R~H
ΔHor R~
H
ΔH
so it can only be bodily transported (CMEs?)
Unless magnetic helicity is not removed, a magnetic system cannot return to the ground, current-free state
currentfreecurrenttotal EEE
~ |Hm|[ Woltjer – Taylor theorem (LFF field state)]
Source: SoHO/LASCO
Helicity rates vs. helicity budgetsHelicity rates vs. helicity budgets
4 / 17Beijing, 12 – 17 Oct. 2009
Calculations of relative magnetic helicity mainly deal with the helicity injection rate, rather than the helicity budget, in active regions:
However:
The helicity injection rate lacks a reference point
Calculation of the velocity field u is non-unique and highly uncertain
What if we tried calculating the budget, rather What if we tried calculating the budget, rather than the rate,of relative magnetic helicity?than the rate,of relative magnetic helicity?
- B uA [2dt
dH
S
pm
Sd] u BAp
Analysis made possible if vector magnetograms Analysis made possible if vector magnetograms are availableare available
5 / 17Beijing, 12 – 17 Oct. 2009
Tic mark separation: 10”
NOAA AR 10930, 12/11/06, 13:53 – 15:15 UT
Continuum intensity
Vertical electric current densityMagnetic field vector
The main magnetic polarity inversion line in
the ARAzimuth disambiguation has been performed using the NPFC method of Georgoulis (2005)
Hinode SOT/SP
Calculation of magnetic energy and relative Calculation of magnetic energy and relative magnetic helicity budgets: I. LFF field approachmagnetic helicity budgets: I. LFF field approach
6 / 17Beijing, 12 – 17 Oct. 2009
Current-free (potential) magnetic energy:
dy'dx' z0,','A0,','B π8
1E g
S
pp yxyx
Total magnetic energy:
p22
t E λFα1E
Free (non-potential) magnetic energy:
p22
np EλFαE Relative magnetic helicity:
p2
m E λ α F π8H where:
x y
ml
x y
ml
n
l
n
m mluu
n
l
n
m mluu
uub
uub
1 1
2/1222,
1 1
2/3222,
/
/
2
1F
Surface-integral calculation (Georgoulis & LaBonte 2007)
NOAA AR 10030
≈ -0.053 ± 0.011 Mm-1
Results of the LFF field approximationResults of the LFF field approximation
7 / 17Beijing, 12 – 17 Oct. 2009
Two active regions tested:
01/25/00, 19:02 UT
NOAA AR 8844
Non-EruptiveNon-Eruptive
NOAA AR 9167
EruptiveEruptive
09/15/00, 17:48 UT
0123456789
10
Force-freeparameter
Magnetic flux Current-freemagneticenergy
Total energy Free energy Relativemagnetichelicity
Force-freeparameter
Magneticflux
Current-freemagneticenergy
Totalmagneticenergy
Freemagneticenergy
Relativemagnetichelicity
Ra
tio
(e
rup
tiv
e /
no
n-e
rup
tiv
e)
For nearly the same force-free parameter, and a ratio of ~ 3.3 in the magnetic flux, current-free, and total magnetic energy, the respective ratios for the free magnetic energy and relative magnetic helicity are ~9.
How realistic is the LFF field How realistic is the LFF field calculation, however?calculation, however?
Calculation of magnetic energy and relative Calculation of magnetic energy and relative magnetic helicity budgets: II. NLFF field approachmagnetic helicity budgets: II. NLFF field approach
8 / 17Beijing, 12 – 17 Oct. 2009
Volume-integral energy-helicity calculation :
Current-free magnetic energy:
dVB π8
1E
V
2pp
Free magnetic energy:
ptnp EEE
Total magnetic energy:
dVB π8
1E
V
2NLFFt
dy'dx',0y',x'B'rr
z'rr
z'rr
z'rr
π2
1rA z
S22g
Relative magnetic helicity:
, wheredVBAH NLFFF
V
gm
e.g, Longcope & Malanushenko (2008)
NLFFF extrapolation for NOAA AR 10930 (Wiegelmann 2004)
Is there any better way than volume integrals? What if Is there any better way than volume integrals? What if we knew the photospheric magnetic connectivity?we knew the photospheric magnetic connectivity?
9 / 17Beijing, 12 – 17 Oct. 2009
Start from the normal magnetic field
Partition the magnetic flux into a sequence of discrete concentrations
Identify the flux-weighted centroids for each partition
ijij L,Φ
Define the connectivity matrices
pnp2p1
2n2221
1n1211
Φ....ΦΦ
................
Φ....ΦΦ
Φ....ΦΦ
M
pnp2p1
2n2221
1n1211
L....LL
................
L....LL
L....LL
L
Which magnetic connectivity?Which magnetic connectivity?
10 / 17Beijing, 12 – 17 Oct. 2009
An alternative connectivity can result in the minimum possible total connection length in the magnetogram
To achieve this, we minimize the functional
ji
ji
ji
ji
ΦΦ
ΦΦ
rr
rr
between any two opposite-polarity fluxes i, j, with vector positions ri, rj
We perform the minimization using the simulated annealing method
Discretized view of the photospheric magnetic flux
Any connectivity (potential, non-potential) can be used with or without flux partitioning
Convergence of the annealing
Calculation of magnetic energy and relative Calculation of magnetic energy and relative magnetic helicity budgets: II. NLFF field approachmagnetic helicity budgets: II. NLFF field approach
11 / 17Beijing, 12 – 17 Oct. 2009
Surface-summation energy-helicity calculation: preliminary analysis (Georgoulis et al., 2010)
Current-free magnetic energy:
dy'dx' z0,','A0,','B π8
1E g
S
pp yxyx
Total magnetic energy:
jicloseij
n
1i
n
ji1,ji
2δi
n
1i
2i
2pt ΦΦLαΦαAλEE
Free magnetic energy:
ptnp EEE
Relative magnetic helicity:
ji
n
1i
n
ji1,jij
2δi
n
1ii
2m ΦΦLΦαAλ π8H
where the VMG has been flux-partitioned into n partitions with fluxes i and alpha-values i
A and are known fitting constants
Mutual term of free energy Lfgclose is
chosen such that free energy is kept to a minimum:
intersectnot do g and f ; 0
intersect g and f ; / Lclose
fgff
Mutual term of relative Lfg is defined following Demoulin et al., (2006):
intersectnot do g f, ; LL
intersect gf, and ; LL
intersect g f, and ; LL
L
gfgf
)((f)closefggf
)((f)closefg
arch
gf
fgarcharch
garch
g
lengthlength
lengthlength
Summary: NLFF magnetic energy and helicity Summary: NLFF magnetic energy and helicity budget calculationbudget calculation
12 / 17Beijing, 12 – 17 Oct. 2009
Volume expressionsVolume expressions Surface expressionsSurface expressions
Current-free magnetic energy:
dVB π8
1E
V
2pp
Current-free magnetic energy:
dy'dx' z0,','A0,','B π8
1E g
S
pp yxyx
Total magnetic energy:
dVB π8
1E
V
2NLFFt
Total magnetic energy:
jicloseij
n
1i
n
ji1,ji
2δi
n
1i
2i
2pt ΦΦLαΦαAλEE
Free magnetic energy:
ptnp EEE
Free magnetic energy:
ptnp EEE
dy'dx',0y',x'B'rr
z'rr
z'rr
z'rr
π2
1rA z
S22g
Relative magnetic helicity:
, wheredVBAH NLFFF
V
gm
Relative magnetic helicity:
ji
n
1i
n
ji1,jij
2δi
n
1ii
2m ΦΦLΦαAλ π8H
where fluxes i and alpha-values i
stem from the analysis of magnetic connectivity
Results: preliminary comparison of free magnetic Results: preliminary comparison of free magnetic energiesenergies
13 / 17Beijing, 12 – 17 Oct. 2009
Limited sample of 9 active regions:
NLFF volume calculation
NLFF surface calculation
LFF calculation
Connectivity matrix has been calculated from line-tracing of a NLFF field extrapolation
Very good agreement between NLFF volume / surface expressions
Acceptable agreement between LFF and NLFF expressions
Results: preliminary comparison of relative Results: preliminary comparison of relative magnetic helicitiesmagnetic helicities
14 / 17Beijing, 12 – 17 Oct. 2009
NLFF volume calculation
NLFF surface calculation
LFF calculation
Connectivity matrix has been calculated from line-tracing of a NLFF field extrapolation
Reasonable agreement between NLFF volume / surface expressions
Fair to poor agreement between NLFF and LFF expressions
Limited sample of 9 active regions:
A quiz: can you identify the eruptive active A quiz: can you identify the eruptive active regions?regions?
15 / 17Beijing, 12 – 17 Oct. 2009
NOAA AR
Mag
neti
c energ
y (
erg
)
8210
9026
9165
1003010930
10953
8210
90269165
1003010930
10953
Potential energy
Free energy
Now we focus on the NLFFF energy / helicity calculations of the entire sample of 22 regions.
Of these active regions, 6 were flaring and eruptive (NOAA ARs 8210, 9026, 9165, 10030, 10930, and 10953)
WHERE ARE THESE SIX WHERE ARE THESE SIX ERUPTIVE REGIONS?ERUPTIVE REGIONS?
In terms of free magnetic energy, the eruptive regions have a noticeable fraction of their total energy being available for release
In terms of relative magnetic helicity, the eruptive regions have clearly larger magnitudes than the non-eruptive ones
An “energy-helicity” eruptive criterion?An “energy-helicity” eruptive criterion?
16 / 17Beijing, 12 – 17 Oct. 2009
Enp > 3 x 1031 erg
Hm > 2 x 1042 Mx2
Eruptive regions tend to have large free magnetic energy (> 3 x 1031 erg) and relative magnetic helicity (> 2 x 1042 Mx2)
The “threshold” helicity magnitude shows excellent agreement with the typical CME helicity budgets (DeVore 2000; Georgoulis et al. 2009)
Summary and ConclusionsSummary and Conclusions
17 / 17Beijing, 12 – 17 Oct. 2009
Adopting that magnetic helicity is an important physical quantity in the solar atmosphere, we attempt a calculation of the relative magnetic helicity and energy budgets from single vector magnetograms of solar active regions
Calculation of the relative helicity budget does not require knowledge of the velocity field and hence avoids its shortcomings. Plus, it provides more information than simply calculating helicity injection rates.
Energy-helicity budget calculation for a LFF field has been achieved. We presented here a more general NLFF field calculation that appears to be working satisfactorily.
For a dataset of 22 active-region vector magnetograms it appears that the 6 eruptive active regions show larger free magnetic energy and larger magnitude of relative magnetic helicity.
An eruptive criterion for an active region may be defined here – there is important physics in the “energy-helicity” diagram for a statistically significant sample
FUTURE PROSPECTS: verify calculations and results, increase the sample of active regions, test different connectivity solutions, detailed uncertainty analysis, etc. etc.
BACKUP SLIDES
Basic mutual helicity configurationsBasic mutual helicity configurations
From Demoulin et al. (2006)
To be consistent with a minimum free magnetic energy, we assume that all the possible configurations collapse to that of picture (a).
Testing the Taylor hypothesisTesting the Taylor hypothesis
After calculating the NLFF field helicity, we can find the -value that would give the same helicity for a LFF field:
P
m2T E F
H
λ π8
1α
Then we can use this -value to calculate a LFF field total energy:
p22
Ttotal E λFα1E(T)
per the Woltjer-Taylor theorem, this energy should be the minimum possible
NLFF surface integral
Min “Taylor” energy
LFF energy estimate
NLFF volume integral
Cross-section of a NLFF field extrapolationCross-section of a NLFF field extrapolation
NLFFF extrapolation for NOAA AR 10930 (Wiegelmann 2004)
Logarithm of the free magnetic energy as a function of altitude – most of it close to the photosphere (< 20 Mm)