Magnetic Helicity and Energetics in Solar Active Magnetic Helicity and Energetics in Solar Active Regions: Can we calculate them – why do we Regions: Can we calculate them – why do we

need them?need them?

Manolis K. GeorgoulisManolis K. Georgoulis

JHU/APLJHU/APL

Whistler, CA, 08/01/07

Partial Support by LWS/TR&T Grant NNG05-GM47G

Why is magnetic helicity physically important to calculate?

Some evidence of the importance of magnetic helicity

Linear force-free calculations of energy/helicity in solar ARs

Preliminary further work

Conclusions / Future work

Whistler, 08/01/07

02/17

Outline

Why is helicity important to calculate?

Whistler, 08/01/07

03/17

Free magnetic energy is due to electric currents

Electric currents imply twisted and tangled (helical) field lines

Phillips et al. (2005)Fan (2005)

Whistler, 08/01/07

04/17

Free magnetic energy vs. magnetic helicityFree magnetic energy and relative magnetic helicity both reflect departures from potentiality. However, they have one fundamental difference:

Magnetic helicity is a signed quantity. Free magnetic energy is

not!

Large total (relative) helicity means large free energy. The opposite is not always true

Both situations are capable of triggering eruptions

Whistler, 08/01/07

05/17

Some evidence of helicity’s involvement in solar eruptions

Nindos & Andrews (2004)

Rust & LaBonte (2005)

Preflare values of constant a are higher for active regions with eruptions

The helical kink instability is viewed to have triggered the above filament eruption

BUT CAN WE CALCULATE HELICITY QUANTITATIVELY TO INVESTIGATE ITS IMPORTANCE?

Whistler, 08/01/07

06/17

Calculation of the helicity’s injection rate …

dS Au B2dt

dHpLCT

S

nm There are concerns about uLCT . As a

result, sufficient statistics might help

Whistler, 08/01/07

07/17

and the statistics of the calculation …

There appears to be a threshold for (dHm/dt) below which an AR does not give a X-class flare

LaBonte, Georgoulis, & Rust (2007)

393 ARs in total57 X-flaring ARs

and the statistics of the calculation …

Whistler, 08/01/07

08/17

LaBonte, Georgoulis, & Rust (2007)

393 ARs in total57 X-flaring ARs

X-flaring ARs accumulate 2 x 1042 Mx2 of helicity within a few hours to a few days. Typical times for non X-flaring ARs range from tens to hundreds of days

Whistler, 08/01/07

09/17

Calculation of the magnetic energy and helicity budgets

cpottot EEE

pot2

c EFd αE

pot2

m EF α d π8H

Georgoulis & LaBonte (2007):Derived expressions for the energy and

helicity budgets (not rates) in the constant- approx.

Applied it to a sequence of VMGs from an eruptive AR (9165) and a noneruptive AR (8844)

Data from Mees/IVM

Whistler, 08/01/07

10/17

Results – energy and helicity budgets

01

23

45

67

89

Magneticflux

alpha Totalmagneticenergy

Freemagneticenergy

Relativemagnetichelicity

Rati

os

(eru

pti

ve /

noneru

pti

ve)

For nearly identical :

The ratio of total magnetic energies scales ~

The ratios of the free magnetic energy and the relative helicity scale ~ 2

THE LF

F APP

ROXIMATI

ON IS

UNREALISTI

C !

BUT MANY OF YOU MIGHT RIGHTFULLY SAY THAT…

Whistler, 08/01/07

11/17

Nonlinear force-free energy/helicity calculation

δ 22c ΦA d α E

δ 22m ΦA α d π8 H

From a single flux tube: To a collection of flux tubes:

2δi

N

1i

2i

2(self)c ΦαdA E

2δi

N

1ii

2(self)m ΦαdA π8 H

THIS IS ONLY THE SELF TERM

Nonlinear force-free energy/helicity calculation (cont’d)

Whistler, 08/01/07

12/17

Demoulin et al. (2006) provide the mutual terms:

jiclose

ji,

N

1i

N

j i 1,ji π8

1(mutual)c ΦΦ L α E

jiarch

ji,close

ji,

N

1i

N

j i 1,j(mutual)m ΦΦ LL H

[from Demoulin et al. (2006)]close

ji,L unknown

where

archji,L known from geometry

To calculate Li,jclose, we keep Ec(mutual)

positive and the minimum possible:

on intersecti no 0,

on intersecti , )α/(α L iiclose

ji,

THIS IS THE MUTUAL TERM

Whistler, 08/01/07

13/17

Preparing a vector magnetogram for calculation

Partition the flux distribution using the MCT model of Barnes et al. (2005)

Calculate the boundary connectivity using simulated annealing (Georgoulis & Rust 2007) Ignore connections closing beyond the FOV – focus on closed connections

For each flux concentration, find average -value from total current – = (4/c) (I/)For two connected concentrations with -values 1 and 2, assign the mean to

the connection 2121 ααδα ; αα2

1α

Data from Mees/IVM

Whistler, 08/01/07

14/17

And applying the calculation to the same ARs

0

12

34

56

7

89

Magneticflux

Totalmagneticenergy

Freemagneticenergy

Relativemagnetichelicity

LFF calculationNLFF calculation

Rati

os

(eru

pti

ve /

noneru

pti

ve)

The ratios of total magnetic energies remain nearly identical

The ratio of the free magnetic energies drops to ~5, but

The ratio of the total helicities remains at ~8

Whistler, 08/01/07

15/17

Actual calculation results

242m Mx 104.62.3H 242

m Mx 104.618.5-H

NOAA AR 8844 NOAA AR 9165

MUTUAL HELICITY DOMINATES OVER SELF HELICITY

Whistler, 08/01/07

16/17

Actual calculation results

242m Mx 104.618.5-H

NOAA AR 8844 NOAA AR 9165

0.25E / E

erg 10 4.915.8E

totc

32tot

0.18E / E

erg 10 0.53.6E

totc

32tot

MUTUAL ENERGY TERMS DOMINATE OVER SELF TERMS

A Quiz, instead of Conclusions

Q: Is it possible to estimate magnetic energy / helicity budgets of ARs?

A: We believe so, adopting the nonlinear force-free approximationQ: Do we need a full-fledged nonlinear force-free field extrapolation for that?

A: From preliminary results, extrapolation may not be necessary, assuming

a minimum magnetic free energy for the ARs

Q: Is knowledge of helicity budgets important in solar ARs?

A: YES – it complements the quantitative knowledge of nonpotentiality

Q: Have we proved that helicity is important in solar eruptions?

A: NOT YET – However, we have brought some evidence in favor of the idea Q: When can we speak with greater confidence?

A: We need to determine whether:

Eruptive ARs can be distinguished from noneruptive ones in terms of their helicity budgets

Significant variations of helicity exist before and after eruptions

Future Work

Q: Is it possible to estimate magnetic energy / helicity budgets of ARs?

A: We believe so, adopting the force-free approximationQ: Do we need a full-fledged nonlinear force-free field extrapolation for that?

A: From preliminary results, extrapolation may not be necessary, assuming

a minimum magnetic free energy for the ARs

Q: Is knowledge of helicity budgets important in solar ARs?

A: YES – it complements the quantitative knowledge of nonpotentiality

Q: Have we proved that helicity is important in solar eruptions?

A: NOT YET – However, we have brought some evidence in favor of the idea Q: When can we speak with greater confidence?

A: We need to determine whether:

Eruptive ARs can be distinguished from noneruptive ones in terms of their helicity budgets

Significant variations of helicity exist before and after eruptions