1

This is how it looks like…This is how it looks like…Magnetic helicityMagnetic helicity at the solar surface at the solar surface

and in the solar windand in the solar wind

Axel Brandenburg(Nordita, Stockholm)

Properties of magn helicityLessons from dynamo theory

What do we see in solar wind?What we can see at solar surface?

How about solar corona?

2

Magnetic helicity measures linkage of fluxMagnetic helicity measures linkage of flux

V

VH d BA1

2

212 H

11

d d1

SL

H SBA

2 d2

S

SA

1S

1

AB

Therefore the unit isMaxwell squared

3

Other fun examplesOther fun examples

Relevance:Slows down decayGrowth at large scalesLarge-scale dynamos

(Candelaresi and B

randenburg (2011))

Trefoil knot H=32

4

Decaying helical fieldsDecaying helical fields)2E(kH(k)k

kE(kkkH(k) )d2d 1 1max

1

1

)d2

d

)d

)d

MM kkE(k

kH(k)

kE(k

kE(kkk

Tevzadze, Kisslinger, Brandenburg, Kahniashvili

(2012, ApJ,)

(i) Transfer toLarge scales

(ii) slow-downof decay

kH(k)

kE(kkM

d

)d2max

5

Dynamos produce bi-helical fieldsDynamos produce bi-helical fields

BAdkkH )(

Magnetic helicity spectrum

031 / bjuω

Pouquet, Frisch, & Leorat (1976)

Southernhemisphere

g. u. a.b

Self-inflicted twist: feedback & CMEsSelf-inflicted twist: feedback & CMEs

N-shaped (north)S-shaped (south)(the whole loop corresponds to CME)

Bla

ckm

an &

Bra

nd

enb

urg

(20

03)

=coronal mass ejection

Magnetic helicity fluxMagnetic helicity flux

• EMF and resistive terms still dominant

• Fluxes import at large Rm ~ 1000

• Rm based on kf

• Smaller by 2

8

m22d

dF BJBBA ε

t

f22d

dF bjBba ε

t

Magnetic helicity fluxMagnetic helicity flux

• EMF and resistive terms still dominant

• Fluxes import at large Rm ~ 1000

• Rm based on kf

• Smaller by 2

9

m22d

dF BJBBA ε

t

f22d

dF bjBba ε

t

Gauge-invariant in steady state!

Del Sordo, Guerrero, Brandenburg (2013, MNRAS 429, 1686)

Northern/southern hemispheresNorthern/southern hemispheres

Cyclones:Down: fasterUp: slower

north

south

g g

0uω 0uω

equator

uω

Northern/southern hemispheresNorthern/southern hemispheres

Cyclones:Down: fasterUp: slower

north

south

g g

0uω 0uω

equator

uω

12

Lessons from dynamo theoryLessons from dynamo theory

• Helicity– Not just a measure of complexity– Critically important in dynamos

• To confirm observationally– Opposite signs at different scales– Opposite signs in different hemispheres

13

(i) Helicity from solar wind: in situ(i) Helicity from solar wind: in situ

)()( xBxBM jiij

)(

)(ˆˆ)(

kHki

kEkkkM

kijk

jiijij

)()( xBxBM jiij

)()()( rxxr jiij BBM

Matthaeus et al. (1982) Measure correlation function

In Fourier space, calculatemagnetic energy and helicity spectra

Should be done with Ulysses data away from equatorial plane

14

Measure 2-point correlation tensorMeasure 2-point correlation tensor

tuRR R 0Taylor hypothesis:

RRNRTR kkBkBkH /)(~

)(~

Im4)( *

u1 u2

15

Ulysses: scaling with distanceUlysses: scaling with distance

022 2/4 BuRL RM

* Fairly isotropic* Falls off faster than R-2

* Need to compensate before R averaging

Power similar to US consumptionEnergy density similar to ISM

Vector helium magnetometer2 sec resolution10 pT sensitivity (0.1 G)

16

Noisy helicity Noisy helicity from Ulyssesfrom Ulysses

• Taylor hypothesis

• Roundish spectra

• Southern latitude with opposite sign

• Positive H at large k

Brandenburg, Subramanian, Balogh,& Goldstein (2011, ApJ 734, 9)

17

Bi-helical fields from UlyssesBi-helical fields from Ulysses

• Taylor hypothesis• Broad k bins• Southern latitude

with opposite sign• Small/large distances• Positive H at large k• Break point with

distance to larger k

1 AU-1 1 Hz

18

Latitudinal scaling and trendLatitudinal scaling and trend

1. Antisymmetric about equator

2. Decline toward minum

19

ComparisonComparison

• Field in solar wind is clearly bi-helical

• ...but not as naively expected

• Need to compare with direct and mean-field simulations

• Recap of dynamo bi-helical fields

Helicity LS SS

Dynamo - +

Solar wind + -

Southernhemisphere

for southern hemisphere

20Strong fluctuations, but positive in north

War

neck

e, B

rand

enbu

rg, M

itra

(20

11, A

&A

, 534

, A11

)Shell dynamos with ~CMEsShell dynamos with ~CMEs

SS: -

To carry negative flux: need positive gradientTo carry negative flux: need positive gradient

f2f

m2m

22d

d

22d

d

FBJB

FBJB

t

t

t

h

t

h

Brandenburg, Candelaresi, Chatterjee(2009, MNRAS 398, 1414)

Sign reversal makes sense!

ff hh F

Similar method for solar surfaceSimilar method for solar surface

22Zhang, Brandenburg, & Sokoloff (2014, ApJ 784, L45)

EE

23

Results & Results & realizabilityrealizability

24

• Isotropy

• Positive hel.

• Expected for south

30,000 G2Mm/(2 6Mm 70,000 G2)=0.04

30,000 G2Mm x (200Mm)2 = 1043 Mx2/100Mm

Radio observations of coronal fields?Radio observations of coronal fields?

Helical field w/positive helicity

dzpQ 2cos-

20

Stokes Q and U parameters

0

1

1

sin

cos

B

kzB

kzB

B

dzpU 2sin-

20

zBKnkz zth ,

21 ,i B

iyx

BeBBB

Intrinsic polarized emission from B

Cancellation condition

iUQP

slope=RM

Bra

nden

burg

& S

tepa

nov

(20

14, A

pJ 7

86, 9

1)

Only works if RM > 0 and k > 0 Only works if RM > 0 and k > 0

But difficult/impossible to recover F()

2

-

22 2 2

1 2

dePF i

RM// 0th21 kBKnk

Peak determined by single parameter

deFP i -

22 2

(Burn 1966)

2

0

22syn 2

2

1 2

dePF i

Positivity:

021

Brandenburg &

Stepanov (2014, A

pJ 786, 91)

Expect bi-helical fieldsExpect bi-helical fields• Magnetic helicity conserved• Inverse cascade produces small-scale waste!• Opposite sign of helicity (or k)

27

Bla

ckm

an &

Bra

nden

burg

(20

03)

28

Galactic Galactic solar sectors solar sectors• RM synthesis: measure magnetic helicity

• Need line of sight component: edge-on galaxy

• Expect polarized intensity only in 2 quadrants

• 2 characteristic peaks: eclipsing binaries??

x.

x.

29

Magnetic Magnetic cross helicity cross helicity

• Large-scale structures from stratified turbulence?

• Application to sunspots?

g.B u.Bg. u. A.B

Sunspot decay

30

Self-assembly of a magnetic spotSelf-assembly of a magnetic spot• Minimalistic model• 2 ingredients:

– Stratification & turbulence

• Extensions– Coupled to dynamo– Compete with rotation– Radiation/ionization

31

ConclusionsConclusions• Magnetic helicity

– Essential for dynamo– Expect bi-helical

• Solar wind: yes, but reversed!

• Galaxies: yes, in theory Brandenburg & Stepanov (2014, ApJ 786, 91)