Chandrasekhar­Kendall functionsChandrasekhar­Kendall functions in astrophysical magnetismin astrophysical magnetism

2

Chandra’s lasting contributions to MHDChandra’s lasting contributions to MHD

3

Accelerated growth!Accelerated growth!

4

Chandrasekhar number in MHDChandrasekhar number in MHD

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Chandra’s interest in MHDChandra’s interest in MHD

6

His most quoted MHD papersHis most quoted MHD papers

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His 1956 papers alone!His 1956 papers alone!

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9

Toward magneto­rotational instabilityToward magneto­rotational instability

( )

xyzy

xzx

yzxy

xzyx

bquBb

uBb

bBuqu

bBuu

Ω−=

=

=Ω−+

=Ω−

'0

'0

'0

'0

2

2

( )[ ] ( ) 02222 22A

2A

22A

24 =Ω−+Ω−+− qq ωωωωω

kvAA =ω

Vertical field B0

Dispersion relation

Alfven frequency:

qrr −∝Ω )(

10

Alfven and slow magnetosonic wavesAlfven and slow magnetosonic waves

Alfven

slowmagnetosonic

qrr −∝Ω )(Degeneracy lifted by q or ‘ à µ 0

11

12

13

Both were very much ahead of their time:No accretion discs were discovered yet!

14

Emphesis on stability – not instabilityEmphesis on stability – not instability

(1976)

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Chandrasekhar­Kendall functionsChandrasekhar­Kendall functionsEigenfunctions of the curl operator: curl B =p B

Fits to solar magnetogramsTheory by B. C. Low

16

CKF functions in plasma contextCKF functions in plasma context

Alladis et al. (2001)Prota­sphere experiment

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18

19

Magnetic helicity measures linkage of fluxMagnetic helicity measures linkage of flux

∫ ⋅=V

VH d BA1Φ

2Φ

212 ΦΦ±=H∫∫ ⋅⋅=

11

d d1SL

H SBA

2 d2

Φ=⋅×∇= ∫S

SA

1S

1 Φ=

AB ×∇=

Therefore the unit isMaxwell squared

20

Magnetic helicity conservationMagnetic helicity conservation

( ) 2221

dd JBJuB η−×⋅−=t

( ) 0dd 2/12/1

21 →=→⋅−×⋅−=⋅ − ηη ηη BJBBuBA

t

kkBJ ∝∝∝ − 2/1ηHow J diverges as η 0

Ideal limit and ideal case similar!

± ∞→=→∇⋅−⋅=⋅ −−− 2/112/1221

dd ννν νν uωωfuωt

21

22

CK functions in periodic spaceCK functions in periodic space

xdett i 3),()( xkk xAA ⋅∫=

−−

++ ++= kkkk hhhA )()()()( ||

|| tatatat

( )−−

++ −= kkk hhB )()()( tatakt

( )( )2222

22

−+

−+

+=

−=⋅

aak

aak

k

kk

B

BA

Fourier space

Expand into longitudinal and polarized contributions

1|| with =±=×∇ ±±±kkk hhh k

so

spectra

( ) ( )( ) 222 /12 kk

ik

ek

ekekkhk⋅−

×××=±

23

Realizability conditionRealizability condition( )

( )2222

22

−+

−+

+=

−=⋅

aak

aak

k

kk

B

BA

kkk BAB ⋅≥ k2

kk HkM 21≥

Spectra

( )kkk kHMM 21

21 ±=±

Shell­integrated spectra

Realizability condition

Energies in positively and negatively polarized waves

(Obtained just from the spectra)

24

Cartesian box MHD equationsCartesian box MHD equations

JBuA η−×=∂∂

t

visc2 ln

DD FfBJu ++×+∇−=

ρρsc

t

u⋅−∇=tD

lnD ρ

ABBJ

×∇=×∇=Induction

Equation:

Magn.Vectorpotential

Momentum andContinuity eqns

( )ρυ ln2312

visc ∇⋅+⋅∇∇+∇= SuuF

Viscous force

forcing function += kk hf 0f (eigenfunction of curl)

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Inverse cascade of magnetic helicityInverse cascade of magnetic helicity

kqp MMM =+ |||||| kqp HHH =+and

||2 pp HpM = ||2 qq HqM =Initial components fully helical: and

( )||||||2|||| qpkkqp HHkHkMHqHp +=≥=+

),max(||||

||||qp

HH

HqHpk

qp

qp ≤++

≤

argument due to Frisch et al. (1975)

k is forcedto the left

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Decaying fully helical turbulenceDecaying fully helical turbulence

Initial slope M~k4

Christensson et al.(2001, PRE 64, 056405)

helical vsnonhelical

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Forced turbulenceForced turbulence

kkT αηλ ±−= 2

Bra

nden

burg

(200

1, A

pJ 5

50, 8

24)

BBB 2∇+×∇=∂ Tt ηα

)0,,1(

i

i

e t

=

= +⋅

B

BB xk

@

λTkdkd ηαλ 2/ 0/ =⇒=

Expected from mean­field theory

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CK functions in linear/nonlinear regimesCK functions in linear/nonlinear regimes

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Slow­down explained by Slow­down explained by magnetic helicity conservationmagnetic helicity conservation

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Slow­down explained by Slow­down explained by magnetic helicity conservationmagnetic helicity conservation

2f

21

211 22 bBB kk

dtd

k ηη +−=−

[ ])(2

1

22 s211 ttkf e

k

k −−−= ηbB

molecular value!!

BJBA ⋅−=⋅ η2dtd

31

Effect of helicity fluxesEffect of helicity fluxesB

rand

enbu

rg (2

005,

ApJ

)

1046 Mx2/cycle

FBJBA ⋅∇−⋅−=⋅ η2ddt

32

3­D simulations in spheres3­D simulations in spheres

(i)(i) The right dynamo regime?The right dynamo regime?(ii)(ii) Or a small scale dynamo?Or a small scale dynamo?

Brun, Miesch, & ToomreBrun, Miesch, & Toomre(2004, ApJ 614, 1073)(2004, ApJ 614, 1073)

33

Takes many turnover timesTakes many turnover times

1

f

rms1t

1

ff

12

31

31

1t

kk

uU

kU

C

kk

kkC

CCD

∆=∆=

=⋅

==

=

Ω

Ω

η

εττ

ηα

α

α

uuω

Rm

=121

, By,

512

^3

LS dynamo notalways excited

34

αα ­effect dynamos (large ­effect dynamos (large scale)scale)

Differential rotation(surface layers: faster inside)

Cyclonic convection;Buoyant flux tubes

Equatorwardmigration

New loop

α ­effect

( ) BBBUB 2)( ∇+++××∇=∂∂

ttηηα

35

How do magnetic helicity losses How do magnetic helicity losses look like?look like?

N­shaped (north)S­shaped (south)(the whole loop corresponds to CME)

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37

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ConclusionsConclusions

• He was close to getting a dynamo• Very much immersed into numerics• Always ahead of his time• Still gaining more citations every year!