ChandrasekharKendall functions in astrophysical magnetism · ChandrasekharKendall functions in...
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ChandrasekharKendall functionsChandrasekharKendall functions in astrophysical magnetismin astrophysical magnetism
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Chandra’s lasting contributions to MHDChandra’s lasting contributions to MHD
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Accelerated growth!Accelerated growth!
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Chandrasekhar number in MHDChandrasekhar number in MHD
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Chandra’s interest in MHDChandra’s interest in MHD
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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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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 )(
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Alfven and slow magnetosonic wavesAlfven and slow magnetosonic waves
Alfven
slowmagnetosonic
qrr −∝Ω )(Degeneracy lifted by q or ‘ à µ 0
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Both were very much ahead of their time:No accretion discs were discovered yet!
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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
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CKF functions in plasma contextCKF functions in plasma context
Alladis et al. (2001)Protasphere experiment
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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
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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
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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⋅−
×××=±
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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)
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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
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Effect of helicity fluxesEffect of helicity fluxesB
rand
enbu
rg (2
005,
ApJ
)
1046 Mx2/cycle
FBJBA ⋅∇−⋅−=⋅ η2ddt
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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)
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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
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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
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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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ConclusionsConclusions
• He was close to getting a dynamo• Very much immersed into numerics• Always ahead of his time• Still gaining more citations every year!