Astrophysical MagnetismAstrophysical Magnetism

Axel BrandenburgAxel Brandenburg

((Nordita, StockholmNordita, Stockholm))

2

Similar physics on different scalesSimilar physics on different scales

Galaxies: radius 10 kpc (=3x10Galaxies: radius 10 kpc (=3x102020 m), 2-20 m), 2-20 GGGalaxy cluster: radius 1 Mpc (=3x10Galaxy cluster: radius 1 Mpc (=3x102222 m), 0.1-1 m), 0.1-1 GG

Sun: radius 700 Mm (=7x10Sun: radius 700 Mm (=7x1088 m), 20-2000 G m), 20-2000 GEarth: radius 60 Mm (=6x10Earth: radius 60 Mm (=6x1088 m), 0.5 G m), 0.5 G

3

Importance of solar interiorImportance of solar interior

4

Large scale coherenceLarge scale coherence

Active regions, bi-polaritysystematic east-west orientationopposite in the south

5

Solar cycleSolar cycle• Longitudinally averaged radial field

• Spatio-temporal coherence– 22 yr cycle, equatorward migration

Poleward branch orpoleward drift?

butterfly diagram

6

7

Karlsruhe dynamo experiment (1999) Karlsruhe dynamo experiment (1999)

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Cadarache experiment (2007)Cadarache experiment (2007)

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Dynamos: kinetic Dynamos: kinetic magnetic energy magnetic energy

thermalenergy

kineticenergy

magneticenergy

Nuclearfusion

surfaceradiation

viscousheat

Ohmicheat

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Faraday dynamoFaraday dynamo

But we want to make it self-exciting, without wires,But we want to make it self-exciting, without wires,and without producing a short circuit!and without producing a short circuit!

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MHD equations (i)MHD equations (i)

BuEJ

002/ ,

1

0 ,

EJBE

BEB

tc

t

/ JBuB

t

12

MHD equations (ii)MHD equations (ii)

uQBJu

AJBuA

21

2

D

D ,

D

D

,

sct

hh

t

ctt

sThp 1

ut

chD

lnD ,ln2

s

Momentum and continuity eqns (usual form)

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Vector potentialVector potential

• B=curlA, advantage: divB=0• J=curlB=curl(curlA) =curl2A• Not a disadvantage: consider Alfven waves

z

uB

t

b

z

bB

t

u

00 and ,

uBt

a

z

aB

t

u02

2

0 and ,

B-formulation

A-formulation 2nd der onceis better than1st der twice!

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Comparison of A and B methodsComparison of A and B methods

2

2

02

2

2

2

0 and ,z

auB

t

a

z

u

z

aB

t

u

2

2

02

2

0 and ,z

b

z

uB

t

b

z

u

z

bB

t

u

15

Kolmogorov spectrumKolmogorov spectrumnonlinearitynonlinearity 2

1212 2coscos kxkx

kk 2

constant flux constant flux cmcm22/s/s33

kk

EE((kk)) 2

21 udkkE cmcm33/s/s22

baK kCkE

123 :cm a

a32 :s aa=2/3, =2/3, bb= = 5/35/3

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Hyperviscous, Smagorinsky, normalHyperviscous, Smagorinsky, normal

Inertial range unaffected by artificial diffusionHau

gen

& B

rand

enbu

rg (

PR

E, a

stro

-ph/

0402

301)

height of bottleneck increased

onset of bottleneck at same position

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Small-scale vs large-scale dynamosSmall-scale vs large-scale dynamos

B-scale larger than U-scale

B-scale smaller than U-scale

Wavenumber=1/scale

energy

injectionscale

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Small scale and large scale dynamosSmall scale and large scale dynamosnon-helically forced turbulence helically forced turbulence

Scale separation :==There is room on scalesLarger than the eddy scale

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Dynamo in kinematic stage –Dynamo in kinematic stage –no large-scale field?no large-scale field?

Fully helical turbulence, periodic box, resistive time scale!

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-effect dynamos (large scale)-effect dynamos (large scale)

Differential rotation(prehelioseism: faster inside)

Cyclonic convection;Buoyant flux tubes

Equatorwardmigration

New loop

-effect?need meridional circulation

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Revised theory for Revised theory for -effect-effect

1st aspect: replace triple correlation by quadradatic

2nd aspect: do not neglect triple correlation

3rd aspect: calculate

rather than

ub

buu

uubbuuu Similar in spirit to tau approx in EDQNM

bubuBubUb

neglected!not t

bubuε t/

'd)'( ttbuε

(Heisenberg 1948, Vainshtein & Kitchatinov 1983, Kleeorin & Rogachevskii 1990, Blackman & Field 2002, Rädler, Kleeorin, & Rogachevskii 2003)

22

Implications of tau approximationImplications of tau approximation

1. MTA does not a priori break down at large Rm.

(Strong fluctuations of b are possible!)

2. Extra time derivative of emf

hyperbolic eqn, oscillatory behavior possible!

4. is not correlation time, but relaxation time

εε

JB

~

~t

new

t

εε JB

231

31

31

~ ,

~

~ ,~

u

bjuω

with

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Kinetic and magnetic contributionsKinetic and magnetic contributions

lKillkljijkii BuBu ~

, uBubu

lkjijkKil uu ,

~ uω ikjijkKii uu ,

~

Kij

Kij ~~

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lMilklilijkii BbbB ~

, bbBbu

likijkMil bb ,

~ bj ijkijkMii bb ,

~

Mij

Mij ~~

31

uω

bj

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22-effect calculation-effect calculation

BB

2

2

2

ii

ii

ii

kkk

kkk

kkk

Txy

xTz

yzT

BBkB

2i ki T

BBB 2 Tt te ii xkBB

kkT 2

Im

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Connection with Connection with effect: effect: writhe with writhe with internalinternal twist as by-product twist as by-product

clockwise tilt(right handed)

left handedinternal twist

031 / bjuω both for thermal/magnetic

buoyancy

JBB

T dt

d2

T

BBJ

effect produces

helical field

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Paradigm shiftsParadigm shiftsi) 1980: magnetic buoyancy (Spiegel & Weiss)

overshoot layer dynamos

ii) 1985: helioseismology: d/dr > 0 dynamo dilema, flux transport dynamos

iii) 1992: catastrophic -quenching Rm-1 (Vainshtein & Cattaneo) Parker’s interface dynamo Backcock-Leighton mechanism

April 20, 2023

(i) Is magnetic buoyancy a problem?(i) Is magnetic buoyancy a problem?

Stratified dynamo simulation in 1990Expected strong buoyancy losses,but no: downward pumping Tobias et al. (2001)

April 20, 2023

(ii) Before helioseismology(ii) Before helioseismology• Angular velocity (at 4o latitude):

– very young spots: 473 nHz– oldest spots: 462 nHz– Surface plasma: 452 nHz

• Conclusion back then:– Sun spins faster in deaper convection zone– Solar dynamo works with d/dr<0: equatorward migr

Yoshimura (1975) Thompson et al. (1975)Brandenburg et al. (1992)

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Near-surface shear layer:Near-surface shear layer:spots rooted at spots rooted at r/Rr/R=0.95?=0.95?

Benevolenskaya, Hoeksema, Kosovichev, Scherrer (1999) Pulkkinen & Tuominen (1998)

nHz 473/360024360

/7.14

ds

do

o

=AZ=(180/) (1.5x107) (210-8)

=360 x 0.15 = 54 degrees!

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(iii) Problems with mean-field theory?(iii) Problems with mean-field theory?

• Catastrophic quenching?– ~ Rm

-1, t ~ Rm-1

– Field strength vanishingly small?

• Something wrong with simulations– so let’s ignore the problem

• Possible reasons:– Suppression of lagrangian chaos?– Suffocation from small scale magnetic helicity?

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Revisit paradigm shiftsRevisit paradigm shiftsi) 1980: magnetic buoyancy

counteracted by pumping

ii) 1985: helioseismology: d/dr > 0 negative gradient in near-surface shear layer

iii) 1992: catastrophic -quenching overcome by helicity fluxes in the Sun: by coronal mass ejections

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Upcoming dynamo effort in Upcoming dynamo effort in StockholmStockholm

Soon hiring:Soon hiring:• 4 students4 students• 4 post-docs4 post-docs• 1 assistant professor1 assistant professor• Long-term visitorsLong-term visitors

April 20, 2023

Pencil CodePencil Code

• Started in Sept. 2001 with Wolfgang Dobler

• High order (6th order in space, 3rd order in time)

• Cache & memory efficient

• MPI, can run PacxMPI (across countries!)

• Maintained/developed by ~20 people (SVN)

• Automatic validation (over night or any time)

• Max resolution so far 10243 , 256 procs

• Isotropic turbulence– MHD, passive scl, CR

• Stratified layers– Convection, radiation

• Shearing box– MRI, dust, interstellar– Self-gravity

• Sphere embedded in box– Fully convective stars– geodynamo

• Other applications– Homochirality– Spherical coordinates

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Increase in # of auto testsIncrease in # of auto tests

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Evolution of code sizeEvolution of code size

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Simulations showing large-scale fieldsSimulations showing large-scale fieldsHelical turbulence (By) Helical shear flow turb.

Convection with shear Magneto-rotational Inst.

1t

21t

kc

k

Käp

yla

et a

l (20

08)

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Convection with shear and Convection with shear and

Käpylä et al (2008)

with rotation without rotation

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How do they work?How do they work?

Interlocked poloidal and toroidal fields

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Magnetic helicityMagnetic helicity V

VH d BA

1

2

212 H

11

d d1

SL

H SBA

2 d2

S

SA

1S

1

AB

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How do they work?How do they work?

effect

Produce interlockedfield at large scale(of positive helicity, say)

… by generating interlockedsmall-scale field of

opposite helicity

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Effect of helicityEffect of helicityB

rand

enbu

rg (

2005

, ApJ

)

1046 Mx2/cycle

April 20, 2023 42

ConclusionConclusion• 11 yr cycle• Dyamo (SS vs LS)• Problems

– -quenching– slow saturation

• Solution– Modern -effect theory– j.b contribution– Magnetic helicity fluxes

• Location of dynamo– Distrubtion, shaped by– near-surface shear

1046 Mx2/cycle