Critical issues to get right Critical issues to get right about stellar dynamosabout stellar dynamosAxel BrandenburgAxel Brandenburg (Nordita, Copenhagen) (Nordita, Copenhagen)

Shukurov et al. (2006, A&A 448, L33)Schekochihin et al. (2005, ApJ 625, L115)

Brandenburg & Subramanian (2005, Phys. Rep., 417, 1)Brandenburg (2001, ApJ 550, 824; and 2005, ApJ 625, 539)

1.1. Small scale dynamo and LESSmall scale dynamo and LES2.2. Pm=0.1 or less (to elim. SS dyn)Pm=0.1 or less (to elim. SS dyn)3.3. Magn. Helicity transport and lossMagn. Helicity transport and loss

2

Long way to go: what to expect?Long way to go: what to expect?

• Turbulent inertial range somewhere• Departures from k -5/3

– k -0.1 correction– Bottleneck effect

• Screw-up from MHD nonlocality?– EM(k) and EK(k) parallel or even overlapping?– Or peak of EM(k) at resistive scales?– Does small scale dynamo work for Pm<<1 and Rm>>1?– How is large scale dynamo affected by small scales?

• Implications for catastrophic -quenching

April 21, 2023 3

Hyperviscous, Smagorinsky, normalHyperviscous, Smagorinsky, normal

Inertial range unaffected by artificial diffusion

Hau

gen

& B

rand

enbu

rg (

Phy

s. F

luid

s, a

stro

-ph/

0412

66)

height of bottleneck increased

with hyper

onset of bottleneck at same position

CCSS=0.20=0.20

4

Allow for Allow for BB: small scale dynamo action: small scale dynamo action

PrM=

non-helicallyforced turbulence

PrM=

5

Peaked at small scales?Peaked at small scales?

PrM=

During saturation peak moves to smaller k.

PrM=

Can we expect inertial range at (much) larger resolution?

k+3/2 Kazantsev spectrum, even for Pm=1

6

Looks like Looks like kk-3/2-3/2 at 1024 at 102433

Still not large enough?!

Spectra not on top of each other??

Different from case with imposed field!

7

Help from LES and theoryHelp from LES and theory

converging spectra at large converging spectra at large kk ?? ??

Can be reproduced with predictionCan be reproduced with predictionby Mby Müüller & Grappin (2005, PRL) :ller & Grappin (2005, PRL) :

|E|EMM((kk)-E)-EKK((kk)| ~ )| ~ kk-7/3-7/3

EEMM((kk)+E)+EKK((kk) ~ ) ~ kk-5/3-5/3

8

Maybe no small scale “surface” dynamo?Maybe no small scale “surface” dynamo?

Small PrM=: stars and discs around NSs and YSOs

Here: non-helicallyforced turbulence

SchekochihinHaugenBrandenburget al (2005)

k

When should we think of extrapolating to the sun?When should we think of extrapolating to the sun?Implications for global models (w/strong SS field)Implications for global models (w/strong SS field)

9

Large scale dynamosLarge scale dynamos

• Dynamo number for 2 dynamo

• May C and/or CC not big enough

• Catastrophic quenching– Suppression of lagrangian chaos?– Suffocation from small scale magnetic helicity?

• Applies also to Babcock-Leighton

• Most likely solution: magnetic helicity fluxes

1

ff

12

31

f31

1t k

k

kkC

u

uω

10

Penalty from Penalty from 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

11

Slow saturationSlow saturationB

rand

enbu

rg (

2001

, ApJ

550

, 824

)

s212

1

f22 1 ttkek

k bB

1.1. Excellent fit formulaExcellent fit formula2.2. Microscopic diffusivityMicroscopic diffusivity

12

Slow-down explained by magnetic helicity conservation

2f

21

211 22 bBB kk

dt

dk

)(2

1

22 s211 ttkf e

k

k bB

molecular value!!

BJBA 2dt

d

13

Helical dynamo saturation with Helical dynamo saturation with hyperdiffusivityhyperdiffusivity

23231 f

bB kk

for ordinaryhyperdiffusion

42k

221 f

bB kk ratio 53=125 instead of 5

BJBA 2d

d

t

PRL 88, 055003

14

Scale separation: inverse cascadeScale separation: inverse cascade

No inverse cascade in No inverse cascade in kinematic regimekinematic regime

Decomposition in terms of Decomposition in terms of Chandrasekhar-Kendall-Waleffe functionsChandrasekhar-Kendall-Waleffe functions

00kkkkkkk hhhA aaa

t2

peakk

Position of the peak compatible with

0

sin

cos

z

z

B LS field: force-free Beltrami

15

Periodic boxPeriodic box, no shear, no shear: resistively limited saturation: resistively limited saturation

Significant fieldalready after

kinematicgrowth phase

followed byslow resistive

adjustment

0 bjBJ

0 baBA

0221 f

bB kk

021211 f

bB kkBlackman & Brandenburg (2002, ApJ 579, 397)

Brandenburg & SubramanianPhys. Rep. (2005, 417, 1-209)

BJBA 2dt

d

16

Boundaries instead of periodicBoundaries instead of periodic

Brandenburg & Subramanian (2005, AN 326, 400)

0 :b.c. yx BB

17

Revised nonlinear dynamo theoryRevised nonlinear dynamo theory(originally due to Kleeorin & Ruzmaikin 1982)(originally due to Kleeorin & Ruzmaikin 1982)

BJBA 2d

d

t

BJBBA 22d

dE

t

bjBba 22d

dE

t

Two-scale assumption

JB t E

031 / bjuω

Dynamical quenching

M

eqmfM B

Rkt

2

22d

d BE

Kleeorin & Ruzmaikin (1982)

22

20

/1

/

eqm

eqmt

BR

BR

B

BJ

Steady limit algebraic quenching:

( selectivedecay)

18

General formula with current helicity fluxGeneral formula with current helicity flux

22

2ft

2SSC

2f2

1

/1

2/

/

eqm

eqmK

BR

kt

BkR

B

BJ

F

Rm also in thenumerator

SSCt

F

cebj 2

jc

beje 2SSCF

Advantage over magnetic helicity1) <j.b> is what enters effect2) Can define helicity density

19

Significance of shearSignificance of shear

• transport of helicity in k-space• Shear transport of helicity in x-space

– Mediating helicity escape ( plasmoids)

– Mediating turbulent helicity flux

kjikji BBu 4 ,C F

Expression for current helicity flux (first order smoothing, tau approximation)

Vishniac & Cho (2001, ApJ 550, 752)Subramanian & Brandenburg (2004, PRL 93, 20500)

Expected to be finite on when there is shear

Arlt & Brandenburg (2001, A&A 380, 359)

Schnack et al.

20

Helicity fluxes at large and small scalesHelicity fluxes at large and small scales

Negative current helicity:net production in northern hemisphere

SJE d2 Sje d21046 Mx2/cycle

Brandenburg & Sandin (2004, A&A 427, 13)

Helicity fluxes from shear: Vishniac & Cho (2001, ApJ 550, 752)Subramanian & Brandenburg (2004, PRL 93, 20500)

21

Forced LS dynamo with Forced LS dynamo with nono stratification stratification

geometryhere relevantto the sun

no helicity, e.g.

azimuthallyaveraged

neg helicity(northern hem.)

...21

JWBB

a

t

Rogachevskii & Kleeorin (2003, 2004)

22

Examples ofExamples ofhelical structures helical structures

23

Mean field model with advective fluxMean field model with advective flux

Shukurov et al. (2006, A&A 448, L33)Shukurov et al. (2006, A&A 448, L33)

24

Saturation behavior with advective fluxSaturation behavior with advective flux

Shukurov et al. (2006, A&A 448, L33)Shukurov et al. (2006, A&A 448, L33)

April 21, 2023 25

ConclusionsConclusions

• LES & DNS EM(k) and EK(k) overlap (?)

• SS dynamo may not work in the sun• Only LS dynamo, if excited

– and if CMEs etc.

1046 Mx2/cycle