Large-scale dynamos at low Large-scale dynamos at low magnetic Prandtl numbersmagnetic Prandtl numbers

Axel Brandenburg (Axel Brandenburg (Nordita, StockholmNordita, Stockholm))

• Small-scale dynamos– Progressively harder to excite at low PrM

– But may level off …• Large-scale dynamos

– Independent of PrM– Low PrM can be used to “filter” out SS dynamo– Most of energy dissipated Ohmically– Can decrease even further

above, below, and inside the lab: PrM=10-5

2

Winter School 11-22 JanuaryWinter School 11-22 January

3

Small-scale vs large-scale dynamoSmall-scale vs large-scale dynamo

4

Low PrLow PrMM results results

• Small-scale dynamo: Rm.crit=35-70 for PrM=1 (Novikov, Ruzmaikin, Sokoloff 1983)

• Leorat et al (1981): independent of PrM (EDQNM)

• Rogachevskii & Kleeorin (1997): Rm,crit=412

• Boldyrev & Cattaneo (2004): relation to roughness

• Ponty et al.: (2005): levels off at PrM=0.2

5

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

Small PrM=: stars and discs around NSs and YSOs

Boldyrev & Cattaneo (2004)

Schekochihin et al (2005)

k

6

Levels off for Taylor-Green flowLevels off for Taylor-Green flow

• Confirmation for finite Rm for SS dynamo?

• Or effect of LS dynamo?

7

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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Re-appearence at low PrRe-appearence at low PrMM

Iskakov et al (2005)

Gap between0.05 and 0.2 ?

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Fully helical turbulenceFully helical turbulence

Brandenburg (2001, ApJ)Here: Rm=urmsl/

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ABC flow dynamoABC flow dynamo

• Rm,crit varies still by factor 2

• Spectral magnetic energy peaks at k=1

Mininni et al. (2007, PRE)

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Cartesian box MHD equationsCartesian box MHD equations

JBuA

t

visc2 ln

D

DFf

BJu

sc

t

utD

lnD

AB

BJ

Induction

Equation:

Magn.Vectorpotential

Momentum andContinuity eqns

ln2312

visc SuuF

Viscous force

forcing function kk hf 0f (eigenfunction of curl)

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Growth rateGrowth rate

• Growth rate scaling for large Rm as for SS dynamo• Helical dynamo still excited for low Rm

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Kinematic regime

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Kinematic vs saturated regimeKinematic vs saturated regime

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Spectra in kinematic regimeSpectra in kinematic regime

• Kazantsev scaling for PrM=1

• Progressively more energy at large scale

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Compensated spectraCompensated spectra

same thestays ~)(

or ~)(3/5

3/52/32/3

kkH

kkkkM

kinematic saturated

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Low PrLow PrMM dynamos dynamos

with helicity do workwith helicity do work• Energy dissipation via Joule• Viscous dissipation weak• Can increase Re substantially!

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PrM=1, saturated case

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U and B fields: minor changesU and B fields: minor changes

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ConclusionsConclusions

• LS dynamo must be excited

• SS dynamo too dominant, swamps LS field

• Dominant SS dynamo: artifact of large PrM=

1

f

rms1t

1

ff

12

31

31

1t

k

k

u

U

k

UC

k

k

kkC

CCD

u

uω

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

1) low PrM helps to distinguish LS and SS dynamos1) low PrM helps to distinguish LS and SS dynamos

2) Important also for accretion disc dynamos2) Important also for accretion disc dynamos