Large-scale dynamos at low magnetic Prandtl numbers
description
Transcript of Large-scale dynamos at low magnetic Prandtl numbers
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
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Winter School 11-22 JanuaryWinter School 11-22 January
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Small-scale vs large-scale dynamoSmall-scale vs large-scale dynamo
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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
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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
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Levels off for Taylor-Green flowLevels off for Taylor-Green flow
• Confirmation for finite Rm for SS dynamo?
• Or effect of LS dynamo?
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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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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
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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