Critical issues to get right about stellar dynamos

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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
• Screwup 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 19, 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=
nonhelicallyforced 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 kk3/23/2 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) :
EEMM((kk)E)EKK((kk) ~ ) ~ kk7/37/3
EEMM((kk)+E)+EKK((kk) ~ ) ~ kk5/35/3
8
Maybe no small scale “surface” dynamo?Maybe no small scale “surface” dynamo?
Small PrM=: stars and discs around NSs and YSOs
Here: nonhelicallyforced 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 BabcockLeighton
• 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 byproduct twist as byproduct
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
Slowdown 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 ChandrasekharKendallWaleffe functionsChandrasekharKendallWaleffe functions
00kkkkkkk hhhA aaa
t2
peakk
Position of the peak compatible with
0
sin
cos
z
z
B LS field: forcefree 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, 1209)
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
Twoscale 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 kspace• Shear transport of helicity in xspace
– 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 19, 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