dynamo action in shear flow turbulence axel brandenburg (nordita, copenhagen) collaborators: nils...
DESCRIPTION
Dynamos & shear flow turbulence3 (i) Turbulence in ideal hydro Porter, Pouquet, Woodward (1998, Phys. Fluids, 10, 237)TRANSCRIPT
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Dynamo action in Dynamo action in shear flow turbulenceshear flow turbulence
Axel BrandenburgAxel Brandenburg (Nordita, Copenhagen) (Nordita, Copenhagen)Collaborators:Collaborators:
Nils Erland HaugenNils Erland Haugen (Univ. Trondheim) (Univ. Trondheim)Wolfgang DoblerWolfgang Dobler (Freiburg (Freiburg Calgary) Calgary)
Tarek YousefTarek Yousef (Univ. Trondheim) (Univ. Trondheim)Antony MeeAntony Mee (Univ. Newcastle) (Univ. Newcastle)
• Ideal vs non-ideal simulations• Pencil code• Application to the sun
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Dynamos & shear flow turbulence 2
Turbulence in astrophysicsTurbulence in astrophysics
• Gravitational and thermal energy– Turbulence mediated by instabilities
• convection• MRI (magneto-rotational, Balbus-Hawley)
• Explicit driving by SN explosions– localized thermal (perhaps kinetic) sources
• Which numerical method should we use?
Korpi et al. (1999), Sarson et al. (2003)Korpi et al. (1999), Sarson et al. (2003)
no dynamo here…
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Dynamos & shear flow turbulence 3
(i) Turbulence in ideal hydro(i) Turbulence in ideal hydro
Porter, Pouquet, Woodward(1998, Phys. Fluids, 10, 237)
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4
Direct vs hyperDirect vs hyper at 512 at 51233
Withhyperdiffusivity
Normaldiffusivity
Biskamp & Müller (2000, Phys Fluids 7, 4889)
u2u4
4
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Dynamos & shear flow turbulence 5
Ideal hydroIdeal hydro: should we be worried?: should we be worried?
• Why this k-1 tail in the power spectrum?– Compressibility?– PPM method– Or is real??
• Hyperviscosity destroys entire inertial range?– Can we trust any ideal method?
• Needed to wait for 40963 direct simulations
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Dynamos & shear flow turbulence 6
33rdrd order hyper: inertial range OK order hyper: inertial range OK
Different resolution: bottleneck & inertial range
SS 12)(
nn
Traceless rate of strain tensor
uuF 431631
visc 1n
3rd order dynamical hyperviscosity 3 22
32 S
Hyperviscous heatHau
gen
& B
rand
enbu
rg (P
RE
70, 0
2640
5)
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Hyperviscous, Smagorinsky, normalHyperviscous, Smagorinsky, normal
Inertial range unaffected by artificial diffusion
Hau
gen
& B
rand
enbu
rg (P
RE
70, 0
2640
5, a
stro
-ph/
0412
66)
height of bottleneck increased
onset of bottleneck at same position
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Dynamos & shear flow turbulence 8
Bottleneck effect: Bottleneck effect: 1D vs 1D vs 3D3D spectra spectra
Compensated spectra
(1D vs 3D)
Why did wind tunnels not show this?
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Dynamos & shear flow turbulence 9
Relation to ‘laboratory’ 1D spectraRelation to ‘laboratory’ 1D spectra2222
3 )(4)( kuku kdkE kD yxkyxkE zzD d d ),,(2)( 2
1 u
kkkkkkkzk
z d )(4d ),(4 2
0
2
uu
kk
E
zk
D d 3
0zk
222zkkk
Dobler, et al(2003, PRE 68, 026304)
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Dynamos & shear flow turbulence 10
(ii) Energy and helicity(ii) Energy and helicity22
21 2
dd Sufuu pt
2221
dd ωufu t
2/112/1221
dd uωωfuωt
Incompressible:
kkuω 2/1How diverges as 0
Inviscid limit different from inviscid case!
surface termsignored
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Dynamos & shear flow turbulence 11
Magnetic caseMagnetic case
2221
dd JBJuB t
0dd 2/12/1
21 BJBBuBA
t
kkBJ 2/1How J diverges as 0
Ideal limit and ideal case similar!
2/112/1221
dd uωωfuωt
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Dynamos & shear flow turbulence 12
Dynamo growth & saturationDynamo growth & saturation
Significant fieldalready after
kinematicgrowth phase
followed byslow resistive
adjustment
0 bjBJ
0 baBA
0221 f
bB kk
021211 f
bB kk
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Dynamos & shear flow turbulence 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 2ddt
PRL 88, 055003
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Dynamos & shear flow turbulence 14
Slow-down explained by magnetic helicity conservation
2f
2m
21m 22 bBB kk
dtdk
molecular value!!
BJBA 2dtd
)(2
m
f22 s2m1 ttke
kk bB
ApJ 550, 824
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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
Yousef & BrandenburgA&A 407, 7 (2003)
031 / bjuω both for thermal/magnetic
buoyancy
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Dynamos & shear flow turbulence 16
(iii) Small scale dynamo: Pm dependence??(iii) Small scale dynamo: Pm dependence??
Small Pm=: stars and discs around NSs and YSOs
Here: non-helicallyforced turbulence
SchekochihinHaugenBrandenburget al (2005)
k
Cattaneo,Boldyrev
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(iv) Does compressibility affect the dynamo?(iv) Does compressibility affect the dynamo?
Direct simulation, =5 Direct and shock-capturing simulations, =1
Shocks sweep up all the field: dynamo harder?-- or artifact of shock diffusion?
Bimodal behavior!ψ u
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Dynamos & shear flow turbulence 18
OverviewOverview• Hydro: LES does a good job, but hi-res important
– the bottleneck is physical– hyperviscosity does not affect inertial range
• Helical MHD: hyperresistivity exaggerates B-field• Prandtl number does matter!
– LES for B-field difficult or impossible!
Fundamental questions idealized simulations important at this stage!
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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 (CVS!)• 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
• Sphere embedded in box– Fully convective stars– geodynamo
• Other applications– Homochirality– Spherical coordinates
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Dynamos & shear flow turbulence 20
(i) Higher order – less viscosity(i) Higher order – less viscosity
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Dynamos & shear flow turbulence 21
(ii) High-order temporal schemes(ii) High-order temporal schemes
),( 111 iiiii utFtww
Main advantage: low amplitude errors
iiii wuu 1
3)1()(
0 , uuuu nn
2/1 ,1 ,3/11 ,3/2 ,0
321
321
1 ,2/12/1 ,0
21
21
10
1
1
3rd order
2nd order
1st order
2N-RK3 scheme (Williamson 1980)
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Cartesian box MHD equationsCartesian box MHD equations
JBuA
t
visc2 ln
DD FfBJu
sct
utD
lnD
ABBJInduction
Equation:Magn.Vectorpotential
Momentum andContinuity eqns
ln2312
visc SuuF
Viscous force
forcing function kk hf 0f (eigenfunction of curl)
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Dynamos & shear flow turbulence 23
Vector potentialVector potential• B=curlA, advantage: divB=0• J=curlB=curl(curlA) =curl2A• Not a disadvantage: consider Alfven waves
zuB
tb
zbB
tu
00 and ,
uBta
zaB
tu
02
2
0 and ,
B-formulation
A-formulation 2nd der onceis better than1st der twice!
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Dynamos & shear flow turbulence 24
Comparison of Comparison of AA and and BB methods methods
2
2
02
2
2
2
0 and ,zauB
ta
zu
zaB
tu
2
2
02
2
0 and ,zb
zuB
tb
zu
zbB
tu
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Dynamos & shear flow turbulence 25
256 processor run at 1024256 processor run at 102433
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Dynamos & shear flow turbulence 26
Structure function exponentsStructure function exponents
agrees with She-Leveque third moment
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Dynamos & shear flow turbulence 27
Wallclock time versus processor #
nearly linearScaling
100 Mb/s showslimitations
1 - 10 Gb/sno limitation
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Dynamos & shear flow turbulence 28
Sensitivity to layout onSensitivity to layout onLinux clustersLinux clusters
yprox x zproc4 x 32 1 (speed)8 x 16 3 times slower16 x 8 17 times slower
Gigabituplink 100 Mbit
link only
24 procsper hub
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Why this sensitivity to layout?
0 1 2 3 4 5 6 7 8 9 0 1 2 3 4 56 7 8 9 0 1 2 3 4
All processors need to communicatewith processors outside to group of 24
16x8
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Dynamos & shear flow turbulence 30
Use exactly 4 columns
0 1 2 34 5 6 78 9 10 1112 13 14 1516 17 18 1920 21 22 230 1 2 34 5 6 78 9 10 1112 13 14 15
Only 2 x 4 = 8 processors need to communicate outside the group of 24 optimal use of speed ratio between 100 Mb ethernet switch and 1 Gb uplink
4x32
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Dynamos & shear flow turbulence 31
Pre-processed data for animationsPre-processed data for animations
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Dynamos & shear flow turbulence 32
Simulating solar-like differential rotation Simulating solar-like differential rotation
• Still helically forced turbulence• Shear driven by a friction term• Normal field boundary condition
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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)
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Wasn’t the dynamo supposed to work at the bottom?Wasn’t the dynamo supposed to work at the bottom?
• Flux storage• Distortions weak• Problems solved with
meridional circulation• Size of active regions
• Neg surface shear: equatorward migr.• Max radial shear in low latitudes• Youngest sunspots: 473 nHz• Correct phase relation• Strong pumping (Thomas et al.)
• 100 kG hard to explain• Tube integrity• Single circulation cell• Too many flux belts*• Max shear at poles*• Phase relation*• 1.3 yr instead of 11 yr at bot
• Rapid buoyant loss*• Strong distortions* (Hale’s polarity)• Long term stability of active regions*• No anisotropy of supergranulation
in favor
against
Tachocline dynamos Distributed/near-surface dynamo
Brandenburg (2005, ApJ 625, June 1 isse)
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Dynamos & shear flow turbulence 35
In the days before In the days before helioseismologyhelioseismology
• 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
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Dynamos & shear flow turbulence 36
Application to the sun: spots rooted at Application to the sun: spots rooted at r/Rr/R=0.95=0.95B
enev
ole n
skay
a, H
oeks
ema,
Ko s
ovic
h ev ,
Sc h
e rre
r (1 9
99) Pulkkinen &
Tuominen (1998)
nHz 473/360024360
/7.14
dsd
o
o–Overshoot dynamo cannot catch up
=AZ=(180/) (1.5x107) (210-8)
=360 x 0.15 = 54 degrees!
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Is magnetic buoyancy a problem?Is magnetic buoyancy a problem?
compressible stratified dynamo simulation in 1990expected strong buoyancy losses, but no: downward pumping
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Lots of surprises…Lots of surprises…• Shearflow turbulence: likely to produce LS field
– even w/o stratification (WxJ effect, similar to Rädler’s xJ effect)• Stratification: can lead to effect
– modify WxJ effect– but also instability of its own
• SS dynamo not obvious at small Pm• Application to the sun?
– distributed dynamo can produce bipolar regions– perhaps not so important?– solution to quenching problem? No: M even from WxJ effect
1046 Mx2/cycle