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
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…
Dynamos & shear flow turbulence 3
(i) Turbulence in ideal hydro(i) Turbulence in ideal hydro
Porter, Pouquet, Woodward(1998, Phys. Fluids, 10, 237)
4
Direct vs hyperDirect vs hyper at 512 at 51233
Withhyperdiffusivity
Normaldiffusivity
Biskamp & Müller (2000, Phys Fluids 7, 4889)
u2u4
4
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
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)
7
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
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?
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)
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
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
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
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
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
15
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
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
17
(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
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!
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
Dynamos & shear flow turbulence 20
(i) Higher order – less viscosity(i) Higher order – less viscosity
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)
22
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)
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!
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
Dynamos & shear flow turbulence 25
256 processor run at 1024256 processor run at 102433
Dynamos & shear flow turbulence 26
Structure function exponentsStructure function exponents
agrees with She-Leveque third moment
Dynamos & shear flow turbulence 27
Wallclock time versus processor #
nearly linearScaling
100 Mb/s showslimitations
1 - 10 Gb/sno limitation
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
29
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
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
Dynamos & shear flow turbulence 31
Pre-processed data for animationsPre-processed data for animations
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
33
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)
34
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)
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
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!
Is magnetic buoyancy a problem?Is magnetic buoyancy a problem?
compressible stratified dynamo simulation in 1990expected strong buoyancy losses, but no: downward pumping
38
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