dynamo action in shear flow turbulence axel brandenburg (nordita, copenhagen) collaborators: nils...

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…


(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


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


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


Bottleneck effect: Bottleneck effect: 1D vs 1D vs 3D3D spectra spectra

Compensated spectra

(1D vs 3D)

Why did wind tunnels not show this?


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)


(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


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


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


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


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


(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


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


(i) Higher order – less viscosity(i) Higher order – less viscosity


(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)


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!


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


256 processor run at 1024256 processor run at 102433


Structure function exponentsStructure function exponents

agrees with She-Leveque third moment


Wallclock time versus processor #

nearly linearScaling

100 Mb/s showslimitations

1 - 10 Gb/sno limitation


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


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


Pre-processed data for animationsPre-processed data for animations


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)


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


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

dynamo action in shear flow turbulence axel brandenburg (nordita, copenhagen) collaborators: nils...

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