Dynamo action in shear flow turbulence

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Dynamo action in shear flow turbulence

• Axel Brandenburg (Nordita, Copenhagen)
• Collaborators
• Nils Erland Haugen (Univ. Trondheim)
• Wolfgang Dobler (Freiburg ? Calgary)
• Tarek Yousef (Univ. Trondheim)
• Antony Mee (Univ. Newcastle)

• Ideal vs non-ideal simulations
• Pencil code
• Application to the sun

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Turbulence 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)
no dynamo here
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(i) Turbulence in ideal hydro
Porter, Pouquet, Woodward (1998, Phys. Fluids,
10, 237)
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Direct vs hyper at 5123
Biskamp Müller (2000, Phys Fluids 7, 4889)
Normal diffusivity
With hyperdiffusivity
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Ideal hydro 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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3rd order hyper inertial range OK
Different resolution bottleneck inertial range
Haugen Brandenburg (PRE 70, 026405)
Traceless rate of strain tensor
Hyperviscous heat
3rd order dynamical hyperviscosity m3
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Hyperviscous, Smagorinsky, normal
height of bottleneck increased
Haugen Brandenburg (PRE 70, 026405,
astro-ph/041266)
onset of bottleneck at same position
Inertial range unaffected by artificial diffusion
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Bottleneck effect 1D vs 3D spectra
Why did wind tunnels not show this?
Compensated spectra (1D vs 3D)
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Relation to laboratory 1D spectra
Dobler, et al (2003, PRE 68, 026304)
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(ii) Energy and helicity
surface terms ignored
Incompressible
How w diverges as n?0
Inviscid limit different from inviscid case!
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Magnetic case
How J diverges as h?0
Ideal limit and ideal case similar!
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Dynamo growth saturation
Significant field already after kinematic growth
phase
followed by slow resistive adjustment
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Helical dynamo saturation with hyperdiffusivity
for ordinary hyperdiffusion
ratio 53125 instead of 5
PRL 88, 055003
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Slow-down explained by magnetic helicity
conservation
molecular value!!
ApJ 550, 824
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Connection with a effect writhe with internal
twist as by-product
clockwise tilt (right handed)
W
? left handed internal twist
Yousef Brandenburg AA 407, 7 (2003)
both for thermal/magnetic buoyancy
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(iii) Small scale dynamo Pm dependence??
Small Pmn/h stars and discs around NSs and YSOs
Schekochihin Haugen Brandenburg et al (2005)
Cattaneo, Boldyrev
k
Here non-helically forced turbulence
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(iv) Does compressibility affect the dynamo?
Shocks sweep up all the field dynamo harder?
-- or artifact of shock diffusion?
Direct and shock-capturing simulations, n/h1
Direct simulation, n/h5
? Bimodal behavior!
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Overview

• 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 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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(i) Higher order less viscosity
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(ii) High-order temporal schemes
Main advantage low amplitude errors
2N-RK3 scheme (Williamson 1980)
2nd order
3rd order
1st order
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Cartesian box MHD equations
Magn. Vector potential
Induction Equation
Momentum and Continuity eqns
Viscous force
forcing function
(eigenfunction of curl)
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Vector potential

• BcurlA, advantage divB0
• JcurlBcurl(curlA) curl2A
• Not a disadvantage consider Alfven waves

B-formulation
A-formulation
2nd der once is better than 1st der twice!
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Comparison of A and B methods
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256 processor run at 10243
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Structure function exponents
agrees with She-Leveque
third moment
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Wallclock time versus processor
nearly linear Scaling 100 Mb/s
shows limitations 1 - 10 Gb/s no limitation
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Sensitivity to layout onLinux clusters
Gigabit uplink
100 Mbit link only

• yprox x zproc
• 4 x 32 ? 1 (speed)
• 8 x 16 ? 3 times slower
• 16 x 8 ? 17 times slower

24 procs per hub
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Why this sensitivity to layout?
16x8
All processors need to communicate with
processors outside to group of 24
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Use exactly 4 columns
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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Pre-processed data for animations
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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 no stratification
azimuthally averaged
no helicity, e.g.
Rogachevskii Kleeorin (2003, 2004)
geometry here relevant to the sun
neg helicity (northern hem.)
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Wasnt the dynamo supposed to work at the bottom?
Tachocline dynamos
Distributed/near-surface dynamo

• 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.)

in favor
against

• 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 (Hales polarity)
• Long term stability of active regions
• No anisotropy of supergranulation

Brandenburg (2005, ApJ 625, June 1 isse)
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In the days before helioseismology

• 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 dW/drlt0 equatorward migr

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Application to the sun spots rooted at r/R0.95
Benevolenskaya, Hoeksema, Kosovichev, Scherrer
(1999)
Pulkkinen Tuominen (1998)

• Overshoot dynamo cannot catch up
• DftAZDW(180/p) (1.5x107) (2p 10-8)
• 360 x 0.15 54 degrees!

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Is magnetic buoyancy a problem?
compressible stratified dynamo simulation
in 1990 expected strong buoyancy losses, but no
downward pumping
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Lots of surprises

• Shearflow turbulence likely to produce LS field
• even w/o stratification (WxJ effect, similar to
  Rädlers WxJ effect)
• Stratification can lead to a 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
• a perhaps not so important?
• solution to quenching problem? No aM even from
  WxJ effect

1046 Mx2/cycle