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Dynamo action in shear flow turbulence

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Nils Erland Haugen (Univ. Trondheim) Wolfgang Dobler ... In the days before helioseismology. Angular velocity (at 4o latitude): very young spots: 473 nHz ... – PowerPoint PPT presentation

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Title: Dynamo action in shear flow turbulence


1
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

2
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
3
(i) Turbulence in ideal hydro
Porter, Pouquet, Woodward (1998, Phys. Fluids,
10, 237)
4
Direct vs hyper at 5123
Biskamp Müller (2000, Phys Fluids 7, 4889)
Normal diffusivity
With hyperdiffusivity
5
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

6
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
7
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
8
Bottleneck effect 1D vs 3D spectra
Why did wind tunnels not show this?
Compensated spectra (1D vs 3D)
9
Relation to laboratory 1D spectra
Dobler, et al (2003, PRE 68, 026304)
10
(ii) Energy and helicity
surface terms ignored
Incompressible
How w diverges as n?0
Inviscid limit different from inviscid case!
11
Magnetic case
How J diverges as h?0
Ideal limit and ideal case similar!
12
Dynamo growth saturation
Significant field already after kinematic growth
phase
followed by slow resistive adjustment
13
Helical dynamo saturation with hyperdiffusivity
for ordinary hyperdiffusion
ratio 53125 instead of 5
PRL 88, 055003
14
Slow-down explained by magnetic helicity
conservation
molecular value!!
ApJ 550, 824
15
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
16
(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
17
(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!
18
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!
19
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

20
(i) Higher order less viscosity
21
(ii) High-order temporal schemes
Main advantage low amplitude errors
2N-RK3 scheme (Williamson 1980)
2nd order
3rd order
1st order
22
Cartesian box MHD equations
Magn. Vector potential
Induction Equation
Momentum and Continuity eqns
Viscous force
forcing function
(eigenfunction of curl)
23
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!
24
Comparison of A and B methods
25
256 processor run at 10243
26
Structure function exponents
agrees with She-Leveque
third moment
27
Wallclock time versus processor
nearly linear Scaling 100 Mb/s
shows limitations 1 - 10 Gb/s no limitation
28
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
29
Why this sensitivity to layout?
16x8
All processors need to communicate with
processors outside to group of 24
30
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
31
Pre-processed data for animations
32
Simulating solar-like differential rotation
  • Still helically forced turbulence
  • Shear driven by a friction term
  • Normal field boundary condition

33
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.)
34
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)
35
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

36
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!

37
Is magnetic buoyancy a problem?
compressible stratified dynamo simulation
in 1990 expected strong buoyancy losses, but no
downward pumping
38
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
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