Bern, MHD, and shearBern, MHD, and shear

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• Near-surface shear layer

2

(i) Can we trust (i) Can we trust idealideal hydro? hydro?

Porter, Pouquet, Woodward(1998, Phys. Fluids, 10, 237)

Majority of public astrophysics codes are “inviscid”

3

Direct vs hyperDirect vs hyper at 512 at 51233

Withhyperdiffusivity

Normaldiffusivity

Biskamp & Müller (2000, Phys Fluids 7, 4889)

u2u4

4

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

5

Hyperviscous, Smagorinsky, normalHyperviscous, Smagorinsky, normal

Inertial range unaffected by artificial diffusion

Hau

gen

& B

rand

enbu

rg (

PR

E 7

0, 0

2640

5, a

stro

-ph/

0412

66)

height of bottleneck increased

onset of bottleneck at same position

6

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 ),(42

0

2

uu

kk

E

zk

D d 3

0zk

222zkkk

Dobler, et al(2003, PRE 68, 026304)

7

256 processor run at 1024256 processor run at 102433

Haugen et al. (2003, ApJ 597, L131)

8

(ii) Helical dynamo saturation (ii) Helical dynamo saturation with hyperdiffusivitywith hyperdiffusivity

23231 f

bB kk

for ordinaryhyperdiffusion

42k

221 f

bB kk ratio 53=125 instead of 5

BJBA 2d

d

t

PRL 88, 055003

9

(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

10

(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

Supersonic shock turblenceSupersonic shock turblence

Gustafsson et al. (2006, A&A, in press)

12

LES conclusionsLES conclusions• 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!

13

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

14

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

15

(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 ,3/1

1 ,3/2 ,0

321

321

1 ,2/1

2/1 ,0

21

21

1

0

1

1

3rd order

2nd order

1st order

2N-RK3 scheme (Williamson 1980)

16

Cartesian box MHD equationsCartesian box MHD equations

JBuA

t

visc2 ln

D

DFf

BJu

sc

t

utD

lnD

AB

BJ

Induction

Equation:

Magn.Vectorpotential

Momentum andContinuity eqns

ln2312

visc SuuF

Viscous force

forcing function kk hf 0f (eigenfunction of curl)

17

Vector potentialVector potential

• B=curlA, advantage: divB=0• J=curlB=curl(curlA) =curl2A• Not a disadvantage: consider Alfven waves

z

uB

t

b

z

bB

t

u

00 and ,

uBt

a

z

aB

t

u02

2

0 and ,

B-formulation

A-formulation 2nd der onceis better than1st der twice!

18

Comparison of Comparison of AA and and BB methods methods

2

2

02

2

2

2

0 and ,z

auB

t

a

z

u

z

aB

t

u

2

2

02

2

0 and ,z

b

z

uB

t

b

z

u

z

bB

t

u

19

Wallclock time versus processor #

nearly linearScaling

100 Mb/s showslimitations

1 - 10 Gb/sno limitation

20

Forced LS dynamo with Forced LS dynamo with nono stratification stratification

•Open bc critical•Helicity losses•Shear relevant to the sun

azimuthallyaveraged

Actual field:(i) kinematicphase

(ii) late phase

21

Current helicity and Current helicity and magn. hel. fluxmagn. hel. flux

Bao & Zhang (1998),neg. in north, plus in south

(also Seehafer 1990)

Berger & Ruzmaikin (2000)

cycle/Mx104 246S

NDeVore (2000)

cycle/Mx10 246

(for BR & CME)

22

Helicity fluxes at large and small scalesHelicity fluxes at large and small scales

Negative current helicity:net production in northern hemisphere

SJE d2 Sje d21046 Mx2/cycle

Brandenburg & Sandin (2004, A&A 427, 13)

Helicity fluxes from shear: Vishniac & Cho (2001, ApJ 550, 752)Subramanian & Brandenburg (2004, PRL 93, 20500)

23

Application to the sun: spots rooted at Application to the sun: spots rooted at r/Rr/R=0.95=0.95

Benevolenskaya, Hoeksema, Kosovichev, Scherrer (1999) Pulkkinen & Tuominen (1998)

nHz 473/360024360

/7.14

ds

do

o

=AZ=(180/) (1.5x107) (210-8)

=360 x 0.15 = 54 degrees!

24

Simulations of near-surface shearSimulations of near-surface shear

• Unstable layer in 0<z<1• 0o latitude• 4x4x1 aspect ratio• 512x512x256

Prograde pattern speed, but rather slow(Green & Kosovichev 2006)

25

Simulations of near-surface shearSimulations of near-surface shear

4x4x1 aspect ratio512x512x256

0o lat

15o latnegative uyuz stress negative shear

Convection with rotationConvection with rotation

Inv. Rossby Nr. 2d/urms=4

Horizontal flow patternHorizontal flow pattern

Stongly retrograde motionsPlunge into prograde shock

yx

28

Arguments against and in favor?Arguments against and in favor?

• 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, 539)

Is magnetic buoyancy a problem?Is magnetic buoyancy a problem?

Stratified dynamo simulation in 1990Expected strong buoyancy losses,but no: downward pumping Tobias et al. (2001)

Is magnetic buoyancy a problem?Is magnetic buoyancy a problem?

Stratified dynamo simulation in 1990Expected strong buoyancy losses,but no: downward pumping Brandenburg et al. (1996)

Is magnetic buoyancy a problem?Is magnetic buoyancy a problem?

Stratified dynamo simulation in 1990Expected strong buoyancy losses,but no: downward pumping Tobias et al. (2001)

32

ConclusionsConclusions• 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