Helical MHD and a -effect

Helical MHD and Helical MHD and -effect-effect

Axel Brandenburg (Nordita, Copenhagen)

Kandaswamy Subramanian (IUCAA, Pune)

arXiv:astro-ph/0405052Phys. Rept. (244 pages, 62 figs)

Brandenburg: Helical MHD 2

MHD equationsMHD 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


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!


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 many people (CVS!)

• Automatic validation (over night or any time)

• Max resolution so far 10243


Helical versus nonhelicalHelical versus nonhelical

Inverse cascade only when Inverse cascade only when scale separationscale separation

Kida et al. (1991)Kida et al. (1991)helical forcing, but no inverse cascadehelical forcing, but no inverse cascade


Allowing for scale separationAllowing for scale separation

No inverse cascade in No inverse cascade in kinematic regimekinematic regime Decomposition in terms of Decomposition in terms of

Chandrasekhar-Kendall-Waleffe functionsChandrasekhar-Kendall-Waleffe functions

00kkkkkkk hhhA aaa

t2

peakk

Position of the peak compatible with


Kazantsev spectrum (kinematic)Kazantsev spectrum (kinematic)

Kazantsev spectrum Kazantsev spectrum confirmed (even for confirmed (even for =1) =1)

Spectrum remains highly Spectrum remains highly time-dependenttime-dependent

Opposite limit, no scale separation, forcing at kf=1-2

8

256 processor run at 1024256 processor run at 102433

EM(k) not peaked at resistive scale, as previously claimedinstead: kpeak~Rm,crit

1/2 kf ~ 6kf


Structure function exponentsStructure function exponents

agrees with She-Leveque third moment


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

Compensated spectra

(1D vs 3D)


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


Bottleneck in the literatureBottleneck in the literature

Porter, Pouquet, & Woodward (1998) using PPM, 10243 meshpoints

Kaneda et al. (2003) on the Earth simulator, 40963 meshpoints


Helical MHD turbulenceHelical MHD turbulence• Helically forced turbulence (cyclonic events)

• Small & large scale field grows exponentially

• Past saturation: slow evolution

Explained by magnetic helicity equation

14

AnimationsAnimations

15

Effects of magnetic helicity conservation

BJBA 2dt

d

0const BAEarly times:=0 important

Late times: steady state 0BJ

0 bjBJBJ

0 baBABA

flux term

02f

2m baBA kk

012f

2m

BABA

k

k

By the time a steady state is reached: net magnetic

helicity is generated


Slow-down explained by magnetic helicity conservation

2f

2m

21m 22 bBB kk

dt

dk

molecular value!!

BJBA 2dt

d

)(2

m

f22 s2m1 ttke

k

k bB


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)


Internal twist as feedback on Internal twist as feedback on (Pouquet, Frisch, Leorat 1976)(Pouquet, Frisch, Leorat 1976)

031 / bjuω

How can this be used in practice?

Need a closure for <j.b>


Rm dependence of PFL formulaRm dependence of PFL formula

031 / bjuω

St = urms kf not

suppressed in Rm dependent fashion

is suppressed in Rm dependent fashion


Revised nonlinear dynamo theoryRevised nonlinear dynamo theory(originally due to Kleeorin & Ruzmaikin 1982)(originally due to Kleeorin & Ruzmaikin 1982)

BJBA 2d

d

t

BJBBA 22d

dE

t

bjBba 22d

dE

t

Two-scale assumption JB t E

Production of large scale helicity comes at the priceof producing also small scale magnetic helicity


Express in terms of Express in terms of

bjBba 22d

dE

t

M

eqmfM B

Rkt

2

22d

d BE

Dynamical -quenching (Kleeorin & Ruzmaikin 1982)

22

20

/1

/

eqm

eqmt

BR

BR

B

BJ

Steady limit: consistent with VC92

no additional free parameters

(algebraicquenching)


Is Is tt quenched? quenched?can be in models with shearcan be in models with shear

Larger mean field

Slow growthbut short cycles:

Depends onassumption aboutt-quenching!


Additional effect of shear

Negative shear

Positive shear

Consistent with g=3 andeq

t0t /1 Bg B

Kitchatinov et al (1996), Kleeorin & Rogachevskii (1999)


Effect of surface losses of Effect of surface losses of current helicitycurrent helicity

• Large scale (LS) field:– Drainage on LS dynamo

• Rm-dependent cutoff

– Shortens saturation time

• Small scale (SS) field– Enhancement of LS dynamo

SJE d2

Sje d2

25

The need for small scale lossesThe need for small scale losses

Numerical experiment:

remove field for k>4every 1-3 turnover

times

2) higher saturation level3) still slow time scale

1) large scale losses:lower saturation level

2ff

2mm

21m 22 bBB kk

dt

dk

)(2

mm

ff22 s2mm1 ttke

k

k

bB

2smff

2 )(2 bB ttkk initial slope


How do magnetic helicity losses How do magnetic helicity losses look like?look like?

N-shaped (north)S-shaped (south)(the whole loop corresponds to CME)


Sigmoidal filamentsSigmoidal filaments

(from S. Gibson)


Examples ofExamples ofhelical structures helical structures


Simulating solar-like Simulating solar-like differential rotationdifferential rotation


Results for current helicity fluxResults for current helicity flux

kjikji BBuF ,C

First order smoothing, and tau approximation

Vishniac & Cho (2001

Expected to be finite on when there is shear


ConclusionsConclusions• Homogeneous dynamos saturate resistively

– Entirely magnetic helicity controlled

• Inhomogeneous dynamo– Open surface, equator– Still many issues to be addressed– Current helicity flux important

• Finite if there is shear

– Avoid magnetic helicity, use current helicity

Helical MHD and a -effect

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