Numerical simulations of Numerical simulations of astrophysical dynamosastrophysical dynamos

Axel Brandenburg (Axel Brandenburg (Nordita, StockholmNordita, Stockholm))

• Dynamos: numerical issues• Alpha dynamos do exist: linear and nonlinear• Constrained by magnetic helicity: need fluxes• How high do we need to go with Rm?• MRI and low PrM issue

2

Struggle for the dynamoStruggle for the dynamo• Larmor (1919): first qualitative ideas• Cowling (1933): no antidynamo theorem• Larmor (1934): vehement response

– 2-D not mentioned• Parker (1955): cyclonic events, dynamo waves• Herzenberg (1958): first dynamo

– 2 small spinning spheres, slow dynamo (~Rm-1)

• Steenbeck, Krause, Rädler (1966): dynamo– Many papers on this since 1970

• Kazantsev (1968): small-scale dynamo– Essentially unnoticed, simulations 1981, 2000-now

3

Mile stones in dynamo researchMile stones in dynamo research

• 1970ies: mean-field models of Sun/galaxies

• 1980ies: direct simulations

• Gilman/Glatzmaier: poleward migration

• 1990ies: compressible simulations, MRI– Magnetic buoyancy overwhelmed by pumping– Successful geodynamo simulations

• 2000- magnetic helicity, catastr. quenching– Dynamos and MRI at low PrM=

4

Easy to simulate?Easy to simulate?

• Yes, but it can also go wrong

• 2 examples: manipulation with diffusion

• Large-scale dynamo in periodic box– With hyper-diffusion curl2nB

– ampitude by (k/kf)2n-1

• Euler potentials with artificial diffusion– D/Dt=, D/Dt=gradxgrad

Dynamos withDynamos withEuler PotentialsEuler Potentials

• B = grad x grad• A = grad, so A.B=0• Here: Robert flow

– Details MNRAS 401, 347

• Agreement for t<8– For smooth fields, not for -correlated initial fields

• Exponential growth (A)• Algebraic decay (EP)

6

Reasons for disagreementReasons for disagreement

• because dynamo field is helical?

• because field is three-dimensional?

• none of the two: it is because is finite

7

Is this artificial diffusion kosher?Is this artificial diffusion kosher?

R2222

D

D

D

D

tt

R

Make very small, it is artificial anyway,Surely, the R term cannot matter then?

8

Problem already in 2-D nonhelicalProblem already in 2-D nonhelical

• Works only when and are not functions of the same coordinates

0

coscos

sinsin

,

0

sin

0

sincos ,cos

yx

yx

y

yxy

R

)sinsin,0,0( 2 yxB

xyy cos ,2sin41

21 Alternative

possible in 2-D

Remember:

9

Method of choice? No, thanks

• It’s not because of helicity (cf. nonhel dyn)

• Not because of 3-D: cf. 2-D decay problem

• It’s really because (x,y,z,t) and (x,y,z,t)

10

Other good examples of dynamosOther good examples of dynamosHelical turbulence (By) Helical shear flow turb.

Convection with shear Magneto-rotational Inst.

1t

21t

kc

k

Käp

yla

et a

l (20

08)

11

One big flaw: slow saturation (explained by magnetic helicity conservation)

2f

21 , bbjBBJ

bjBJBJ

kk

bjBJBJ

2f

21

211 22 bBB kk

dt

dk

)(2

1

22 s211 ttkf e

k

k bB

molecular value!!

BJBA 2dt

d

2f

21 , bbjBBJ kk

12

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 2d

d

t

PRL 88, 055003

13

Test-field results for Test-field results for and and tt kinematic: independent of Rm (2…200)kinematic: independent of Rm (2…200)

1frms3

10

rms31

0

ku

u

Sur et al. (2008, MNRAS)

1frms

231

0

31

0

ku

u

uω

14

RRmm dependence for B~B dependence for B~Beqeq

(i) is small consistency(ii) 1 and 2 tend to cancel(iii) making small(iv) 2 small

15

Boundaries instead of periodicBoundaries instead of periodic

Brandenburg, Cancelaresi, Chatterjee, 2009, MNRAS

FbjBEba 22t

16

Boundaries instead of periodicBoundaries instead of periodic

FbjBEba 22t

Hubbard &

Brandenburg, 2010, G

AF

D

17

Connection with MRI turbulenceConnection with MRI turbulence

5123

w/o hypervisc.t = 60 = 2 orbits

No large scale field(i) Too short?(ii) No stratification?

18

Low PrLow PrMM issue issue

• Small-scale dynamo: Rm.crit=35-70 for PrM=1 (Novikov, Ruzmaikin, Sokoloff 1983)

• Leorat et al (1981): independent of PrM (EDQNM)

• Rogachevskii & Kleeorin (1997): Rm,crit=412

• Boldyrev & Cattaneo (2004): relation to roughness

• Ponty et al.: (2005): levels off at PrM=0.2

19

Re-appearence at low PrRe-appearence at low PrMM

Iskakov et al (2005)

Gap between0.05 and 0.2 ?

20

Small-scale vs large-scale dynamoSmall-scale vs large-scale dynamo

21

Low PrLow PrMM dynamos dynamos

with helicity do workwith helicity do work• Energy dissipation via Joule• Viscous dissipation weak• Can increase Re substantially!

22

Comparison with Comparison with stratifiedstratified runs runsB

rand

enbu

rg e

t al.

(199

5)

<By >

Dynamo wavesVersus

Buoyant escape

23

Phase relation in dynamosPhase relation in dynamos

2

2

23

2

2

z

BB

t

B

z

B

z

B

t

B

y

Tx

y

x

T

yx

kkc

ctzkB

ctzkB

Tk

cx

x

4/3

sin2

sin

43

Mean-field equations: Solution:

24

Unstratified: also LS fields?Unstratified: also LS fields?L

esur

& O

gilv

ie (

2008

)B

rand

enur

g et

al.

(200

8)

25

Low PrLow PrMM issue in unstratified MRI issue in unstratified MRI

Käpylä & Korpi (2010): vertical field condition

26

LS from Fluctuations of LS from Fluctuations of ijij and and ijij

Incoherent effect(Vishniac & Brandenburg 1997,Sokoloff 1997, Silantev 2000,Proctor 2007)

27

ConclusionsConclusions

• Cycles w/ stratification• Test field method robust

– Even when small scale dynamo

– 0 and t0

• Rotation and shear: *ij

– WxJ not (yet?) excited– Incoherent works