1

Simulations and understanding

large-scale dynamos

• Issues with global models

• Possibility of smaller scales

• Consequences of this

• Magnetic flux concentrations

• Unusual dynamo effects

Axel Brandenburg

(Nordita, Stockholm CU Boulder)

2

Global models suggest

• Distributed dynamo action

– Difference to flux transport dynamos

– Would require smaller turb. diff.

ht=urms/3kf=urmsl/3

• Surface flux from upper layers

– Difference to deeply rooted tube picture

– Surface flux reamplification needed

– NEMPI: works best for large kfHp

• Mostly cylindrical W-contours

– Anti-solar differential rotation

•

Hanasoge

Do we need to rethink?

• In mixing length theory: l=Hp only hypothesis

– cf. Nick Featherstone’s talk

• Simulations: subgrid scale diffusion, viscosity

• Envisage reasons for (i) smaller scale flows

and/or (ii) deeper parts subadiabatic?

• But depth of convection zone still 200 Mm

Spruit97 A changing paradigm

Entropy rain

Stein & Nordlund (1998) simulations

Filamentary, nonlocal shown: entropy fluctuations pos neg

Tau approximation

upii

sjj

Ncsgu

NSus

/

supijjiiii NcsgSuususu

t

F

/2

i

su

FN

Closure

hypothesis

Deardorff1

Deardorff2

Physical meaning?

lnln/ 1 pcs p

z

S pert coasting…

0 0 , 0 suus zz

Physical meaning?

lnln/ 1 pcs p

z

S

pert

0 0 ,0 suus zz

Why should only the top be unstable

constrad dz

dTKF

const3

16 3

TKe.g. if const

dz

dT

Power law baT 0

nTT ab

13

Polytropic index n

Deeper parts intrinsically stable

nTT ab

13

Polytropic index n

Kramers opacity

(interior): a=1, b=-7/2

n=3.25

Entropy gradient positive (stable) for n > 3/2

Solar opacities

n << -1 n = 3.25

Hydrostatic

reference

solutions

Thickness only

~1Mm

111 KrH

Double Kramers-like

Early work in the 1930s

Original mixing length model

surface interior

unstable

stable

stable

weakly

unstable

Su rms31

conv Fassume

New solutions

with Deardorff flux

)( adconv F

Dadconv )( F

Entropy gradient

old

new

pd

Td

ln

ln

pp HdzcSd /)(/)/( ad

arXiv:1504.03189v2

20

Consequences of small scales

• Larger kf less turb. Diffusion: ht=urms/3kf

• Applications to dynamos: stronger, less turb diffusive

– Helps flux transport dynamos

• Two other important effect:

– Lambda effect differential rotation (Co smaller, Ta larger)

– Baroclinic term stronger?

– Negative effective magnetic pressure spots

21

Flux emergence in global simulations

Nelson, Brown, Brun,

Miesch, Toomre (2014)

22

3 scenarios

• Rising flux tubes?

• Hierachical convection?

• Self-organization as part

of the dynamo

g.B u.B g.W u.w A.B

Sunspot decay

23

Self-assembly of a magnetic spot • Minimalistic model

• 2 ingredients:

– Stratification & turbulence

• Extensions

– Coupled to dynamo

– Compete with rotation

– Radiation/ionization

24

25

A possible mechanism

2

2122

312

212

312

21 BBUBUB

const

ijijijjiji BBUU

Breakdown of quasi-linear theory

ReM here based on forcing k

Here 15 eddies per box scale

ReM=70 means 70x15x2p=7000

based on box scale

Brandenburg et al (2011,ApJ 740, L50)

26

Negative effective magnetic pressure instability

• Gas+turbulent+magnetic pressure; in pressure equil.

• B increases turbulence is suppressed

• turbulent pressure decreases

• Net effect?

Kleeorin, Rogachevskii, Ruzmaikin (1989, 1990)

Sunspot

formation

that sucks

27

Typical

downflow

speeds

Ma=0.2…0.3

Mean-field

simulation:

Neg pressure

parameterized

Brandenbur et al (2014)

Bi-polar regions in simulations with corona

28

Warn

ecke et al. (2

013, A

pJL

777, L

37)

Coronal loops?

Warnecke et al. (2013, ApJL 777, L37)

First dynamo-generated bi-polar regions

30

Mitra et al. (2

01

4, arX

iv)

Still negative effective

magnetic pressure?

Or something new?

31

Mitra et al. (2014, arXiv)

Global models

32

Jabb

ari

et a

l. (

20

15

, ar

Xiv

)

33

New aspects in mean-field concept

buBUEJ

...t JBbu

...... 2

p21

s,,t BqBBqUUuu ijjiijjiji

Ohm’s law

Theory and simulations: a effect and turbulent diffusivity

Turbulent viscosity and other effects in momentum equation

34

Calculate full ij and ij tensors

• Imposed-field method

– Convection (Brandenburg et al. 1990)

• Correlation method

– MRI accretion discs (Brandenburg & Sokoloff 2002)

– Galactic turbulence (Kowal et al. 2005, 2006)

• Test field method

– Stationary geodynamo (Schrinner et al. 2005, 2007)

JBUA ε

tbuε

jijjijj JB *

turbulent emf

effect and turbulent

magnetic diffusivity

35

Calculate full ij and ij tensors

JBUA

t

JbuBUA

t

jbubuBubUa

t

pqpqpqpqpqpq

tjbubuBubU

a

Original equation (uncurled)

Mean-field equation

fluctuations

Response to arbitrary mean fields

36

Test fields

0

sin

0

,

0

cos

0

0

0

sin

,

0

0

cos

2212

2111

kzkz

kzkz

BB

BB

pq

kjijk

pq

jij

pq

j BB ,

kzkkz

kzkkz

cossin

sincos

11311

21

1

11311

11

1

21

1

11

1

113

11

cossin

sincos

kzkz

kzkz

k

213223

113123

*

22

*

21

*

12

*

11

Example:

37

Kinematic and t

independent of Rm (2…200)

1

frms31

0

rms31

0

ku

u

Sur et al. (2008, MNRAS)

1

frms

2

31

0

31

0

ku

u

uω

38

Nonlocality: convolution

• Multiplication convolution

• Babcock-Leighton effect is an example

• Sharp structures in mean-field dynamos artifacts

• Convolution in x-space multiplication in k

39

The 4 Roberts flows

IV flow: negative eddy diffusivity dynamo

But positive diffusion at small scales

Devlen et al. (2013)

40

Time-delay dynamo for

Roberts II and III flows

xzxzxt BBB 2

2kik decay

oscillatory

With time delay

xzxzxt BtBB 2)(

xzxtxzxt BBBB 2

41

Time-delay dynamo for

Roberts II and III flows

Rheinhardt et al. (2014)

Growth when

xzxtxzxt BBBB 2

])(1[

)(Re

2

22

k

kk

2/

31

frms ku

42

Conclusions • Small scale deep convection

• Deep convective flux: Deardorff

• Thus marginally stable (not unstable)

• Such flows yield weaker turb diffusion

• Favor spot formation by NEMPI

• Dynamo effect from time delay