1 Forced – decaying Helical – nonhelical. 2 Points of the talk Resistive effects during inverse...

1

Forced – decayingForced – decayingH

elic

al –

non

helic

alH

elic

al –

non

helic

al

2

Points of the talkPoints of the talk

• Resistive effects during inverse transfer

• B-field does not care about irrotational part

• Magnetic helicity from 1-D data sets

3

(i)(i) Small scale dynamoSmall scale dynamo

(i)Exponential growth

(ii)Growth rate proportional to Rm1/2

(iii)Kazantsev spectrum

4

(ii) Large-scale dynamo(ii) Large-scale dynamo

• Similar to SS dynamo at early times

• Inverse cascade/transfer behavior

• Resistively slow saturation (!)

5

(iii) Non-helical decay(iii) Non-helical decay

• Fast decay at small scales

• Slight increase for small k and strong B

6

(iv) Helical decay(iv) Helical decay

• Inverse cascade on large scales

Christensson et al.(2001, PRE 64, 056405)

Initial slope E~k4

7

Revised helical decay lawRevised helical decay law

HkH H22

sI tHE 22/1/||

H not exactly constant

rHH ttkk 00 /Assume power law, not const

H follows power law iff r=1/2; thenstH 2

2diff

20

20 / HHH tktks

M. Christensson, M. Hindmarsh, A. Brandenburg: 2005, AN 326, 393M. Christensson, M. Hindmarsh, A. Brandenburg: 2005, AN 326, 393

8

Dynamos: small-scale vs large-scaleDynamos: small-scale vs large-scale

B-scale larger than U-scale

B-scale smaller than U-scale

Wavenumber=1/scale

energy

injectionscale

9

SS and LS dynamos

10

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

11

Inverse cascadeInverse cascade

12

Resistive effects on inverse transfer

2/1frms

2/1

1rms

2/1

2/1

tkut

tuRk

tk

mm

m

13

Application to phase transitionsApplication to phase transitions

• Forcing purely potential• No vorticity production?

ln2

2

Sωωuω

BBuB

t

t

14

Gaussian expansion wavesGaussian expansion waves

15

No dynamo from potential flowsNo dynamo from potential flows

No dynamo actionin nearly potentialflows (at least notfo far)

16

No vorticity eitherNo vorticity either

ln2

2

Sωωuω

BBuB

t

t

17

Vorticity productionVorticity production

• if 2/urms kf > 1

• if Ma > 0

18

B field ignores irrotational partB field ignores irrotational part

19

Helicity from 1-D data setsHelicity from 1-D data sets

)()( xBxBM jiij

)(

)()(

kHki

kEkkkM

kijk

jiijij

)()( xBxBM jiij

)()()( rxxr jiij BBM

Matthaeus et al. (1982) Measure correlation function

In Fourier space, calculatemagnetic energy and helicity spectra

Should be done with Ulysses data away from equatorial plane

20

Bi-helical fields from UlyssesBi-helical fields from Ulysses

• Taylor hypothesis• Broad k bins• Southern latitude

with opposite sign• Small/large distances• Positive H at large k• Break point with

distance to larger k

21

ConclusionsConclusions

• Resistive effects during inverse transfer

• B-field does not care about irrotational part

• Magnetic helicity from 1-D data sets