Shell Model Approaches to the Problem of Galactic MHD

Rodion Stepanov, Peter Frick, ICMM, Perm, Russia

Dmitry Sokoloff , MSU, Russia

Franck Plunian, LGIT, Grenoble, France

1. Why and what is a shell model?2. MHD turbulence with rotation and applied magnetic field3. Forced MHD turbulence with cross-helicity injection4. Magnetic helicity in a free decay of MHD turbulence5. Combined thin disk dynamo model

109

106

1012

1

103

103 106 109 10121 1015

Liquid coreThe Earth

Liquid coreJupiter

Convective zoneThe Sun

Interstellar mediumGalaxies

DNS ExperimentsLiquid Na

Accretion discs

P m=10+

3

1 10-3

10-6

1015

10+12

Re=UL/

10+6

10+9

Rm=UL/

Kolmogorov 41

0

)( 2

u

fupuuut

kkF

k

Energy cascade

Inertial range

Dissipatio

n

Large Small

Spectral flux

Mean field equation

Shell model of MHD turbulence

Generic equations:

)(),(ˆ),(

)(),(ˆ),(

tBtt

tUtt

n

n

kBrB

kUrU

kL

kd

k

kn+1

E

kn

nnn kS

1

22 |||| nnn BUE

Coefficients c are derived from conservation laws (total energy, cross-helicity, magnetic helicity)

Brandenburg, A. et al, 1996; Frick, P. and Sokoloff, D. 1998; Basu, A. et al 1998

Triadic interaction:

k p

q

nnnt

nnnnt

BkUBQBUQBd

FUkBBQUUQUd2

2

),(),(

),(),(

qp

qpn YXQikYXQ,

),(),(

How much do we gain ?

3/23/11

3/13/1

)()(

)(:range Inertial

kkukt

kku

k

uk

kFk

k -1/3

Fkk 4/3Re:scale Viscous

Number of gridpointsper unit volume:

4/9

3

Re

Fk

kN

Number of logarithmic shells: Fnn

Fk

k 4/3Re Reln FnnN

Number of time steps:2/1

3/2

Re

F

F

k

k

t

tM

RelnRe 2/1NM

4/11ReNMNumber of timesteps x gridpoints:

3D Kolmogorov

Number of timesteps x gridpoints:

Shell model

Pm=10-3 Re=109

Small scale dynamo at low Pm

Eu

Eb

Stepanov R. and Plunian F., 2006, J. Turbulence

Phenomenology of isotropic turbulencewith applied rotation and field

Zhou 1995

vA

RK

R RA

0

RKA

()1/4

()1/2

vA

2

vA2 /

vA

k

(/)1/2

E(k)

vA3

k

vAk

R K A

(/)1/2

E (k)

k

R

k

(/)1/2

E(k)

3)1/4

k

R K

vA

E(k)

vAk

A

k

R

/vA

k

(/)1/2

E(k)

3)1/4

k

R K

vA

k

(/)1/2

E(k)

vA3

k

vAk

R K A

vA

E(k)

vAk

A

k

R

/vA Plunian F. and Stepanov R. 2010, PRE

Energy spectra for different cross helicity input rate

Normalized spectra Spectral index versus

Mizeva I.A. et al, 2009, Doklady Physics

Energy injection rate – Cross-helicity injection rate –

The stationary input of cross-helicity strongly affects the small-scale turbulence: the spectral energy transfer becomes less efficient and the turbulence accumulates the total energy.

Frick & Stepanov 2010

Long-term free decay 128 runs Re=Rm=105

Evolution up to t=105

Initial conditions

Normalized cross-helicity C=Hc/E

First scenario: C=±1Secondscenario

Cross-helicity vs time Cross-helicity spectra for different time

Inverse cascade of cross-helicity

Long-term free decay

- at t=100- at t=1000- at t=10000

Normalized magnetic helicity Cb=Hb/Eb

Second scenario: Cb=±1

Long-term free decay

k<1 is available

E

GRID-SHELL MODEL OF TURBULENT DISK DYNAMO

Mean fields (large-scale)

Turbulent fields (small-scale)

large smallCoupling: small large

Numerical results

αu≠0 αb=0 D=-1 αu≠0 αb=0 D=-20

αu≠0 αb≠0 D=-200αu≠0 αb=0 D=-100

large-scale poloidal/toroidal magnetic field small-scale kinetic/magnetic field vs timeEnergy of

Dynamical alpha-quenching

log <B>

log α

Global reversals

• Shell models have passed many tests• Shell models can be used to check phenomenological predictions• Shell models are able to give something new about MHD turbulence

Conclusion remarks