Shell Model Approaches to the Problem of Galactic MHD
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Transcript of Shell Model Approaches to the Problem of Galactic MHD
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