Saturation scenario for the turbulent dynamo Shell model approach

1. Saturation scenario for the turbulent dynamo Shell model approach Rodion Stepanov, ICMM, Perm, Russia and Franck Plunian, LGIT, Grenoble, France • What is, why, and how to derive, a shell model? • A non local shell model of MHD turbulence • Forced MHD turbulent dynamo: kinematic regime and saturation

2. MHD equations

3. 1015 1 103 106 109 1012 1015 Shell models Rm=UL/h Interstellar medium Galaxies 10+12 1012 Convective zone The Sun 10+9 Accretion discs 109 10+6 10-3 10-6 1 Pm=10+3 106 Liquid core Jupiter 103 Liquid core The Earth DNS Experiments Liquid Na 1 Re=UL/n

4. Fourier mode triads exchange

5. How to derive a shell model of turbulence ? -k3 Taking three vectors satisfying k2 p Assume a given shell Sn and a given shell Sp and k1 with q given by the grey squares below Then q What are the possible shells interacting together (possible triads )? The shells Sp and Sq corresponding to white squares cannot interact with Sn. True for any logarithmically spaced shell model with a space ratio greater than the gold number.

6. Why a shell model of turbulence ? What we lose: - all informations in real space - details within each shell - inhomogeneity - anisotropy What we still have: - energy transfers between shells - energy and helicity balances - spectra, fluxes in Fourier space - time dependency What we gain: - System of ODE’s to solve => accurate statistics - large range of viscosity and magnetic diffusivity - extended inertial range of scales (10 decades).

7. A non local shell model of MHD turbulence Generic equations: All possible non-diagonal triads: Applying of conservation laws:

8. Shell model Dynamo at Pm << 1 n=10 -7, h=10 - 4

9. uk k -1/3 k kn k=1 Phenomenology Navier-Stokes Dissipation scale:

10. Shell model Magnetic energy spectrum peaks at small scale |u(k)|2 |b(k)|2 kmax kF

11. k u(k) k2/3 k kn k=1 Phenomenology Dynamo Condition for dynamo Growth rate

12. Phenomenology Dynamo Condition for dynamo Growth rate k u(k) k2/3 b(k) k kn kh k=1

13. k u(k) k2/3 b(k) k kn kh k=1 Phenomenology Kinematic regime at Pm << 1

14. Phenomenology Kinematic regime at Pm << 1 Growth rate spectra gmax kmax

15. Shell model Kinematic regime at Pm << 1 |u(k)|2 |b(k)|2 -0.46

16. |u(k)|2 |b(k)|2 Shell model Saturation regime Pm << 1 1 k 100 |bk(t)|2

17. |u(k)|2 |b(k)|2 Shell model Saturation regime Pm << 1 t g(k)

18. k k+dk Saturation scenario Phenomenology bk+dk (t+dt) bk(t) ~ k -1/3 ~ k a bk+dk(t) k+dk k Saturation of bk at time t: bk(t) = u(k) Infra-red spectrum ~ k a: bk+dk(t)=bk(t)((k+dk)/k)a bk+dk(t+dt)=bk+dk(t) exp(gk+dk t) Exponential growth: bk+dk(t+dt)= u(k+dk) Saturation of bk+dkat time t+dt: u(k+dk) = u(k) (1 + dk / k)-1/3 Saturation level :

19. gmax kmax Phenomenology Saturation regime dk2 + dt2 << 1

20. Phenomenology Saturation regime Phenomenology: Shell model: Pm=10-3 a=2 reduction of dissipation scale

21. Shell model Saturation regime Phenomenology: Shell model: Pm=10-3

22. TG kmax kF ABC Minnini et al. (2005) Schekochihin et al. (2007) kmax kF kF Is it the same for DNS ?

23. Shell model Dynamo at Pm >> 1 n=10 - 4, h=10 - 8

24. Phenomenology k u(k) b(k) k2/3 kn k=1 kh

25. Shell model |u(k)|2 |b(k)|2 -0.46

26. Shell model |u(k)|2 |b(k)|2 1 k 1000 |bk(t)|2

27. Route to saturation |u(k)|2 |b(k)|2 t g(k)

28. Route to saturation Phenomenology: Shell model: Pm=104 Pm=10-3 a=2 a=2

29. Route to saturation Phenomenology: Shell model: Pm=104 Pm=10-3

30. Conclusion and perspective Saturation regime The growth rate of largest scale obeys quadratic equation of type : dg / dt = - C g 2 Pm=104 Pm=10-3 Numerical / experimental verification is desirable.

31. It is agree with Is there any disagreement? Please, tell me.

32. Some large-scale dynamo model + shell model

33. Thank you!