cattaneo@flash.uchicago

1. Astrophysical dynamos Fausto Cattaneo Center for Magnetic Self-Organization Computations Institute Department of Mathematics University of Chicago cattaneo@flash.uchicago.edu INI 2004

2. Introduction Small-scale dynamo action Rough velocities and small magnetic Prandtl number Large scale dynamo action Transport coefficient and averages Large Rm regime Turbulent vs laminar dynamos Role of turbulence vs large scale flows Essentially non-kinematic dynamos Content INI 2004

3. Dynamos in “astrophysics” • Liquid metal experiments • Typical size 0.5 – 2.0 m • Turbulence driven by propellers • Almost there • Geodynamo • Size: 6,400 Km • Turbulence driven by compositional convection in the liquid core • Evidence for magnetic reversals INI 2004

4. Dynamos in astrophysics • Sun (late-type stars) • Size 600,000 Km • Turbulence driven by thermal convection • Evidence for activity cycles • Accretion disks • Typical size varies • Turbulence driven by MRI INI 2004

5. Dynamo in astrophysics • Galaxy • Typical size: 1020 m • Turbulence driven by supernovae explosions • Field mostly in the galactic plane • Radio galaxies --IGM • Typical size: 30 Kpc wide, 300 Kpc long • Turbulence in central object driven by gravitational/rotational energy of SMBH • Evidence for expulsion of magnetic helices in lobes INI 2004

6. Most sophisticated model is for the Geodynamo (Glatzmeier, Roberts and coworkers) Dynamo models for stars, galaxies and exotic objects, over the last 3 decades have relied mostly on Mean Field Electrodynamics difficulties with nonlinear extensions Problems with a direct numerical approach because of very large magnetic Reynolds numbers ( Rm > 106 ) Dynamo models INI 2004

7. Prescription for advecting velocity Synthetic velocity Random Deterministic Incompressible NS Compressible NS Other Requires accurate solution of induction equation Dynamo equations INI 2004

8. How high can we go? inertial range diffusive sub-range • Spectral methods: 3 collocation point / wavenumber Smooth velocity Kolmogorov velocity • Finite differences: 3-7 collocation points / wavenumber INI 2004

9. Concerned with the problem of field generation on scales comparable with or smaller than some characteristic velocity scale (Batchelor 1950; Schlüter & Biermann 1950; Saffman 1963; Kraichnan & Nagarajan 1967) Generation of magnetic fields by non-helical (reflectionally symmetric) turbulence Typical examples: Randomly stirred flows Convectively driven flows Dynamo action easily excited ( Pm1 ) Strongly intermittent field Cattaneo Small-scale dynamo action INI 2004

10. In dense plasmas and liquid metals Many report that dynamo action disappears as Pm decreases through unity (Nordlund et al. 1992; Brandenburg et al. 1996; Nore et al. 1997; Christensen et al 1999; Schekochihin et al. 2004; Yousef 2004) Can (s-s) dynamo action be maintained when Pm <<1? Small-magnetic Prandtl number effects Cattaneo& Emonet Pm=5, Rm=1000 Pm=1, Rm=550 Pm=0.5, Rm=550 INI 2004

11. Decreasing Pm through unity corresponds to a transition from smooth to rough velocities Dynamo action by rough velocities (assuming stationarity, homogeneity, isotropy) Where  is the roughness exponent. <1  rough velocity p<3  rough velocity Equivalently INI 2004

12. Exactly solvable model of kinematic dynamo action in random velocity field (stationary, isotropic, Gaussian, zero correlation time) Velocity described by single spatial correlator Schrödinger like equation for magnetic field correlator Critical system size (in units of the diffusion length) that contains growing eigenfunction increases sharply with roughness exponent Kazantzev model (1968) Dynamo action is always possible. Numerical resolution required to describe dynamo solution increases sharply as velocity becomes rougher (Boldyrev & Cattaneo 2004; Vainshtein & Kichatinov 1986 ) INI 2004

13. Extensions • Most physical flows have nontrivial phase correlations, i.e. contain coherent structures • Coherent structures can affect the cancellation exponent, and hence the dynamo growth rate • Morphology of resulting field very different from random case (more and better folding) synthetic (3-D) simulated (2.5-D) velocity Simulations by Tobias & Cattaneo B-field INI 2004

14. Generation of magnetic field on scales larger than velocity correlation length --- generation of flux Associated with turbulence lacking reflectional symmetry – helical flows Possibly associated with inverse cascade of (magnetic) helicity Often described in terms of Mean Field Electrodynamics Average induction --- -effect Average diffusion --- β-effect Average advection --- γ-effect Large scale dynamos INI 2004

15. At large Rm difficulties may arise with proper definition of turbulent transport coefficients Consider kinematic regime with uniform mean field Mean field transport Definition of turbulent  Definition of kinematic regime By definition Because of S-S dynamo action Lots of O(1) fluctuations should average to a small number INI 2004

16. Averages Averages may not be well behaved on any computationally sensible scale What do they mean? emf: volume average Cattaneo & Hughes emf: volume average and cumulative time average time INI 2004

17. Difficulties may arise between a S-S dynamo with an extended eigenfunction and a large scale field generated by an inverse cascade Difficulties with inverse cascade. Example: Rotating convection Mean field generation Cattaneo & Hughes Non rotating Rotating INI 2004

18. Memory effects (2D) In 2D induction equation becomes scalar transport equation With suitable boundary conditions we have • In order to maintain “turbulent” behaviour as Rm   gradients of A must diverge • Generation of small scale fluctuations increases magnetic field energy • Reasonable energetic constraint <B2> ≤ u2 , gives estimate INI 2004

19. Memory effects (2D) • With diffusivity given by (Taylor 1921) from Cattaneo Turbulence develops a memory INI 2004

20. What is the role of turbulence in dynamos? Turbulence (chaoticity) provides exponential stretching in S-S dynamos Turbulence enhances effective diffusivity (β-effect) Turbulence can give rise to mean advective effects (γ-effect) Turbulence gives rise to mean induction effects (-effect) It is possible to have dynamo action associated with large scale organization of magnetic field Dynamo works irrespective of turbulence as opposed to as a result of it Flux-tube dynamos Babcock-Leighton type dynamos Role of turbulence in dynamo action INI 2004

21. Interaction between localized velocity shear and weak background poloidal field generates intense toroidal magnetic structures Magnetic buoyancy leads to complex spatio-temporal beahviour By + Shear driven system Cline, Brummell & Cattaneo INI 2004

22. Slight modification of shear profile leads to sustained dynamo action System exhibits cyclic behaviour, reversals, even episodes of reduced activity By - By + Shear driven dynamo INI 2004

23. Computing resources are available for 3D moderate Reynolds number simulations Simulation of turbulent dynamos (i.e. driven by rough velocities) remain challenging Meaning of turbulent transport coefficients should be clarified What is the role of turbulence in specific models of astrophysical dynamos? Conclusion INI 2004

24. The end INI 2004

25. Kazantsev model Velocity correlation function Isotropy + reflectional symmetry Solenoidality Magnetic correlation function Kazantsev equation Renormalized correlator Funny transformation Funny potential Sort of Schrödinger equation INI 2004