INI 2004 Astrophysical dynamos Fausto Cattaneo Center for Magnetic Self-Organization Computations Institute Department of Mathematics University of Chicago

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

INI 2004 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 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 Dynamo in astrophysics Galaxy Typical size: 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 Dynamo models 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 > 10 6 )

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

INI 2004 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 Small-scale dynamo action 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 ( P m  1 ) –Strongly intermittent field Cattaneo

INI 2004 Small-magnetic Prandtl number effects 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? Pm=5, Rm=1000Pm=1, Rm=550Pm=0.5, Rm=550 Cattaneo& Emonet

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

INI 2004 Kazantzev model (1968) 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 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 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 B-field Simulations by Tobias & Cattaneo

INI 2004 Large scale dynamos 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

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

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

INI 2004 Mean field generation 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 Non rotatingRotating Cattaneo & Hughes

INI 2004 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 ≤ u 2, gives estimate

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

INI 2004 Role of turbulence in dynamo action 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

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

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

INI 2004 Conclusion 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?

INI 2004 The end

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