1. INI 2004 cattaneo@flash.uchicago.edu Small-scale dynamos Fausto Cattaneo Department of Mathematics University of Chicago

2. INI 2004 What is a small-scale dynamo? 50’s - 60’s Is MHD turbulence self excited? (Batchelor 1950; Schlüter & Biermann 1950; Saffman 1963; Kraichnan & Nagarajan 1967) 80’s -90’sFast dynamo problem (Childress & Gilbert 1995) NowLarge scale numerical simulations A hydromagnetic dynamo is a sustained mechanism to convert kinetic energy into magnetic energy within the bulk of an electrically conducting fluid Concentrate on magnetic field generation on scales comparable with or smaller than the velocity correlation length (cf. mean field theory).

3. INI 2004 Astrophysically speaking… L > Ro –Differential rotation –Helical turbulence –Large-scale dynamo action (mean field electrodynamics) L< Ro –Non helical flows –Small-scale dynamo action In a rotating body, the Rossby radius, Ro, defines the spatial scale above which rotation becomes important Earth:Ro  50 km SunRo  50,000 km

4. INI 2004 Stellar examples G. Scharmer SVST, 07/10/97 From Schrijver et al. 1989 Small scale photospheric fields Solar granulation –Size 1,000 km –Lifetime 5-10 mins Basal flux Chromospheric emission increases with rotation Residual emission at zero rotation Possibly acoustic

5. INI 2004 Physical parameters 10 7 Re Rm Pm =1 Stars Liquid metal experiments simulations IM 10 2 10 3 With the exception of numerical simulations, everything has either Pm>>1 or Pm <<1 (typical !!)

6. INI 2004 Rough vs smooth velocities Introduce longitudinal (velocity) structure functions (assuming stationarity, homogeneity, isotropy) Where  is the roughness exponent.  <1  rough velocity Equivalently with Turnover time shorter at small scales Reynolds number deceases with scale p<3  rough velocity Define scale δ for which R m (δ)=1 diffusion dominates advection dominates Similarly for R e

7. INI 2004 Magnetic Prandtl number dependence Smooth velocity case Rough velocity case Both cases have comparable values of R m

8. INI 2004 Rough vs smooth The difference between the smooth and rough cases (for dynamo action) is related to the behaviour of the velocity at the reconnection scale δ η –Slowly varying –Strongly fluctuating Smooth and rough velocities have very different dynamo properties As P m decreases through unity dynamo velocity changes from smooth to rough Begin with smooth case (easier)

9. INI 2004 Dynamo action Magnetic field grows if on average rate of field generation exceeds rate of field destruction Magnetic field generation due to line stretching by fluid motions Magnetic field destruction due to enhanced diffusion Dynamo growth rate depends on competition between these two effects. In a chaotic flow both effects proceed at an exponential rate. Why can’t we have a gradient dynamo for finite η?

10. INI 2004 Chaotic flows Fluid trajectory given by Follow deformation of cube of fluid of initial size  x over a short time  t. New size If deformation proceeds at exponential rate on average, flow is chaotic. 1, 2, 3 are the Lyapunov exponents. (incompressibility: 1 + 2 + 3 = 0 ) 1 Rate of stretching 3 Rate of squeezing

11. INI 2004 Example of 2-D chaotic flow Simple example of smooth solenoidal flow with chaotic streamlines. Streamlines Finite time Lyapunov exponents Red and yellow correspond to trajectories with positive (finite time) Lyapunov exponents Cattaneo &Hughes Planar flow  2 = 0, and 1 = - 3

12. INI 2004 Line stretching in a chaotic flow In a chaotic flow the length of lines increases exponentially (on average) is the topological entropy. It satisfies

13. INI 2004 What about dissipation? | 3 | local rate of growth of gradients  dissipation also increases exponentially. In two dimensions 1 = - 3. Magnetic field is destroyed as rapidly as it is generated. Two-dimensional dynamo action is impossible (Zel’dovich 1957) In 2D magnetic flux behaves like a scalar. Thus Do (fast) dynamos operate at the rate λ T ?

14. INI 2004 Introduce third dimension Simple modification leads to dynamo action (Galloway & Proctor 1992) Three-dimensional but still y- independent. However still have λ 1 = - λ 3

15. INI 2004 Enhanced diffusion Diffusion of magnetic field is determined by two processes: Growth in the magnitude of the gradients Geometry of the sign reversals of the field lines Effectiveness of diffusion of a vector field depends also on how the field lines are arranged Effective Ineffective

16. INI 2004 Enhanced diffusion Effective diffusion of vector quantities depends on magnitude of gradients and orientation. Packing becomes important. k is the cancellation exponent. Measures the singular nature of sign reversals (Du & Ott 1994 ) from Cattaneo

17. INI 2004 (Fast) dynamo growth rate Enhanced diffusion depends on the (exponential) growth of gradients and on field alignment Conjecture by Du & Ott (1995) for foliated fields as Rm   Local stretching Local contraction

18. INI 2004 The flux problem Fractal dimensions can be constructed similarly to κ (Hentschel & Procaccia 1983) Define averages (Cattaneo, Kim, Proctor & Tao 1995) For smooth velocities m is the cancellation exponent (Bertozzi et al 1994)

19. INI 2004 Convectively driven dynamos g hot cold Plane-parallel layer of fluid Boussinesq approximation Simulations by Emonet & Cattaneo time evolution

20. INI 2004 top middle volume Convectively driven dynamos: field structure detail Ra=500,00; Pm=5 kinematically efficient (i.e. growth rate) nonlinearly efficient (i.e. total magnetic energy)

21. INI 2004 PDF is exponential in the interior. Super equipartition fields possible. PDF is stretched exponential near the surface  extreme intermittency. Relatively strong fields can be found everywhere; strongest near the surface. Convectively driven dynamos: field intensity P(56) = 0.5 P(U/2) = 0.2 P(U) = 0.03

22. INI 2004 Extrapolation to Pm << 1 Dynamo action may disappear as Pm becomes small

23. INI 2004 Pm= 1 Pm=0.5 Re=1900, Rm=950, S=1 Re=1100, Rm=550, S=0.5 yes no Kinematic and dynamical issues Kinematic issue: Does the dynamo still operate? (Batchelor 1950; Nordlund et al. 1992; Brandenburg et al. 1996; Nore et al. 1997; Christensen et al 1999; Schekochihin et al. 2004; Yousef 2004) Dynamical issue: Dynamo may operate but become extremely inefficient Simulations by Cattaneo & Emonet

24. INI 2004 Dynamo action in rough velocity fields (schematic) Is dynamo action always possible provided L η, and hence R m, are sufficiently large? Does the dynamo growth rate become independent of kν, and hence Pm, as k ν   (P m  0)?

25. INI 2004 Kazantsev-Kraichnan models Previous questions can be answered within the framework of an analytical model of dynamo action by Kazantsev (1968) Velocity: incompressible, homogeneous, isotropic, stationary, Gaussian random process with zero correlation time. Longitudinal velocity correlator Closed form equation for magnetic field correlator Equation similar to Schrodinger equation with 1/r 2 potential in inertial range Existence of bound states (dynamo instability) depends on  Several extensions: Vainshtein & Kichatinov (1986), Kim & Hughes (1997), Kleorin & Rogachevskii (1997) Also: Zel’dovich, Ruzmaikin & Sokoloff (1990)

26. INI 2004 Properties of Kazantsev dynamos Magnetic field correlator H is localized near 1/k η Correlator decreases exponentially with r for r > 1/k η Localization decreases with decreasing β (i.e. correlators for rough velocities are more “spread out”) Dynamo action is always possible. Numerical resolution required to describe dynamo solution increases sharply as velocity becomes rougher (Boldyrev & Cattaneo 2004) Prescribe outer boundary conditions at r = L η. Increase L η until growing eigenfunction appears (i.e. dynamo instability)

27. INI 2004 Peak amplification for steady axi-symmetric flux rope maintained by convection (Galloway et al.) Expression derived assuming α=1 (laminar velocity) Assumes inertia terms are negligible (not true if Re>>1) Nonlinear dynamo efficiency depends on generation of moderately strong fluctuations Dynamical effects: convectively driven dynamos Pm= 1 Pm=0.5

28. INI 2004 Relax requirement that magnetic field be self sustaining (i.e. impose a uniform vertical field) Construct sequence of simulations with externally imposed field, 8 ≥ Pm ≥ 1/8, and S =  = 0.25 Adjust Ra so that Rm remains “constant” Easier case: magneto-convection Simulations by Emonet & Cattaneo 768512 256 Nx, Ny 2.80E+061.40E+067.04E+053.50E+052.00E+051.40E+059.20E+04Ra 0.1250.250.51.02.04.08.0Pm

29. INI 2004 B-field (vertical)vorticity (vertical) Pm = 8.0 Pm = 0.125 Magneto-convection: results

30. INI 2004 Energy ratio flattens out for Pm < 1 PDF’s possibly accumulate for Pm < 1 Evidence of regime change in cumulative PDF across Pm=1 Possible emergence of Pm independent regime Magneto-convection: results

31. INI 2004 Conclusion Huge difference between dynamo action driven by smooth and rough velocities Varying the magnetic Prandtl number across unity in a turbulent environment changes the dynamo from smooth to rough In all cases dynamo action is very likely if R m is large enough Some evidence of a P m independent regime at small (but not miniscule) values of P m

32. INI 2004 The end

33. INI 2004 Vectors and gradients Induction equation Scalar advection-diffusion Scalar variance equation Scalar gradient equation Dynamo action as a way to reduce diffusion

34. 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