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cattaneo@flash.uchicago

Small-scale dynamos. Fausto Cattaneo Department of Mathematics University of Chicago. cattaneo@flash.uchicago.edu. 50’s - 60’s Is MHD turbulence self excited? (Batchelor 1950; Schl ü ter & Biermann 1950; Saffman 1963; Kraichnan & Nagarajan 1967)

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

  2. 50’s - 60’s Is MHD turbulence self excited? (Batchelor 1950; Schlüter & Biermann 1950; Saffman 1963; Kraichnan & Nagarajan 1967) 80’s -90’s Fast dynamo problem (Childress & Gilbert 1995) Now Large scale numerical simulations What is a small-scale dynamo? 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). INI 2004

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

  4. Stellar examples 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 G. Scharmer SVST, 07/10/97 From Schrijver et al. 1989 INI 2004

  5. IM Stars Physical parameters Rm Pm =1 103 simulations 102 Liquid metal experiments Re 107 103 With the exception of numerical simulations, everything has either Pm>>1 or Pm <<1 (typical !!) INI 2004

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

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

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

  9. 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? INI 2004

  10. 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, 3are the Lyapunov exponents . (incompressibility: 1 + 2 +3 = 0 ) 1 Rate of stretching 3 Rate of squeezing INI 2004

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

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

  13. What about dissipation? Do (fast) dynamos operate at the rate λT? |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. In 2D magnetic flux behaves like a scalar. Thus Two-dimensional dynamo action is impossible (Zel’dovich 1957) INI 2004

  14. Introduce third dimension Simple modification leads to dynamo action (Galloway & Proctor 1992) Three-dimensional but still y-independent. However still haveλ1= - λ3 INI 2004

  15. 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 INI 2004

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

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

  18. 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) INI 2004

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

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

  21. Convectively driven dynamos: field intensity P(56) = 0.5 P(U/2) = 0.2 P(U) = 0.03 • 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. INI 2004

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

  23. Pm=1 Pm=0.5 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 yes Re=1100, Rm=550, S=0.5 Simulations by Cattaneo & Emonet no Re=1900, Rm=950, S=1 INI 2004

  24. Dynamo action in rough velocity fields (schematic) • Is dynamo action always possible provided Lη, and hence Rm, are sufficiently large? • Does the dynamo growth rate become independent of kν, and hencePm, askν  (Pm 0)? INI 2004

  25. 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/r2 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) INI 2004

  26. Prescribe outer boundary conditions at r = Lη. Increase Lηuntil growing eigenfunction appears (i.e. dynamo instability) Properties of Kazantsev dynamos • Magnetic field correlator H is localized near 1/kη • Correlator decreases exponentially with r for r > 1/kη • 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) INI 2004

  27. Pm=1 Pm=0.5 Dynamical effects: convectively driven dynamos Nonlinear dynamo efficiency depends on generation of moderately strong fluctuations • 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) INI 2004

  28. Pm 8.0 4.0 2.0 1.0 0.5 0.25 0.125 Ra 9.20E+04 1.40E+05 2.00E+05 3.50E+05 7.04E+05 1.40E+06 2.80E+06 Nx, Ny 256 256 256 512 512 512 768 Easier case: magneto-convection • 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” Simulations by Emonet & Cattaneo INI 2004

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

  30. Magneto-convection: results • 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 INI 2004

  31. 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 Rm is large enough Some evidence of a Pm independent regime at small (but not miniscule) values of Pm Conclusion INI 2004

  32. The end INI 2004

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

  34. 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

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