cattaneo@flash.uchicago.edu

1. Chicago 2003 Turbulent transport of magnetic fields Fausto Cattaneo Center for Magnetic Self-Organization in Laboratory and Astrophysical Plasmas cattaneo@flash.uchicago.edu

2. Chicago 2003 Motivation/Objective • Typically: present physical motivation for research (not the case here) • Rather: Decide if subject is suitable area of research for center activity • Apparently, of some interest to several neighbouring communities • Geophysics • Planetary physics • Dynamo theorists • Generic astrophysics • Present some of the issues

3. Chicago 2003 Turbulent transport • Important effect of turbulence is greatly to enhance transport over collisional values. • Reynolds, Peclet, magnetic Reynolds numbers are ratios of turbulent to collisional diffusivities • Magnetic fields affect turbulence, hence its transport properties • Can consider transport of • temperature fluctuations • energetic particles-chemical species • angular momentum • magnetic fields

4. Chicago 2003 Mean-field-electrodynamics Two-scale approach. Consider the kinematic problem Introduce averages Mean-field equation

5. Chicago 2003 Closure of mean-field equations Closure requires equations for mean emf Consider evolution of fluctuations Linearityestablishes relationship betweenmean field and mean emf • Isotropic case • αmean induction • βeddy diffusion

6. Chicago 2003 Transport coefficients • In kinematic regime α and β should be determined solely by Rm and the statistics of u • α is a pseudo-scalar (tensor) requires lack of reflectional symmetry • Simple solutions of dynamo equation with In large Rm situation α and β should have turbulent values. i.e. independent of Rm Dynamo sets in at small, rather than large scales

7. Chicago 2003 Nonlinear effects: 2D diffusion 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 • Energetic constraint <B2> ~ u2 , gives estimate

8. Chicago 2003 Effect on velocity field • Assume diffusive behaviour of large scale component of A • Assume diffusive behaviour of large scale component of A

9. Chicago 2003 Back to α • Possible problems with definition of α as a sensible statistical quantity as Rm • Also possible problems with small-scale dynamo action. Look at simple case when B is uniform Assume large scale field is “kinematic”. Then in turbulent situation With b typically strongly non-Gaussian

10. Chicago 2003 Nonlinear effects Most nonlinear treatments rely on two statements • geometrical • dynamical Assume suitable boundaries Stationary, uniform mean field From EDQNM, say, get dynamical relationship

11. Chicago 2003 Nonlinear effects Combine to get saturation effects (as before)

12. Chicago 2003 Problems Three types of criticisms: • Formula is incorrect • Agrees with DNS but only at small (moderate) Rm • “The Sun does it stuff therefore it must be wrong” • Formula is correct but derivation is wrong/suspect • Assumptions about correlation time need justification • Second formula neglects intermittency (possibly strong in 3D) • Second formula just plain wrong • Formula is correct but irrelevant (should not be used in physical models) • First formula neglects time dependence • First formula neglect large scale gradients • First formula assumes special boundary conditions (no flux of helicity)

13. Chicago 2003 Conclusion • After about a decade controversy continues • Is it likely to be settled by theoretical arguments alone? (Probably not) • Can Center activity help resolve situation? • Better theories (Yeah right…) • More fancier simulations? (Possibly) • Experiments? (Possibly) • What about β? Is it suppressed in 3D? • What about non MHD-effects? • Hall effect • ambipolar diffusion