advertisement

MRI Driven turbulence and dynamo action Fausto Cattaneo University of Chicago Argonne National Laboratory [email protected] ITP 2008 MRI in cylindrical annulus • Incompressible fluid Finite viscosity and magnetic diffusivity Periodic in the vertical • • • Pm = 0.5 Not a dynamo at Re=6,000 Probably a dynamo at Re=60,000 inner cylinder • • velocity Simulations by Obabko, Fischer, FC ITP 2008 What happens if Pm 0? Two issues: • Dynamo action becomes impossible. Turbulence decays. No enhanced transport • Dynamo action remain possible but amplitude of fluctuations decreases with decreasing Pm. Asymptotically enhanced transport becomes ineffective ITP 2008 Dynamo action at small Pm Two possibilities: • There exist a critical value of Pm below which dynamo action is impossible for any Re • Dynamo action is always possible: – Asymptotically Rmcrit independent of Pm (σ=1) – Asymptotically Rmcrit increases with decreasing Pm Pm 1.0 Turbulence Re crit Pm Re ITP 2008 Small Pm regime Assume Re>>1. • Dynamo action driven by strongly fluctuating velocity. • Introduce roughness exponent u 2 () u(x l) u(x) el 2 2 • • <1 rough velocity ( =1/3 corresponds to K41) Related to energy spectrum: E(k)k-p p=1+2 • At small scales ( O() ) action of viscosity is to smooth out velocity 0 Re 1 1 0 Rm 1 1 1 Pm ITP 2008 Structure functions • • • Shearing box simulation Isothermal EOS Finite vertical flux • • Simulations using PLUTO No explicit dissipation x 0.002 ny 512 inertial u 2 () 5x 50x Simulations by Bodo, Mignone, FC ITP 2008 Kazantsev model Resolution parameter From Boldyrev & FC dynamo No dynamo 1 increasing roughness ITP 2008 What do we need? • Asymptotic regime is reached when η 50 x • If =1/3 dynamo action requires 30 η 1500 x ( 2 for periodic systems) • Requirement to reach asymptotic regime with =1/3 ν /η 0.1 Pm 0.14/3 0.046 x 0.002 ny 512 u () 2 5x 50x ITP 2008 Amplitude effects Effective transport ( - BBr ) depends on: • Correlations • Amplitude of fluctuating quantities (mostly B) For a flux rope in equilibrium with an axi-symmetric stagnation point flow, peak field satisfies (Galloway, Proctor & Weiss) 2 u B*2 ( / ) ln( Rm ) Valid for Pm >>1 ITP 2008 What happens if Pm < 1? B-field (vertical) vorticity (vertical) Magneto-convection Pm = 8.0 cold g Simulations by Emonet & FC hot Pm = 0.125 ITP 2008 From magneto-convection ITP 2008 Conclusion • Present simulations not suitable to inform us on small Pm regime • At present there is no result, theoretical or numerical indicating that dynamo action becomes impossible at small Pm • Results for Pm > 1 not representative of Pm <<1 • Informative simulations are possible, but very demanding ITP 2008