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An Isaac Newton Institute Workshop

MAGNETOHYDRODYNAMICS OF STELLAR INTERIORS <br> Supported by the European Commission, Sixth Framework Programme - <br> Marie Curie Conferences and Training Courses - MSCF-CT-2003-503674

Small-scale dynamos

13th September 2004

Author: Fausto Cattaneo (University of Chicago)


Small-scale dynamo action describes the generation of magnetic fields on scales comparable with, or smaller than the characteristic scale of the velocity. It is believed to occur quite naturally in turbulent fluids when the magnetic Reynolds number --- a dimensionless measure of the electrical conductivity --- is sufficiently high. In general terms dynamo action succeeds if, on average, field amplification exceeds field destruction. In a turbulent system, field generation is due to the stretching of field lines by the flow, while field destruction is due to enhanced diffusivity. In these lectures I will review some of the efforts to provide a quantitative description of these two processes. I will introduce ideas like the Lyapunov exponents and the topological entropy to measure the stretching rate, and the cancellation exponent to measure the rate of enhanced diffusion. I will also distinguish between dynamos driven by smooth velocities and dynamos driven by rough velocities, i.e. velocities that are strongly fluctuating on the scale at which magnetic reconnection occurs. Physically these two cases correspond to fluids whose magnetic Prandtl number---the ratio of the viscosity to the magnetic diffusivity---is larger (smooth case), or smaller (rough case) than unity.