Growth of Turbulent Magnetic Fields

The evolution of a weak, random initial magnetic field in a highly conducting, isotropically turbulent fluid is discussed with the aid of the exact expression for initial growth of the magnetic energy spectrum. Equipartition arguments, the vorticity analogy, and the known turbulence approximations all are found inadequate for predicting whether the magnetic energy eventually dies away or grows exponentially. This is true for any ratio of magnetic diffusivity λ to kinematic viscosity ν. The possibilities of eventual growth and eventual decay are therefore both admitted, and, for each, the shape of the magnetic-energy spectrum in the case λ » ν is estimated by simple dynamical arguments. If there is growth, it is concluded that the magnetic spectrum below the Ohmic cut-off eventually reaches equipartition with the kinetic-energy spectrum roughly in the fashion predicted by Biermann and Schlüter, with the principal exceptions that the spectrum of kinetic energy in the equipartition inertial range evolves to the form k−3/2 and that equipartition is maintained, with rapidly falling spectrum, through part of the Ohmic dissipation range. The evolution of the magnetic spectrum in the weak-field λ » ν regime is also computed numerically with a simplified transfer approximation suggested by the Lagrangian-history direct-interaction equations. This calculation turns out to yield an eventual very weak exponential growth of magnetic energy.