We report the results of an extensive numerical study of the small-scale turbulent dynamo. The primary focus is on the case of large magnetic Prandtl numbers Pr m , which is relevant for hot low-density astrophysical plasmas. A Pr m parameter scan is given for the model case of viscosity-dominated (low-Reynolds-number) turbulence. We concentrate on three topics: magnetic-energy spectra and saturation levels, the structure of the magnetic-field lines, and intermittency of the field-strength distribution. The main results are as follows: (1) the folded structure of the field (direction reversals at the resistive scale, field lines curved at the scale of the flow) persists from the kinematic to the nonlinear regime; (2) the field distribution is self-similar and appears to be lognormal during the kinematic regime and exponential in the saturated state; and (3) the bulk of the magnetic energy is at the resistive scale in the kinematic regime and remains there after saturation, although the magnetic-energy spectrum becomes much shallower. We propose an analytical model of saturation based on the idea of partial two-dimensionalization of the velocity gradients with respect to the local direction of the magnetic folds. The model-predicted saturated spectra are in excellent agreement with numerical results. Comparisons with large-Re, moderate-Pr m runs are carried out to confirm the relevance of these results and to test heuristic scenarios of dynamo saturation. New features at large Re are elongation of the folds in the nonlinear regime from the viscous scale to the box scale and the presence of an intermediate nonlinear stage of slower-than-exponential magnetic-energy growth accompanied by an increase of the resistive scale and partial suppression of the kinetic-energy spectrum in the inertial range. Numerical results for the saturated state do not support scale-by-scale equipartition between magnetic and kinetic energies, with a definite excess of magnetic energy at small scales. A physical picture of the saturated state is proposed. Subject headings: magnetic fields -MHD -plasmas -turbulence -methods: numerical 1 Present-time address: DAMTP/CMS,

We simulate incompressible, MHD turbulence using a pseudo-spectral code. Our major conclusions are as follows. 1) MHD turbulence is most conveniently described in terms of counter propagating shear Alfvén and slow waves. Shear Alfvén waves control the cascade dynamics. Slow waves play a passive role and adopt the spectrum set by the shear Alfvén waves. Cascades composed entirely of shear Alfvén waves do not generate a significant measure of slow waves.2) MHD turbulence is anisotropic with energy cascading more rapidly along k ⊥ than along k , where k ⊥ and k refer to wavevector components perpendicular and parallel to the local magnetic field. Anisotropy increases with increasing k ⊥ such that excited modes are confined inside a cone bounded by k ∝ k γ ⊥ where γ < 1. The opening angle of the cone, Θ(k ⊥ ) ∝ k −(1−γ) ⊥