  Cosmic structure formation is characterized by the complex interplay between
gravity, turbulence, and magnetic fields. The processes by which gravitational
energy is converted into turbulent and magnetic energies, however, remain
poorly understood. Here, we show with high-resolution, adaptive-mesh
simulations that MHD turbulence is efficiently driven by extracting energy from
the gravitational potential during the collapse of a dense gas cloud.
Compressible motions generated during the contraction are converted into
solenoidal, turbulent motions, leading to a natural energy ratio of E_sol/E_tot
of approximately 2/3. We find that the energy injection scale of gravity-driven
turbulence is close to the local Jeans scale. If small seeds of the magnetic
field are present, they are amplified exponentially fast via the small-scale
dynamo process. The magnetic field grows most efficiently on the smallest
scales, for which the stretching, twisting, and folding of field lines, and the
turbulent vortices are sufficiently resolved. We find that this scale
corresponds to about 30 grid cells in the simulations. We thus suggest a new
minimum resolution criterion of 30 cells per Jeans length in
(magneto)hydrodynamical simulations of self-gravitating gas, in order to
resolve turbulence on the Jeans scale, and to capture minimum dynamo
amplification of the magnetic field. Due to numerical diffusion, however, any
existing simulation today can at best provide lower limits on the physical
growth rates. We conclude that a small, initial magnetic field can grow to
dynamically important strength on time scales significantly shorter than the
free-fall time of the cloud.
