Coexistence of Magneto-Rotational and Jeans Instabilities in an
  Axisymmetric Nebula
  We analyze the magneto-rotational instability (MRI) effects on gravitational
collapse and its influence on the instability critical scale. In particular, we
study an axisymmetric nonstratified differentially rotating cloud, embedded in
a small magnetic field, and we perform a local linear stability analysis,
including the self gravity of the system. We demonstrate that the linear
evolution of the perturbations is characterized by the emergence of an
anisotropy degree of the perturbed mass densities. Starting with spherical
growing overdensities, we see that they naturally acquire an anisotropy of
order unity in their shape. Despite the linear character of our analysis, we
infer that such a seed of anisotropy can rapidly grow in a nonlinear regime,
leading to the formation of filament-like structures. However, we show how such
an anisotropy is essentially an intrinsic feature of the Jean instability, and
how MRI only plays a significant role in fixing the critical scale of the mode
spectrum. We then provide a characterization of the present analysis in terms
of the cosmological setting, in order to provide an outlook of how the present
results could concern the formation of large-scale structures across the
Universe.
