  Comparisons of the integrated thermal pressure support of gas against its
gravitational potential energy lead to critical mass scales for gravitational
instability such as the Jeans and the Bonnor-Ebert masses, which play an
important role in analysis of many physical systems, including the heuristics
of numerical simulations. In a strict theoretical sense, however, neither the
Jeans nor the Bonnor-Ebert mass are meaningful when applied locally to
substructure in a self-gravitating turbulent medium. For this reason, we
investigate the local support by thermal pressure, turbulence, and magnetic
fields against gravitational compression through an approach that is
independent of these concepts. At the centre of our approach is the dynamical
equation for the divergence of the velocity field. We carry out a statistical
analysis of the source terms of the local compression rate (the negative time
derivative of the divergence) for simulations of forced self-gravitating
turbulence in periodic boxes with zero, weak, and moderately strong mean
magnetic fields (measured by the averages of the magnetic and thermal
pressures). We also consider the amplification of the magnetic field energy by
shear and by compression. Thereby, we are able to demonstrate that the support
against gravity is dominated by thermal pressure fluctuations, although
magnetic pressure also yields a significant contribution. The net effect of
turbulence in the highly supersonic regime, however, is to enhance compression
rather than supporting overdense gas even if the vorticity is very high. This
is incommensurate with the support of the highly dynamical substructures in
magneto-turbulent fluids being determined by local virial equilibria of volume
energies without surface stresses.
