  We present the results of eighteen magnetohydrodynamical calculations of the
collapse of a molecular cloud core to form a protostar. Some calculations
include radiative transfer in the flux-limited diffusion approximation while
others employ a barotropic equation of state. We cover a wide parameter space,
with mass-to-flux ratios ranging from $\mu = 5$ to $20$; initial turbulent
amplitudes ranging from a laminar calculation (i.e. where the Mach number,
$\mathscr{M} = 0$) to transonic $\mathscr{M} = 1$; and initial rotation rates
from $\beta_\mathrm{rot} = 0.005$ to $0.02$. We first show that using a
radiative transfer scheme produces warmer pseudo-discs than the barotropic
equation of state, making them more stable. We then `shake' the core by
increasing the initial turbulent velocity field, and find that at all three
mass-to-flux ratios transonic cores are weakly bound and do not produce
pseudo-discs; $\mathscr{M} = 0.3$ cores produce very disrupted discs; and
$\mathscr{M} = 0.1$ cores produce discs broadly comparable to a laminar core.
In our previous paper (arXiv:1701.08741), we showed that a pseudo-disc coupled
with sufficient magnetic field is necessary to form a bipolar outflow. Here we
show that only weakly turbulent cores exhibit collimated jets. We finally take
the $\mathscr{M} = 1.0$, $\mu = 5$ core and `stir' it by increasing the initial
angular momentum, finding that once the degree of rotational energy exceeds the
turbulent energy in the core the disc returns, with a corresponding (though
slower), outflow. These conclusions place constraints on the initial mixtures
of rotation and turbulence in molecular cloud cores which are conducive to the
formation of bipolar outflows early in the star formation process.
