We use smoothed particle hydrodynamics simulations of massive protostellar discs to investigate the predicted broadening of molecular lines from discs in which self-gravity is the dominant source of angular momentum transport .
The simulations include radiative transfer , and span a range of disc-to-star mass ratios between M _ { d } / M _ { * } = 0.25 and M _ { d } / M _ { * } = 1.5 .
Subtracting off the mean azimuthal flow velocity , we compute the distribution of the in-plane and perpendicular peculiar velocity due to large scale structure and turbulence induced by self-gravity .
For the lower mass discs , we show that the characteristic peculiar velocities scale with the square root of the effective turbulent viscosity parameter , \alpha , as expected from local \alpha -disc theory .
The derived velocities are anisotropic , with substantially larger in-plane than perpendicular values .
As the disc mass is increased , the validity of the \alpha approximation breaks down , and this is accompanied by anomalously large in-plane broadening .
There is also a high variance due to the importance of low- m spiral modes .
For low-mass discs , the magnitude of in-plane broadening is , to leading order , equal to the predictions from \alpha disc theory and can not constrain the source of turbulence .
However , combining our results with prior evaluations of turbulent broadening expected in discs where the magnetorotational instability ( MRI ) is active , we argue that self-gravity may be distinguishable from the MRI in these systems if it is possible to measure the anisotropy of the peculiar velocity field with disc inclination .
Furthermore , for large mass discs , the dominant contribution of large-scale modes is a distinguishing characteristic of self-gravitating turbulence versus MRI driven turbulence .