  We perform numerical analyses of the structure induced by gravitational
instabilities in cooling gaseous accretion discs. For low enough cooling rates
a quasi-steady configuration is reached, with the instability saturating at a
finite amplitude in a marginally stable disc. We find that the saturation
amplitude scales with the inverse square root of the cooling parameter beta =
t_cool / t_dyn, which indicates that the heating rate induced by the
instability is proportional to the energy density of the induced density waves.
We find that at saturation the energy dissipated per dynamical time by weak
shocks due is of the order of 20 per cent of the wave energy. From Fourier
analysis of the disc structure we find that while the azimuthal wavenumber is
roughly constant with radius, the mean radial wavenumber increases with radius,
with the dominant mode corresponding to the locally most unstable wavelength.
We demonstrate that the density waves excited in relatively low mass discs are
always close to co-rotation, deviating from it by approximately 10 per cent.
This can be understood in terms of the flow Doppler-shifted phase Mach number
-- the pattern speed self-adjusts so that the flow into spiral arms is always
sonic. This has profound effects on the degree to which transport through
self-gravity can be modelled as a viscous process. Our results thus provide (a)
a detailed description of how the self-regulation mechanism is established for
low cooling rates, (b) a clarification of the conditions required for
describing the transport induced by self-gravity through an effective
viscosity, (c) an estimate of the maximum amplitude of the density perturbation
before fragmentation occurs, and (d) a simple recipe to estimate the density
perturbation in different thermal regimes.
