  Across a large range of scales, accreting sources show remarkably similar
patterns of variability, most notably the log-normality of the luminosity
distribution and the linear root-mean square (rms)-flux relationship. These
results are often explained using the theory of propagating fluctuations in
which fluctuations in the viscosity create perturbations in the accretion rate
at all radii, propagate inwards and combine multiplicatively. While this idea
has been extensively studied analytically in a linear regime, there has been
relatively little numerical work investigating the non-linear behaviour. In
this paper, we present a suite of stochastically driven 1-d $\alpha$-disc
simulations, exploring the behaviour of these discs. We find that the eponymous
propagating fluctuations are present in all simulations across a wide range of
model parameters, in contradiction to previous work. Of the model parameters,
we find by far the most important to be the timescale on which the viscosity
fluctuations occur. Physically, this timescale will depend on the underlying
physical mechanism, thought to be the magnetorotational instability (MRI). We
find a close relationship between this fluctuation timescale and the break
frequency in the power spectral density (PSD) of the luminosity, a fact which
could allow observational probes of the behaviour of the MRI dynamo. We report
a fitting formula for the break frequency as a function of the fluctuation
timescale, the disc thickness and the mass of the central object.
