  We analyzed the growth of non-spherical perturbations in supersonic accretion
flows. We have in mind the application to the post-bounce phase of
core-collapse supernovae (CCSNe). Such non-spherical perturbations have been
suggested by a series of papers by Arnett, who has numerically investigated
violent convections in the outer layers of pre-collapse stars. Moreover, Couch
& Ott (2013) demonstrated in their numerical simulations that such
perturbations may lead to a successful supernova even for a progenitor that
fails to explode without the fluctuations. This study investigated the linear
growth of perturbations during the infall onto a stalled shock wave. The
linearized equations are solved as an initial and boundary value problem with
the use of Laplace transform. The background is a Bondi accretion flow whose
parameters are chosen to mimic the 15 $\mathrm{M_\odot}$ progenitor model by
Woosley & Heger (2007), which is supposed to be a typical progenitor of CCSNe.
We found that the perturbations that are given at a large radius grow as they
flow down to the shock radius; the density perturbations can be amplified by a
factor of 30, for example. We analytically showed that the growth rate is
proportional to $l$, the index of the spherical harmonics. We also found that
the perturbations oscillate in time with frequencies that are similar to those
of the standing accretion shock instability. This may have an implication for
shock revival in CCSNe, which will be investigated in our forthcoming paper in
more detail.
