  Massive elliptical galaxies harbor large amounts of hot gas
($T\gtrsim10^6~\mathrm{K}$) in their interstellar medium (ISM) but are
typically quiescent in star formation. Active-galactic nuclei (AGNs) jets and
Type Ia supernovae (SNIa) inject energy into the ISM which offsets its
radiative losses and keeps it hot. SNIa deposit their energy locally within the
galaxy compared to the larger few$\times10~\mathrm{kpc}$-scale AGN jets. In
this study, we perform high-resolution ($512^3$) hydrodynamic simulations of a
local ($1~\mathrm{kpc}^3$) density-stratified patch of massive galaxies' ISM.
We include radiative cooling and shell-averaged volume heating, as well as
randomly exploding SNIa. We study the effect of different fractions of
supernova heating (with respect to the net cooling rate), different initial ISM
density/entropy (which controls the thermal-instability growth time
$t_\mathrm{ti}$) and different degrees of stratification (which affects the
free-fall time $t_\mathrm{ff}$). We find that the SNIa drive predominantly
compressive turbulence in the ISM with a velocity dispersion $\sigma_v$ up to
$40~\mathrm{km}s^{-1}$ and logarithmic density dispersion
$\sigma_s\sim0.2$--$0.4$. These fluctuations trigger multiphase condensation in
regions of the ISM where $\min(t_\mathrm{ti})/t_\mathrm{ff}\lesssim 0.6\exp(6
\sigma_s)$, in agreement with theoretical expectations that large density
fluctuations efficiently trigger multiphase gas formation. Since the SNIa rate
is not self-adjusting, when the net cooling drops below the net heating rate
the SNIa drive a hot wind which sweeps out most of the mass in our local model.
Global simulations are required to assess the ultimate fate of this gas.
