Decay of Turbulence in Fluids with Polytropic Equations of State
  We present numerical simulations of decaying hydrodynamic turbulence
initially driven by solenoidal (divergence-free) and compressive (curl-free)
driving. Most previous numerical studies for decaying turbulence assume an
isothermal equation of state (EOS). Here we use a polytropic EOS, $P$ $\propto$
$\rho^{\gamma}$, with polytropic $\gamma$ ranging from 0.7 to 5/3. We mainly
aim at determining the effects of polytropic $\gamma$ and driving schemes on
the decay law of turbulence energy, E $\propto$ $t^{-\alpha}$. We additionally
study probability density function (PDF) of gas density and skewness of the
distribution in polytropic turbulence driven by compressive driving. Our
findings are as follows. First of all, we find that even if polytropic $\gamma$
does not strongly change scaling relation of the decay law, the driving schemes
weakly change the relation; in our all simulations, turbulence decays with
$\alpha$ $\approx$ 1, but compressive driving yields smaller $\alpha$ than
solenoidal driving at the same sonic Mach number. Second, we calculate
compressive and solenoidal velocity components separately and compare their
decay rates in turbulence initially driven by compressive driving. We find that
the former decays much faster so that it ends up having a smaller fraction than
the latter. Third, the density PDF of compressively driven turbulence with
polytropic $\gamma$ $>$ 1 deviates from log-normal distribution: it has a
power-law tail at low density as in the case of solenoidally driven turbulence.
However, as it decays, the density PDF becomes approximately log-normal. We
discuss why decay rates of compressive and solenoidal velocity components are
different in compressively driven turbulence and astrophysical implication of
our findings.
