Does Turbulent Pressure Behave as a Logatrope?

We present numerical simulations of an isothermal turbulent gas undergoing gravitational collapse, aimed at testing for logatropic'' behavior of the form $P_t \sim \log \rho$, where $P_t$ is the turbulent pressure’’ and $\rho$ is the density. To this end, we monitor the evolution of the turbulent velocity dispersion $\sigma$ as the density increases during the collapse. A logatropic behavior would require that $\sigma \propto {\rho }^{-1\mathrm{/}2}$, a result which, however, is not verified in the simulations. Instead, the velocity dispersion increases with density, implying a polytropic behavior of $P_t$. This behavior is found both in purely hydrodynamic as well as hydromagnetic runs. For purely hydrodynamic and rapidly-collapsing magnetic cases, the velocity dispersion increases roughly as $\sigma \propto {\rho }^{1\mathrm{/}2}$, implying $P_t\sim {\rho }^2$, where $P_t$ is the turbulent pressure. For slowly-collapsing magnetic cases the behavior is close to $\sigma \propto {\rho }^{1\mathrm{/}4}$, which implies $P_t \sim \rho^{3/2}$. We thus suggest that the logatropic equation of state'' may represent only the statistically most probable state of an ensemble of clouds in equilibrium between self-gravity and kinetic support, but does not adequately represent the behavior of the turbulent pressure’’ within a cloud undergoing a dynamic compression due to gravitational collapse. Finally, we discuss the importance of the underlying physical model for the clouds (in equilibrium vs. dynamic) on the results obtained.