Does Turbulent Pressure Behave as a Logatrope?

We present numerical simulations of an isothermal turbulent gas undergoing gravitational collapse, with the aim of testing for "logatropic" behavior of the form P t ~ log ρ, where P t is the turbulent pressure and ρ is the density. To this end, we monitor the evolution of the turbulent velocity dispersion σ as the density increases during collapse. A logatropic behavior would require σ ∝ ρ -1/2 , a result that is not, however, 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 and in hydromagnetic runs. For purely hydrodynamic and rapidly collapsing magnetic cases, the velocity dispersion increases roughly as σ ∝ ρ 1/2 , implying P t ~ ρ 2 , where P t is the turbulent pressure. For slowly collapsing magnetic cases, the behavior is close to σ ∝ ρ 1/4 , implying P t ~ ρ 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 as a result of gravitational collapse. Finally, we discuss the importance of the underlying physical model of the clouds (equilibrium versus dynamic) for the results obtained.