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 ~ ρ², 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.