Does Turbulent Pressure Behave as a Logatrope?

We present numerical simulations of an isothermal turbulent gas undergoinggravitational collapse, aimed at testing for ``logatropic'' behavior of theform $P_t \sim \log \rho$, where $P_t$ is the ``turbulent pressure'' and $\rho$is the density. To this end, we monitor the evolution of the turbulent velocitydispersion $\sigma$ as the density increases during the collapse. A logatropicbehavior would require that $\sigma \propto \rho^{-1/2}$, a result which,however, is not verified in the simulations. Instead, the velocity dispersionincreases with density, implying a polytropic behavior of $P_t$. This behavioris found both in purely hydrodynamic as well as hydromagnetic runs. For purelyhydrodynamic and rapidly-collapsing magnetic cases, the velocity dispersionincreases roughly as $\sigma \propto \rho^{1/2}$, implying $P_t\sim \rho^2$,where $P_t$ is the turbulent pressure. For slowly-collapsing magnetic cases thebehavior is close to $\sigma \propto \rho^{1/4}$, which implies $P_t \sim\rho^{3/2}$. We thus suggest that the logatropic ``equation of state'' mayrepresent only the statistically most probable state of an ensemble of cloudsin equilibrium between self-gravity and kinetic support, but does notadequately represent the behavior of the ``turbulent pressure'' within a cloudundergoing a dynamic compression due to gravitational collapse. Finally, wediscuss the importance of the underlying physical model for the clouds (inequilibrium vs. dynamic) on the results obtained.Comment: Accepted in ApJ. 10 pages, 3 postscript figure