Non-adiabatic turbulence driving during gravitational collapse

We investigate the generation of turbulence during the prestellar gravitational contraction of a turbulent spherical core. We define the ratio $g$ of the one-dimensional turbulent velocity dispersion, $\sigma_\mathrm{1D}$ to the gravitational velocity $v_g$, to then analytically estimate $g$ under the assumptions of a) equipartition or virial equilibrium between the gravitational ($E_g$) and turbulent kinetic ($E_\mathrm{turb}$) energies and b) stationarity of transfer from gravitational to turbulent energy (implying $E_\mathrm{turb}/E_g=$cst). In the equipartition and virial cases, we find $g=\sqrt{1/3}\approx0.58$ and $g=\sqrt{1/6}\approx0.41$, respectively; in the stationary case we find $g=\langle v_\mathrm{rad}\rangle L_d/(4\pi\sqrt{3}\eta Rv_g)$, where $\eta$ is an efficiency factor, $L_d$ is the energy injection scale of the turbulence, and $R$ is the core's radius. Next, we perform AMR simulations of the prestellar collapse of an isothermal, transonic turbulent core at two different resolutions, and a non-turbulent control simulation. We find that the turbulent simulations collapse at the same rate as the non-turbulent one, so that the turbulence generation does not significantly slow down the collapse. We also find that a) the simulations approach near balance between the rates of energy injection from the collapse and of turbulence dissipation; b) $g\approx0.395\pm0.035$, close to the "virial" value (turbulence is $\sim35-40\%$ of non-thermal linewidth); c) the injection scale is $L_d\lesssim R$, and d) the "turbulent pressure" $\rho\sigma_\mathrm{1D}^2$ scales as $\sim\rho^{1.64}$, an apparently nearly-adiabatic scaling. We propose that this scaling and the nearly virial values of the turbulent velocity dispersion may be reconciled with the non-delayed collapse rate if the turbulence is dissipated as soon as it is generated.