The Energy Dissipation Rate of Supersonic, Magnetohydrodynamic Turbulence in Molecular Clouds

Molecular clouds have broad line widths, which suggests turbulent supersonic motions in the clouds. These motions are usually invoked to explain why molecular clouds take much longer than a free-fall time to form stars. Classically, it was thought that supersonic hydrodynamical turbulence would dissipate its energy quickly but that the introduction of strong magnetic fields could maintain these motions. A previous paper has shown, however, that isothermal, compressible MHD and hydrodynamical turbulence decay at virtually the same rate, requiring that constant driving occur to maintain the observed turbulence. In this paper, direct numerical computations of uniform, randomly driven turbulence with the ZEUS astrophysical MHD code are used to derive the value of the energy-dissipation coefficient, which is found to be with η_v = 0.21/π, where v_rms is the root-mean-square (rms) velocity in the region, E_kin is the total kinetic energy in the region, m is the mass of the region, and is the driving wavenumber. The ratio τ of the formal decay time E_kin/_kin of turbulence to the free-fall time of the gas can then be shown to be where M_rms is the rms Mach number, and κ is the ratio of the driving wavelength to the Jeans wavelength. It is likely that κ < 1 is required for turbulence to support gas against gravitational collapse, so the decay time will probably always be far less than the free-fall time in molecular clouds, again showing that turbulence there must be constantly and strongly driven. Finally, the typical decay time constant of the turbulence can be shown to be where is the driving wavelength.