The energy dissipation rate of supersonic, magnetohydrodynamic turbulence in molecular clouds

Molecular clouds have broad line widths, which suggests turbulent supersoni c motions in the clouds. These motions are usually invoked to explain why m olecular clouds take much longer than a free-fall time to form stars. Class ically, it was thought that supersonic hydrodynamical turbulence would diss ipate its energy quickly but that the introduction of strong magnetic field s could maintain these motions. A previous paper has shown, however, that i sothermal, compressible MHD and hydrodynamical turbulence decay at virtuall y the same rate, requiring that constant driving occur to maintain the obse rved turbulence. In this paper, direct numerical computations of uniform, r andomly driven turbulence with the ZEUS astrophysical MHD code are used to derive the value of the energy-dissipation coefficient, which is found to b e (E) over dot(kin) similar or equal to -eta(upsilon)m(rms)(3), with eta(upsilon) = 0.21/pi, where upsilon(rms) is the root-mean-square (rm s) velocity in the region, E-kin is the total kinetic energy in the region, m is the mass of the region, and (k) over tilde is the driving wavenumber. The ratio tau of the formal decay time E-kin/(E) over dot(kin) of turbulen ce to the free-fall time of the gas can then be shown to be tau(kappa) = kappa/M-rms 1/4 pi eta(upsilon), where M-rms is the rms Mach number, and Ic is the ratio of the driving wave length to the Jeans wavelength. It is likely that kappa < 1 is required for turbulence to support gas against gravitational collapse, so the decay tim e will probably always be far less than the free-fall time in molecular clo uds, again showing that turbulence there must be constantly and strongly dr iven. Finally, the typical decay time constant of the turbulence can be sho wn to be t(0) similar or equal to 1.0 L/upsilon(rms), where L is the driving wavelength.