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

Mordecai-Mark Mac Low
Max-Planck-Institut für Astronomie

submitted to The Astrophysical Journal, 14 Sept 1998

resubmitted to The Astrophysical Journal, 10 Jan 1999

Molecular clouds have broad linewidths suggesting 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. It has classically been thought that supersonic hydrodynamical turbulence would dissipate its energy quickly, but that the introduction of strong magnetic fields could maintain these motions. In a previous paper it has been 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 uniformly driven turbulence with the ZEUS astrophysical MHD code are used to derive the absolute value of energy dissipation, which is found to be

\dot{E}_{kin} \simeq - \eta_v m k v_{rms}^3,

with \eta_v = 0.21/\pi, where v_{rms} is the root-mean-square velocity in the region, E_{kin} is the total kinetic energy in the region, m is the mass of the region, and k is the driving wavenumber. The ratio of the formal decay time E_{\rm kin}/\dot{E}_{kin} of turbulence to the free-fall time of the gas can then be shown to be

\tau(\kappa) = \frac{\kappa}{M_{rms}} \frac{1}{4 \pi \eta_v},

where M_{rms} is the rms Mach number, and \kappa is the ratio of the driving wavelength to the Jeans wavelength. It is likely that \kappa < 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

t_0 \simeq 1.0 {\cal L} / v_{rms},

where {\cal L} is the driving wavelength.

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Last modified: Mon 14 Sept 1998