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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Mac Low preprints
MPI Astronomie theory group

Mordecai-Mark Mac Low,
mordecai@mpia-hd.mpg.de
Last modified: Mon 14 Sept 1998