Three-dimensional MHD simulations of magnetized cloudfragmentation with turbulence and ion-neutral friction
Takahiro Kudoh1 & Shantanu Basu2
1 National Astronomical Observatory, Japan

2 University of Western Ontario, CanadaWe perform a 3D-MHD simulation of a self-gravitating isothermal gas layer that is initially penetrated by a uniform magnetic
field. The strength of the initial magnetic field is such that the cloud is slightly sub-critical. In this system, we input random supersonic turbulence initially. Ion-neutral friction is also introduced in the magnetized gas so that the magnetic diffusion allows gas togo across the magnetic field and form self-gravitating cores. We found that the time scale of core formation is on the order of 106
years, which is faster than the usual magnetic diffusion time (~ 107 years). The result is consistent with that of 2D-MHD
simulations by Li & Nakamura (2004).

Introduction

Results

When the strength of the magnetic field is weaker than thecritical value, the cloud is fragmented by gravitational
instability in a free-fall time of the cloud (super-critical). On
the other hand, when the magnetic field is stronger than thecritical value, the cloud is gravitationally stable because
magnetic field prevents the contraction of the cloud (subcritical). However, because the molecular cloud contains a
lot of neutrals as well as some ions, magnetic diffusion
induced by ion-neutral friction (amibipolar diffusion) occurs
in the cloud. Due to this effect, gravitational instability
develops gradually over the diffusion time even when thecloud is sub-critical (e.g., Basu & Ciolek 2004).
It has been suggested that sub-critical clouds have a timescale problem: the age spreads of young stars in nearby
molecular clouds is often 1-3 x 106 years, while theambipolar diffusion time is typically ~ 107 years.
Recently, however, Li & Nakamura (2004) have shown thatthe time-scale of mildly sub-critical cloud fragmentation is
reduced by supersonic turbulence to ~106 years byperforming 2D simulations in the thin-disk approximation. In
this paper, we study the 3D extension of the model by
including the self-consistent calculation of vertical structureof the cloud.

Time evolution of maximum density.
subsonic
perturbation

supersonic
perturbationt0 ~ 2 x 105 years

The figure shows the time evolution of maximum density. The blueline shows the case when the initial perturbation is supersonic (3times sound velocity). The black line shows the case when the initial
perturbation is subsonic (0.1 times sound velocity).
The figure shows that the time-scale of cloud fragmentation forsupersonic perturbation is much shorter than that of the subsonic
perturbation. The time-scale of the fragmentation of sub-critical cloudwith supersonic turbulence is ~ 4 x 106 years in this case.
The orange line shows the case when the initial magnetic field is
weaker than the critical value and perturbation is super-sonic. The
fragmentation of super-critical cloud with supersonic turbulence
evolves too quickly (~ 2 x 105 years).

Time evolution of density contour.
t=1.4

t=10.0a cloud coret=20.5The Model
Low density andwarm gas

periodical boundary
symmetrical boundary
Molecular
cloud

3D
simulationbox

gravitational density
stratification

Magnetic field lines

dense gaszy~2.5pc

Initial condition

Magnetic field line
Low density andwarm gas

3D-MHD with ambipolar diffusion

Figures show the time evolution of density contour for a sub-critical
cloud with initial supersonic turbulence. Arrows show velocities.
Upper panels show the cross section at z=0. Lower panels show
the cross section at y=-13.0, y=10.6, and y=-5.9 from left to right,respectively.

X

We input velocity perturbation
perpendicular to magnetic field at t=0.

We assume that the initial cloud is a gas layer that is in a selfgravitational equilibrium along uniform magnetic field lines.
The initial cloud is assumed to be slightly sub-critical as a typicalcase. The magnetic field strength is 2 times larger than thecritical value.
We also assume that the cold dense gas, which corresponds tomolecular cloud, is sandwiched by warm low density gas. Eachgas layer is assumed to be isothermal in Lagrangian coordinate.
For simplicity, we assume a symmetrical boundary condition at
z=0 and solve equations for only half of the sandwiched layer.
In this equilibrium gas layer, we input initial velocity perturbation
perpendicular to magnetic field. When the perturbation is supersonic, we assume |v| ~ k-2, where v is velocity and k is wave
number.

The cloud is first strongly compressed by super-sonic turbulence(t=1.4). However, the dense region rebounds and shows
oscillations (t=10.0) because the cloud is sub-critical. Finally,
ambipolar diffusion allows gravitational collapse and cores to form
(t=20.5).
At the final stage (t=20.5), the magnetic field is dissipated enoughat the center of the densest core. Then, the core undergoes
runaway collapse.

Summary and Discussion
We performed 3D-MHD simulations and confirmed that the time-scaleof sub-critical cloud fragmentation is reduced by supersonicturbulence. The result is consistent with Li & Nakamura’s 2D
simulations. The short time-scale of the fragmentation may be causedby the strong magnetic field gradient created by the super-sonicturbulence.The time scale of the fragmentation is the order of 106years. It may solve the time-scale problem of sub-critical clouds.