cci_d.0500.jpg cci_d.1000.jpg cci_d.1500.jpg hydrostatic, wkb, numeric density vs time and space density vs time and space

Models for molecular clouds:
shock bound slabs and
self-gravitating magnetized slabs

by Doris Folini and Rolf Walder

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Driven supersonic turbulence in shock bound slabs
MHD-waves as structuring agents


Today, molecular clouds are regarded as a part of a larger, turbulent environment. In this new picture, they are dynamical, supersonically turbulent, transient entities resulting from the collision of large scale flows. Picking up this idea of molecular clouds being related to the collision of large scale flows, we use our work on driven supersonic turbulence in shock bound slabs to learn more about the structuring of molecular clouds.

Observations also show the existence of magnetic fields in at least some molecular clouds. Ordered magnetic fields have been observed on scales of about 0.05 pc in pre-stellar cores and on somewhat larger scales in star-forming regions. Coherent velocities in pre-stellar cores are observed on scales of about 0.01 pc. On larger scales, observed line widths indicate supersonic motions. Taken together, these observations suggest high-density condensates, threaded by magnetic fields, to be embedded in a supersonically turbulent environment. Under such conditions, the generation of magentic waves is to be expected. In view of this background, we use high resolution numerical simulations to investigate how Alfvén-waves affect a self-gravitating slab when injected at the slab center.

For all the simulations we use the AMRCART codes from our A-MAZE package.

Driven supersonic turbulence in shock bound slabs

Colliding flows, and the resulting, shock bound interaction zones, are ubiquiteous in astrophysics. We use high resolution numerical simulations to investigate the basic properties of such collision zones. The results are of interest for a wide variety of astrophysical objects. Here we concentrate on some aspects which are probably relevant with regard to molecular clouds. Further results on shock bound slabs we give on a seperate page.

The model we use

We consider a 2D, plane parallel situation where two flows, whose momentum fluxes are balanced, collide head on. In all simulations we use Euler equations together with a polytropic equation of state. In the isothermal simulations we solve the adiabatic Euler equations with gamma = 1.000001. The resulting collision zone accumulates matter as time evolves. The temperature of this collision zone is, in essence, constant and equal to the temperature of the colliding flows. Consequently, the zone becomes more and more spatially extended as it accumulates matter.

Some of our findings

  • The turbulence within the slab is naturally driven by the incoming, colliding flows. Turbulence inevitably develops if either at least one of the confining isothermal shocks is slightly disturbed or if the radiative cooling zone of the confinig shocks is thermally unstable and numerically well resolved. With regard to molecular clouds this finding is interesting as it automatically guarantees driven turbulence throughout the cloud if the cloud corresponds to a flow collision zone.

  • The scale of the density structure within the slab apparently increases with increasing thickness of the slab (see figure below). With regard to molecular cloud this aspect is interesting as the distribution of high density regions, the scale of the density structure, is decisive for where stars could be formed. This may mean that the size of the cloud affects the distribution of the stars that form.
    cci_d.0500.jpg cci_d.1000.jpg
    Evolution of a 2D plane parallel shock bound slab. The colliding flows come from the left and the right. Shown is density. As time increases, from 886 y, to 1773 y, to 2663 y, the slab becomes geometrically thicker and the scale of the density structure within the slab increases.
    (Click on pictures to enlarge or watch an mpeg [8.6MB] or a quicktime movie [115MB])

MHD-waves as structuring agents

The model we use

We consider a 1D (x-direction), plane-parallel, self-gravitating slab which we assume to be symmetric with respect to a central plane at x=0 (yz-plane, infinitely extended). In this geometry, all variables are functions of distance x to the central plane and time t only. Velocities and magnetic fields perpendicular to the x-direction are allowed, but gradients can occur only in x-direction. As initial conditions we use the stationary WKB solution by Martin et al. A & A 326, 1997. To describe the time evolution of this slab, we use ideal, isothermal MHD equations, including a source term to account for self-gravity. The use of isothermal equations implies that we assume instantaneous, optically thin radiative cooling. We consider only one sort of particles and consequently have no wave damping due to ion-neutral friction in our model. At x=0, the inner boundary of our model problem, we assume that a purely monochromatic, circularly polarized, left-handed Alfvén-wave is generated, traveling in positive x-direction.

Some of our findings

  • A single source of monochromatic, circularly polarized Alfvén-waves in the slab center is sufficient to excite supersonic turbulence within the slab. High density sheets and voids develop.
    hydrostatic, wkb, numeric Density distribution as a function of distance to central plane of the slab. Shown is the hydrostatic distribution (green), the WKB distribution (yellow), and the turbulent, numerically obtained distribution (red).
    Model parameters are:
    domain size = 1 = 6pc = 5000 cells;
    magnetic field Bx=20mG and Bt=20mG;
    temperature T=10K;
    particle density N(x=0,t=0) = 1000;
    wave frequency = 2pi/50000
    Click on figure for enlarged picture.

  • The mean spatial separation of the high density sheets as well as the mean spatial extension of the slab increase with increasing background field and with decreasing wave-frequency.
    density vs space and time density vs space and time
    Density as function of space (x-axis, 6pc) and time (y-axis, million years) for the same model parameters as given above (left) and for a five times lower wave frequency (right). Dark colours=low density, bright colours=high density. Click on image to enlarge.

  • To observe both, the structuring and support of the slab, good spatial resolution and a high order of integration are required in the numerical simulations.


R. Walder and D. Folini 1999
The formation of knots and filaments in shocks
Astrophysics and Space Science 260, 215-224
(Available as a 761 KB gzipped ps-file)

R. Walder and D. Folini 2000
On the stability of colliding flows: radiative shocks, thin shells, and supersonic turbulence
Astrophysics and Space Science, 274, 343-352
(Available as a 0.5 MB gzipped ps-file)

R. Walder and D. Folini 1999
Radiative Shocks, Supersonic Turbulence and Structure Formation in Space
Proceedings of the Seventh International Conference on Hyperbolic Problems:
Theory, Numerics, Applications
(Available as a 2.7 MB gzipped ps-file)

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Doris Folini and Rolf Walder
Last Update: October 14, 2002