MHD Clumps with Global Uniform Field

Problems involving magnetized clouds and clumps, especially their interaction with shocks are common in astrophysical environments and have been a topic of research in the past decade.
The magnetic field structure, whether aligned with the shock or perpendicular to the shock, can have profound influence on the shocked behavior and evolution of the clump. In this presentation
we review some basic results of the shocked MHD clumps by past simulations, as well as movies of preliminary 3D simulations produced by our parallel MHD code AstroBEAR. We will also
discuss future directions of numerical simulations on such topic.

The important physics parameters are:
The density contrast:

\chi=\frac{\rho_{clump}}{\rho_{ambient}}

The sonic Mach number:

M=\frac{u_{wind}}{c_s}

The Alfvenic Mach number:

M_A =\frac{u_{wind}}{c_A}

The magnetic beta:

\beta =\frac{2 c_s^2}{\gamma c_A^2}

The clump crushing time (defined by the time for the transmitted shocked to pass through the entire clump):

\tau_{cc} = \frac{\chi^{1/2}r_{clump}}{M c_s}

The paper Jones, T.W., Ryu, Dongsu, Tregillis, I.L. 1996 ApJ, 473, 365 studied the effect of 2-D magnetic field on a bullet passing through a uniform ambient medium.
It also serves as an example on clumps getting shocked with 2-D uniform magnetic field.
The image below shows the clump evolution when the uniform magnetic field is aligned with the shock direction.

In these simulations we notice:

1. No significant difference in terms of density evolution even for β = 1.
   
2. For the magnetic field to suppress the K-H instabilities at the boundary flows, one requires that the Alfven speed to be greater than the velocity difference of the shear layers
   

at the boundary flows. This criterion can be translated to roughly: β < 1 along the clump edge.

3. When the clump is getting shocked, the clump is accelerating along the horizontal x axis, towards the lighter ambient material. This creates R-T instability whose bubbles will flow into
   

and deform the shocked clump material. The magnetic field along the acceleration access has a less dramatic effect in suppressing the R-T instability comparing to that perpendicular
to the acceleration axis. The criterion for the magnetic field to stabilize R-T instabilities is roughly: β < χ/M.

4. The field surrounding the clump is getting amplified due to compressing and stretching. But in the aligned field case, there is no place that the magnetic field becomes energetically
   

dominant despite the amplification; that is, almost everywhere β >> 1. So the K-H and R-T instabilities are hardly suppressed. The clump density evolution is not dramatically different
from the case when there is no field.

5. The stretching is the dominant magnetic field amplification mechanism. The "wing" shaped field encompassing the clump has the most stretching and thus the strongest amplification, which
   

increases the flow coherence.

6. The low beta area is concentrated on the axis, behind the clump, which forms a "wake" of low density, high magnetic pressure region.
   

The image below shows the clump evolution when the uniform magnetic field is perpendicular to the shock direction.

1. The magnetic field is stretched and wrapped around the clump, which effectively confines the clump and prevents its fragmentation, even for moderately strong field β = 4. The clump
   

embedded in the stretched field is compressed, but then, because of the strong confining effect of the field develops a streamlined profile and is not strongly eroded.

2. The field amplification is strong. One can observe some locations where the field strength is amplified by more than two orders of magnitude. The field is concentrated around the clump
   

profile, which serves as a "shell", preventing the clump from fragmentation. The magnetic pressure at the clump head increases due to compression, which acts as a shock absorber.
The magnetic pressure encompassing the clump increases due to stretching, which stabilizes the instabilities and gives the shocked material a more streamlined shape.

3. At later stage, the stretched field around the clump edge has β < 1 even for moderately strong initial field condition, indicating a much stronger amplification effect comparing to the
   

aligned field case.

In AstroBEAR, the clump simulation is done using the clump object, the wind object and the cooling object. We also implement various multiphysics processes to make the situation
more interesting. Below are snapshots of the clump density and magnetic pressure in a 3-D AMR simulation. Notice the field concentration at \tau_{cc} is very different for
the aligned and perpendicular field cases.



​Here is a movie of the mentioned simulation.

The following images show the high resolution shocked clump problem with uniform magnetic field in AMR.

Shock Clumps Interaction with Contained Magnetic Field

Here we study the shock interaction with the clump when the clump has tangled magnetic field contained. We focus on the following 4 cases:


The contained magnetic field has a volume-average beta of 0.25. The volume rendered results for the four cases (the four columns for TA, TP, PA, PP cases, respectively). Each row corresponds to 1, 2 and 3.5 cloud crushing time.


Movies:
TA case:
​http://www.pas.rochester.edu/~shuleli/HedlaMovie/torxvr.gif
TP case:
​http://www.pas.rochester.edu/~shuleli/HedlaMovie/torzvr.gif
PA case:
​http://www.pas.rochester.edu/~shuleli/HedlaMovie/polzvr.gif
PP case:
​http://www.pas.rochester.edu/~shuleli/HedlaMovie/polxvr.gif

The magnetic field intensity map is plotted as follows:

The contained magnetic field geometry can influence the kinetic and magnetic energy evolution. As shown in the following plot:




The wind-clump mixing ratio can be affected also by the contained field geometry, as the toroidal contained field can enhance the clump edges to resist the instability erosion.

Mixing Ratio Evolution Movies:
Strong contained field (average beta of 0.25)
​http://www.pas.rochester.edu/~shuleli/movie0703/mrst.gif

Weak contained field (average beta of 1)
​http://www.pas.rochester.edu/~shuleli/movie0703/mrwk.gif

The mathematical model that can qualitatively explain the line plots is available in the paper:
Magnetohydrodynamic Shock-Clump Evolution with Self-contained Magnetic Fields, Shule Li, Adam Frank, Eric Blackman, Astrophys Journal 774 (2013), 133-149

Random Field Configuration

We have started investigating the contained field interaction with shock when the field is random (with a certain known spectrum).

Presentations

​Presentation: Thermal Conduction Solver Outline

​Presentation: MHD Clumps with Shocks and Thermal Conduction (HEDLA)

​Poster: MHD Clumps with Shocks (AAS)

References

​http://arxiv.org/abs/0707.2616 ​http://arxiv.org/abs/1003.5546