Characterizing the Forces on the Clumps in Pulsed Jets

In this problem, I am considering a magnetic pinching force caused by the toroidal field and the gas pressure force. We have:

The jet starts out in magnetohydrodynamic equilibrium so that the magnetic force and gas pressure force cancel out (F = 0).  When the jet material gets shocked at an internal working surface, the magnetic field increases and the gas pressure increases.  They increase such that the previous equilibrium is no longer preserved.  The clump will expand if F > 0, and if F < 0 the clump will compress.

I used the MHD shock jump conditions to find the resulting F inside a clump. The jump conditions actually depend on radius in this problem. For this reason, it is extremely difficult to write an analytic expression for F. I decided to study F numerically.

Based on the above equations, you would expect to get clump compression as the magnetic field is increased. The first time I ran the numbers, I could not get F < 0. I knew this had to be wrong, so I started playing around with my initial parameters. I found that the sign of F was highly sensitive to the strength of the shock (typically characterized by the mach number M).

In general, it seems that low M shocks are easier to compress. Once the mach number goes below 1.0, the material is no longer shocked, and equilibrium can be reestablished (F = 0).

Below are a few plots for the three cases of beta that I'm running in my simulations (5, 1, 0.4).

The way I like to think of these plots is that the relative velocity plots will tell you about clump evolution. The mach number plots can tell you more general things about field strengths and shock strengths. Several things that we can tell from these plots:

  1. The clumps are only in the compression regime at certain relative velocities. Typically, smaller relative velocities (aka weaker shocks). This means that the clumps will be more susceptible to compression at later times.
  1. The inner regions of the clump are more susceptible to compression first. This means that as the relative velocity decreases, the inner regions of the clump will start compressing first. We could call this "inside-out" compression.
  1. At some point during the compression phase, the outer regions begin to be compressed with greater force than the inner regions. This seems counter to point 2. If the entire clump enters the compression regime quickly, then the "inside-out" compression might look more like "outside-in" compression, or possibly uniform compression.
  1. As the magnetic field strength increases (beta decreases), the compression region shifts to the right on the relative velocity plots. However, this does not mean that stronger fields are capable of compressing stronger shocks. The mach numbers would also be shifted to the right on this relative velocity plot. Despite this fact, the mach number plots show a similar trend. So yes, stronger fields can compress stronger shocks.
  1. At some point, the relative velocity becomes so small that a shock is no longer formed. This is the same as the mach number going to 1.0. This means that the pressure and magnetic field do not follow shock jump conditions, and thus the net force returns to zero.
  1. The peak magnitude of the force increases as beta decreases. This confirms that stronger fields compress with greater force.

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Comments

1. Adam Frank -- 11 years ago

This is a good start Eddie. Your shocks are essentailly isothermal. Are you including this in your analysis. Cooling will rob gas pressure but leave the post shock B unchanged.