Characterizing the Forces on the Clumps in Pulsed Jets II - Isothermal Approx.

As discusses in the meeting today, it would be more accurate to do my calculations in the isothermal limit. So I set gamma = 1.0, and got some interesting results.

Turns out that in this limit, the force is always negative (compression regime). As a reminder, the forces are:

The magnetic field and pressure profiles are as follows:

Where is the beta that I define in my initial conditions. is derived from and . is some intermediate radius, set to 0.6 rjet in my set up. depends on and . These are the profiles for . Outside of the profiles are such that the forces go to zero.

Solving for the net force as is results in zero, but if I apply shock jump conditions first, then I can write

where I have defined as the pressure jump and as the density (or magnetic field) jump. It is important to remember that the jump conditions are radially dependent due to the fact that the pressure is radially dependent. In the adiabatic case, this gets pretty complicated. If I enforce the conditions for an isothermal shock, then it simplifies a bit. To maintain the same temperature across the shock, the pressure jump has to match the density jump, so .

So making the problem isothermal guarantees that the force is always negative; the clump is always in the compression regime. This is consistent with what De Colle and Raga found in their 2008 paper: the non-radiative cases were dominated by thermal pressure forces, and the radiative cases were dominated by magnetic forces.

Even though I made the problem simpler, there are still some interesting things to learn from the plots…

It is easy to make sense of these plots. The force is greater for greater velocities and mach numbers because the magnetic field gets compressed more. More magnetic field means stronger pinching. The other cases of beta follow the same trend, so I will not post them.

If you only looked at the mach number plot, you would naively think that compression is stronger at larger radii. However, the plots follow this trend because mach number increases with radius. At any given moment, the velocity is constant across the shock, so you might think that stronger compression occurs at smaller radii. You would again be wrong. I made a different plot to explore this point.

This plot shows that for a given beta, you will get stronger compression with smaller radii but only to a certain point. Then, the curve turns over and the force starts decreasing. I also noticed in this plot that the different cases for beta were not in order. Strange right? So I made yet another type of plot to check this out.

Indeed, you do not keep getting stronger compression with stronger fields. Could it be that at some point you are "saturated" with magnetic field and the force really stops increasing, or is this a flaw of the isothermal approximation? In reality, the shock always increases the temperature. I played around with gamma in my spreadsheet, and I found that as I increased it to say 1.10, the above plot began to turn the other way.

So yes, for increasing gamma you get weaker compression, but you get compression that always increases with decreasing beta.

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