Issues with NEQ and Z Cooling

I haven't been able to get steady shocks for these types of cooling yet. I originally thought that it had to do with mu (mean molecular weight). I need to calculate mu for every cell, because it is always changing. Heavier species will increase mu, and ionization will decrease mu. However, I just realized this morning that the shocks are not steady for a different reason. I went back to the beginning and just set mu = 1, and the shocks move. I'm guessing that there's a scaling issue somewhere since that has often been the case when I am close but not quite steady. There could also be a typo in the way temperature is being calculated. I'll look into this further and update this post later today.


Why mu is important

The following equation can be derived from the Rankine-Hugoniot conditions in the strong shock limit:

This shows that post-shock temperature is proportional to mu. Temperature in general is actually always proportional to mu because of the ideal gas law. We are just so used to having mu = 1.0 for an ideal hydrogen gas. For my pure hydrogen runs, mu is just 1.0, but Pat's runs have other species. The aforementioned post-shock temperature is immediately after the shock interface and before ionization occurs. Therefore, since the gas has no pre-ionization, the mu for Pat's runs is higher.

Let's look at the temperature for both runs (Pat's is in black, mine is in red):

You can get a rough estimate of what mu should be in Pat's runs by just looking at the difference in temperature. I suspect that Pat has an initial mu of approximately 1.22.

Here is how mu is calculated in the cooling routines: 

 mu = (nvec(iH2)*muH2 + (nvec(iH)+nvec(iHII))*muH + (nvec(iHe)+nvec(iHeII)+nvec(iHeIII))*muHe)/npart

nvec is the number density vector, and it contains the number densities for all the different species. npart is the total number density of particles which includes all the electrons as well. This is why mu significantly decreases with ionization. For example, a fully ionized gas of HII would have mu = 0.5 instead of 1.0 for a neutral gas of all H. In other words, you double the number of particles without adding any mass (electrons are treated as mass-less). Recombination has the opposite effect as it reduces the number of electrons.

So in general, mu is important in calculating temperature. Both for the immediate post-shock temperature, and the decreasing temperature within the cooling region.

Here is what mu looks like throughout the problem domain:

Notice that mu starts off a little less than 1.0. That is because there is a trace amount of pre-shock ionization. Mu then drops rapidly (to about 0.65) after the shock due to ionization,and gradually increases due to recombination.

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