## Virtual Tokamak Exercises

The calculation used to compute the score for the Virtural Tokamak is really quite complex. In fact, fusion scientists actually use a full-fledged (with more adjustable controls) version when they want a quick idea of how well a tokamak reactor design would work.

At the same time, this calculation is based on real fusion experiments. With these exercises, you can use the Virtual Tokamak to unearth that underlying fusion physics.

• Investigate the density limit:
1. Keeping the magnetic field and auxiliary power the same, raise the plasma density in small steps until the plasma disrupts (a score of 0.00 is given). Make a note of the slider values for the highest density you reach before the disruption.
2. Change one of the magnetic field or auxiliary power sliders and again determine the maximum density.
3. Repeat this exercise until you understand how the maximum density depends on the values of the other two sliders.
4. Test your theory by estimating the maximum density for values of the magnetic field and auxiliary power which you haven't tried yet. Then, try them and see if your prediction was correct.
5. Be careful while you're doing this not to get to temperatures which are too high, otherwise the limit on the temperature will come into play, making it more difficult for you to single out the density limit. This is the way real physics works!
We'd like to tell you more about the density limit, but frankly we just don't really know what causes it!

• Having understood the density limit, you can examine temperature limit. Strictly speaking, it is a limit on the plasma pressure, which is the product of the plasma density and temperature. In practice, though, it's usually the temperature that increases too much, producing a plasma too hot for the magnetic field to confine in a stable manner.
1. At this limit, the ratio of the plasma pressure to the pressure of the magnetic field has a certain value which has been determined by experiments and theoretical calculations. The way the Virtual Tokamak is designed, this critical ratio is always the same.
2. First, try to find a point where the plasma disrupts at this pressure limit. Hold the magnetic field and plasma density fixed and slowly raise the auxiliary heating power until you get a score = 0.00. Make a note of the density, temperature, and magnetic field for the auxiliary power value just below the pressure limit.
3. How do you know this is not the density limit?
4. Now, repeat this exercise at one or two other values for the magnetic field. How did the maximum possible plasma pressure (density times temperature) change?
5. The relationship between the maximum plasma pressure and the magnetic field is complicated by the contribution to the plasma pressure made by the high energy helium nuclei which are produced by the fusion reactions. In this case, the pressure which is compared with the limiting value is significantly larger than you will compute just by multiplying the plasma density and temperature.
• Low Temperature Limit. The defining characteristic of a tokamak is the toroidal plasma current, which along with the magnetic field produced by the external magnetic field coils provides the confining and stabilizing forces. As this current runs through the plasma, it generates heat just like electricity does when it runs through a circuit. In this exercise, we first set the auxiliary power slider to 0.00.
1. Then, hold the magnetic field constant, and change the density in small steps. Make a note of the temperature values at each density. What happens to the temperature values as the density is raised?
2. Now, do the same at a different magnetic field. Considering only the lowest densities you tried at each magnetic field, see if you can determine the relationship between the magnetic field and the temperature in this ohmic heating limit.
3. Test your theory by first selecting another magnetic field, predicting the temperature (at low density), and then running the Virtual Tokamak to see he well you did.