# Plasma pressure/plasma beta

Hello all,

I am trying to calculate the plasma beta in a specific plasma temperature. for the plasma pressure i am using the formula p=(ni*kB*Ti)+(ne*kB*ne) and for the beta i am using β=(2*μ0*p)/B^2. I plotted an excel sheet for different values of the magnetic field and realized that the higher the magnetic field the lower the beta. I'v read that for better tokamaks, a beta close to 1 is needed and also a high magnetic field. How is this going to work? Unless I am missing something or misread some information. Any help?

Paul Colby
Gold Member
Wouldn't raising the particle density increase ##\beta## as well?

I am assuming quazi-neutral plasma such that ne=ni=n=10^20 (m^-3). Maybe i have my units wrong for the temperature. I've just encountered an example in which they use the temperature in eV units. And just to clarify in the pressure formula above the second term is (ne*kB*Te).

Paul Colby
Gold Member
Well, the pressure looks to be the sum of ideal electron and an ideal ion gases at two different temperatures. This seems reasonable. I'm asking kind of a naive question just based on the equations given. The beta is the ratio of plasma pressure to magnetic pressure. Is it possible the "better" tokamaks operate at a higher plasma pressure? Is plasma pressure linked to magnetic pressure by some other assumption like confinement in your calculation?

No not really. I just checked the equation for the ITER reactor and it works for its number. I get the same numbers as obtained experimentally. I just have it stuck in my brain that a beta close to one with as high magnetic field as possible is what reactors are trying to achieve. Again maybe I read it wrongly or the source wasn't correct. I know there are limits regarding the electron density (Greenwald limit), Troyon limit etc. that limit the beta. Anyway, thanks for your answers sir!

The fusion power scales with both the temperature and density. This implies that there is a minimum pressure that we have to reach in order to achieve fusion. So one of the goals in fusion research is to maximize the confined pressure.

In turn out that maximum stable pressure in a given configuration scales with the square of the magnetic field strength. So one way to increase the maximum stable pressure is to increase the magnetic field strength. Unfortunately current technology limits the maximum strength of the magnetic field. Additionally we want fusion to be economical. The costs associated with the superconducting magnets increases with the field strength. So we want to find stable configurations that get the most pressure per magnetic field strength. Thus we want to find configurations that optimize beta (=2mu_0p/B^2).

There are many different "betas" used in fusion research. Your definition of beta is total beta and it is usually limited to around 10% in tokamaks. We also talk about the poloidal beta and the normalized beta can often be of order unity. Then there is the engineering beta, which for a magnetically confined plasma cannot be greater than one from equilibrium considerations alone.

I understand but let me give you an example just so you can see what i mean and so that you can correct me if I am wrong. Assume I have a small tokamak device say with a major radius of 6cm and a minor radius 2cm. Let's say that I am using a 1mm wire for my toroidal confinement coils with a fusing current of 83A (for 10sec). This gives a maximum toroidal magnetic field of 0.33T with N=400 coil turns (using the B=(μ0ΝΙ/2πr) formula). As I stated above I am using n=10^20 m-3, T=1.85*10^8 K, and kB=1.38*10^-23 so the pressure is p=2nkBT=512344.32 Pa. If i use the beta formula i get 11.82 which is just not acceptable. The beta should be a number between 0 and 1 (or as a percentage if multiplied by 100). That's where I have my issue.. Unless I am doing something wrong.

Paul Colby
Gold Member
I checked your numbers with Mathematica and all is in order. It seems you've chosen your numbers and design and this is the result. If you need another ##\beta## you need to change something. Why were you expecting these numbers to yield a beta less than one?

Yes all the units are fine (it was a painful process to check all my equations). Well here is my thought process. The ITER has an ion and electron combined temperature of 16.8 keV. At this temperature the pressure is 269108 Pa. Now I know there are limitations regarding the achievable beta but if tokamaks want a beta close to one (that is 100%) then why do they apply such huge magnetic fields? The bigger the magnetic field the lower the beta.. Unless instabilities require big magnetic fields to stabilize them.

Paul Colby
Gold Member
I'm just starting to try to learn plasma physics in my old age. My interest is in computer simulations. My understanding is the ITER is huge compared to the numbers you are playing with. I assume that many many years of simulation and design has gone into it. ITER is not by any stretch a simple device and it's certainly not a simple mater to choose one (or even a handful) of parameters in its design.

Yes this is my understanding as well. I mean I tried the equations I found with the ITER parameters and they work fine. Only when I get to small numbers such as my design, the equations don't work. That's why I believe that there is more into the beta at such big designs than meet the eye.

Paul Colby
Gold Member
My understanding is the real issues in confinement are in controlling plasma instabilities. Plasma instabilities I believe are oscillatory or chaotic modes that have a net gain > 1.

Yes, that's why I believe you have to find the optimum solution for achieving as high beta as possible and to also take care of the many instabilities. Maybe the ITER numbers are in some way the optimum for that design (as well as for fusion power production and for minimizing cost)

Yes all the units are fine (it was a painful process to check all my equations). Well here is my thought process. The ITER has an ion and electron combined temperature of 16.8 keV. At this temperature the pressure is 269108 Pa. Now I know there are limitations regarding the achievable beta but if tokamaks want a beta close to one (that is 100%) then why do they apply such huge magnetic fields? The bigger the magnetic field the lower the beta.. Unless instabilities require big magnetic fields to stabilize them.

You're putting the cart before the horse.

One way to design a reactor is to start by considering plasma physics. When you do so you will learn that things like equilibrium, stability, and transport will set the maximum beta that your configuration can achieve. The next step, once you know the max beta, is then to calculate the strength of the magnetic field needed to support a given pressure. One you know the strength of the magnetic field, you can then design the magnets and calculate the current you need.

Another way to design a experiment is to start with a coil configuration. Then use plasma physics to calculate the maximum beta. Then use these two pieces of information to calculate the maximum pressure your experiment can achieve.

However your design ignores all plasma physics. You picked a pressure. You picked a magnetic field. Then you calculated beta. You found that your beta is way too large! What this should tell you is that your design is not physical. In real life if you tried to operate this experiment, the plasma would disrupt long before you reached your target pressure.

It might help to think of the design of an airplane. An airplane with a given shape and particular set of jet engines will have a maximum lift. Once we know what that lift is we can proceed in two ways. This lift is set by the physics of aerodynamics.

One way to operate the plane is to fill it full of cargo. This is analogous to what you're doing. Once the plane is packed full of cargo we then taxi to the runway and hope it can fly. But if the weight exceeds the maximum lift then the plane we never take off. Oops. The reason why the plane won't take off is simple. You've over loaded the plane.

Or instead, once we know the lift of the plane, we weigh cargo. Then we load the plane up to the maximum weight (with safety margins) that it can carry. This way when we taxi to the runway, we know that the plane will takeoff.

Alternatively, we could start with the goal of designing a plane that can carry a specific cargo weight. To do this we will have modify the shape of the plane, the set of engines, etc until we've met our design goals.

In sense by asserting that you experiment with reach a certain pressure you've overloaded it. The fact that beta is greater than one means that your "plane will never take."

I see what you mean. You are absolutely right. I am picking my numbers. I will revise my research. Thank you for pointing out my wrong thought process!