Talk-Polywell.org

One thing that has puzzled me with respect to Polywell for some time is the meaning of the term "beta". I finally looked it up on Wikipedia, which had this to say about the matter:

Beta is a parameter indicating the relative importance of kinetic to electromagnetic phenomena. In fusion power applications, beta can be thought of as an economic figure of merit. The magnetic field in a fusion device has technological limits such as the critical field of practical superconductors. The fusion power density, which determines the power output of a reactor, scales with the square of the plasma pressure. Therefore the output of a reactor can be said to scale with the square of the average beta. In a tokamak, for example, there is a fairly firm limit on the value beta can have before destructive instabilities occur.

If you take a look at the equation in the article, temperature (T) is in the numerator of the ratio defining beta, so my question: if temperature is supposed to be the wrong thing to look at in a Polywell, how do you define beta?

All I've heard is that it's the ratio of magnetic field energy to plasma pressure energy. Here's something on it from Dr. B's Gyro-loss holes report:

In the fully-filled model of a PolywellT field system, the electron P = I surface at :t Specif
<rb> = (rb/R) is at or near the outer edge of the plasma. At this condition the electrons
are at maximum energy E and speed and provide a pressure p-sub-e =2/3*n*E-sub-zero against the
magnetic field B at that point and its magnetic pressure P= B^2/(8*pi). This electron
pressure is the averaged result of electron gyro motion in the external field envelope at
various angles of incidence for the quasi-non-adiabatic region within the radius rb.

scareduck wrote:If you take a look at the equation in the article, temperature (T) is in the numerator of the ratio defining beta, so my question: if temperature is supposed to be the wrong thing to look at in a Polywell, how do you define beta?

Actually, temperature is in the given definition of plasma pressure (p = n kT). The definition of β is the ratio of plasma pressure to magnetic pressure.

So if you discount temperature as a useful measurement, a better question isn't "how do you define β" but "how do you define plasma pressure"?

To go further, T appears in the definition of plasma pressure in the form kT, where k is Boltzman's Constant, which relates particle kinetic energy to temperature. kT is the (average) energy per particle, and the n in the formula is the number of particles per unit volume.

So overall, plasma pressure works out to p = E/V, regardless of temperature, which gives us a β definition that doesn't involve temperature.

I also am confused about beta.
For example Dr B says in the first sentence of the “Gyro Holes Loss” paper that the β=1 surface is at some point. (This makes sense to me.)
It looks to me like B goes from 0 at the center up to a maximum at a coil with saddles at the cusps.
Further I see that the number of electrons goes from a maximum at the center down to ~0 at the coils.
So there will always be a β=1 surface somewhere in between where the magnetic pressure and the plasma pressure are equal.
This is also required for simple mechanical equilibrium.
And, that surface’s location will move with the state of the machine.

But, people here talk as if there is one beta value applied to the whole machine at a particular state.
I’m guessing that you are using it as a shorthand to describe the beta at a commonly recognized surface or something.
Please enlighten me.

I used to work on theta pinches and FRCs, where a similar question arises. There is a local beta defined as the ratio of plasma pressure to magnetic field pressure at some particular location, but also a global beta defined in terms of either a local pressure or a volume averaged pressure, but the reference magnetic field pressure in the denominator is always calculated using the external magnetic field. In a linear machine the "external" field is unambiguous, but in a 3d machine like the polywell, it is not clear what field should be used as a reference. Because of pressure balance, the global beta cannot be above 1, and a global beta of 1 corresponds to a local beta of infinity.

The way I have been understanding and using the term "beta=1" in relation to the polywell is that there is a region of finite (significant) volume where the plasma pressure is much higher than the magnetic pressure. As tombo pointed out, because of the internal multiple field, there is always a region (possibly vanishingly small) of beta = 1 in a polywell, but sometimes this region can expand to be a large fraction of the machine radius. What doesn't fit is the way Bussard talked about passing through a state of beta = 1 by varying the field strength, and the question of how sharp the transition from high to low plasma pressure is. I believe this second point is important to "wiffle ball" confinement, but it is not identical to beta = 1, and I have not seen it explicitly discussed.

Art Carlson wrote:the way Bussard talked about passing through a state of beta = 1 by varying the field strength

Is that in reference to the use of a PMT to measure when the "beta=1" condition occured? If so, I think his point was that as the b-field was lowered, the beta=1 surface would move further out until it got to the cusp throats, and then the plasma confinement would drop, and the plasma coming out the cusps would glow.

I’m not sure that a global beta for the whole device is really very meaningful.
If it is used it must be clearly defined and agreed upon.
It is sort of like asking “what is the barometric pressure of the earth’s atmosphere?” You have to ask, well, where?
Bussard may have been using the local beta at the throat of the cusps.
That would be consistent with the well”blowing out” at beta=1.

When the local beta=1 surface reaches a saddle point either at a point cusp in a coil center or at a line cusp between the coils the well “blows out”. I think, “spills over” might be a better visualization.
My image is one of rising water spilling over a pass between hills into another watershed.
(Although what that really illustrates is plasma pressure and magnetic field pressures not their ratio, beta. And of course it is really 3-D. But it is hard for me to picture more than 3 dimensions and even harder to document on a 2 dimensional medium. I think I need imagination lessons from my kids.)
I guess Solo just said almost the same thing.

I think that an important point to measure the beta in the device is the local beta at the lowest (B field pressure) type of cusp.
Control-wise I think it is very important to monitor that point (or those points) and to adaptively modulate the B field and electron/ion injection rates to avoid “blow outs”.

Does the instrumentation exist to measure the beta at the cusp throats?

Ok, I've got a related question: what does the plasma pressure mean? Can you use P=nRT/V so that n/V is the number density of ions and T is the temperature?

The ideal gas law only rigorously applies if the distributions are thermal. In genersl, pressure is a tensor. For the nonthermal case, the pressure tensor needs to be closed kinetically ( a rather nasty sort of thing).

Thanks Dr. Nebel! Time for me to learn about tensors! And, I just saw Blaise's post, which seems to help as well.

BlaisePascal wrote:So overall, plasma pressure works out to p = E/V, regardless of temperature

I guess that would be kinetic energy only. Makes sense.

Check out page 4 in this presentation. It gives the general definition, then shows what happens with a thermal distribution of particles.

It gets messy fast.

drmike wrote:Check out page 4 in this presentation. It gives the general definition, then shows what happens with a thermal distribution of particles.

It gets messy fast.

Added your link to the sidebar at IEC Fusion Tech. Gave you a H/T.

Very nice stuff. Too bad my math skills are so atrophied. I do get the drift.

Here's the quick and dirty wiki answer:

http://en.wikipedia.org/wiki/Beta_%28plasma_physics%29

The beta of a plasma, symbolized by β, is the ratio of the plasma pressure (p = n kB T) to the magnetic pressure (pmag = B²/2μ0).

I figured that I'd resurrect an ancient thread, rather than create a new one for this:

I've recently realized that I'm confused about beta, as well. Tokamaks have betas around 5-10%, but here's where I've fallen and can't get up: If the magnetic pressure is less than the plasma pressure, doesn't that mean that the plasma isn't in equilibrium, and the greater plasma pressure will cause it just to expand until it hits the vessel walls?

Since tokamaks seem to have a real live separatrix, I'm assuming that that's not what happens. So what other terms are there other than plasma pressure and magnetic pressure that keep the plasma bottled up?