## The consequences of quasi-neutrality in the cusps

Discuss how polywell fusion works; share theoretical questions and answers.

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Art Carlson
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### The consequences of quasi-neutrality in the cusps

In I presented a calculation (again), showing that the plasma in the cusp (not only in the main plasma ball) must be quasi-neutral. I seem to have most everybody on board on that point. Now I'd like to examine the consequences of that fact.

Radially outward from the cusps, the magnetic field lines go more or less straight to the walls. The field will drop to some extent, so the mirror effect will tend to push the plasma outward, but we can ignore that for now. More important is the potential drop from the cusps to the wall. This drop should be large enough to turn back almost all of the electrons. For the ions, it's a downhill ride.

Qualitatively, if the ions have a small speed (fluid or thermal) in the cusp, the ones near the outside will drop off into the deep, leading to a smaller density on the outside and consequently an electric field that tends to pull additional ions after them.

Physically and mathematically, the situation is very similar to the Debye sheath. Bohm showed that the only stable solution is when the ions enter the region with the sound speed ( c_s = sqrt(kT_e/m_i) ). He assumed Maxwellian electrons and cold ions, two assumptions we may need to examine in more detail before we are through.

So my claim is that the ions, which have a density in the cusp only a small factor lower than the density in the central ball, will also be moving at a speed near c_s. They will smash into the wall with an energy on the order of the magrid potential. If this is all correct, the energy loss rate will be fatal.

P.S. Now that it sounds like Rick Nebel is not interested in grappling with the details of a polywell theory, this thread might be pretty lonely. Tom Ligon seems to still have some arrows in his quiver, so maybe I still have a chance of learning something here.

MSimon
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Art Carlson
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MSimon
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imaginatium
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### Re: The consequences of quasi-neutrality in the cusps

Solo
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I think we agree with you on quasineutrality in the cusps. As to the applicability of the Bohm theory, the ions should be cold since the average ion temperature should be lower than the central potential by a handful of times. Thus the ions flowing through the point of maximum potential will be the energetic tail of the Maxwellian, shifted down in velocity by the potential energy of the well. They probably won't themselves be Maxwellian, for what it's worth. The electron distribution will be pretty skewed in phase space, with the parallel velocity (relative to B-field) much greater than the perpendicular, and, if the lifetimes are low as Dr. Nebel suggests, not really Maxwellian.

At the actual cusp throat, we know there will be a saddle point in the potential distribution. Beyond this point, where the potential decreases, the density of ions is going to be lower, and the phase space distribution is going to be lopsided, since few ionswill have velocity inward through the cusp. The lopsidedness will grow until all the ions at a given point are moving outwards. At that point, the average ion velocity times the density integrated over the area in each cusp should give the loss rate of ions in # particles/second. That seems like a reasonable calculation to make. We can make a pretty good guess at the area already, and the density.

Here's another handy drawing:

I have some misgivings about it though, and I will be disappointed if there is no debate.

One other way to treat this subject is as Dolan does, is to say that the probability of an ion being lost is some function that is zero up til approximately the point where the ions kinetic energy is equal to the well depth; at that point the probablity goes quickly to one as energy increases. It could be a simple step function. Under this condition, the distribution is more or less truncated, and the number of ions lost is proportional to the number that would ordinarily occupy that part of the distribution, dependent on the collision timescale. I've got the equations Dolan comes up with if you'd like me to type them up. This treatment could be applied to electrons too, which would tell us about the electron recycling rate and energy loss rate.

TallDave
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So what do you predict this would look like in WB-7?

Art Carlson
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icarus
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Solo
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Last edited by Solo on Sun Feb 08, 2009 8:03 pm, edited 2 times in total.

Art Carlson
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So you want me to sing for my supper. I am willing to stick with cold ions, even though I believe they won't be cold. This will give a lower limit to the streaming velocity and the energy loss, and it also makes the theory easier.

The Bohm criterion says the ions will have a velocity equal to or greater than sqrt(kT_e/m_i), if the electrons have a Maxwellian distribution with temperature T_e. What we need to do is to generalize the Bohm criterion to a non-Maxwellian electron energy distribution. The general distribution would be an interesting problem, but also a bit of serious plasma theory that I probably don't have time for. I am working on a short-cut, but I don't know if it will work. What I can probably handle is a two-temperature electron distribution. Would that be an acceptable place to start?

Solo
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Well, ok. There are likely to be both cold electrons oscillating in the cusp as well as fast electrons fresh from the electron injector.

I suppose this may show my ignorance, but I'm not sure what the electron speed has to do with the ion distribution. That's what I was getting at. I'll see about doing some more homework.

rnebel
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Here's what we know and what we don't know:

1. We don't have the spatial resolution of the density to see if the cusps are quasi-neutral on the WB-7
2. In one-D simulations the plasma edge (which corresponds to the cusp regions) is not quasi-neutral. Therefore, if the cusps are quasi-neutral it must be a multidimensional effect.
3. Energy confinement on the WB-7 exceeds the classical predictions (wiffleball based on the electron gyro-radius) by a large factor.

Our conclusion is that both the wiffleball and the cusp recycle are working at a reasonable level.

icarus
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