## The consequences of quasi-neutrality in the cusps

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

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icarus
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This is what I learned from Haines. The sheath is spread over the whole surface of the ball. As it converges on the point cusp, the flux tubes gather together and the sheath gets thicker. I got 4*R*rho_e by calculating the flux where the sheath is thinnest. There it is perhaps as thin as rho_e, but it cuts through a circle with a radius related to R.

Ok, sounds logical, almost too simple. There is a thin, even layer on the ball that has all it's flux tubes exiting the cusps. So you are assuming a spherical ball of plasma then and have a picture like this in mind for flux tubes on the surface I take it?

http://www.mare.ee/indrek/ephi/invwb/

If so, then I agree that the total volume of a thin sheath (electron gyro-radius thick) cannot be more than

V_tot = 4*pi*(R^2)*rho_e

correct? So now we have to figure out how that layer would likely shed from the ball by flowing out the cusps. Interesting problem, but not intractable I think.
Last edited by icarus on Tue Feb 10, 2009 10:28 pm, edited 1 time in total.

Art Carlson
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### the potential perturbation extends into the plasma ball

I've been grappling with this question of the structure of the cusp plasma. If I feed the cusp at one end with cold ions and monoenergetic electrons, the sheath propagates inward and the electrons get stuck near the cusp throat. This may result in jubilation by some of you, but the cost is a strong electric field at the cusp throat, which I see as unphysical. I wondered if I could get closer to a reasonable solution by sticking with a 1-D model but making the cusp boundary semi-permiable to electrons, with a transparency of A_cusp/A_surface, but I think this model has the same problem. The large radial electric field at the cusp suggest that the potential might propagate into the plasma ball. This would be analogous to the reduction of gas pressure in a tank in front of a leak. The ions, which we otherwise assumed to have low energy, would not be slowed down completely before they move from the ball into the cusp. Bohm is back!

This potential structure would probably also broaden the electron energy distribution, which should help Bohm even more. If I/we worked on it, we might be able to patch together a 1+ dimensional model, but it would have to make a lot of assumptions and conpromises. I think the essential physics can be captured well in 2-D (a linear cusp).

This problem is starting to look a lot like Langmuir probe theory. Everyone in the business uses a 1-D model, but when you start to look closer, it's full of holes. In some ways it is easier to model a complete tokamak than to model a Langmuir probe in a tokamak.

Where does that leave us. At least I feel that these ideas strengthen my intuition that using Bohm is going to be close to right. Unfortunately - Let's be honest - not everyone here appreciates how good my intuition really is. This doesn't amount to an argument that I can knock an idiot over the head with and make him see the light. The best way to come to an agreement may be to do a 2-D PIC simulation with a simplified magnetic field model and ignoring collisions. If you feel the calling, go ahead and do a realistic 3-D simulation. Neither one will be easy, but we are bound to gain some insight from it.

MSimon
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This problem is starting to look a lot like Langmuir probe theory. Everyone in the business uses a 1-D model, but when you start to look closer, it's full of holes. In some ways it is easier to model a complete tokamak than to model a Langmuir probe in a tokamak.

Yep. Dielectrics that can get a surface charge. Conductors. "Contact" with the plasma at two points.

It is a wonder the tube guys figured out as much as they did given the tools they had to work with. Which is not to say their understanding was correct at all points. It does have the advantage of being practical. You can design tubes with it.
Engineering is the art of making what you want from what you can get at a profit.

icarus
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### Are the cusps quasi-neutral?

The more I look at this, the more I'm beginning to question the quasi-neutral sheath/cusp assumption.

I think we are in agreement that the outermost layer on our spherical plasma ball will be primarily electrons. Although the ions are not attached to electron sheath in any field-continuous manner, there are enough of them whizzing radially back and forward through the layer, as they fall/rise radially due to the electric potential well, so as to result in an almost net zero charge in this spherical sheath region. I guess this is the crux of the non-ambipolar but quasi-neutral argument.

Now when the electrons in the sheath on the surface of the ball flow towards the cusps, i.e. initially azimuthally along the magnetic field lines, the ions are not affected by those lines but by the electric potential well, which is largely spherical. Further along, as the electrons follow the cusp mag. field hyperbolic lines radially outward, then the ions that populate that region do not necessarily follow them by having orbits azimuthally to and fro through the cusps.

The result being that the cusp sheath becomes increasingly negatively charged with the increase in radius.

Solo
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Well, if we could hash out a plan for a simulation, there's a chance I could do that as my senior thesis! I've got some other possibilities (NSF internships I might be accepted to) but I'm trying to find a back up plan. If we could figure out a reasonably-sized project, I might just spend this summer coding it up and running it!

Art Carlson wrote:- Let's be honest - not everyone here appreciates how good my intuition really is.

Yeah, and not everyone here appreciates how humble you are either. Just kidding! I couldn't pass that up.

Ok, so if we are looking at A=50*R*rho_e and c_s=3e6m/s, assume density is about 1e20/m^3, what kind of current are we looking at for WB7? Say R=0.18 m, rho_e=1e-3 m, then that's 2.7e25 ions/s, or 432,00 amps. Hmm. There's no way that's right.

What do you mean by the electrons getting stuck at the cusp?

icarus
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Okay, how's this for a starting point for a cusp solution.

We take the spherical wiffleball model that Indrek has coded and extract the field lines from a circle around one of the simple point cusps and see where they go to, their shape, etc.

Take a layer of field lines about 2*rho_e thick above the surface of the sphere and begin from a circle about midway between the stagnation point on the sphere and the nearest cusp point neighbours. Follow them in azimuthally towards the cusp point and then out radially , they will be hyperbolic like, away from the sphere to some point not far outside the physical coil radius.

It'll give us a visualisation of what region of the field we are playing in, convergences, divergences. Of course, we could do this for all 3 of the different types of singularities occurring on the sphere but lets just begin with the easiest of them.

Art Carlson
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### Re: the potential perturbation extends into the plasma ball

Art Carlson wrote:This potential structure would probably also broaden the electron energy distribution, which should help Bohm even more.

Aw for chrissake, learn some physics, will ya? If mono-energetic electrons are injected into a static potential structure of any form, the distribution will be mono-energetic everywhere and always (until collisions kick in).

Still, bouncing around in there should at least broaden the angular distribution. (And of all collisional processes in a plasma, the one with the fastest timescale is the isotropisation of the electron velocities.) What consequences will that have? Might this even be enough to make it possible to derive a modified Bohm condition? Stay tuned.

imaginatium
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### Re: the potential perturbation extends into the plasma ball

Art Carlson wrote:
Art Carlson wrote:This potential structure would probably also broaden the electron energy distribution, which should help Bohm even more.

Aw for chrissake, learn some physics, will ya?

Art,
Do you usually speak to yourself, with such harsh tones?

chrismb
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### Re: the potential perturbation extends into the plasma ball

Art Carlson wrote:Aw for chrissake

Don't bring me into it!

I'm waiting for experimental results, I don't see the value in speculating on this stuff.

TallDave
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Do you usually speak to yourself, with such harsh tones?

It's usually a sign of intellectual honesty.

alexjrgreen
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TallDave wrote:
Do you usually speak to yourself, with such harsh tones?

It's usually a sign of intellectual honesty.

Conscientious practice of self-criticism...
Ars artis est celare artem.

TallDave
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Here's Bussard's argument on electrons being Maxwellianized:

If electrons live sufficiently long in the machine they could
become Maxwellianized (thermalized) and develop high
energy loss distributions. However, this has been found not
to be the case. The same arguments have been found for the
ions, as well. Detailed analyses show that Maxwellianization
of the electron population will not occur, during the lifetime
of the electrons within the system. This is because the
collisionality of the electrons varies so greatly across the
system, from edge to center. At the edge the electrons are all
at high energy where the Coulomb cross-sections are small,
while at the center they are at high cross-section but occupy
only a small volume for a short fractional time of their
transit life in the system. Without giving the details, analysis
shows that this variation is sufficient to prevent energy
spreading in the electron population before the electrons are
lost by collisions with walls and structures. Similarly, for
ions, the variation of collisionality between ions across the
machine, before these make fusion reactions, is so great that
the fusion reaction rates dominate the tendency to energy

Art Carlson
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### the Carlson sheath

icarus wrote:We could call it a "Carlson sheath" if you can pull it off.

Hey, I can do that! Ladies and gentlemen, hold on to your hats. For the first time in the free world, never before done, never before attempted, the death-defying mono-energetic electron sheath! Oh, heck, let's just call it the Carlson sheath for simplicity, OK?

The trick is I have found a mono-energetic distribution that is preserved as the electrons move through potential variations:
f(v_x, v_y, v_z) = (1/4pi)*n_0*(1-e*phi/W_0)*delta(v)*|v_z|/v^3

where n_0 and W_0 are the electron density and energy upstream, where the reference voltage is also defined. v = sqrt(v_x^2+v_y^2+v_z^2), and the E field is in the z direction.

I'll leave the proof for another day. Better yet, I'll try this business model: Anyone who wants to see the proof send me 2 Euros and I'll mail it to them.

The neat thing, aside from the self-similarity, is the linear dependence on the potential. Actually, that's not so important, just neat. I think the derivation could be generalized to other distributions as well.

Anyway, at this point we can call Bohm back in. Basically, his condition says that, as the potential starts to drop, the electron density must drop faster than the ion density. Otherwise you would get an excess of electrons and the potential would curve in the wrong direction. When the electron distribution is Maxwellian, d n_e / d phi is n_e*e / T_e. Here it is n_e / W_0. For the ions, n_i = Gamma / v = Gamma / sqrt( v_z0^2 - 2*e*phi/m_i ), so d n_i / d phi = n_i*e/m_i. To require that d (n_e-n_i) / d phi > 0 is equivalent to v_z0 > sqrt(W_0/m_i).

Thus, even though the electron energy distribution is nowhere near Maxwellian, the Bohm condition still applies, with the electron energy playing the role of the electron temperature. This is the idea behind my claim that the ions, even if they are born with little energy, will be accelerated by the electric field in the plasma until they are zipping along when the leave through the cusps, and that this means a hardy energy drain.

Tom Ligon
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Art said "This problem is starting to look a lot like Langmuir probe theory. Everyone in the business uses a 1-D model, but when you start to look closer, it's full of holes. In some ways it is easier to model a complete tokamak than to model a Langmuir probe in a tokamak."

And thanks for that, Art. I built and operated a couple of Langmuir probes, but was totally perplexed with any attempt to analyze the readings they produced. First off, most texts decline to describe what they do in a magnetic field. Secondly, mine did make an inflection on one side of the curve, never the other (I think I mentioned this to you before, and you confirmed it was a likely result).

Measuring the voltage of the plasma seemed straightforward enough, if not as data-rich. That did reveal some insights.

icarus
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Art said:

Better yet, I'll try this business model: Anyone who wants to see the proof send me 2 Euros and I'll mail it to them.

Okay, I'll take the bait. It sounds interesting enough that I'll take a look at it. I can offer three forms of payment: i) I'll proof read the proof for the discount rate of 2 euros and we call it square, ii) you can call it in lieu for services for being the namer of the name "Carlson sheath" and we call it square or iii) I'll buy you "ein stein bier" next time I'm over your way (preferably at Oktoberfest).

Seriously though, I would be really interested to look at it, don't know if you have it in Latex or some other symbolic electronic form, since I find these line text equations hard to read and sometimes open to interpretation. Is it possible we can use it to estimate the thickness of the layer, i.e., the "Carlson length",, for the mono-energetic electrons?

I'm thinking we might use that sheath of electrons (known thickness and density?) covering the surface of the quasi-neutral sphere as an improvement to the spherical wiffleball magnetostatic model. Basically, we use the charge of the sheath and the Magrid to introduce electrostatics potentials into our existing magnetostatic approximation. Step by step we maybe able to build something useful for analysis.