New FAQ - What are Cusps and what kind does a Polywell Have?

[icarus]

Suppose some poorly understood physics completely blocks off the exodus from the line cusps. The sheath in that region must still be at least rho_e thick, so we can map the flux tubes back to the point cusp, where the plasma exhausts through the normal cusp mechanism. That model would lead to the same numerical result as loss through line cusps with a thickness of rho_e.

Okay, but at the point cusps we are more likely to achieve the critical pitch angles required for mirroring just from the hyperbolic geometry of the point cusps .... so just maybe that's how it's working (taking rnebel at his word) ... the line cusps don't exhaust because god only knows why and the point cusps mirror a fat sheath because the electrons have a large field-orthogonal component of velocity at the point cusps .... I think we've arrived back at Polywell nirvana!

[Art Carlson]

Suppose some poorly understood physics completely blocks off the exodus from the line cusps. The sheath in that region must still be at least rho_e thick, so we can map the flux tubes back to the point cusp, where the plasma exhausts through the normal cusp mechanism. That model would lead to the same numerical result as loss through line cusps with a thickness of rho_e.

Okay, but at the point cusps we are more likely to achieve the critical pitch angles required for mirroring just from the hyperbolic geometry of the point cusps .... so just maybe that's how it's working (taking rnebel at his word) ... the line cusps don't exhaust because god only knows why and the point cusps mirror a fat sheath because the electrons have a large field-orthogonal component of velocity at the point cusps .... I think we've arrived back at Polywell nirvana!

Nirvana. Freedom from the bonds of reality. Yes.

Moving dully back to physics, I have trouble grasping the argument "god only knows why ". Frankly I don't do a lot better with "because the electrons have a large field-orthogonal component of velocity at the point cusps" either. Your argument seems especially tenuous since I was suggesting that some kind of mirror effect might conceivably occur beyond the high-field cusps, and you suddenly invoke this (still unspecified) mechanism at the low-field cusps.

[icarus]

Back to reality, sigh. Now did we have an expression for the cross-field drift of electrons out of the plasma and onto field lines? i.e. normal to the beta=1 surface over the bulk of the surface, the boring part away from the cusps.

I think I have an expression for the fraction of those electrons that will have velocity vectors in the loss cone (i.e lower than the critical angle). But if we don't know the electron diffusive flux out of the plasma then knowing what fraction will not get mirrored when they make it to the cusps is a tad moot.

[Art Carlson]

Back to reality, sigh. Now did we have an expression for the cross-field drift of electrons out of the plasma and onto field lines? i.e. normal to the beta=1 surface over the bulk of the surface, the boring part away from the cusps.

I think I have an expression for the fraction of those electrons that will have velocity vectors in the loss cone (i.e lower than the critical angle). But if we don't know the electron diffusive flux out of the plasma then knowing what fraction will not get mirrored when they make it to the cusps is a tad moot.

My simplistic way of looking at it is that the electrons come out of the field-free plasma and hit the field. They swing around half of a gyro-orbit and zip back the way they came from. That half a gyro-orbit is what smears the electron (and ion) density out at the edge.

I think a more sophistaced analysis also takes micro-instabilities into account. If you calculate the drift velocity of the electrons in a sheath just one rho_e thick, it turns out to be comparable to the thermal speed. This results in two-stream instabilities that would immediately broaden the sheath if it tried to get any thinner than rho_e. (Actually, there is a strong case that the lower-hybrid-drift instability is even more virulent and prevents the sheath from getting thinner than an ion gyro-radius.) The growth rate of these instabilities is proportional to some power of the density gradient, so it is not particularly useful to think in terms of a diffusivity constant. A better picture is that there is a critical gradient where they turn on and off.

All of this should make the sheath thickness one of the highest priority goals on Rick Nebel's list of diagnostics. Unfortunately, I can't think of any really simple and reliable way to measure it in the plasma ball. My choice would be to measure the plasma coming out of the cusps with either Langmuir probes or possibly thermography.

[D Tibbets]

I'm not equiped to argue on this level, but just for comic relief let me add some questions. It's a given that the Wiffle Ball effect and resultant cusp confinement is inadiquit by itself. That is why the claimed recirculation is so important. I don't know the numbers except I recall that electron transits increased from a few thousand (?) to ~100,000 with recirculation. Will that baseline confinement without recirculation fit your arguments?

Again, I'm not shure what you are refering to concerning sheeths (Debye?), but if sheeth lengths compared to Wiffle Ball size is an issue, then increasing the size of the machine (eg: from 30 cm to 300 cm diameter) would decrease this ratio, unless competing parematers keeps the ratio near a constant (?).

Dan Tibbets

[MSimon]

My choice would be to measure the plasma coming out of the cusps with either Langmuir probes or possibly thermography.

I like the idea of high speed ion gauges to check the plasma coming out of the cusps. Response times in the 1 to 10 MHz range. I have a cute design that uses COTs parts designed for fiber optic communications. Of course it could be adapted to Langmuir probes.

I think there may be something to be learned from watching the high speed twitches.

[TallDave]

I'm not equiped to argue on this level, but just for comic relief let me add some questions. It's a given that the Wiffle Ball effect and resultant cusp confinement is inadiquit by itself.

Yes, though I think it goes mirror<cusp<Wiffle-Ball<WB+recirc.

Will that baseline confinement without recirculation fit your arguments?

From that we might expect results like WB-5, which did not recirc.

Again, I'm not shure what you are refering to concerning sheeths (Debye?),

I'm a bit confused on this too. I'm not sure we even agree yet whether the electrons collapse into a Debye sheath, or the system is more dynamic than that.

This is not argument from experiment, it is argument from vague comments by an experimentalist.

I agree, it's sub-optimal. Unfortunately we have to argue with the data we have, not the data we might wish to have, or might have in the future. I'm going to trust the vague comments for the nonce. You should feel free to doubt them.

[Art Carlson]

I'm not talking about any Debye sheath here. I'm talking about the transition layer between the beta = 1 plasma and the the near vacuum outside the ball.

[icarus]

Art said:

I think a more sophisticated analysis also takes micro-instabilities into account. If you calculate the drift velocity of the electrons in a sheath just one rho_e thick, it turns out to be comparable to the thermal speed. This results in two-stream instabilities that would immediately broaden the sheath if it tried to get any thinner than rho_e. (Actually, there is a strong case that the lower-hybrid-drift instability is even more virulent and prevents the sheath from getting thinner than an ion gyro-radius.) The growth rate of these instabilities is proportional to some power of the density gradient, so it is not particularly useful to think in terms of a diffusivity constant. A better picture is that there is a critical gradient where they turn on and off.

Now this makes me think turbulent entrainment velocity. It is an effective tool for analysing fluxes across stratified layers that have impinging or resident turbulence at the interface between the layers. In fact, your earlier comments about Rayleigh-Taylor flows had already pricked up the antennas but I have yet to see how the plasma-field interface would be in bulk "effective gravitationally" (accelerationally?) unstable in that sense. One thing I've learnt about fluid dynamic turbulence is this, put aside the messy, seductive vortices, cascades and instabilities for now and figure out the physics that drives them and their implications.

I'm considering the simple case of the ideal mono-energetic, single-species electron plasma. I have a proposition that I think can help explain simply the much talked about "annealing" at the beta=1 plasma-field interface. Any spreading out of velocity distribution in this region has two effects;
i) the electrons that lose kinetic energy slow down and drop down back in towards the plasma core ... end of story, they will soon collide with higher energy electrons down in there and get pumped back up to the monoenergetic level
ii) the electrons that have gained enough extra energy are able to rise up across the interface (as unstable R-T eddies?) and get onto field lines that are headed to the cusps. Some calculable proportion of these higher energy electrons, that make it out of the plasma and onto field lines, will have velocity directions that lie in the loss cone of the magnetic mirror attributable to the point cusps.

So the implications from such a process are that the beta=1 surface effectively "caps" off the electron energy distribution, at the monoenergetic level, by expelling higher energy electrons out onto field lines. Also, the rate of transport of higher energy electrons out the cusps must be related to the proportion of electrons with velocity direction lying in the loss cone. The higher energy electrons that escape the plasma and onto field lines but do NOT have velocity direction lying on the loss cone must be mirrored back in to the plasma, in the region of the cusps.

A first order estimate of the proportion of electrons having velocity direction lying on the loss cone can be given by a simple argument that the electrons in the plasma will have velocity directions uniformly distributed in 3-spatial directions, i.e. velocity direction distribution evenly occupies the solid angle 4*pi. The loss cone half-angle, alpha_m, is given by
sin^2 alpha_m = B_min/B_max
where B_min is the field strength at the beta=1 surface, i.e.,
B_min=sqrt(2 mu P),
and B_max is the field strength at the cusp, i.e. for a current loop,
B_max = (mu I)/(2R),
I- current in the loop, R - radius of loop.

So the proportion of higher energy electrons that are "transported" across the beta=1 interface and onto field lines that can actually escape the cusps is simply
alpha_m/2*pi
.... there is a consideration of electrons with loss cones facing diametrically opposed to the ones escaping the point cusps but they are heading for the line cusps and we already agreed they are completely blocked ... but god only knows how.

Now, if we could come up with an expression of the rate at which electrons at the beta=1 surface are "thermalising" into higher energies above the monoenergetic cap we could estimate the "transport" losses ... no?

[icarus]

Okay, if we come up with an expression for the relaxation rate of the electron velocity distribution from the mono-energetic state it will help solve our electron loss "transport" problem.

And that work has some references out there so it's probably already done .... I can't pay for articles on the hermit's begging bowl salary but if any of you institutionalised folks could throw us a crumb ...

http://www.iop.org/EJ/article/0741-3335 ... 0209.ps.gz

http://adsabs.harvard.edu/abs/1966JPSJ...21..515H

Edit:

found another good one, this guy looks at the "runaway rate" for high energy electrons in crossed electric and magnetic fields, appears to very relevant ... only the intro page is free though ...
http://www.springerlink.com/content/q28 ... pdf?page=1

[Betruger]

Please check your PMs, Icarus.

[Art Carlson]

'Pon my word, icarus, you are coming along wonderfully. You have really done very well indeed. It is true that you have missed everything of importance, but you have hit upon the method.

I would have a good deal to say about your latest posts, but no time to say it. In a rare display of self-discipline, I plan to do some of the work I get paid to do first.

I'll be back.

[icarus]

icarus wrote:

A first order estimate of the proportion of electrons having velocity direction lying on the loss cone can be given by a simple argument that the electrons in the plasma will have velocity directions uniformly distributed in 3-spatial directions, i.e. velocity direction distribution evenly occupies the solid angle 4*pi.

An approach for a second order correction to this estimate becomes immediately obvious after reading the third reference in the above posting ... the conclusion being that in the case of the electrons being in an electric field orthogonal to a strong magnetic field (as is the case at the beta=1 surface) the runaway rate drops markedly.

We can think of it like this, the electron velocity vector direction distribution is not evenly distributed in 3-space as it might be in field-free gas, but due to the influence of the electric field the velocity vector direction distribution is skewed towards the normal direction to the plane of the plasma surface, and thus orthogonal to the magnetic field, and out of the loss cones!

For a mental picture, an evenly distributed vector direction field will have a perfectly spherical shape, if all the vectors are drawn originating from the same point with the same length (as they happen to be with a mono-energetic speed distribution). Now turn on the electric field and the spherical direction distribution becomes a prolate spheroidal distribution with the longer dimension aligning with the electric field. In conjugation, the loss cones radiating from the origin have an axis aligned with the magnetic field, i.e. in the plane of the narrower diameter of the prolate spheroid and orthogonal to the electric field.

I would posit that having the electric field orthogonal to the magnetic field at the plasma-field beta=1 surface is a crucial key to the enhanced electron confinement. The conformal cans for the MaGrid has done the trick, not only because of the reduced intersection of field lines out at the physical magnets, but critically down at the plasma surface it will keep the electron velocity vectors pointing away from the loss cones, resulting in an enhanced mirroring effect that is better than simple cusp confinement.

Increased electric field will increase confinement but only up until the point that the electron velocity vector distribution becomes so prolate that there are effectively no longer any electrons with velocity directions lying in the loss cones: either pointing towards the point cusp directions or the line cusp directions.

[icarus]

Art being not so charitable:

It is true that you have missed everything of importance, but you have hit upon the method.

Sometimes a blind pig finds an acorn .... I think the method has yielded something of great import that even you may have missed heretofore. See most recent post above.

[Art Carlson]

Art being not so charitable:

I'm sorry, that might have come over a bit rough if you didn't realize I was quoting Sherlock Holmes (A Case of Identity). I was recently frustrated by a discussion here where the physics was so vaguely expressed that I didn't know how to deal with it. I meant to compliment you because your ideas at least have enough physics content that they can be intelligently discussed. And I will do so as soon as I can find the time.