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

[Art Carlson]

Would you agree that at some radius that this cusp line will terminate on the beta=1 surface of the electron plasma? (And probably as a normal to that surface.)

Yes, certainly in this very symmetrical case. The field strength will also decrease as you get closer to the plasma and vanish at the termination point.

[icarus]

Art:

Yes, certainly in this very symmetrical case. The field strength will also decrease as you get closer to the plasma and vanish at the termination point.

Hmmm, I agree that the field strength decreases as we go radially inwards towars the plasma along the cusp line. However, at the termination point on the beta=1 surface wouldn't it be given by the relationship with the kinetic pressure, .... unless the pressure were to vanish there identically.

[Art Carlson]

Art:

Yes, certainly in this very symmetrical case. The field strength will also decrease as you get closer to the plasma and vanish at the termination point.

Hmmm, I agree that the field strength decreases as we go radially inwards towars the plasma along the cusp line. However, at the termination point on the beta=1 surface wouldn't it be given by the relationship with the kinetic pressure, .... unless the pressure were to vanish there identically.

Oops. I had the wrong model in mind. You're at least half right. Maybe more. I'll post a correction when I have a few minutes.

[Art Carlson]

Even though I have been harping about how the plasma must be spikey, I was thinking here of the earlier and wrong-er model of a spherical plasma. With a beta=1 plasma, the central field line of a point cusp will terminate on a spike with the same field strength as everywhere else on the surface of the plasma. At least on scales larger than any relevant gyro-radii.

[icarus]

Art said:

With a beta=1 plasma, the central field line of a point cusp will terminate on a spike with the same field strength as everywhere else on the surface of the plasma. At least on scales larger than any relevant gyro-radii.

Okay, good, we're on the same page, I should have made it clearer I was referring to the quasi-spherical spiky beta=1 surface than the spherical model surface.

Now in fluid dynamics that line would be a stagnation streamline and the streamfunction, that is constant along that line, can be a reference stream function value for the entire flow-field, being as the minimum it is. I suppose for a magnetic field the function that is constant along that cusp line is some scalar related to the vector potential function, (magnitude?)

The point being that since it terminates on the plasma beta=1 surface, then that line will have the magnetic field strength given by the kinetic pressure at the termination point. So now we can reference the field, say by setting the vector potential to zero on the beta=1 plasma surface, and then take that as a starting point for an analysis ....

E.g. a value for the gyro-radius about that cusp line is well-defined at the beta=1 surface (and can be shown to be related to the inverse of the electron-density^1/2) and decreases as we go radially outwards from there, until we reach the plane of the coil (the gyro-radius magnetic throat) where it is a minimum.

Crank those kinds of relations through rnebel's contention that the loss fraction goes like (r_g/R)^2 and one can come up with a scaling law that has some interesting implications .... like that larger drive voltages (or bigger wiffle-balls) reduce Q .... something like Q ~ B^6 R^5 /V .... of course it depends on the assumptions.

[MSimon]

I would expect larger drive voltages to have lower Q due to the slope of the reaction rate drive voltage curve. i.e. drive power vs fusion power without respect to reaction rates. The ultimate Q at the .1 barn resonance peak of pB11 is about 22. At the broad 1.2 barn peak it is about 8 or 9. Lower net out but a more compact reactor.

The fact that you get that from a different way of attacking the problem is interesting.

[icarus]

MSimon:

The ultimate Q at the .1 barn resonance peak of pB11 is about 22. At the broad 1.2 barn peak it is about 8 or 9. Lower net out but a more compact reactor.

Yes, the further implication being that since the drive voltage is set by the requirement for fusion energies then B and R have minimum values to achieve Q>1. Then there's the consideration that the drive voltage to achieve fusion may not be directly proportional to the energies of fusion but some fraction of it that could also be dependent on B and R also.

[TallDave]

So by this am I to assume you think added sphericity is bad?

Yes, he thinks it will be unstable for some important MHD modes, but some of us question whether the shape is important as long as long the forces are increasing in the right directions.

I think you didn't follow my math. Using Ampere's Law I showed that the forces increase in the right direction if and only if the surface is concave.

I follow the math (more or less), I'm just not convinced it applies given the way the field deforms.

icarus put it thus:

Art, this is not exactly so but only mostly, it's a rule of thumb of vectors you are using, in actuality MHD stability in general has to do with gradients and curvatures of the surfaces of scalar magnetic field strength, not just curvature of vector field lines. We covered it already a bit back here,

Maybe Rick can better elucidate on this, but it seems intuitively obvious the fields must increase going toward the coils. My understanding is the shape changes due to the motion of the electrons, but the gradients become much steeper; i.e., a convex field squeezed into a concave shape by electrons isn't the same as a concave field and may be more stable for those MHD modes. I agree they would have to prove this.

Mostly I'm just assuming the Valencia diagram is accurate, though. It seems to indicate a spherical plasma.

In fact, it seems to require it, in order to get the Wiffle-ball effect of squeezing the cusps closed, which we know is real now. I suppose they could just be getting less convex.

[Art Carlson]

I follow the math (more or less), I'm just not convinced it applies given the way the field deforms.

You don't have a lot of room to wiggle. Resistive ballooning modes are unstable whenever the gradient of the plasma pressure is opposed to the gradient of the field modulus. ( http://en.wikipedia.org/wiki/Resistive_ ... rowth_rate ) The derivation is straightforward and is based on consideration of two small, adjacent flux tubes that change places. The derivation does not depend on how these gradients are produced. All you need is the equation of state of the plasma and of the magnetic field. No rocket science.

Mostly I'm just assuming the Valencia diagram is accurate, though. It seems to indicate a spherical plasma.

I wonder how you choose which statement of Bussard to assume is true when he contradicts himself. In fact, using your intuitive argument that the field pushes harder against the plasma when the plasma is closer to the coils, it would be natural to expect the plasma tries to squeeze a bit into the holes in the centers of the coils and the cracks between them.

In fact, it seems to require it, in order to get the Wiffle-ball effect of squeezing the cusps closed, which we know is real now. I suppose they could just be getting less convex.

We don't know any darn (Hey, this is some cool software. I didn't actually type "darn".) such thing. Bussard might have claimed this, but even Nebel does not take all of Bussard's conclusions at face value. And Nebel has made some hints that the confinement is anomolously good, but apparently doesn't have the diagnostics to conclude in detail why that is so. - But aside from my rant, I don't why you see such a connection between the "wiffle-ball effect" (if we really know what that means) and the surface curvature.

[TallDave]

You don't have a lot of room to wiggle.

But Polywell geometry might mean I can wriggle at least most of the way out of this one. If the fields are being pushed back against each other and have a fair distance between coil and plasma, they don't need much concavity on the interior to enclose an area at least roughly spherical. They might still have good overall curvature.

Mostly I'm just assuming the Valencia diagram is accurate, though. It seems to indicate a spherical plasma.

I wonder how you choose which statement of Bussard to assume is true when he contradicts himself.

Usually the appearance of contradiction has turned out to mean we didn't understand him or the physics.

In fact, it seems to require it, in order to get the Wiffle-ball effect of squeezing the cusps closed, which we know is real now. I suppose they could just be getting less convex.

We don't know any darn (Hey, this is some cool software. I didn't actually type "darn".) such thing. Bussard might have claimed this, but even Nebel does not take all of Bussard's conclusions at face value. And Nebel has made some hints that the confinement is anomolously good, but apparently doesn't have the diagnostics to conclude in detail why that is so.

I think we DO know such a darn thing. Rick has stated confinement was much better than cusp and that WB mode was "easy to see," and also stated the whole Polywell concept doesn't work without it. Not knowing the precise details of the wiffle-ball effect is not the same as not knowing it exists. I don't think we'd be building WB-8 if we didn't know the WB effect was real.

But aside from my rant, I don't why you see such a connection between the "wiffle-ball effect" (if we really know what that means) and the surface curvature.

Because the WB effect is supposedly caused by the magnetic fields being squeezed together by the plasma. This same deformation would presumably tend to make the area enclosed by the fields more spherical.

[TallDave]

Bussard certainly seemed to think the end result was spherical, fwiw, or at least found it a useful shorthand description:

Initially, when the electron density is small, internal B field
trapping is by simple “mirror reflection“ and interior
electron lifetimes are increased by a factor Gmr, proportional
linearly to the maximum value of the cusp axial B field.
This trapping factor is generally found to be in the range of
10-60 for most practical configurations. However, if the
magnetic field can be “inflated“ by increasing the electron
density (by further injection current), then the thus-inflated
magnetic “bubble“ will trap electrons by “cusp confinement“
in which the cusp axis flow area is set by the electron gyro
radius in the maximum central axis B field. Thus, cusp
confinement scales as B2. The degree of inflation is
measured by the electron “beta“ which is the ratio of the
electron kinetic energy density to the local magnetic energy
density, thus beta = 8(pi)nE/B2. Figure 16 shows two
means of reaching WB beta = one conditions
Figure 16. Two different ways of achieving wiffleball
The highest value that can be reached by electron density is
when this ratio equals unity; further density increases simply
“blow out“ the escape hole in each cusp. And, low values of
this parameter prevent the attainment of cusp confinement,
leaving only Gmr, mirror trapping. When beta = unity is
achieved, it is possible to greatly increase trapped electron
density by modest increase in B field strength, for given
current drive. At this condition, the electrons inside the
quasi-sphere “see“ small exit holes on the B cusp axes,
whose size is 1.5-2 times their gyro radius at that energy and
field strength. Thus they will bounce back and forth within
the sphere, until such a —hole“ is encountered on some
bounce. This is like a ball bearing bouncing around within a
perforated spherical shell, similar to the toy called the
“Wiffle Ball“. Thus, this has been called Wiffle Ball (WB)
confinement, with a trapping factor Gwb (ratio of electron
lifetime with trapping to that with no trapping).
Analyses show that this factor can readily reach values of
many tens of thousands,

[Art Carlson]

Bussard certainly seemed to think the end result was spherical, fwiw, or at least found it a useful shorthand description:

He also seemed to think it had the curvature needed for MHD stability. You may think he's such a genius that he understood some deep and complicated truths that he was unfortunately unable to communicate to us because we are too stupid. I think this is simple physics and he didn't understand it. As an exercise try drawing some field lines on the surface of his cartoon wiffle ball. Some of those holes should be cusps with all the field lines pointing away and some should be cusps with all the field lines pointing in. That is topologically impossible without having some field nulls on the surface.

[MSimon]

Bussard certainly seemed to think the end result was spherical, fwiw, or at least found it a useful shorthand description:

He also seemed to think it had the curvature needed for MHD stability. You may think he's such a genius that he understood some deep and complicated truths that he was unfortunately unable to communicate to us because we are too stupid. I think this is simple physics and he didn't understand it. As an exercise try drawing some field lines on the surface of his cartoon wiffle ball. Some of those holes should be cusps with all the field lines pointing away and some should be cusps with all the field lines pointing in. That is topologically impossible without having some field nulls on the surface.

Art,

Nicholas Krall and Bussard were friends. In fact they authored a paper together.

Is it possible that you are mistaken?

Maybe this isn't simple physics. Maybe it is complicated physics.

OTOH I'm in no position to judge. But I have talked with other physicists who seemed to think it might work.

I am in a position to judge Dr. B's engineering and I found it to be balanced and nuanced and well integrated.

[KitemanSA]

...As an exercise try drawing some field lines on the surface of his cartoon wiffle ball. Some of those holes should be cusps with all the field lines pointing away and some should be cusps with all the field lines pointing in. That is topologically impossible without having some field nulls on the surface.

Here we go with foot in mouth disease again, but I thought those "field nulls on the surface" were called funny cusps.

[TallDave]

Art:

Heh, well, I don't know if Bussard was some kind of hypergenius, but he was clearly aware of the problem.

However, the low energy electrons are heated by fast
collisions with incoming fast injected electrons. The
Coulomb energy exchange time for this process is also about
1 usec. Thus the device will reach beta = one conditions
when the mean electron energy is about 2.5 keV, in ca. 20
usec. Beyond this point excess electron density will be
driven out beyond the beta = one limit; the field will have
expanded as far as it can within MHD stability limits.