Shape of Wiffleball

[jmc]

We have now established that if the wiffleball was spherical there would be no line cusps, however the magnitude of the magnetic field outside, was by no mean homogeneous, and to first approximation (if only to make life easier) the plasma inside is an isobaric gas. Therefore it would be expected to be pushed in where the magnetic field is stronger than average and expand out where the magnetic field is weaker than average.

I haven't read much of the thread on the Vlasov solver, but I'm guessing it is pretty hard to progamme right and be sure its working properly once you've finished.

Perhaps to next approximation it would be of interest to model an outward point (or small spherical) explosion of plasma from the centre of the device out until the shock wave is successfully resisted by the magnetic field. This might be accomplished in a real experiment by firing lasers at a pellet inside a Polywell field.

I think the simplest way to do this is to have a small superconducting bag with a certain number of energetic particles inside (not necessary to model their tragectories), make the bag say, 1/10 of the radius of the polywell in the centre of the polywell.

Set up the vacuum field.

For each surface element, if the vacuum field pressure (taken to be B^2/(2{mu_0}), where be is the total magnitude of the field regardless of the direction it is pointing in) is greater than the pressure in the bag move the surface element of the bag a step inward, if the vacuum field is less, move it a step outward. Since we are taking the density and temperature inside to be homogeneous, the mass density and sound speed are also homogeneous. Thus for each timestep set the rate at which each volume element contracts or expands proportional to the magnitude of the pressure difference.

This will give you the new surface.

Indrek, is it possible to generate image currents for an irregular non-spherical surface? If it is possible, then the next thing to do would be to generate the image currents, regenerate the field taking the new image currents into account.

Calculate the new volume of the bag.

Use the adiabatic relation: P(n+1)=P(n)*(V(n)/V(n+1))^(gamma) to calculate the new pressure inside the wiffleball. And repeat the entire procedure iteratively until the answer converges to a steady state where at every point the pressure inside the wiffleball is equal to the pressure of the magnetic field immediately next to it.

[kcdodd]

Yeah, vlasov was too ambitious i think. but i did learn a lot, which is the primary goal i think. But anyway, i posted this update in the other thread as well for the superconducting bag solver: http://www.andromedaspace.com/files/polyg0001.png.

[Roger]

the superconducting bag solver: http://www.andromedaspace.com/files/polyg0001.png.

You just poured green plaster into the inside of a polywell, that is so excellent.

Wheres that other thread, with this in it.

[kcdodd]

viewtopic.php?t=650

[jmc]

Are the edges of the superconducting bag line cusps or is there a tangetial insulating component there aswell?

[kcdodd]

The little black lines represent the direction of the magnetic field. They are all pointed straight out. The tangential part seems to disappear as it expands. And it's all (mostly) green because that's the magnitude of the magnetic field.

[TallDave]

Says the lazy empiricist: can't we just measure it?

[kcdodd]

go ahead. lol. but i don't have any fusion reactors sitting around my house.

[KeithChard]

I think that the plasma is unlikely to be much like an isobaric gas because the electrons are constrained by powerful magnetic fields caused by reflections of the magrid coils in the mirror. When a more realistic shape for the wiffleball has been found there will still be a strong but fuzzy unfocused reflection of the coils with a corresponding magnetic field. If a new shape could be estimated it would lead to a new reflected image of the coils and an iteration could be continued until the shape of the wiffleball hopefully converges to a boundary that is identical on successive iterations. I have not yet thought out the details but there would need to be a rational plan for selecting the next shape and a means of building the blurred image of the coils that results from the image being formed by reflections arising from the distorting mirror. The blurred image would, of course, be the sum of weak images caused by the reflecton arising from various parts of the mirror and the magnetic field would be the sum of the corresponding weak fields.

BTW I do not think that the ions will have much effect on the shape because I assume that they will take up a reasonably uniform circumferential distribution after crashing into the centre of the well and back out to the boundary again a few dozen times.

[KeithChard]

I have omitted the most important steps which are, of course, to introduce electrons to the wiffleball, circulating around the field lines, and to calculate the effect of these currents on the fields inside and outwith the wiffleball. This provides the mechanism for finding the new wiffleball boundary and is the start of the iteration process.

[KeithChard]

I am beginning to think that a large portion of the electrons in the well circulate in the image coils (probably a fuzzy version). It could be that there are an infinite number of infinitessimally spaced image coils along radial lines as well as a similar circumferential distribution caused by a non spherical image surface. Electrons circulating in the field lines between the wiffleball surface and the outermost image coils may not produce any net current because there is little, if any, reason to circulate in the same direction. It could be that by integrating the fields of the radial distribution of images that we arrive at a non spherical wiffleball.

It seems to me that the integral of the patterns of weak images that I have been attempting to describe would be a more likely representation of the reality than a simple mirror solution. The problem that remains is to find the correct distribution of position and strength of these image coils. Can anyone come up with a solution to that?

IIRC Tom Ligon said that there were probably some very large currents circulating in the wiffleball, and this whole approach of images is confirming that and is giving insight into the structure of the currents.

[Solo]

To continue with the questions about the shape of the plasma, what I'd like to do (or see someone do, anyrate) is figure out what the flux surfaces look like that bound the plasma in a polywell. Despite the fact that the coil cans are supposed to be conformal, there must be some flux surfaces which intersect the coils, and from which electrons are lost once they diffuse far enough cross-field. This will form the outside boundary of the plasma.

Based on Dolan's paper on MEIC machines, I conclude that the inside boundary of the sheath will follow the flux surface that goes through a circle about 1-2 times the electron Larmour radii at the cusps. (The rationale for this: inside the plasma sheath is the non-adiabatic central plasma. The width of the channel of non-adiabatic plasma going through the cusp determines the ion loss rate, which needs to be small. If the channel is wide, the plasma will hemorrage until the sheath expands and the channel shrinks.)

From this point, all we have to do to figure out the shape of the whiffle-ball is estimate the diamagnetic re-shaping of the vacuum field and we've got a pretty good picture of the whiffleball.

I'm particularly interested in seeing what kind of drop there will be in the saddle point of electrostatic potential in the cusps due to the unneutralized electrons in the sheath. I'm afraid that there will be practically no confinement without doing something to intentionally limit the outer flux surface of sheath, as Dolan's Jupiter-1 and 2M and the ATOLL toroidal cusp did.

[KeithChard]

Despite the fact that the coil cans are supposed to be conformal, there must be some flux surfaces which intersect the coils, and from which electrons are lost once they diffuse far enough cross-field.

The coil cans will be quite thick in the real thing because they have to contain the cooling system, particularly on the faces visible to the alphas. What has to be done is to calculate the field lines for the coil cross section used, which is not necessarily circular, and then make the cross section of the cooling system conformal to these bfield lines. Unfortunately some of the strongest field regions are going to be lost inside the cooling system.

[MSimon]

Despite the fact that the coil cans are supposed to be conformal, there must be some flux surfaces which intersect the coils, and from which electrons are lost once they diffuse far enough cross-field.

The coil cans will be quite thick in the real thing because they have to contain the cooling system, particularly on the faces visible to the alphas. What has to be done is to calculate the field lines for the coil cross section used, which is not necessarily circular, and then make the cross section of the cooling system conformal to these bfield lines. Unfortunately some of the strongest field regions are going to be lost inside the cooling system.

This has its good and bad points. In the very center of the coils there is flux canceling. At the edges the fields are irregular. And a little farther away they are nearly circular.

Look up some of Indrek's simulations.

[KeithChard]

Between the adjacent coils the field lines are straight, radial and densely packed. The fields are stronger than anywhere else, see Indrek's linear plots as opposed to logarithmic.

I envisage a radially elongated coil cross section with almost flat faces where the coils are adjacent. So far nobody has done any field calculations for such shapes, nor have coil spacings that leave room for cooling been investigated. Until such calculations are done we can only guess at the likely configuration. However, with careful design, I see no reason why a geometry that is conformal to the field lines cannot be achieved.