All here
http://www.mare.ee/indrek/ephi/icarus1/
Possible wiffle-ball analytical solution
Well finally got back to this and put the math into Indrek's code to locate the image system correctly so that the wiffle ball surface is perfectly spherical (the ideal case). Have run it at spherical wiffle ball radii of 0.2m, 0.15m, 0.1m and 0.05m. If the interior were a superconducting, superfluid it would look like this (we know it isn't but it's a starting point) Consider a spherical cow! Let the perturbation analysis begin. These are valid magneto-static solutions for an interior "inverted" field.
All here
http://www.mare.ee/indrek/ephi/icarus1/
All here
http://www.mare.ee/indrek/ephi/icarus1/
Nice Work!
Now, that we are not flying so blindly, I have a question.
Look at the 5 O'clock (for example) radial line connecting the outer coil x-sec and the inner coil x-sec.
Now, look where that where the line crosses the surface (spherical) where the field from the outer coil pushes against the field from the inner coil.
That surface looks concave toward the plasma.
This is easiest to see in the bottom plot but shows up in all of them.
How does this square with drmike's reference to the Krall & Trivelpiece textbook here:
See the bottom 4 lines on this page:
http://www.eskimo.com/~eresrch/Fusion/K ... Pic002.JPG
Picture on this page:
http://www.eskimo.com/~eresrch/Fusion/K ... Pic003.JPG
drmike's index on this page:
http://www.eskimo.com/~eresrch/Fusion/Krall_5.12/
This is the section where he argues that the plasma must be convex toward the plasma for stability.
Do these results mean that the polywell is unstable at those points?
Or, is that shape an artifact of using a sphere for the field reflecting surface?
Now, that we are not flying so blindly, I have a question.
Look at the 5 O'clock (for example) radial line connecting the outer coil x-sec and the inner coil x-sec.
Now, look where that where the line crosses the surface (spherical) where the field from the outer coil pushes against the field from the inner coil.
That surface looks concave toward the plasma.
This is easiest to see in the bottom plot but shows up in all of them.
How does this square with drmike's reference to the Krall & Trivelpiece textbook here:
See the bottom 4 lines on this page:
http://www.eskimo.com/~eresrch/Fusion/K ... Pic002.JPG
Picture on this page:
http://www.eskimo.com/~eresrch/Fusion/K ... Pic003.JPG
drmike's index on this page:
http://www.eskimo.com/~eresrch/Fusion/Krall_5.12/
This is the section where he argues that the plasma must be convex toward the plasma for stability.
Do these results mean that the polywell is unstable at those points?
Or, is that shape an artifact of using a sphere for the field reflecting surface?
-Tom Boydston-
"If we knew what we were doing, it wouldn’t be called research, would it?" ~Albert Einstein
"If we knew what we were doing, it wouldn’t be called research, would it?" ~Albert Einstein
Unless there's some oversight here, this concavity would have to be compensated for by recirculation or some other mechanism... you can't be convex at all points. It would have to be a perfect sphere or some mix of convex and concave surfaces.tombo wrote:Nice Work!
That surface looks concave toward the plasma.
Do these results mean that the polywell is unstable at those points?
The inner coils are not real.
They are the virtual coils that would be required to create spherical surface. (I think)
This is a standard E&M technique.
Elsewhere there are images posted that show fields always convex toward the plasma.
This is one of the attractions of the polywell design.
The opposite curvature required to get a quasi-spherical total result is concentrated at the cusps.
That is why the cusps are where most of the problems and arguments take place.
I just hope that we don't have the same problem at the peaks of the field arches as is shown in this picture.
I think that this is one of the failure modes at high plasma pressures.
I just hope that the pressures required are much higher than those causing the cusp failures.
They are the virtual coils that would be required to create spherical surface. (I think)
This is a standard E&M technique.
Elsewhere there are images posted that show fields always convex toward the plasma.
This is one of the attractions of the polywell design.
The opposite curvature required to get a quasi-spherical total result is concentrated at the cusps.
That is why the cusps are where most of the problems and arguments take place.
I just hope that we don't have the same problem at the peaks of the field arches as is shown in this picture.
I think that this is one of the failure modes at high plasma pressures.
I just hope that the pressures required are much higher than those causing the cusp failures.
-Tom Boydston-
"If we knew what we were doing, it wouldn’t be called research, would it?" ~Albert Einstein
"If we knew what we were doing, it wouldn’t be called research, would it?" ~Albert Einstein
tombo: if you read your pictured text carefully you'll see that the equations say that the plasma is unstable in the direction of decreasing magnetic field strength. The author then goes on to erroneously extend that the general rule of thumb is that the plasma stability is related to the curvature of field lines, all based on his single cusp example.
In the images I've posted above, the colour map depicts magnetic field strength, it increases away from the wiffle ball surface even when the field lines are curved towards it. Watch out for sloppy generalisations.
In the images I've posted above, the colour map depicts magnetic field strength, it increases away from the wiffle ball surface even when the field lines are curved towards it. Watch out for sloppy generalisations.
Last edited by icarus on Fri Jul 18, 2008 9:11 pm, edited 1 time in total.
Some more replies:
drmike: I'm trying to do a write-up of the math, I do latex but have real job and mortgage also. The important thing was to scratch out the bare bones enough to get it into a code. The concept is sound as you'll see in the Jerry Kevorkian reference I gave above and can be derived from that very general solution technique for any distribution of sources, sinks, dipoles, etc.
kcdodd: yes, I think this is exactly the solution technique that Carlson seems to be alluding towards. Also, I think where you have streamlines on your simulation that end on the sphere but are not stagnation streamlines (i.e. cusp lines) then you probably have the image system not correctly located so you won't be generating the closed spherical surface exactly. That electron in the magnetic well simulation is encouraging, wonder how we can verify that?
JohnP: you obviously haven't read the earlier pieces of the thread, maybe you'd like to edit out your questions to save clutter when you figure out what we're discussing.
drmike: I'm trying to do a write-up of the math, I do latex but have real job and mortgage also. The important thing was to scratch out the bare bones enough to get it into a code. The concept is sound as you'll see in the Jerry Kevorkian reference I gave above and can be derived from that very general solution technique for any distribution of sources, sinks, dipoles, etc.
kcdodd: yes, I think this is exactly the solution technique that Carlson seems to be alluding towards. Also, I think where you have streamlines on your simulation that end on the sphere but are not stagnation streamlines (i.e. cusp lines) then you probably have the image system not correctly located so you won't be generating the closed spherical surface exactly. That electron in the magnetic well simulation is encouraging, wonder how we can verify that?
JohnP: you obviously haven't read the earlier pieces of the thread, maybe you'd like to edit out your questions to save clutter when you figure out what we're discussing.
OK, I will try to re-read it more carefully.
I was under the impression that the gradient and the curvature were intimately related.
The color gradient is very ambiguous on my monitor.
The local color noise (i.e. the mottling along the field lines) is much greater than the gradient from the coil toward the center.
The color gradient is a little more clear in the views of the smallest well where the adverse curvature is much less.
In the largest well it is quite invisible.
Perhaps that one is showing the well near failure.
So, you think that this picture shows a stable well?
Does you model also predict that the virtual coil currents are real plasma currents?
(perhaps actually spread out more than the idealized concentrated currents?)
How much of the detail inside of the sphere do you think represents real fields?
I was under the impression that the gradient and the curvature were intimately related.
The color gradient is very ambiguous on my monitor.
The local color noise (i.e. the mottling along the field lines) is much greater than the gradient from the coil toward the center.
The color gradient is a little more clear in the views of the smallest well where the adverse curvature is much less.
In the largest well it is quite invisible.
Perhaps that one is showing the well near failure.
So, you think that this picture shows a stable well?
Does you model also predict that the virtual coil currents are real plasma currents?
(perhaps actually spread out more than the idealized concentrated currents?)
How much of the detail inside of the sphere do you think represents real fields?
-Tom Boydston-
"If we knew what we were doing, it wouldn’t be called research, would it?" ~Albert Einstein
"If we knew what we were doing, it wouldn’t be called research, would it?" ~Albert Einstein
tombo:
In 2-d yes, in the 3D most likely but exactly how, well that is one of the great open questions for mathematical physics I think. How do you relate a 3-d curl to a twisted, curved streamsurface?I was under the impression that the gradient and the curvature were intimately related.
Pertaining to the field strength gradient criteria, just maybe. Thinking out loud, I think when the wiffle-ball will fail will be when the surface starts to pinch up into plasma droplets at the cusps and leak out there, a full blown failure it will be blasting out the cusps. Google "G.I. Taylor's four-roller experiment" for analogous 2-d cylindrical flows trapped in cusps and instability regimes for some more ideas, particularly relevant if there was identified a physical source for an analogous effect to surface tension between the interior and exterior. Then it would become a balance between magnetic pressure, plasma pressure, volume of ball and interfacial "tension".So, you think that this picture shows a stable well?
To some extent I think if it is stable and somewhere near these stationary solutions the "plasma" interior solution to the first-order will match continuity and field direction across the wiffle-ball surface. It is not superconducting or inviscid but those approximations are a good place to begin, in my opinion. Down in the very interior who knows, particularly with the electrostatics coming heavily into it.Does you model also predict that the virtual coil currents are real plasma currents?
(perhaps actually spread out more than the idealized concentrated currents?)
How much of the detail inside of the sphere do you think represents real fields?
I don't think the technique of images really gives much information on actual structure. In fact images may not even be placed inside the object. It just happened this time they are inside the sphere. They're just a way to get the value you want at a boundary, as long as they aren't placed in the domain you are interested in finding the field. So by that you can't really depend on anything you see from them inside the sphere.
I will have to look at your equations to see where I went wronge.
I will have to look at your equations to see where I went wronge.
Carter
I think you have to tune the spacing between the inner coils so that the field lines don't terminate inside. I tried this manually here:kcdodd wrote: I decided to attempt this as well just guess the value should solve for the sphere radius. ...
The blue line starts at the top inside the inner sphere but slightly offset from the center.
The red line starts from the bottom but outside and again slightly offset from the center.
Getting the right spacing between coils took some trial and error. But it seems it's possible to get a solution where internal and external field lines don't mix.
- Indrek
EDIT: I think I figured out the exact formula. No need to adjust the spacing - instead you have to compensate with the current in the inner coils - as they are smaller. I'll let icarus brief you as this inversion was his idea.
Last edited by Indrek on Sat Jul 19, 2008 7:07 am, edited 2 times in total.
I've been mulling over icarus's and Indrek's images for some time. It seems that the lowest containment field strength points are at the point and line cusps where they intersect with the "imaginary" superconducting sphere. If I understand it aright, these points would represent the places where species would preferentially gather (if they are of low enough energy).
Since polywell works on the understanding that an electrostatic potential well is needed to confine positive ions, it might be that these 6 + 8 points are the lowest points of the electric well, providing us with not one but 14 centres to the polywell. Secondly, the point cusps are probably stellate shapes, and the line cusp centres are probably flattened banana shapes in 3D. It would be very good to see the Meridian45 sections (at present we have seen the MeridianZero sections).
I realise that the sim does not say anything about the space on the inside of the superconducting sphere. So these colour gradients only imply mild confinement (in the absence of knowing what field strength the colour represents). It might be that the centre of the polywell is indeed a great hole, and the stuff I note here is just some fringing round the edges.
Kudos to icarus and Indrek for their work.
Regards,
Tony Barry
Since polywell works on the understanding that an electrostatic potential well is needed to confine positive ions, it might be that these 6 + 8 points are the lowest points of the electric well, providing us with not one but 14 centres to the polywell. Secondly, the point cusps are probably stellate shapes, and the line cusp centres are probably flattened banana shapes in 3D. It would be very good to see the Meridian45 sections (at present we have seen the MeridianZero sections).
I realise that the sim does not say anything about the space on the inside of the superconducting sphere. So these colour gradients only imply mild confinement (in the absence of knowing what field strength the colour represents). It might be that the centre of the polywell is indeed a great hole, and the stuff I note here is just some fringing round the edges.
Kudos to icarus and Indrek for their work.
Regards,
Tony Barry
Here's the first visualization of the field lines on the ball.
Unfortunately I didn't find any way in the gnuplot to hide the lines on the other side of the sphere so looking at it takes a bit of getting used to. For bearings: two fixed faces up and down, 4 rotating on the center line.
And animation:
The data file for the lines is also available at
http://www.mare.ee/indrek/ephi/invwb/
- Indrek
Unfortunately I didn't find any way in the gnuplot to hide the lines on the other side of the sphere so looking at it takes a bit of getting used to. For bearings: two fixed faces up and down, 4 rotating on the center line.
And animation:
The data file for the lines is also available at
http://www.mare.ee/indrek/ephi/invwb/
- Indrek
