Possible wiffleball analytical solution
One thing I just realized is that we apply inversive geometry here and so the square coils you use don't really invert to image coils that are square. In this sense circular coils are much easier to model.kcdodd wrote:I decided to attempt this as well just guess the value should solve for the sphere radius. I had it crop the fields inside the sphere since, well, there aren't supposed to be any there due to the electrons. In the pic some of the field lines actually terminate on the very surface of the sphere. That's bad but I am thinking that is just from the interpolation matlab has to do and it lost track of the field. I'm not sure though maybe the crop actually cropped a little too much, i hope not.
 Indrek
Oh, here you are:kcdodd wrote:The render of them is square because matlab does not have a "draw circle" function, at least that I found, and I was lazy so I just drew squares. In math they are circles. I don't have the visual skills you do, hehe.
Code: Select all
function [x, y, z] = circle3d(center, normal, radius)
p = cross(normal, [1 0 0]);
if (dot(p, p) < 0.3 )
p = cross(normal, [0 1 0]);
if (dot(p, p) < 0.3 )
p = cross(normal, [0 0 1]);
endif
endif
p = p ./ norm(p);
q = cross (normal, p);
t = (0:pi/16:2*pi)(:);
ret = repmat(center, size(t)) + radius * cos(t) * p + radius * sin(t) * q;
x = ret(:,1);
y = ret(:,2);
z = ret(:,3);
endfunction
 Indrek
icarus,
OK, you’re right, the field curvature into the plasma is a sufficient but not necessary condition for plasma stability. (Krall & Trivelpiece 5.8.12)
But if the field curves away from the plasma locally the analysis must be carried further to find some compensating stability from the next term.
Can you have the computer model zoom in on the area of inflected field lines and turn up the color contrast to clearly show the field gradient in that small area?
Krall does not develop it from the cusp geometry and it is not just a “rule of thumb”.
He clearly states after the derivation that the cusp is just a simple example. (5.12.2)
The previous independent derivation (section 5.8 ) is actually based on a linear (mirror) configuration, but it looks to me like he is trying to be careful to keep it generalized.
You gentlemen are working this math at a level way above my head, but:
I would like you to go back and double check to make sure that you are not engaged in a very sophisticated exercise in “begging the question.”
I.e. you assumed a sphere to begin with and now your result is a sphere.
The sphere shape does not agree with Indrek’s star shaped core.
I’m really enjoying the good work in this thread.
OK, you’re right, the field curvature into the plasma is a sufficient but not necessary condition for plasma stability. (Krall & Trivelpiece 5.8.12)
But if the field curves away from the plasma locally the analysis must be carried further to find some compensating stability from the next term.
Can you have the computer model zoom in on the area of inflected field lines and turn up the color contrast to clearly show the field gradient in that small area?
Krall does not develop it from the cusp geometry and it is not just a “rule of thumb”.
He clearly states after the derivation that the cusp is just a simple example. (5.12.2)
The previous independent derivation (section 5.8 ) is actually based on a linear (mirror) configuration, but it looks to me like he is trying to be careful to keep it generalized.
You gentlemen are working this math at a level way above my head, but:
I would like you to go back and double check to make sure that you are not engaged in a very sophisticated exercise in “begging the question.”
I.e. you assumed a sphere to begin with and now your result is a sphere.
The sphere shape does not agree with Indrek’s star shaped core.
I’m really enjoying the good work in this thread.
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:
The question was, IF a wiffleball forms and it is spherical what would the field look like? Particularly near line and point cusps, etc. Hasn't this been helpful enough on it's own? From here we can perturb the ideal spherical shape, given whatever wild assumptions anyone wants to make, and find best geusses ... and then we'll get some real data and see who knows what they're talking about. It was an "ansatz" used to make some progress towards better understanding in the absence of physical data.
If you have a better idea of what the field solution for a wiffleball might look like in the very center put it forward and let's see what it looks like.
I won't be going back to do that because it was stated right at the beginning that it was a spherical analytical solution. Of course, we knew were going to get a sphere. Noone is "begging any questions", however sophisticated.You gentlemen are working this math at a level way above my head, but:
I would like you to go back and double check to make sure that you are not engaged in a very sophisticated exercise in “begging the question.”
I.e. you assumed a sphere to begin with and now your result is a sphere.
The sphere shape does not agree with Indrek’s star shaped core.
The question was, IF a wiffleball forms and it is spherical what would the field look like? Particularly near line and point cusps, etc. Hasn't this been helpful enough on it's own? From here we can perturb the ideal spherical shape, given whatever wild assumptions anyone wants to make, and find best geusses ... and then we'll get some real data and see who knows what they're talking about. It was an "ansatz" used to make some progress towards better understanding in the absence of physical data.
If you have a better idea of what the field solution for a wiffleball might look like in the very center put it forward and let's see what it looks like.

 Posts: 794
 Joined: Tue Jun 24, 2008 7:56 am
 Location: Munich, Germany
You guys have been doing impressive work here while I've been twiddling with analytical models. Would you mind if I placed an order?
1. I am convinced that all the edges of the cube constitute cusps in the sense that a field line starting on an edge and going inward will land on the sphere (in our approximation of a high beta polywell). Unfortunately, this is not a topological necessity, so I don't know how to prove it. Could you show the magnetic field (one of your beautiful iron filing pictures) in the plane that passes through the center of the configuration and through the diagonal of a face? In other words, in a plane of symmetry that is tilted 45 degrees to the ones you have shown. That should provide a good answer to whether the edges are technically cusps or not (although no one seems to doubt this point).
2. The next thing I would like to see done with these magnetic models is to follow particle trajectories in them. I would start electrons with random position and velocity on the the surface of the sphere and follow them until they either return to the sphere or exit through a cusp. I would then divide the outer surface into a few regions, circular ones with a radius of several gyroradii around the corners, midpoints of the edges, and centers of the faces; bands with a width of several gyroradii along the edges of the cube; and all the rest. It should be possible to add up the number of particles that pass through these regions to determine the effective loss areas for the point and line cusps. (I can help you with more detail on the math, if you don't quite follow me.) We should be able to verify the theoretical estimates, and in particular establish whether a highbeta polywell acts like line cusps or like a discrete number of point cusps. (You might have to/want to vary the gyroradius and look at the scaling.)
3. The third thing that would interest me is what happens to the particles that pass through a cusp. I'm afraid this will require addition of electric charge/potential associated with both the central sphere and the coils. (Might be tricky to get this exactly right, but a decent approximation should not be too hard.) I am interested in the question of whether single particles (collective effects will come later) can really be reflected (recirculated) back into a cusp. I suspect that a good fraction of them won't make it.
Thanks, and keep up the good work.
Art
1. I am convinced that all the edges of the cube constitute cusps in the sense that a field line starting on an edge and going inward will land on the sphere (in our approximation of a high beta polywell). Unfortunately, this is not a topological necessity, so I don't know how to prove it. Could you show the magnetic field (one of your beautiful iron filing pictures) in the plane that passes through the center of the configuration and through the diagonal of a face? In other words, in a plane of symmetry that is tilted 45 degrees to the ones you have shown. That should provide a good answer to whether the edges are technically cusps or not (although no one seems to doubt this point).
2. The next thing I would like to see done with these magnetic models is to follow particle trajectories in them. I would start electrons with random position and velocity on the the surface of the sphere and follow them until they either return to the sphere or exit through a cusp. I would then divide the outer surface into a few regions, circular ones with a radius of several gyroradii around the corners, midpoints of the edges, and centers of the faces; bands with a width of several gyroradii along the edges of the cube; and all the rest. It should be possible to add up the number of particles that pass through these regions to determine the effective loss areas for the point and line cusps. (I can help you with more detail on the math, if you don't quite follow me.) We should be able to verify the theoretical estimates, and in particular establish whether a highbeta polywell acts like line cusps or like a discrete number of point cusps. (You might have to/want to vary the gyroradius and look at the scaling.)
3. The third thing that would interest me is what happens to the particles that pass through a cusp. I'm afraid this will require addition of electric charge/potential associated with both the central sphere and the coils. (Might be tricky to get this exactly right, but a decent approximation should not be too hard.) I am interested in the question of whether single particles (collective effects will come later) can really be reflected (recirculated) back into a cusp. I suspect that a good fraction of them won't make it.
Thanks, and keep up the good work.
Art
Here is some very early work. I didn't follow particles to escape, just gave them time limits. All masses are electrons, I don't remember what the I:V ratio was (grid current to voltage), but something like 2e5 ampturns and 50kV was what I recollect.
http://www.eskimo.com/~eresrch/Fusion/squiggle_osc.png
http://www.eskimo.com/~eresrch/Fusion/vwiggle2.png
http://www.eskimo.com/~eresrch/Fusion/vwiggle3.png
For this last one I put an oscillating potential on the MaGrid.
http://www.eskimo.com/~eresrch/Fusion/vwiggle_osc.png
I'm more of an experimentalist though, but this stuff is fun.
http://www.eskimo.com/~eresrch/Fusion/squiggle_osc.png
http://www.eskimo.com/~eresrch/Fusion/vwiggle2.png
http://www.eskimo.com/~eresrch/Fusion/vwiggle3.png
For this last one I put an oscillating potential on the MaGrid.
http://www.eskimo.com/~eresrch/Fusion/vwiggle_osc.png
I'm more of an experimentalist though, but this stuff is fun.
For 1 here's a picture:
For the second item on your shopping list there are a few ambiguities that should be hashed out:
Current in the coils and coil sizes, spacings, etc? What I've used so far is 200K amps, 15cm radius coils, 8cm spacing.
Electron random velocity and position. How random velocity (eV range?) and what direction?
And finally the big question: what should the field within the sphere look like for the particles?
Also what does "returning to the sphere" mean in this context?
 Indrek
PS: I'll be probably off the grid for the rest of the week.
For the second item on your shopping list there are a few ambiguities that should be hashed out:
Current in the coils and coil sizes, spacings, etc? What I've used so far is 200K amps, 15cm radius coils, 8cm spacing.
Electron random velocity and position. How random velocity (eV range?) and what direction?
And finally the big question: what should the field within the sphere look like for the particles?
Also what does "returning to the sphere" mean in this context?
 Indrek
PS: I'll be probably off the grid for the rest of the week.

 Posts: 794
 Joined: Tue Jun 24, 2008 7:56 am
 Location: Munich, Germany
Just what the doctor ordered, but I'm not sure what to make of it. It's not what I expected. It looks like (almost) all field lines starting in this plane just outside the sphere and going outward all end up at the corner cusp, whereas (almost) all field lines starting on the edge of the cube and going inward end up on the sphere at the line through the midpoint of the side of the cube. I think that confirms that the edges are line cusps, but they are just a bit strange. I have no idea if that is caused by our use of a spherical plasma. I guess it means that most of the particles come out the corners, but I don't know if that will result in rho^2 scaling or R*rho scaling. I guess we need to move on to step 2 if we want more definitive answers.Indrek wrote:For 1 here's a picture: ...
* The coil geometry is fine with me, but you should ask the boys in the lab if that is what they think of as a wellproportioned polywell.Indrek wrote:For the second item on your shopping list there are a few ambiguities that should be hashed out:
Current in the coils and coil sizes, spacings, etc? What I've used so far is 200K amps, 15cm radius coils, 8cm spacing.
Electron random velocity and position. How random velocity (eV range?) and what direction?
And finally the big question: what should the field within the sphere look like for the particles?
Also what does "returning to the sphere" mean in this context?
* The physical parameter of interest should be R/rho. Let's see, B (and 1/rho) should scale like I/R, so R/rho should scale like I, independent of R. That sounds strange, but I think it's right. So for these simulations, it doesn't matter what you choose for R, but we will probably want to use the actual current in WB6, plus another simulation with a design value of the current for a reactor (100 times bigger?).
* I think we agree that a typical electron energy at the surface of the sphere will be somewhere between 10 keV and 100 keV. I would expect the velocities to follow a Boltzmann distribution, but I think the polywell guys have something more exotic in mind. If I were you, I would start with a thermal distribution in velocity and direction, since we will surely want to see that sooner or later, and then modify it if you catch flak.
* In my understanding of "beta = 1", there will be no magnetic field inside the sphere so you don't need to follow the orbits there. I was thinking of starting a trajectory calculation with an electron at r = R_sphere and v_r > 0, and aborting the calculation if at some time r < R_sphere. If you prefer, you could trace the path as a straight line inside the sphere and continue the calculation where it comes out the other side.
Midpoint singularities
kcdodd has pointed out the midpoint singularities earlier in the thread. These midpoint singularities are very, very interesting and not what anyone was expecting.
On the surface of the sphere, they appear to be saddle points in terms of field line topology but minimums of field strength. Moving radially away from the sphere in the Meridian45 plane, the field lines converge to the midpoint singularity. I'm not even sure if these singularities have a mathematical classification. A saddle point on a spherical surface but with a radial convergence.
I think for this topology wiffleball (sphere) we can say then there are NO line cusps but there are these weird midpoint singularities and the corner cusps.
8 corner cusps, 12 midpoint singularities and 6 facecentered cusps., no lines is my take on it.
Now my question has to be, if we are getting results that noone expects just looking at basic topologies, why would we waste time doing detailed electronic analysis around "cusps" that may or may not exist and may or may not look like we are showing??
ART: In all fairness, I think doing 2 and 3 on your shopping list is a waste of time at this stage until the actual topology of the plasma can be nailed down.
On the surface of the sphere, they appear to be saddle points in terms of field line topology but minimums of field strength. Moving radially away from the sphere in the Meridian45 plane, the field lines converge to the midpoint singularity. I'm not even sure if these singularities have a mathematical classification. A saddle point on a spherical surface but with a radial convergence.
I think for this topology wiffleball (sphere) we can say then there are NO line cusps but there are these weird midpoint singularities and the corner cusps.
8 corner cusps, 12 midpoint singularities and 6 facecentered cusps., no lines is my take on it.
Now my question has to be, if we are getting results that noone expects just looking at basic topologies, why would we waste time doing detailed electronic analysis around "cusps" that may or may not exist and may or may not look like we are showing??
ART: In all fairness, I think doing 2 and 3 on your shopping list is a waste of time at this stage until the actual topology of the plasma can be nailed down.
Yeah, I think they are more funny point cusps then any line cusps. I mean, look at the field it is nearly parallel to the "line cusp". Any electron trying to go that way will feel a VxB force. In the pictures so far it looks like the only radially bfield lines in actual contact with the plasma are at the midpoints, corner points, and face points. Unless I missed a line somewhere.
Carter
Good. I just wanted to make sure the assumptions were all explicit, especially in light of the wide variation in the training of the members.The question was, IF a wiffleball forms and it is spherical what would the field look like?..................
From here we can perturb the ideal spherical shape...
That elliptical Blue (low field) patch located on the sphere directly under the cube edge looks to me like it could be a short segment of a line cusp where the effect is stronger for some reason.
Counting contour lines in Indrek's last color+line plot shows the depth to be the same as the triangular corner cusp.
It also shows them both to be shallower than the coil center (point) cusps which agrees with Dr. B's assertion that the coil center point cusps are the worst case locations.
It is surprising to see it right where we expected the field to be the strongest.
It would be easier to visualize its 3D structure if we could see the field levels on spheres closer and closer to the coils. (Without changing the conductive sphere diameter or the fields.)
We might see it connecting to the corner cusps in a banana shape as we look closer to the coils.
A possible explanation:
Visualize the field around the 2 coils at closest approach.
Simplify the visualization to 2 straight conductors with opposite direction currents.
Between them in their plane the fields reinforce as we expect.
But, as you move perpendicularly out of the plane the fields from the two conductors begin to curve away from each other and by the time you are their spacing above the plane they begin to cancel.
(That’s my best guess at this point.)
P.S. I have a problem interpreting the iron filing plots.
They create optical illusions of 3D fields that make sense to my eye.
But I don't think the illusion is the correct interpretation of the data.
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