MHD is not a good idea (just like it isn't for a Field Reversed Configuration) becasue there is a field null at r=0 and the wuiffle-ball effect (expansion of the plasma against the field) makes this low field region fill almos the entire plasma. Besides, the field line curvature is good everywhere so MHD stability isn't an issue.
New FAQ - What are Cusps and what kind does a Polywell Have?
FWIW, here was Nebel way back when:
Art:
If the cusps' field lines end on that beta=1 surface, we can legitimately make the potentials zero on that surface. This is not to say that the magnetic field is zero on that surface, in fact it must be given by the kinetic pressure of the contained plasma there.
What it means though, is that the reference magnetic field strength at regions of field nulls is not zero, although the potential there is zero. More simply, we have a reference value for magnetic field strength that is given by its relationship with the plasma kinetic pressure in regions where there are no magnetic field vector lines (the field nulls).
The whole of the magnetic field, in the regions where there are field lines, are then measured against this reference value of strength in the null field line region which is greater than zero.
It is the analogue of the stagnation pressure (Bernoulli head) for the streamlines of a potential fluid flow.
That's interesting, I was thinking that the whole beta=1 surface is best thought of as zero point of the vector potential field. Read my post above about referencing the potential field to the beta=1 surface.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.
If the cusps' field lines end on that beta=1 surface, we can legitimately make the potentials zero on that surface. This is not to say that the magnetic field is zero on that surface, in fact it must be given by the kinetic pressure of the contained plasma there.
What it means though, is that the reference magnetic field strength at regions of field nulls is not zero, although the potential there is zero. More simply, we have a reference value for magnetic field strength that is given by its relationship with the plasma kinetic pressure in regions where there are no magnetic field vector lines (the field nulls).
The whole of the magnetic field, in the regions where there are field lines, are then measured against this reference value of strength in the null field line region which is greater than zero.
It is the analogue of the stagnation pressure (Bernoulli head) for the streamlines of a potential fluid flow.
-
Art Carlson
- Posts: 794
- Joined: Tue Jun 24, 2008 7:56 am
- Location: Munich, Germany
-
Art Carlson
- Posts: 794
- Joined: Tue Jun 24, 2008 7:56 am
- Location: Munich, Germany
Sorry, I can't make any sense out of this. What I know is that what Bussard claimed, that the beta = 1 plasma is surrounded by a confining (tangential) magnetic field except at a discrete number of point cusps, is topologically not possible. Whether he was being deliberately misleading or whether he simply didn't understand this fact, I can't say.icarus wrote:What it means though, is that the reference magnetic field strength at regions of field nulls is not zero, although the potential there is zero. ...
Art:
I agree that the "tangential magnetic field except at discrete number of point cusps" is topologically impossible ..... however a "tangential field except at discrete number of point cusps and a network of continuous line cusps" is topologically possible, lossiness of such cusps to be determined.
Now if the "strength" of the line cusps were to vary along it's length, such that the corners (where lines meet) were stronger to an extent that the pieces in between were negligible then it may approximate as a model of distinct point cusps .... how weak are we allowed to make the perforated pieces of the line cusps between the corners and mid-points before it becomes topologically nonsensical?
I think that the line cusps are a red herring argument, until they can be shown to be more lossy than the face-centered cusps .... and those we can analyse to some extent....
Hmmm, am I gonna have to write it up, I hate that.Sorry, I can't make any sense out of this. What I know is that what Bussard claimed, that the beta = 1 plasma is surrounded by a confining (tangential) magnetic field except at a discrete number of point cusps, is topologically not possible.
I agree that the "tangential magnetic field except at discrete number of point cusps" is topologically impossible ..... however a "tangential field except at discrete number of point cusps and a network of continuous line cusps" is topologically possible, lossiness of such cusps to be determined.
Now if the "strength" of the line cusps were to vary along it's length, such that the corners (where lines meet) were stronger to an extent that the pieces in between were negligible then it may approximate as a model of distinct point cusps .... how weak are we allowed to make the perforated pieces of the line cusps between the corners and mid-points before it becomes topologically nonsensical?
I think that the line cusps are a red herring argument, until they can be shown to be more lossy than the face-centered cusps .... and those we can analyse to some extent....
I think you have a point there because even with toroidal magnets the field line density in the "virtual faces and line cusps" is going to be stronger than in the center of the real faces. And that without any WB effect.icarus wrote:Art:Hmmm, am I gonna have to write it up, I hate that.Sorry, I can't make any sense out of this. What I know is that what Bussard claimed, that the beta = 1 plasma is surrounded by a confining (tangential) magnetic field except at a discrete number of point cusps, is topologically not possible.
I agree that the "tangential magnetic field except at discrete number of point cusps" is topologically impossible ..... however a "tangential field except at discrete number of point cusps and a network of continuous line cusps" is topologically possible, lossiness of such cusps to be determined.
Now if the "strength" of the line cusps were to vary along it's length, such that the corners (where lines meet) were stronger to an extent that the pieces in between were negligible then it may approximate as a model of distinct point cusps .... how weak are we allowed to make the perforated pieces of the line cusps between the corners and mid-points before it becomes topologically nonsensical?
I think that the line cusps are a red herring argument, until they can be shown to be more lossy than the face-centered cusps .... and those we can analyse to some extent....
Indrek's stuff pretty much proved that for the initial state.
How does it evolve? I'd love to have an answer.
Engineering is the art of making what you want from what you can get at a profit.
-
Art Carlson
- Posts: 794
- Joined: Tue Jun 24, 2008 7:56 am
- Location: Munich, Germany
Right. Put the line cusps back in that Bussard thought he could take out and I'm with you all the way. The experimental and theoretical literature claims that the effective width of a line cusp when it comes to leakage is a gyro-radius, with some discussion of whether you can do as good as the electron gyro-radius or only as good as the ion gyro-radius. My impression is that Bussard agrees with this. Flux conservation in the sheath suggests that the point cusps all together will have an overall loss similar that of the line cusps all together. There might be a few subtleties here, so let me make the following proposal: Let's
- neglect the losses from the point cusps altogether,
- calculate the effective width of the line cusps using the electron gyro-radius, and
- use the maximum magnetic field in the system to calculate the gyro-radius.
I don't like arguments from authority either, but I do like arguments from experiment, and I haven't heard a physical scenario that explains the WB-7 results Rick has talked about. It was easier to dismiss WB-6 data on the grounds we didn't really know how much of what Bussard claimed was true, but that's less applicable now.
I generally don't find arguments that contradict the experimental evidence interesting enough to spend the requisite time delving into the physics, but you should certainly feel free to make them since your viewpoint is skeptical anyway (plus, it's fun!). I'll just stipulate I don't feel bound by them.
I certainly would be interested in a physical model that does explain the WB-7 results but is a good deal more specific than what we have in Valencia. I guess I can only hope Rick gets enough time away from WB-8 and has sufficient leeway in his contract to enlighten us.
I generally don't find arguments that contradict the experimental evidence interesting enough to spend the requisite time delving into the physics, but you should certainly feel free to make them since your viewpoint is skeptical anyway (plus, it's fun!). I'll just stipulate I don't feel bound by them.
I certainly would be interested in a physical model that does explain the WB-7 results but is a good deal more specific than what we have in Valencia. I guess I can only hope Rick gets enough time away from WB-8 and has sufficient leeway in his contract to enlighten us.
Last edited by TallDave on Mon Jun 15, 2009 12:55 am, edited 1 time in total.
Art:
By saying we use the total length of the line cusps that could markedly overestimate losses if the line cusps are actually heavily perforated and only total up to a series of points. But by then using the max. magnetic field strength you allow a generous account of the losses. Do those two balance each other out, who knows?
I'd rather just calculate what we can agree on and note the uncertainties and unknowns as we go along.
So, the face-centered cusps have gyro-radius (are we ever going to get Tex on this forum?)
r_g = (m_e U_e)/(e B)
where B = sqrt(2 mu P) since cusp hole is on the beta=1 surface and
2W_o = m_e (U_e)^2 = 2 e V is electron kinetic energy as related to drive voltage, V,
also for the electron plasma kinetic pressure P = (2 n_e W_o)/3, n_e being electron number density at the beta=1 surface, so that
r_g^2 = (m_e W_o)/(mu e^2 P)
or
r_g^2 = (3 m_e)/(2 mu e^2 n_e)
If we take B as being the value at the center of the coil, in the plane of the coil and use a value got from calculating the field with the plasma occupying the core (as per kcdodd's spiky blob or Indrek's sphere) then I think we are in agreement for this part of the field.
The line cusps we can work up to, maybe. It will be more work, but we should be able to trace around the line cusps at the level midway between the coils (region of strongest B-field on the line cusps) for the 1/48th model and extract B, thus r_g. Then assume the beta=1 surface extends out in a sheath to such a "zipper" that closes off the plasma bag.
Art, that's quite clever, you've given back with the right hand and taken away with the left,There might be a few subtleties here, so let me make the following proposal: Let's
* neglect the losses from the point cusps altogether,
* calculate the effective width of the line cusps using the electron gyro-radius, and
* use the maximum magnetic field in the system to calculate the gyro-radius.
Those are three hefty concessions that should give you the benefit of the doubt on all the points you raised. Can you accept that?
By saying we use the total length of the line cusps that could markedly overestimate losses if the line cusps are actually heavily perforated and only total up to a series of points. But by then using the max. magnetic field strength you allow a generous account of the losses. Do those two balance each other out, who knows?
I'd rather just calculate what we can agree on and note the uncertainties and unknowns as we go along.
So, the face-centered cusps have gyro-radius (are we ever going to get Tex on this forum?)
r_g = (m_e U_e)/(e B)
where B = sqrt(2 mu P) since cusp hole is on the beta=1 surface and
2W_o = m_e (U_e)^2 = 2 e V is electron kinetic energy as related to drive voltage, V,
also for the electron plasma kinetic pressure P = (2 n_e W_o)/3, n_e being electron number density at the beta=1 surface, so that
r_g^2 = (m_e W_o)/(mu e^2 P)
or
r_g^2 = (3 m_e)/(2 mu e^2 n_e)
If we take B as being the value at the center of the coil, in the plane of the coil and use a value got from calculating the field with the plasma occupying the core (as per kcdodd's spiky blob or Indrek's sphere) then I think we are in agreement for this part of the field.
The line cusps we can work up to, maybe. It will be more work, but we should be able to trace around the line cusps at the level midway between the coils (region of strongest B-field on the line cusps) for the 1/48th model and extract B, thus r_g. Then assume the beta=1 surface extends out in a sheath to such a "zipper" that closes off the plasma bag.
Art:
Every point cusp has a corresponding line cusp that forms a complete circuit on the boundary between the field associated with that point cusp and the adjacent point cusps. That boundary lies on the edges of the face of a quasi-cube, if you will. There are 6 such line cusps, as there are six faces of the cube, six point cusps etc, it happens that each edge is actually made up of two line cusps forming opposing halves of the sheath that leaves the interior between the coils.
So, using your flux conservation argument we only need to calculate the losses from the point cusps then double it to account for the line cusps, right?
The same radial flux through the end of cylindrical part of the sheath around the point cusps must be flowing radially through the sheath lying next to the surrounding line cusp. So while the line cusp may be long in extent it must have a much thinner sheath to account for flux conservation. Right?
Hey Art, I read this bit too quickly last time, it's good, real good.Flux conservation in the sheath suggests that the point cusps all together will have an overall loss similar that of the line cusps all together.
Every point cusp has a corresponding line cusp that forms a complete circuit on the boundary between the field associated with that point cusp and the adjacent point cusps. That boundary lies on the edges of the face of a quasi-cube, if you will. There are 6 such line cusps, as there are six faces of the cube, six point cusps etc, it happens that each edge is actually made up of two line cusps forming opposing halves of the sheath that leaves the interior between the coils.
So, using your flux conservation argument we only need to calculate the losses from the point cusps then double it to account for the line cusps, right?
The same radial flux through the end of cylindrical part of the sheath around the point cusps must be flowing radially through the sheath lying next to the surrounding line cusp. So while the line cusp may be long in extent it must have a much thinner sheath to account for flux conservation. Right?
-
Art Carlson
- Posts: 794
- Joined: Tue Jun 24, 2008 7:56 am
- Location: Munich, Germany
Don't blame me. I don't know how many MW Rick measured leaving WB7. Or what the density was. Or what the magnetic field was. Or what the ion or the electron temperature was. Or how any of these things were measured. Or how the instruments were calibrated or what consistency checks were made. How am I supposed to come up with a scenario? This is not argument from experiment, it is argument from vague comments by an experimentalist.TallDave wrote:I don't like arguments from authority either, but I do like arguments from experiment, and I haven't heard a physical scenario that explains the WB-7 results Rick has talked about.
-
Art Carlson
- Posts: 794
- Joined: Tue Jun 24, 2008 7:56 am
- Location: Munich, Germany
icarus, you're learning. You seem to have realized that the topology does not allow a "perforated" line cusp, and you seem to have grasped why the losses from the point cusps and from the line cusps should be similar. Now think through the business of the sheath thickness one more time. Gyro-orbits always smear things out, so nothing much in a magnetized plasma can change on a scale smaller than the smallest gyro-radius, usually that of the electrons. That is true everywhere, not just near the point cusps, so flux conservation doesn't allow you to make the line cusp thinner, it forces you to make the point cusp fatter.
Except that the gyro-radius is dependent on B .... and B at the point immediately between coils of the line cusp is going to be quite a bit stronger than for point cusp in the planar center of coils, no?That is true everywhere, not just near the point cusps, so flux conservation doesn't allow you to make the line cusp thinner, it forces you to make the point cusp fatter.
-
Art Carlson
- Posts: 794
- Joined: Tue Jun 24, 2008 7:56 am
- Location: Munich, Germany
That's the subtlety I alluded to. I believe the loss rate based on sheath thickness assumes that the field decreases monotonically moving away from the cusp point. If not, I imagine that some sort of mirror effect might further reduce the losses, but I don't know by how much. For a number of reasons, I am pretty sure this won't change my conclusions. Maybe the best argument is to bring the point cusps back into the game. 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.icarus wrote:Except that the gyro-radius is dependent on B .... and B at the point immediately between coils of the line cusp is going to be quite a bit stronger than for point cusp in the planar center of coils, no?That is true everywhere, not just near the point cusps, so flux conservation doesn't allow you to make the line cusp thinner, it forces you to make the point cusp fatter.