Talk:Aneutronic fusion/Archive 2

Archive 1

Archive 2

Archive 3

Archive 4

More magnetic field effects

Eric, I see that Todd Rider on p. 135 discusses (briefly) magnetic fields, concluding: Unfortunately, detailed studies of this issue [55, 82, 83] revealed that magnetic fields actually increase the energy transfer rate. (He also mentions the issue of synchrotron radiation.) Is he talking about the same effect as you are? If not, did you include this negative effect in your calculation along with your positive effect? --Art Carlson 13:52, 6 July 2006 (UTC)

I don't have a complete copy of Rider at my office. What are the references?Elerner 00:21, 7 July 2006 (UTC)

"once suggested":

• J.R.McNally, Jr., Physics of Fusion Fuel Cycles, Nuclear Technology/Fusion 2, 9-28 (1982).
• J.R.McNally, Jr., Simple Physical Model for the Effect of a Magnetic Field on the Coulomb Logarithm for Test Ions Slowing Down on Electrons in a Plasma. Nuclear Fusion 15, 344-346 (1975).

"detailed studies":

• J.D.Galambos, Effects of Nuclear Elastic Scattering and Modifications of Ion-Electron Equilibration Power on Advanced-Fuel Burns (Ph.D. thesis, University of Illinois at Urbana/Champaign, 1982).
• J. Galambos and G.H. Miley, Effects of Enhanced Ion/Electron Equilibration Power on Cat-D Tokamak Ignition. Nuclear Technology/Fusion 4, 241-245 (1983).
• S. Ichimaru and M.N. Rosenbluth, Relaxation Processes in Plasmas with Magnetic Field. Temperature Relations. Physics of Fluids 13, 2778-2789 (1970).

--Art Carlson 07:42, 7 July 2006 (UTC)

No authors other than I have considered the quantum version of the magnetic field effect in fusion applications because they were all looking at tokamaks, where the fields are enormously lower. The quantum effects were only studied in the context of neutron stars.

The classical effects studied in, for example, Galambos and Miley, are quite small—less than 15%, compared with the 20-fold decrease in the quantum case, and deal with fields that are several orders of magnitude less than those I focused on.

My calculations, and those in the neutron star studies I partially relied on, integrate over all the relevant velocities, so provide an accurate picture of the net effect.Elerner 04:04, 8 July 2006 (UTC)

What you say about the other studies seems rasonable. I am still surprised that you claim a quantum effect much larger than about a factor of two because the energy levels parallel to the field are not constrained. Still, it could be right since most collisions in a plasma are small-angle and in a super-strong field the direction of motion of the electrons cannot be changed. If only collisions near 180 degreees are effective, then the energy transfer might be suppressed by a factor up to something like the Coulomb logarithm (10 to 20). Is this the right way to think about it? I reserve the right to be skeptical, but maybe only at level yellow, not level red. What would give me some confidence in your result is if neutron star calculations also talk about a factor of 20 (although it is quite a stretch to compare a DPF with a neutron star!). Do they? --Art Carlson 10:13, 8 July 2006 (UTC)

Yes, they do. The difference is that the ion energy there is 1000 times higher and the B field also 1000 times stronger, so the dimensionless T is about the same.Elerner 04:25, 9 July 2006 (UTC)

After giving the problem some more thought, I have identified processes that are equivalent to small-angle scattering, so I would like to call back my suspension of disbelief expressed above. That apparently puts me at odds with the neutron stars theories. Could you give me a reference to check this out? --Art Carlson 07:51, 10 July 2006 (UTC)

G. S. Miller, E.E. Salpeter, and I. Wasserman, 1987, Deceleration of infalling plasma, ApJ, 314, 215 Elerner 03:20, 11 July 2006 (UTC)

For the record: This paper gives an example where "[t]he stopping length in the B = 0 case is then ~y = 4.3 g cm^-2", while "increasing the field to B = 7 × 10^12 G gives us y = 52 g cm^-2". This is a factor of 12, so Eric is right here that quite large reductions in the transfer of energy from ions to electrons are possible. The paper goes into some detail explaining the physics behind the effect, but I couldn't follow it. --Art Carlson 08:02, 12 July 2006 (UTC)

End of the road

Sorry, Eric. All your latest comments either don't speak to the real issues or illustrate a lack of any real understanding of plasma physics. After this lengthy exchange I must conclude that you will never allow yourself to be convinced that you are wrong, even by the best physical arguments. I imagine you will say exactly the same thing about me. The depth of the problem is illustrated by the fact that you reverted all my last edits, even those I thought we could compromise on, like reducing the detail on cyclotron radiation since we both agree that is not a fundamental issue here. My ideal solution is that you just disappear. More likely is a revert war. Better would be if one of us could bring in outside expertise, though I'm not sure that would help. --Art Carlson 07:52, 11 July 2006 (UTC)

Right, Art, you really know more plasma physics than Alfven, who I was quoting. You did not answer anything that I wrote, so you just go back to the same games of reverting that Joshua played.Elerner 23:13, 11 July 2006 (UTC)

Art is still reverting without responding to my arguments. It is particularly egregious to continually bring up radioactive waste when I demonstrated that the waste produced is insignificant. To claim that waste comparable to the radioactivity in a human being must be "desposed of" is unprincipled scare-mongering. Art has not pointed to any errors in my calculations above. Nor has he responded to my arguments about magnetic confinement.Elerner 23:35, 12 July 2006 (UTC)

A competent reader will recognize that it is the other way around. You have written some things in after I did, but what you wrote does not challenge the validity of my arguments.

For the sake of anyone entering the fray, the brief summary is this:

• Radioactivity: Eric believes that radioactivity in a p-B11 reactor will be negligible, I believe it will be manageable. He presented a calculation for the reaction Be10 + n -> Be11. I insist that a serious calculation must include a concrete engineering design, consider many materials including isotopes and impurities, and calculate many reactions in addition to neutron capture.

A plasma focus device, unlike a tokomak, does not require a large variety of materials. The main unavoidable material inside the neutron shield is beryllium for the electrodes. Given the low energy of the neutrons from side-reactions, there is no energetically possible reaction with Be-9 other than neutron absorption and there is no other naturally occurring isotope. Other than the electrodes and the thin Be vacuum chamber, the only other structure inside the neutron shield is the x-ray conversion device, which includes extremely thin layers of aluminum, copper and tungsten, each with a total thickness of a fraction of a mm. After passing through a final aluminum wall, the neutron enters the water shield to be moderated and then the boron-10 absorber. No long-lived isotopes can be produced in the aluminum wall. Thus my calculation is right. Art would have to point to some concrete way that waste that is orders of magnitude more radioactive would be produced before there is enough to need “management.” Our bones contain long-lived radioactive materials, but no one talks about the need to “manage radioactive waste” at a cemetery. To say that radioactive waste is a problem for pB11 DPF reactors is not only wrong, it is scare-mongering.Elerner 14:53, 14 July 2006 (UTC)

I am not qualified to do a radiological analysis. I only know how difficult they can be. Has a detailed radiological analysis of a p-B11 reactor design ever been published? The easiest, maybe the only way out of this disagreement is to leave behind original research and to find verifiable sources. --Art Carlson 08:35, 17 July 2006 (UTC)

When a rough calculation shows that the radioactivity is orders of magnitude too small to be a problem, it is up to you to prove that there is a problem.Elerner 15:07, 17 July 2006 (UTC)

I would interpret that to mean that you know of no radiological studies that are published or otherwise verifiable. Is that correct? --Art Carlson 19:14, 17 July 2006 (UTC)

If someone wants to say, in a wikipedia article, something as absurd as "pB11 reactors will create enough radioactive waste as to require management" then is is they who need verifiable studies. No one would do a study to prove something that scarcely needs proving. Scientific studes of the sun rising in the East are a bit scarce too. I might add that much of what you have writtten on this page is not cited to verifiable sources, either. The page might be considerably shorter, and better, if we stuck to the verifiable source rule.Something to consider.Elerner 22:35, 17 July 2006 (UTC)

I realize that certain facts cannot be expected to be stated in the peer-reviewed literature. I would be satisfied with a simple statement by someone with credentials in the business. Do you know anyone? (BTW, can you learn to indent? It's hard enough keeping track of comment and response as it is.) --Art Carlson 07:46, 18 July 2006 (UTC)

• Confinement: Eric believes that megatesla fields can be produced and maintained in a DPF for times that are relevant for fusion power. I believe that any configuration of particles and fields with a pressure much higher than the surroundings will disassemble in a time close to the Alfven transit time.

This contradicts the existence of plasma filaments, which are "configuration(s) of particles and fields with a pressure much higher than the surroundings" and which have been observed and analyzed in hundreds if not thousands of papers. They last far longer than the Alfven transit time. Examples are lightning strokes, and filaments in solar prominences.Elerner 13:56, 14 July 2006 (UTC)

Example of pinch confinement of solar coronal loops: magnetic fields about 100 Gauss, particle density around 10^10/cc, radius of filaments about 5,000 km, Alfven velocity 2,000 km/s. This means the Alfven crossing time is a few seconds. But the lifetimes of loops is typically thousands of seconds—-thousands of Alfven crossing times.[[User:Elerner|Elerner]

This is a good example of a plasma which is not confined. There is radial confinement, sure, but the loop as a whole expands at millions of km/h. --Art Carlson 13:23, 15 July 2006 (UTC)

He cites papers that, at least on first blush, give lifetimes that are 10 or 15 times longer than the acoustic transit time (which is closely related to the Alfven time). I cite the virial theorem as a general proof and some toy models involving magnetic energy or jXB forces as paedogogical aids. Eric seems to think that the virial theorem depends in some way on the velocity distribution or the collisionality. It doesn't. The one point which might bear some discussion to tie up the argument is the surface integrals in the virial theorem.

• There are other arguments, too. The one that most efficiently demolishes his credibility as a physicist is his refusal to recognize that $\omega _{p}>\omega _{c}/2$ is mathematically equivalent to $nkT/(B^{2}/2\mu _{0})>kT/(2m_{e}c^{2})$,

The problem is that Art, in his version of the article, says that the ratio of plasma pressure to magnetic pressure, beta, is equivalent to the the ratio of plasma frequency to cyclotron frequency. That is wrong. That would be true if kT appeared only on the left of the above inequality. But kT appears on both sides of that inequality, so Art's statement in his version is wrong. The correct way to state the inequality is to cancel out the common term on both sides, kT. In that case the correct English translation is "If the magnetic field energy per electron does not exceed twice the elctron rest energy, cyclotron radiation is trapped in a plamsa." Elerner 13:56, 14 July 2006 (UTC)

Do you or don't you acknowledge that $\omega _{p}>\omega _{c}/2$ is mathematically equivalent to $nkT/(B^{2}/2\mu _{0})>kT/(2m_{e}c^{2})$? --Art Carlson 13:11, 15 July 2006 (UTC)

but his claim that you can confine a plasma without a jXB force comes close. I will re-engage in the discussion if someone new shows up or if Eric shows any sign of coming to reason. When I find time I also plan to extend the article on the virial theorem to cover charged particles in electromagnetic fields in detail.

--Art Carlson 09:58, 14 July 2006 (UTC)

I stole some time from more important things and added the magnetic field version to the virial theorem article, including the estimate that the expansion time is equal to the acoustic (or Alfven) transit time. --Art Carlson 13:23, 15 July 2006 (UTC)

Loops tend to kink, which is the first stage in forming plasmoids in a DPF. I had forgotten, but good description of the formation of plasmoids at much lower densities and larger scales than in a DPF is at http://ve4xm.caltech.edu/webpub/hsu03prl-preprint.pdf. , http://arxiv.org/abs/physics/0411089 and especially http://ve4xm.caltech.edu/webpub/yeebellan.pdf

The first two papers by Hsu and Bellan show experimentally the formation of a magnetically confined plasmoid, with no external confining magnetic field. (They refer to the plasmoid as a spheromak configuration.) The third paper by Yee and Bellan study the stability of the plasmoid. The fact that it is magnetically confined can best be seen from their data that shows that the plasmoid is moving at a velocity of about 8cm/microsecond but is expanding at a velocity of 0.5cm/microsecond. The Alfven velocity of this plasmoid with 800 G magnetic field is around 17cm/microsecond. So the expansion time is about 30 times longer than the Alfven time.

It should be added that in this experiment the plasmoid is not trapped within the axial current of the device, while in a plasma focus, the axial current continues during the whole plasmoid lifetime, which contributes to the stability of the plasmoid against expansion. As I emphasized above, it is incorrect to view the plasmoid as entirely self-contained because of the current running along the axis, which extends much beyond the plasmoid radius. This is ONE reason why Art's simple-minded appeal to the virial theorem is wrong. But in any case, the experimental results are clear and theory never can be used to contradict experiment, only the other way around..Elerner 15:00, 15 July 2006 (UTC)

Still no response from Art Carlson justifying his reverts on radioactive waste and confinement.Elerner 22:19, 16 July 2006 (UTC)

perhaps he's realized he's already wasted too much time on your nonsense?--Deglr6328 16:35, 17 July 2006 (UTC)

It's more that I don't like repeating myself, especially when nobody is paying attention anyway:

• "A promising place to start would be the question of whether the size, time, and temperature quoted were really measured simultaneously."
• "That is, expansion is energetically favorable and can only be prevented if the boundary resists it, that is, exerts a pressure approximately equal to the energy density of the configuration."
• "[A] strong component of the current running along the axis ... [w]on't give you a net radial j×B."

I have wasted too much time on his nonsense, but I'm not sure I've realized that yet. --Art Carlson 19:39, 17 July 2006 (UTC)

Actually, supposing that Eric actually wants to give a serious response, I would suggest that we try to clear up the theoretical issue before tackling the experimental one.

That is exactly the opposite of the scientifc methodElerner 22:35, 17 July 2006 (UTC)

Eric, would you please help me understand your point of view by answering these simple questions?

• Do you agree that nature in general and the DPF in particular are adequately described by Maxwell's equations and the Lorentz force?
• Do you agree that the virial theorem is a correct mathematical derivation from these equations?
• When the virial theorem is applied to the special case of the DPF, do you believe that the conclusion that the expansion time is on the order of the Alfven time is improper?
• Can you specify what you believe the DPF configuration to be in sufficient detail to check this claim mathematically?

Thanks very much. --Art Carlson 20:08, 17 July 2006 (UTC)

Plasmoids have been studied for decades, both theoretically and experimentally, in many contexts, even if Art is ignorant of all this. He should start by reading and replying to the references above(Hsu, Yee and Bellan). But while he is looking those up, I will get some more. Some of this will be integrated into this and other articles shortly.Elerner 22:35, 17 July 2006 (UTC)

I should have stuck to my "end of the road" resolution. Not only do I not know why Eric thinks an azimuthal j and an azimuthal B produce a radial j×B force, I can't even figure out if he believes there is such a thing a j×B force! --Art Carlson 07:56, 18 July 2006 (UTC)

Unfortuntately I know very little on this subject. But isn't it just a matter of finding some suitable references that support one side or the other? After all, we shouldn't be debating the issue, but describing it. --Iantresman 13:01, 18 July 2006 (UTC)

I agree with Ian. It is ridiculous to be debating this back and forth. It is absurd that Art asks for evidence from “someone with credentials in the field” and obviously excludes me, even though I’ve been funded for years by two governments to do research in this field and have given invited presentations on it at international fusion conferences. The whole article needs to have all unverifiable (unsourced) material eliminated from it. That will take some work, but will happen in time.

Art, I entirely disagree with your method. In science, you can’t use theory to refute observations. Many researchers have observed long-lived plasmoids for decades. If you can prove theoretically they don’t exist, you’ve just proved that your use of theory is wrong. And, I’ve explained several times that your use of the viral theorem is wrong since force-free toroidal configurations are not confined within a surface. The force fee lines have larger and larger radii of curvature as they approach the axis until they run along it. But leakage out along the axis is limited to the beams, which evacuate the plasmoid in a time much longer than the Alfven crossing time.Elerner 14:27, 18 July 2006 (UTC)

Thanks, Ian. Relying on verifiable sources is neither as easy nor as desirable as it sounds, but it is certainly the ideal. How can you even start to get an article on plasma physics right if you don't know how to calculate j×B? On the radiation issue, Eric doesn't have an verifiable sources, so he resorts to calling my position "absurd" (twice). On the equilibrium issue, it is verifiable that the virial theorem is commonly understood to imply that all plasmoids are unstable on an Alfven time. We should report this. If it is verifiable that a significant minority of experts believe that this conclusion is false in the case of the DPF, then we should report that, too. Do you think you can suggest some appropriate language? --Art Carlson 15:19, 18 July 2006 (UTC)

Art, the thing you seem not to understand is that the fields and currents of a force-free toroidal plasmoid are not contained in a surface and therefore the surface integral is never zero. I mentioned this above, but you never replied to it. This is the mathematical reason your derivation about plasmoids on the virial theorem page is wrong. Your definition of a plasmoid is not the one everyone else uses. The fact that plasma is confined in a finite space is not at all the same as saying that the field lines and currents are also all confined in the same finite space. They are not.Elerner 13:35, 20 July 2006 (UTC)

I should add that Art's over-enthusiastic application of the virial theorem also would forbid the existence of solids and liquids, which are held together by purely electromagnetic forces. Help, I'm exploding!Elerner 14:20, 20 July 2006 (UTC)

Hey, maybe this discussion will get interesting after all. I am pleased to learn, Eric, that you do not question the virial theorem per se or its fundamental applicability to the DPF. Good move. It's your best shot. And since you're a special friend, I'll give you another tip. The surface integral is only one place to look for a loophole in the virial theorem. The other - and the more promising one - is in the difference between mean values and maximum values. I have a lot more to say about this, but won't have time to go into details before next week. (Besides, some of it is probably crossing the line to original research.) The question about solids and liquids is good, too. I remember thinking about it and answering it to my satisfaction several years ago. Unfortunately I don't remember what that answer was. I promise to give it some thought. --Art Carlson 20:12, 20 July 2006 (UTC)

References are not so hard, they just take a bit of time. How about letting this stand until Friday and hitting the library in the meantime? I have already found some relevent papers.Elerner 23:58, 18 July 2006 (UTC)

Art, it is fortunate that we don’t have to strain our brains about this, since the problem has been studied by others for the last fifty years. An examination of my files and notes and a quick trip to the library gives the following results:

In 1958, S. Chandrasekhar and L. Woltjer showed that force-free stable configurations can exist in space. They concluded that ”...the maintenance of a stationary state requires the appearance of surface currents at the boundaries” of the configuration. They showed that this condition leads both to no contradiction to the virial theorem and no necessity for expansion. (Proc. Nat Acad Sci., 44, 285) (Since it is not possible to have force free surface currents without also having some currents that cross the boundary, this is connected with my point that the current system is not bounded.)

It doesn't matter for the current version of the article, but for the record, you have not read this paper carefully. They do not consider a solution where "some currents ... cross the boundary" but explicitly note (at the top of page 287): In solving equation (9) we may certainly use the vanishing of the normal component of the current, j.dS, as a boundary condition. Furthermore, they explain that the surface currents are required to counteract the universal tendency of a magnetic field configuration to expand (i.e. the virial theorem), that is, the force on the current is outwards and must be taken up by something outside the system. One possibility they mention explicitly is that an asymptotically uniform external field could provide this containment. --Art Carlson 08:58, 28 July 2006 (UTC)

My parenthetical comment relating to my own arguments above is, as you say, incorrect, but that does not affect the overall conclusion. Chandrasekar and Woltjer’s solution is for, as they describe, an ”isolated spherical plasma”—one surrounded by nothing. You have misquoted the paper. The paragraph mentioning asymptotic fields begins ” We can avoid surface currents on a plasma with a force-free magnetic field by fitting suitable external fields.” External fields are explicitly posed as an ALTERNATIVE to the surface currents that maintain stability for a totally isolated sphere. The surface currents contain an ISOLATED plasma with no need for external fields.

There are two ways I can try to explain this to you. The first is to understand why the authors speak of an (apparent) "paradox":

1) There are, by assumption, no forces on the plasma inside the sphere. (j×B = 0)

2) There are, by assumption, no forces outside the sphere. (j = B = 0)

3) And yet, by the virial theorem, there must be an outward acceleration.

The resolution is that there are forces on the surface, neither inside nor outside the sphere, and these forces, to resolve the paradox, must obviously be in the outward direction required by the virial theorem.

The second way to explain this is to look directly at the fields, currents, and forces. We have a sphere, with no field outside, and a surface current j.

1) In what direction is the field just inside the sphere? (Answer: in the direction of r×j, by Ampere's Law)

2) In what direction is the Lorentz force? (Answer: in the direction of j×B ~ j×(r×j) ~ r, that is, outwards)

It doesn't get any clearer than that. --Art Carlson 21:11, 28 July 2006 (UTC)

The Chandrasekhar and other references describe "stable", non-expanding plasmoids.Elerner 05:44, 29 July 2006 (UTC)

Do they? The way I read it, Chandrasekhar et al. describe "stationary" states within a finite volume, and go to great efforts to point out that such states can only exist if currents or fields are imposed outside that volume (possibly in the surface). If the system of internal currents plus surface currents were force-free, then you could just draw the boundary a bit farther out and you would have no surface currents!

You don't have anything at all to say about my two physics arguments here (or the physics arguments I have given before relating to this point)?! I get the impression you are not too comfortable talking about physics. --Art Carlson 11:24, 30 July 2006 (UTC)

It is absurd for you to argue that “external forces” are needed. What external forces are needed to stabilize an atom? Or do you think atoms are inherently unstable because of the virial theorem?

This argument does matter for the article, because you persist in inserting sentences that imply only solid objects can contain plasma. I am again removing this un-physical idea.Elerner 14:12, 28 July 2006 (UTC)

I think it is important to point out that a megatesla is the mother of all magnetic fields, not just the garden variety. Two numbers that we can agree on to describe the conditions are the magnetic pressure and the electron density. Since few people have a feel for MPa and number densities, I thought it was a good idea to give them a quick reference point. This, and not the idea that you have to put a plasma in a bottle, was my primary motivation to mention the strength of steel and the density of solids. With this background, are you willing to put the statements back in? Do you have an alternative suggestion to help the reader understand the conditions we are talking about? --Art Carlson 16:35, 28 July 2006 (UTC)

The relevent comparison is to what has been obtained in the lab. I provide that now. We don't need "wow" comparisons and we can leave mother out of it.Elerner 05:44, 29 July 2006 (UTC)

I'm not entirely happy with the section, but I can live with it. As a compromise, I put the info on pressures and densities into footnotes. That might not be a bad idea anyway: I feel we are getting into too much detail for a general article on aneutronic fusion. Some of these things (and other things I want to say but haven't had time to) might better be placed in Virial theorem or Dense plasma focus. If you can live at all with the idea that information in footnotes will not irrevocably poison the minds of youth, please leave it alone. Wouldn't it be cool if we actually had a stable version for a few weeks? Oh, and one more thing: Do you think you can provide footnotes for the fields produced by lasers and DPF? --Art Carlson 11:31, 29 July 2006 (UTC)

However, I am glad to see some progress on the radioactive issues.

Much later, in 1987, D. Wells and L.C. Hawkins showed mathematically that self-confined quasi-force-free plasma configurations do have a confining force, and put this explicitly in the context of explaining experimental results where hot plasma are confined for many Alfven transit times without an external magnetic field (J. Plasma Physics, 38, 263)

The most interesting statement in this paper is the following: It has been observed experimentally that plasmas with very high ion temperatures and densitites can be contained for times corresponding to many Alfven transit times by the use of magnetic fields. In some experiments, magnetic pressures too low to support the corresponding plasma pressure are recorded. It is a shame that the authors do not elaborate or give references to support either of these statements! It is obvious that they want to say that there are plasmoid configurations that are isolated and do not expand. As far as I can tell, they don't actually say that. They do deduce the existence of "non-zero containment forces" and "finite containment forces". (What distinction are they trying to make between the two?) It is not at all clear whether they really derive the sign of these forces, i.e. whether they are inwardly-directed/containing or outwardly-directed/disruptive. This paper be a good starting point to follow up on references, but it is not a good citation to support the hypothesis of isolated plasmoids that don't expand. --Art Carlson 11:42, 30 July 2006 (UTC)

There are still other approaches. In 2000, L. Fadeev and A.J. Niemi demonstrated mathematically the existence of stable knotted self-confined configuration of plasma filaments could exist(Phys. Rev Lett 85,3416 and arXiv:physics/0003083). They argued that the use of the pressure term in the standard virial theorem argument was incorrect, and that nonlinear effect that arise from the interaction of individual particles with the electromagnetic field lead to an entirely different conclusion, namely that self-confined plasma configurations are possible.

This is probably the most interesting paper. In particular, the assumptions and limits of the standard derivation and of this one are laid out in some detail. The authors, if I understand them correctly, argue that there are special configurations with statistical properties not captured by the averages used in the derivation of the virial theorem. What is not entirely clear is whether they are talking about strictly isolated plasmoids. See p. 3418 (my emphasis):

The field ρ is a measure of the particle density in the bulk of the plasma. If its average (asymptotic) value <ρ²> = ρ₀² becomes too small, .... Consequently we select the averge ρ₀² so that it aquires a sufficiently large value in the medium. ... From this we conclude that |ρ(x)| never vanishes; it is bounded from below by a nonvanishing positive value.

I don't see any way to read this other than that the plasmoids being discussed are - indeed must be - imbedded in an infinite plasma. It is the current and field "fluctuations" that exist only within a bounded region. (In addition it is not always clear whether the structures considered are bounded in three dimensions or only in two (helical filaments), but as I read it the authors are not only talking about structures of infinite length.) The authors admit that it is not clear to what extent their solutions exist or can exist in the real world. My prejudice is that collisions and instabilities in actual plasmas will quickly destroy the correlations required to maintain such a state, but of course there can be no proof of such a statement.

I believe that special correlations of the type considered in this paper are also the reason that Eric Lerner (unfortunately?) does not explode. (See the first paragraph in the right-hand column on p. 3417.) The standard derivation assumes the particles react with the macroscopic fields resulting from macroscopic charge and current densities. Momentum transfer through collisions is not a problem either. In condensed matter, however, there are microscopic correlations between charges that cannot be ignored. As Faddeev and Niemi point out, "at high enough temperature the bound state degrees of freedom decouple" and a plasma description becomes appropriate.

--Art Carlson 13:02, 30 July 2006 (UTC)

I did not check anybody’s math, but Chandraskhar and Woltjer did have a remarkably good reputation for getting things right. In any case for wiki purposes, I surely don’t have to. What is key is that there is an abundant peer-reviewed literature showing 1) that self-confined plasmoids are possible and, more importantly, 2) that they are experimentally observed to exist for periods much longer than the Alfven crossing time. Therefore you and your pal Delgr really have no excuse to continually revert to your paragraph that states:” A magnetic field of this magnitude cannot, however, be maintained for an appreciable time in practical devices”. Even if you two think you are so much smarter than Chandraskhar, you can’t put your unsourced personal opinions in when they contradict stuff in the literature.Elerner 03:39, 21 July 2006 (UTC)

Thanks for the references. I've printed them out and will chew on them. (Please be a little patient.) --Art Carlson 14:18, 26 July 2006 (UTC)

Until you have, don't keep eliminating perfectly vaid, verifiable phsyics from the article.Elerner 18:41, 26 July 2006 (UTC)

• I'm sure you can use yourself as a source, though conflicts of interest would require them to be publishsed in reliable sources, tedious to find perhaps, but probably necessary for particularly contenious issues.
• As for j×B calculations, they may or not be relevent depending on the context. One thing I've learnt about plasmas is that they do some weird, wonderful and often unpredictable things; but at least these should be documented somewhere. --Iantresman 08:58, 19 July 2006 (UTC)

Hahaha! wow, next thing you know we'll be hearing arguments such as "F=-g((m1*m2)/r^2) may or may not be relevant to understanding gravity but who cares since it does some weird/cool stuff!".--Deglr6328 20:32, 19 July 2006 (UTC)

Art, you are reverting even though you have no reply to the references I've cited.Elerner 15:16, 23 July 2006 (UTC)

I didn't revert the article, for a change, I blanked it. Even if your references say what you think they do, your version is not acceptable for a number of reasons.

What reasons?Elerner 18:32, 23 July 2006 (UTC)

I thought leaving the sections out for a few days until we agree on a neutral, verifiable version would be a way out of the silly revert war.

We don't have to agree. We just both have to cite verifiable sources.Elerner 18:33, 23 July 2006 (UTC)

If you prefer simple reversions, I can oblige. --Art Carlson 16:08, 23 July 2006 (UTC) (P.S. I'm sorry I can't deal with this faster. I have a daytime job.)

Art, you have yet to answer the references on the virial theorem and self-contained plasmas, yet you continually revert to your version on this. While we try to settle the radioactivity question you should at least leave my version of the magnetic effect alone. You have no excuse not to.Elerner 14:24, 27 July 2006 (UTC)