Talk:Aneutronic fusion/Archive 1

Problems and proposed solutions

According to the Rider paper it seems that these fuels would work in a non-equilibrium plasma. [1] But the whole point of his paper is to prove that such things cannot realistically exist. Still, it should be mentioned as the reason why people pursue them. - Omegatron 02:56, May 12, 2005 (UTC)

The reason people pursue aneutronic fusion is that are green techies. They love science fiction and futuristic technology, but they also think small-is-beautiful and hate nuclear power. Then people come and rain on their parade and say aneutronic fusion can't be done for this reason or that, e.g. Bremsstrahlung. The natural and admirable reaction is to look for a way around the problem, e.g. non-Maxwellian electrons. Rider and I are saying, as much as we agree that aneutronic fusion is a dream that was worth dreaming, there is just no sign of a passage through the mountain range blocking the way. My point is that non-equilibrium plasmas are not anybody's reason to pursue aneutronic fusion but rather a last-ditch effort to save it. And there's not just one such obstacle, there are several: Bremsstrahlung, confinement requirements, power density, direct conversion, neutron production from side reactions. And if you ever found the miracles to solve all these, you would still have to ask whether it is not better to apply the miracles to a reactions with a cross section a thousand times bigger, like D-T. Art Carlson 08:15, 2005 May 12 (UTC)

Art, respectfully, many people working on aneutronic fusion aren't exactly green techies. This is pointedly biased. At the least, Sam Cohen, my advisor, who used to be fairly high up in the ITER hierarchy, and now works on confinement concepts which may turn out to be amenable to anuetronic fusion. There are potential answers to all of these problems: "Bremsstrahlung, confinement requirements, power density, direct conversion, neutron production from side reactions"

• Bremsstrahlung: Heat ions preferentially, possibly using orbit resonances. Non-maxwellian electrons are another possibility. Run at a lower density. Achieve better confinement times...

Fusion products tend to heat the electrons more strongly because the velocity is more closely matched. Non-maxwellian electrons were considered in several forms by Rider. Bremsstrahlung power density scales with n^2, but so does fusion power density. If the Bremsstrahlung is higher than the fusion then all the confinement time in the world won't help you. On the contrary, with better energy confinement you will almost certainly have better particle confinement. The fusion products will build up and make your Bremsstrahlung more severe. This is almost a problem for D-T fusion (within a factor of three). I've never seen a calculation for p-B fusion, but I suspect it would be a show-stopper.

Okay, I didn't think of the velocity being more closely matched. You should be able to construct a region where the parallel velocity of the ions and the fusion products are matched more closely though, yes? This would at least help.

As I highlighted earlier, I think there might be a problem with Rider's claims, but I can't examine them in detail as of yet. at the least the more promising idea is to have a higher temperature for the ions than electrons, because electron-electron collisions are going to be quite imperative in relaxing the electron distribution to maxwellian. If I recall correctly, the time scale for the electrons to reach the same temperature as the ions is roughly M/m times larger than the relaxation time for electrons to maxwellian. One can think of there being a phase space flow which scales like this. Then the redistribution energy for ions being hotter electrons should scale around m/M times less, yes?

I'm pretty sure that Rider included this factor as it is rather obvious and standard. As a conservative simplification, you can assume that all the fusion energy goes to the ions (somehow), that the ions lose their energy to the electrons (only) classically, and that the electrons lose their energy (only) by bremsstrahlung. If you artificially cool the electrons so they don't radiate so much, the ions lose energy even faster. If you heat them so the ions stay hotter, then you have to pay a higher price to heat the electrons than you save from the ions. Take your time to read Rider, then we can get into details. (I'll chew on Rostocker in the meantime.) --Art Carlson 11:02, 5 March 2006 (UTC)

The problem with the build up of fusion products is an area of active research I think. Nat Fisch keeps talking about 'alpha channelling' in Tokamaks or mirror machines. The problem in FRC's is probably a bit tougher, but there should be some kind of flexibility. LDX also claims that they can get a higher energy confinement with a worse particle confinement.

• Confinement Requirements: I'm not sure what exactly you're talking about here. Are you just saying hotter => faster diffusion?

No, simply that the Lawson requirement for p-B is 500 times higher than for D-T (which is already damned hard to reach).

• Power Density: So the power density is much lower compared to D-T reactions. We're not concerned which power density, we're concerned with economics. It's still possible to make an economical reactor with a low power density. The sun has a low power density...

The sun? Not in my backyard! When the power density is a factor of 2500 lower, believe me, we are concerned about it. Assuming 10 billion dollars allows you to produce the same volume of plasma, then a factor of 2500 in power density is the difference between producing power for, say, 20 cents/kW-hr and 500 dollars/kW-hr.

Your first assumption is far from clear, though, and you also have to weigh capital costs versus maintainance. Granted, with these kind of numbers, a general statement is an easy one to make...

• Direct Conversion: There have been operating direct conversion systems for some time now, at albeit much lower fluxes. What part of direct conversion do you regard as an impassible mountain range?

If you look at the plans for a fusion reactor with direct conversion, it is hard to find the plasma vessel because it is dwarfed by the convertor. Direct conversion is possible and is economical and elegant in principle, but it would be a major scientific and engineering effort to actually get one working at the many MW level.

I think direct conversion also admits the possibility of a smaller reactor. One on the range of KW. If you can shink your heating and driver systems along with the direct convertor, you might get a small reactor without as many maintainance issues as D-T reactors. Of course, that's also a tall order, since nobody is sure how to heat these things to 123 keV, but still...

• Neutron production for side reactions: This is at the least a significantly less challenging problem than a D-T reactor must deal with. I recognize the problems you outline in potential neutron production, but it's not as if the neutron flux is impossible to deal with. You add shielding, you protect against hard x-rays. The lifetimes of materials will go up tremendously versus D-T plasmas, yes? That's good, right? I also think your 'simple calculation' is a bit sketchy on assumptions. The neutronic reactions are nearly all endothermic: thus having 0.1% of the reactions create neutrons does not imply that the neutrons have 0.1% of the energy: it's likely much less. Less energetic neutrons are easier on materials and easier to handle. I can't find the paper online -- I'll look in the library tomorrow.

That is all correct. I just want us to keep in mind that "aneutronic" just means a heck of a lot fewer neutrons, not none at all.

--Art Carlson 14:52, 4 March 2006 (UTC)

I don't think the characterization of non-Maxwellian plasmas as a 'last-ditch effort' to save aneutronic fusion is really correct. Or fair. The same basic ideas are being tried out in other areas of magnetic confinement. You have people working on alpha channelling, or heating up ions preferentially, or polarizing nuclei to improve cross-sections, or people trying to stem the high energy tail by various damping processes, etc. etc. These are all decent ideas. A successful implementation would help fusion in that device. It's hardly a 'last-ditch effort'.

You make the point that if these 'miracles' were discovered, why wouldn't you just use them on D-T fusion. Many wouldn't exactly be applicable (eg. direct conversion probably wouldn't help too much. Bremstrahlung is also much less of an issue...). But you're right, many would be applicable to D-T fusion. That's -good-. But -eventually-, aneutronic fusion might yield a more economical, compact reactor. You might be able to put it on planes, or submarines. Many of the proposed reactors could be much less of an engineering nightmare -- it's the physics that isn't developed enough yet to know if this is a decent path or not. It's not like the idea has no potential. I think that you have far too much conviction here. I don't entirely, but mostly, agree with the tone of the articles, but I may go over the calculations and edit a bit here and there. Is this ok? Danielfong 08:13, 4 March 2006 (UTC)

Cyclotron radiation, radiation dose, and suppression of bremsstrahlung

The Carlson argument about cyclotron radation is just plain wrong. If the plasma frequency is more than half the synchrotron frequency the radiaton can't escape except from a very thin surface layer. In the plasmoid of a plasma focus, radiation can also escape from the very small core area at the initation of the beam, but this loss is also small and brief. This has been discussed extensively for over 40 years in the plasma focus community. I also corrected the shielding part, which implies that workers are inside the reactor--that's absurd. Finally I corrected the part on my own work, which substitutes the factor two for the correct one of five--a significant difference.Elerner 01:01, 27 June 2006 (UTC)

Cyclotron radiation (1)

1) I suppose you are right about the plasma frequency. I would like to read about it. Could you supply a reference? I find the calculation of maximum beta useful anyway. I would want to put it back in, with mention of any appropriate restrictions.

try any standard textbook. The calcuation is not at all valid.68.39.247.3 02:47, 28 June 2006 (UTC)

The criterion $\omega _{p}>\omega _{c}/2;$ is equivalent to $nkT/(B^{2}/2\mu _{0})>kT/(2m_{e}c^{2})=kT/(1\,{\mbox{MeV}})$. Is it possible you meant $\omega _{p}^{2}>\omega _{c}^{2}/2$? That is what would make sense looking at the dispersion relation of the X wave (or the L wave). Either way, a substantial, but not obviously prohibitive beta would be required. Although this restriction might be somewhat more severe for aneutronic fusion because it generally requires a higher temperature, I think it is rather too detailed - and difficult to get the details right - to include on this page. Maybe I can put it under Electron cyclotron resonance, but even then... --Art Carlson 09:00, 29 June 2006 (UTC)

Your formulas are entirely wrong. Please refer to a text-book for the correct ones. What is right is that the magnetic field energy per electron is 1 Mev. kT can be anything.Elerner 01:05, 5 July 2006 (UTC)

Should I make an equally helpful contribution to this discussion? You are entirely wrong. Go read an (unspecified) elementary text book. (See below for a less emotional response.) --Art Carlson 09:05, 5 July 2006 (UTC)

1 MeV per electron is a really interesting number, too. In a perfect system, you first supply magnetic energy of 6 MeV for each p-B11 pair (one electron for p and 5 for B), plus a bit for thermal energy. When they fuse - and let's assume they all do -- each pair releases 8.7 MeV in the form of fast alphas. Now you have about 15 MeV of energy, partly in the form of magnetic fields and partly in the form of fast particles. This energy has to be extracted, converted to electricity, and reinserted with an total efficiency of 6/15 = 40% to break even. That includes the efficiency of the direct convertor, the efficiency of the capacitor banks and magnetic coils, the production of the plasmoid, bremsstrahlung losses, and losses due to heat conduction and instabilities. Oh, and I almost forgot, we want to have a little bit of electricity left over to sell! --Art Carlson 09:27, 6 July 2006 (UTC)

Radiation dose

2) I differentiated between operators and maintenance personell. Somebody will have to go into the reactor sometime, and they will have to worry about residual radiation from activated materials. You don't mention material damage or accidental release of inventory, so I don't know why you deleted that, too.

Activation is trivial. The beryllium electrodes after one year's use will have radioactivity equivlent to a class full of children--the radioactivity in thier bones. Material damage is similarly trivial.

Maintenance will require allowing several hours for decay of carbon 11(half-life 20.3 min) Carbon will be deposited as a solid coating--how can it escape?Elerner 02:49, 28 June 2006 (UTC)

As cited, we are talking about neutronicity on the order of 0.1%. A thousandth of an ITER is still a hell of a lot of neutrons. Even in ASDEX Upgrade (deuterium fill, density and temperature below reactor conditions, duty cycle < 0.01) we have to wear radiation badges inside the meter thick concrete walls for several hours after operation has stopped, and we wait a few weeks after shutdown before doing any serious maintenance inside the vessel. The biggest problems are usually from the activation of impurities in the materials, and materials on surfaces like to oxidize and become volatile, not to mention what happens in a fire or loss-of-flow accident. --Art Carlson 09:50, 29 June 2006 (UTC)

You are ignoring the difference between a huge device which absorbs nearly all the neutrons produced and a very small one, where only a tiny fraction of the neutrons will be absorbed before they hit the boron-10. A tokamak has very significant stored energy, while a DPF does not. TFTR used gigajoules per pulse while a DPF uses less than 100 kilojoules typically.Elerner 00:58, 5 July 2006 (UTC)

So let the amount of activated material per neutron be a factor of 10 or 100 or 1000 smaller. You will still need a license from the NRC. My statements are not exactly inflammatory. I clearly say that radiation is not a show-stopper and is a tiny problem compared to radiation with D-T fusion. Isn't that reasonable? --Art Carlson 08:43, 5 July 2006 (UTC)

I have tried to create a compromise on the radiation question.Elerner 00:38, 6 July 2006 (UTC)

I appreciate the effort to get specific and quantitative (although considering the speculative nature of fusion reactor design, I am not sure it is appropriate here). Some of the wording ("trivial", "classroom of children") is POV. The biggest problem is that I am still skeptical of the content and will need to see where the numbers came from.--Art Carlson 07:44, 6 July 2006 (UTC)

Calculation of long term radioactivity induced in Be electrodes

Be neutron x-sect 10^-28 cm^2 for broad range of neutron energy

50% of neutron encounter average 1 cm Be 9

sp. Gr. 1.85

1.2x10^23 atoms/cm^2

1.2x10^-5 abs rate

6.5x10^-3 neutrons per pB11 reaction

7.8x10^-10 Be10/reaction

5.6x10^4Be10/J

4.8x10^11Be10/s

1.4x10^19Be10/yr

1.9x10^5Bq (decays/s)

5 microCuries

1 microCurie/MW-year

So I am restoring my remarks.Elerner 02:28, 11 July 2006 (UTC)

Eric Lerner's version of the radioactivity story is unacceptable for two reasons:

1. It is reasonable to expect a problem, and
2. even if the problem could be eliminated, it is difficult to prove that.

If we have to start from scratch and are not experts, the most reasonable thing to do is to look for a similar system that has been analyzed by experts. The closest thing to an aneutronic fusion reactor is a "conventional" fusion reactor. The advantage of this approach is that the reference point has a high degree of realism, because people have taken a serious look into the details of the problem and have tried hard to fix as much as they can. There the biggest radiological problems are tritium (which we can ignore here) and activation and damage of materials by neutrons. The biggest single difference is that the fraction of energy carried by neutrons in an aneutronic reactor is approximately 1000 times smaller. Therefore the simplest estimate is that the problems in an aneutronic reactor are 1000 times smaller than in a D-T reactor of the same size. There may be some factors that shift this result. Eric mentions, for example, the low average energy of the neutrons. Since we normalized to neutron power, lower energy means a higher flux. It may help some to spread the energy over a larger number of neutrons, but I don't know how to quantify the effect.

The other approach to estimating the radiological problems is to do it ourselves from scratch. Eric made a first cut by considering neutron absorption reactions in the primary materials Be, Al, Cu, and W. He concluded that the most difficult reaction is Be9 (n,gamma) Be10, and that the radioactivity produced would be negligible. This is in blatant contradiction to the estimate above, so Lerner's method requires a closer look. Be10 has a half-life of 1.5 million years. To illustrate the necessity of going into detail, suppose there is an impurity in the beryllium or a minor structural material at a level of 0.1%, whose neutron absorption product has a half-life of 100 years. The activity due to this impurity would be 15 times larger than the activity of Be10. (Math available on request, if that went by too fast.) Among the questions that would have to be answered (An expert would certainly have a different list and probably a longer one.) are:

• What are the weigth, pressure, and magnetic forces expected, and what are the structural materials used to contain them?

Once again you are ASSUMING that there are powerful magnetic fields that need to be contained by strong strcutural materials. Once again, decades of experience with the plasma focus show that the powerful fields are in the plasmoids and that the fields affecting the structure are much smaller.Elerner 18:41, 26 July 2006 (UTC)

• What are the primary impurities in industrial sources of the matrials used (Be, Al, Cu, and W, at the least)?

Just out of curiosity, I looked up the imputities in ultra-high purity commercial Be. No long-lived(more than 30 y lifetime) impurity radioisotopes will contribute even 10% of the radioactivity of the Be-10. Cd112, 14 y lifetime, will about double the radioactivity initially. But the anode would still be well within NRC safety regulations and would not be considred as radioactive waste. It could be safely recycled into new Be products.Elerner 01:21, 28 July 2006 (UTC)

• Have all (n,p), (n,alpha), (n,2n), and (gamma,n) reactions been considered?
• Is isotopic tailoring of the boron fuel, the hydrogen fuel, the water coolant, or any of the solid materials required to suppress unwanted reactions?

This, of course, is getting into way too much detail for interested laymen writing an encyclopedia article. We should be trying to simply state the current state of knowledge, not trying to extend it. I would like to have a statement summarizing the current state of knowledge at the end of the section, but maybe we would do better to refrain. Maybe we need to get the opinion of an expert in materials and radiation. (Sorry Eric, you may have "credentials" in plasma physics or cosmology, but I don't believe you have ever been funded to do research on nuclear materials engineering.) Maybe we need a statement along the lines of this: While no radiological study of a detailed reactor design has been published, it appears that radiation will not present any problems that cannot be overcome. Make your own suggestion, Eric, but I won't let you leave your version as it is. I hope you now understand why. --Art Carlson 12:50, 26 July 2006 (UTC)

The difficulty only arises because you have devoted a large section of the article to trying to prove that radiation is a problem.Elerner 18:41, 26 July 2006 (UTC)

Suppression of bremsstrahlung

3) Your numbers for the 1.5 MA conditions are fusion/bremsstrahlung = 2.1 with suppression of bremsstrahlung by the field and 0.97 without. That is my factor of 2 (actually 2.16). Where does your factor of 5 come from? For that matter, the ratio of 0.97 with no quantum effect is suspect. Rider got a maximum ratio of 0.57. (If I recall, he also took credit for a monoenergetic velocity at the resonance and spin polarization, neither of which you claim.) Anyway, the bottom line is that you published a ratio of 2.1 (whether it is right or wrong).

--Art Carlson 08:05, 27 June 2006 (UTC)

Why don't we arrive at consensus offline by email-more effcient? Rider made a few mistakes, which are not really worthwhile correcting. I see where you got the factor of 2. But that was based on earlier calculations and an arbitrary T of 300keV. I am about to submit for publication simulations that show much higher T being reached.Elerner 02:49, 28 June 2006 (UTC)

At any rate, the key point is that ,without the magnetic effect, ignition is either not reached or barely reached, while with it, it is achieved by a good margin.Elerner 02:49, 28 June 2006 (UTC)

Another question, if I may: Is my impression correct that you do not envision any mechanism of confinement parallel to the field (other than inertial)? --Art Carlson 15:30, 27 June 2006 (UTC)

No, that's wrong. The plasmoid field is a force free toroid. The only escape is along the axis in the beam. Ions makes tens of thosuands of orbits before exiting in the beam.Elerner 02:49, 28 June 2006 (UTC)

A toroid in free space will rapidly expand. If you don't have another trick, your confinement won't be much better than inertial. --Art Carlson 09:10, 29 June 2006 (UTC)

Please Art, don't be a Joshua. You were once involved in the field--read the literature on plasma focus before making more changes. Force-free toroids are quite stable. This is not just theory--they have been observed for 40 years to have lifetimes tens of thousands of times longer than the Alfven velocity crossing-time. They are not formed by shocks, but by a kink instability in the vortex filaments. The critical parameter in whether or not cyclotron (synchrotron is more accurate even at 50keV) radiation escapes is the ratio of cyclotron frequency to plasma frequency, which is completely different than the ratio of plasma pressure to magnetic pressure. So everything you wrote was wrong. Please, check the literature before trying again. Why are you so intent on attacking this effort? Read up on it before you draw your conclusions.Elerner 00:58, 5 July 2006 (UTC)

1) It is a well-established and well-known result of magnetohydrodynamics - found in many textbooks - that any configuration of plasma and fields in a vacuum will expand on the Alfven time.

Cyclotron radiation (2)

2) The statement $\omega _{p}>\omega _{c}/2;$ is mathematically equivalent to $nkT/(B^{2}/2\mu _{0})>kT/(2m_{e}c^{2})=kT/(1\,{\mbox{MeV}})$. If you can't do the math yourself, you have disqualified yourself as a physicist.

According to the NRL plasma formulary, p.29, the ratio of electron plasma frequency to gyrofrequency is proportional to n^1/2/B. You have asserted, in your addition to this article, that the ratio is that of plasma pressure to magnetic field pressure, which is proportional to nT/B^2. Your error is that you have introduced a factor of T which does not belong there.Elerner 00:38, 6 July 2006 (UTC)

Look again. I also have a factor of T on the right hand side. Your mistake. --Art Carlson 07:49, 6 July 2006 (UTC)

Right, you can introduce T into any equation by multiplying both sides of the equation by T. We all learned this in high school. But that does not miraculously create a real physical dependence on T. If you take T back out, you can see that the criteria depends on the plasma density, not the plasma pressure. Different physical quantity. In dimensionless terms, the ratio is the square root of the ratio of the rest energy density of the plasma to the magnetic energy density.Elerner 00:21, 7 July 2006 (UTC)

I found it helpful to express the relation in terms of β and T. For some purposes, using n and B as the variables is more useful. The physics is always the same no matter what set of variables are chosen. You seem to realize that now, although before you refered to my "error" and wrote, Your formulas are entirely wrong. I don't expect it, and I won't hold a grudge, but a simple "I'm sorry I called you stupid" would seem appropriate here. --Art Carlson 07:55, 7 July 2006 (UTC)

You continually put back in the incorrect formula that the ratio of plasma frequency to cyclotron frequency depends on the ratio of plasma pressure to magnetic pressure or beta. But the first ratio is proportional to nT/B^2, while the second is proportional to n^1/2/B. Depending on T, beta could be high or low and still have plasma frequency exceeding cyclotron frequency. I took out the incorrect formula again.Elerner 03:20, 11 July 2006 (UTC)

Equilibrium and time scales

3) You're the one pushing the DPF. You must have current references (other than your own) supporting your positions. Just give them to me, instead of demanding me to do a literature search as if I were getting paid to do it.

Bostick, W.H. et al, Ann. NY Acad. Sci., 251, 2 (1975).
Brzosko, J.S. et al, Physics Letter A, 192, 250 (1994).
G.R.Neil, R.S. Post, Plasma Phys., 14, 425 (1988).
I.Volobuev et al, Sov. J., Plasma Phys., 14, 401 (1988)
Brzosko, J.S. and Nardi, V., Physics letters A, 155, 162 (1991)
Brzosko, J.s. et al, Phys. Plasmas, 2, 1259(1995)

Elerner 00:38, 6 July 2006 (UTC)

Thanks. I'll be back. --Art Carlson 07:49, 6 July 2006 (UTC)

OK. I looked through these papers (except the first one: the IPP library does not carry the Annals of the New York Academy of Sciences). I didn't find a single mention of the Alfven time (hardly any mention of magnetic fields), plasmoids, equilibrium, or the virial theorem. The times, dimensions, and temperatures mentioned usually seemed compatible with thermal disassembly. I found one phrase in Brzosko (1994) especially vivid. He refers to "the explosive decay of the magnetic structure of organized plasma filaments". So, having now "read the literature", I can see no support for your statements Force-free toroids are quite stable. This is not just theory--they have been observed for 40 years to have lifetimes tens of thousands of times longer than the Alfven velocity crossing-time. I guess I am now obligated to revert your edits. --Art Carlson 21:51, 6 July 2006 (UTC)

You can find substantially the same material in Colloques internationaux CNRS no. 242, p.129-138. Also J. Plasma Physics 8, 7(1972) To summarize the numbers reported in this work, which has been know for a very long time, B~10^8 G, radius~200 microns n `10^20/cc, plasmoid lifetime~10 ns. It is easy to calculate that for these numbers the Alfven velocity is about 1.5cm/ns and the crossing time is then 26 ps, 400 times shorter than the lifetime. (The plasmoids in the experiments we performed were a lot smaller, so the lifetime is thousands, not hundreds of times longer than the crossing time.) The later sources confirm the high density (in fact observe higher density) and the confinement times of tens of ns. I notice you did not reply on the pinch effect. That makes it no mystery why confinement times have no relationship to the Alfven crossing time.Elerner 00:21, 7 July 2006 (UTC)

Sorry, I could not find either of these references in the library or on line. Numbers like 200 microns, 10^20 cm^-3 (or higher), a few keV, and 10 ns showed up in several of the papers, apparently refering to simultaneous conditions in the hot spots, so we can play with those numbers for now (although a DPF discharge is a complex process taking place on several disparate time- and length-scales, so caution is in order). The number I haven't seen there is 10^8 G = 10 kT. At a radius of 200 microns, such a field could be produced by a current of 10 MA, which I guess is higher than the total current by an order of magnitude or so. I could conceive of the field getting this high in a sausage instability, but not staying that way for very long. Therefore: Please, Eric, send me an electronic copy of one or both papers, if you have them, and tell me whether the measurement of magnetic field is time-resolved and the high values persist for the 10 ns you mention. --Art Carlson 11:44, 7 July 2006 (UTC)

Even in the absence of magnetic field estimates, the data you correctly cited indicates that confinement is occurring for times much longer than the Alfven crossing time. The plasma energy density even with a few KeV (and some of the sources speak of tens of KeV) and 10^20/cc (and some speak of 10^21) is at least 5x10^11 erg/cc. So the confining magnetic field, even with a beta of unity, is at least 3.5MG. Even with this unrealistically conservative estimate, the confinement time is still at least 15 times the Alfven crossing time.Elerner 04:04, 8 July 2006 (UTC)

That's a round-about way to talk about the ion acoustic wave. You don't need either the density or the field for that calculation.

$c_{s}={\sqrt {\frac {ZKT_{e}}{m_{i}}}}$

I have taken gamma_e=1 and assumed T_e>>T_i. For most fully ionized gases (e.g. deuterium) Z/μ is about 1/2. Then c_s at 1 keV is about 22 mm/μs (check my math) and the transit time of 0.2 mm is 10 ns. That fits the data real well, although calculations of this kind can have lots of factors of two flying around. --Art Carlson 11:26, 8 July 2006 (UTC)

I did check your math. That's 22cm/μs, so your calculation is off by a factor of 10. Factors of ten need to be watched even more than those pesky factors of two.;)Elerner 04:34, 9 July 2006 (UTC)

Shucks. I was afraid of that. (Excuse: I was tired and had a bad conscience because there are other things I should rather be doing.) That's close to your calculation and, admittedly, makes my position less comfortable. I need at least three factors of two, or two factors of pi. Faced with the choice between finding them and giving up the theory of MHD, I would rather dig around in rough numbers, experimental errors, and imprecise statements. A promising place to start would be the question of whether the size, time, and temperature quoted were really measured simultaneously. (Obviously not, in this case, since I fabricated the 1 keV temperature myself.) Of course, if you can offer solid measurements to back up your claim of a factor of thousands instead of a measly factor of ten, we would have a different discussion. I would also dearly love to see a discussion of these numbers by a scientist who is aware of the potential conflict with virial theorem and the consequences that would imply. --Art Carlson 08:16, 10 July 2006 (UTC)

I see that the one common element between your comments here and on plasma cosmology is the entirely un-scientific approach that theory trumps observation. Plato thought that too. But scientists believe in observation over theory. This is especially the case as you are appealing to an approximation of EM theory that is not relevant here. In MHD, you have, by definition, a collisional plasma in which force-free configurations get wiped out by collisions. But we are dealing with non-collisional plasma here and if you apply MHD to such plasma you get what Alfven called “pseudo-plasma”—hypothetical entities that don’t bear any resemblance to the real plasma. Also, if you were really right that all pressure gradients had to be balanced by the walls, there would be no pinch effect, since the filaments created are very far from walls. Where are the walls in the solar corona that confines those nice filaments in prominences? Elerner 03:20, 11 July 2006 (UTC)

4) Can we go back to a civilized tone now?

--Art Carlson 09:05, 5 July 2006 (UTC)

P.S. Point (1) can be proved quantitatively and in elegant generality, but there is also a Physics 101 version: Take any configuration of plasma and magnetic field that is confined to a finite volume. Expand all dimensions by a factor R, keeping the magnetic flux through any surface constant. Since the area goes with R^2, B must go with 1/R^2. The magnetic energy thus drops with B^2*V ~ (1/R^2)^2*(R^3) ~ 1/R, and, of course, with adiabatic expansion the plasma energy per particle drops, too. 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. (If you follow me this far but don't believe that the disruption time is close to the Alfven transit time, I can continue the calculation.) --Art Carlson 14:42, 5 July 2006 (UTC)

Reference added in proof: In a PPPL report, Hammett, Jardin and Stratton, citing both Shafronov and Freidberg, write One common use of the virial theorem is to take the inner product of this equation with the position vector x and integrate over all space to show that an isolated MHD equilibrium can not exist by itself (unless there are physical coils or gravity to provide overall force balance). (In case anybody thinks I'm just making this stuff up.) --Art Carlson 12:36, 10 July 2006 (UTC)

First, the MHD approximation is not appropriate in non-collisional plasmas where the gyrofrequency greatly exceeds the collision frequency. Alfven, the inventor of MHD approximation, pointed this out repeatedly, including in his Nobel address. Second, it would be incorrect to describe a force-free toroid as strictly isolated, since the magnetic field lines, and current runs along the axis out to large distances. Elerner 03:20, 11 July 2006 (UTC)

In the well-known pinch effect, the magnetic field created by electric currents through a plasma creates inward-directed forces which confine the plasma. While pinches can be disrupted by instabilities, they can also remain stable for far longer than the Alfven crossing time. An example familiar to all is a lightning stroke, where plasma heated strongly by the lightning current is also prevented from expanding by the pinch forces of the current for an appreciable fraction of a second, much longer than the Alfven crossing time. Thunder occurs when the heated plasma suddenly expands after the current ends.

In a DPF, a kink instability within the pinched filament creates the plasmoid, which is then metastable for ns to tens of ns, far longer than the Alfven crossing time of the order of 1 ps. The plasmoid is confined by the pinch forces of the current flowing through it.

A good description of the pinch effect and the formation of force free-filaments (in an astrophysical context) is in Alfven and Falthammar, Cosmical Electrodynamics, p. 192-199. This book is out of print, but available in any physics library. Force-free configurations are minimum energy configurations.Elerner 00:38, 6 July 2006 (UTC)

(Sorry. I overlooked this section before.) The question is the boundary conditions. A pinch provides radial confinement but not axial. If you pinch a toroid off a filament, it may have confinement in the minor radius, but it will expand in major radius. (The toroidal currents on opposite sides of the loop will repel each other.) The "force-free-configurations" in a Reversed field pinch or a Spheromak are confined by a conducting wall. (I will check out your experimental references.) --Art Carlson 08:24, 7 July 2006 (UTC)

In a force-free toroid, there is a strong component of the current running along the axis, which provides a confining force against expansion of the major radius, perpendicular to the axis. There is escape along the axis. This is in the form of an accelerated beam which leads to both the heating and eventual evacuation of the plasmoid. But this takes a great deal longer than an Alfven crossing time.The toroid is metastable, not stable, but lasts quite long.Elerner 04:04, 8 July 2006 (UTC)

Sorry, doesn't fly. Won't give you a net radial j×B. --Art Carlson 10:50, 8 July 2006 (UTC)

In a force-free configuration there never is a jxB force. J is always colinear with B. That's what makes it force-free. The ions travel along the lines of force.Elerner 04:25, 9 July 2006 (UTC)

That is quite a trick of yours: In a force-free toroid, there is ... a confining force. No j×B, no force, no confinement. --Art Carlson 08:47, 9 July 2006 (UTC)

If there is stable confinement, of course there is no net force. If a particle, however, hit by a collision, moves radially,it has to move across the field lines and then there is a jxB force until it returns to the direction of the field lines.Elerner 22:29, 9 July 2006 (UTC)

Sometimes I almost start to take you seriously, and then you come up with something like this that makes me wonder if you understand anything about plasmas. You probably don't mean "stable confinement" ("stable" means there is a restoring force on a perturbation, "confinement" refers to any mechanism that slows down loss of energy or particles) but "equilibrium", which is the lack of net forces. That lack of net forces refers to the sum of jXB and pressure gradients. If you claim that there is neither a net force nor a jXB force in a DPF equilibrium, then you are claiming that there is no pressure gradient. That means either you have no pressure at all - not a very interesting situation - or the pressure at the wall is the same as the maximum pressure, in which case the pressure you claim is not compatible with limits of consturction materials. j is always the macroscopic current density. If you want to talk about individual particles without having to face uncomfortable singularities, it is better to talk about the Lorentz force qv×B. It also sounds like you think that this force tends to restore motion parallel to the field. It doesn't. If you are only talking about gyro-motion, that, in the absense of a pressure gradient, does not result in a macroscopic current density. I am willing to overlook sloppy language, but only if I can detect physical understanding behind it. --Art Carlson 07:43, 10 July 2006 (UTC)

See above on walls, etc.Elerner 03:23, 11 July 2006 (UTC)