I have read about various variations of magnetic bottle fusion reactors for 50 years. No one seems to be able to predict the behavior of the plasmas well enough to get them to work. Since there is plenty of computers and money in the projects, obviously the equations are wrong.
I do not what but one thing is: Some books I looked at on the subject feel they do not need worry about relativity or the axial Doppler shift becuse they do not need relativistics temperatures to get fusion. But lot the products will be for a time at relativistic velocities.
The attached papers, in the form of PDF files, that the resultant of the axial transverse Doppler shifts and will have a big effects on Maxwell's equations. Even changing the dimensions.
Samuel,
A couple of quick comments off the cuff. Maybe they will provide pointers for you to continue to investigate.
1. "No one seems to be able to predict the behavior of the plasmas well enough to get them to work." is not correct. It is more of:
We have learned to predict the behavior well enough to show that they will not work.
But that does not completely describe the situation. It is more that there are parameters that show that practical configurations will probably have either instabilities or poor transport.
2. Relativistic and Quantum corrections are interesting but probably are NOT the main issues that make mirror confinement problematic.
3. A good place to start to examine the Vlasov equation for two or more species. This kinetic approach reveals many of the core issues. The single particle Vlasov equation can be rigorously derived from the N-particle distribution function (or N-particle wavefunction) and each approximation can be quantified as to their validity and for fusion applications, these approximations are valid for fusion related plasmas. The single particle wavefunction is a function of 6 variables, 3 spatial dimensions and 3 velocity dimensions. The problems creep in because the open field lines mean that the distribution function is zero or near zero for regions of velocity space where the particles will escape along the field lines. That is, for particles with low velocities perpendicular to the magnetic field, they can escape along the field lines. This leads to a situation where the derivative of the phase space distribution function with respect to velocity becomes very, very large and can drive instabilities and/or multiply transport loss rates. Look at the structure of the Vlasov equation to see how this can drive dynamics. THIS is the core problem with mirror concepts.
4. There is always more to learn. The fact that mirror concepts are not seen as a current leading contender is not necessarily indicative of where breakthrough ideas might occur. This justifies why ITER is a Tokamak rather than a mirror. If you want to get to ignition with today's ideas (and "only" a few tens of billions of dollars) then it is feasible with a Tokamak and unrealistic with a mirror. But innovation continues. For example, the Elmo Bumpy Torus arose from a mirror concept, or really a multiple mirrors where the ends were wrapped around together. Or take the Field Reverse Configuration (FRC). Much of the interest in it came from the Livermore mirror program initially where it was noticed that there was a tendency for the mirror to reverse the field for high density plasmas. Externally, the FRC is topologically simple like a mirror, but internally the plasma is contained toroidally. So when people "poo, poo" ideas, don't let them get you down but don't ignore the information they are giving even if the critique is not encouraging.
Best of luck
Hi Samuel, by "Magnetic Bottle" do you mean what is commonly known as Magnetic mirrors, or more generally magnetic confinement? In the case of Magnetic Mirrors, the more practical toroidal configurations (ITER being the most advanced example) took the scene since early '80s. We can discuss wether single particles species MHD is sufficient to drive a long-lasting discharge, but they put quite a lot of money in this experiment, and in all the "satelite" experiments.
As concering Vlasov equations and other tentative configurations, I do agree with John Cobb. You can read the papers by Boozer and his team. PLease note that magnetic field lines in such complex systems do have chaotic behavior, so it is not possible at present state of knowledge to build a device perfectly matching expectations!!
With or without relativity, Dr. Cobb pointed out there is a big problem with unstable regions. Adding the axial Doppler shift may appear at first not to add anything, because when averaged over all angles it is zero. But the variance is not zero. It is 1/3 of e maximum, causing entropy to go up. Also sections of a plasma with axial velocity components to the electromagnetic fields see full axial Doppler effect. Note: Absorption impacts have only a blue shift no averaging. Partial energy impacts also have manly blue shifts. So if and only if there is significant fusion there will significant Doppler shift effects.
But at lest one reference of the Vlasov-Maxwell equations shows relativistic terms in them without the observation angle. By observation angle, I mean: the angle between the closing velocity and the line of sight between the observer and the observed. If it takes into axial shift (which should affect mass) that angle would be in it. The error omission of the axial Doppler shift is common to works on relativity, as I pointed out in my paper (the added file, I cannot miss a chance to cite it).
But with or without relativity, Dr. Cobb pointed out there is a big problem with unstable regions. Turbulent fluid flow is difficult to solve for in common fluids like air and water so plasmas present a bigger problem.
Look at the attached files to see the effects and proofs.
Note files in the original question has been replaced by the ones below.
https://drive.google.com/open?id=1o9mW38Vyx3C7IuN6xazdS7SSYF_JRdJY
https://drive.google.com/open?id=1LiG8oeVm9eHiSYcbDJxwDvaewM4WFZWz
https://drive.google.com/open?id=1vU9ADVkG70s_UjV5MmDsrzcb7ME1XEkj
https://drive.google.com/open?id=1pj_0oKjZwGqmFbU5-yL9OrKtYI84SITS
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