Quasi-axisymmetric magnetic fields: weakly non-axisymmetric case in a vacuum
An asymptotic expansion is performed to obtain quasi-axisymmetric magnetic configurations that are weakly non-axisymmetric. A large space of solutions is identified, which satisfy the condition of quasi-axisymmetry on a single magnetic flux surface, while (non-axisymmetric) globally quasi-axisymmetric solutions are shown to not exist, agreeing with the conclusions of previous theoretical work. The solutions found are shown to be geometrically constrained at low aspect ratio or high toroidal period number. Solutions satisfying the more general condition of omnigeneity (generalized quasi-axisymmetry) are also shown to exist, and it is found that quasi-axisymmetric deformations can be superposed with an omnigenous solution, while preserving the property of omnigeneity, effectively extending the space of "good" configurations. A numerical solution of the first order quasi-axisymmetry problem is demonstrated and compared with solutions found with a widely used MHD equilibrium solver, independently verifying that quasi-axisymmetry is satisfied at the appropriate order. It is thereby demonstrated that approximately quasi-axisymmetric solutions can be directly constructed, i.e. without using numerical search algorithms.
Three‐dimensional equilibrium of the anisotropic, finite‐pressure guiding‐center plasma: Theory of the magnetic plasma
Theoretical and numerical methods now give a complete solution to the problem of finite‐β plasma equilibrium in mirror magnetic wells and toroidal devices. The equilibria can be made consistent on all of the progressively longer time scales of the guiding‐center fluid model, including the particle magnetic drifts and the Coulomb scattering equilibrium of a neutral injected plasma. The theory of equilibrium in the guiding‐center fluid model of a finite‐β plasma with an arbitrary, anisotropic pressure tensor can be formulated as a classical magnetostatic system: ∇⋅B = 0, ∇×H = 0, B = H + 4πM(B). The plasma magnetization is found explicitly in terms of three physically distinct components related to the laws of conservation of magnetic moment, of longitudinal invariant, and of the sign of the velocity along B of particles that do not undergo mirror reflection. A condition is derived upon the field geometry whereby a large class of special equilibria can be found in which all particles on a given line drift o...
The unfolded universe of elementary particles. A geometric explanation of the standard model structure
We propose a geometric explanation of the standard model of Glashow, Weinberg and Salam for the known elementary particles. Our model is a generic Quantum Field Theory in dimension four, obtained by developing along a Lorentz sub-manifold the lagrangian of Einstein and Dirac in dimension twelve. The main mechanism which gives birth to the standard model is a certain gauge fixing of triality, which permits to identify the multiplicity of fermions, as seen from the four dimensional world, with the eight unseen dimensions of the generating universe. In this way we get the known tables of particles, explaining the series of fermions and the gauge bosons. We suggest that the Higgs field dynamics could appear through a bosonization of the right handed neutrino and correspond to a displacement in the unseen dimensions. We also propose hypotheses for dark matter, and perhaps dark energy. Then we suggest predictions to go beyond the standard model.