Particle motion in 3D MHD equilibria versus relaxed states

The complex interplay between electromagnetic fields and the motion of particles is at the heart of collective plasma phenomena. In toroidal devices such as tokamaks or stellarators, part of the challenge to achieve fusion is to nurture a magnetic configuration in which the particles composing the fusing plasma cannot easily escape while they gain enough energy to reach ignition temperatures. A great variety of instabilities (linear modes), non-linear processes and collisional phenomena spoil the absolute confinement of particles and degrade the conditions for fusion. The trajectory of charged particles in magnetic fields can be pictured as a succession of infinitesimal rotations about a moving point called the guiding-centre. The fascinating relation between the topology of field-lines and particle orbits transpires as the Aharonov–Bohm effect in the quantum world. Classically, the motion is polarised in a way that the parallel dynamics detach from the perpendicular. In order to understand particle dynamics within plasmas, it is thus important to dissect the structure of magnetic fields. It is insightful to notice that magnetic field-lines are analogous to Hamiltonian systems; chaoticity/stochasticity is a general feature of field-lines and their integrability only arises upon a symmetry condition. The symmetry, leading to the existence of flux-surfaces, can either be explicit in the case of translational symmetric, axisymmetry or helical symmetry, or implicit (hidden), e.g. in the case of three-dimensional equilibrium with nested flux-surface. Such stationary solutions refer to magnetic fields B which, in addition to being solenoidal ∇ · B = 0, respect the Magneto-Hydrodynamics (MHD) force balance equation j × B = ∇p. This apparently simple equation reflects something quite fundamental, namely the containment of pressure, i.e. thermal agitation of particles, via magnetism. MHD, as the description of a state of plasma by a magnetised fluid, embeds the microscopic/kinetic phenomena into the macroscopic layout of the magnetic field. The MHD model also suggests that, although the plasma remains a neutral fluid, the pressure coexists with plasma current (internal flow of electric charges), which is a curious deviation from thermodynamic equilibrium. Taylor relaxation is the process through which a plasma reaches an energy-minimising arrangement while preserving its initial helicity, a fundamental invariant (of topological origin, one might say). Such states obey the Beltrami equation ∇×B = μB and form a class of “force-free” magnetic fields, whose confinement properties remain vastly misunderstood. Can we say something about the motion of particles within relaxed plasmas ?