Theory of plasma confinement in non-axisymmetric magnetic fields

Abstract

The theory of plasma confinement by non-axisymmetric magnetic fields is reviewed. Such fields are used to confine fusion plasmas in stellarators, where in contrast to tokamaks and reversed-field pinches the magnetic field generally does not possess any continuous symmetry. The discussion is focussed on magnetohydrodynamic equilibrium conditions, collisionless particle orbits, and the kinetic theory of equilbrium and transport. Each of these topics is fundamentally affected by the absence of symmetry in the magnetic field: the field lines need not trace out nested flux surfaces, the particle orbits may not be confined, and the cross-field transport can be very large. Nevertheless, by tailoring the magnetic field appropriately, well-behaved equilibria with good confinement can be constructed, potentially offering an attractive route to magnetic fusion. In this article, the mathematical apparatus to describe stellarator plasmas is developed from first principles and basic elements underlying confinement optimization are introduced.

A rotating and magnetized three-dimensional hot plasma equilibrium in a gravitational field

“…We present a three-dimensional analytic equilibrium solution for hot, toroidally rotating, gravitationally and magnetically confined plasma. In addition to satisfying Ampere's law, quasineutrality, continuity, and momentum conservation, our solution is consistent with the kinetic constraints on a drifting Maxwellian that are imposed by the Fokker-Planck equation (Hinton and Wong 1985;Catto et al 1987;Helander 2014).…”

“…Unlike Blandford and Payne (1982), our results do not indicate any restriction on the angle that the poloidal field makes with the equatorial plane. However, their treatment allows poloidal flow along the magnetic field lines that is not allowed by our kinetic constraints (Hinton and Wong 1985;Catto et al 1987;Helander 2014). …”

“…The relation between p and n defines the effective plasma temperature T. This temperature must be a flux function based on the kinetic constraint B · ∇T = 0 that has to be satisfied for a rotating Maxwellian to satisfy the gyro-averaged or drift kinetic equation in the frozen-in limit c∇Φ = V × B (Hinton and Wong 1985;Catto et al 1987;Helander 2014). …”

Existence of three-dimensional ideal-magnetohydrodynamic equilibria with current sheets

“…The properties and numerical computation of three-dimensional, ideal-MHD equilibria is of fundamental importance for understanding the behaviour of both magnetically confined fusion and astrophysical plasmas [1][2][3]. In particular, singular current densities at rational surfaces are predicted in equilibria with continuously-nested flux-surfaces, with and without pressure [1,[4][5][6][7]. These singular currents play a crucial role in describing (i) the plasma response to non-axisymmetric boundary perturbations [8][9][10], (ii) ideal and resistive stability [11,12], and (iii) the dynamics of reconnection phenomena, such as sawteeth [13].…”

Variational formulation of relaxed and multi-region relaxed magnetohydrodynamics

“…the 'entropy freezing' constraint prevents dynamical temperature equilibration along magnetic field lines, and the 'flux-freezing' constraint prevents changes in the topology of magnetic field lines, thus preventing magnetic reconnection, island formation or formation of chaotic lines. These constraints also give rise to mathematical problems due to a tendency for singularities to form when systems are perturbed away from simple geometries with a continuous symmetry, Grad (1967), Cary & Kotschenreuther (1985), Hegna & Bhattacharjee (1989), Bhattacharjee et al (1995), Hudson et al (2012), Helander (2014), Loizu et al (2015a,b). It is the aim of this paper to formulate an alternative self-consistent, non-dissipative single-fluid model for toroidal plasmas that is simpler than IMHD yet is less physically restrictive and better posed mathematically in general geometries.…”