Magnetic Force-Free Theory: Nonlinear Case

In this paper, a theory of force-free magnetic field useful for explaining the formation of convex closed sets, bounded by a magnetic separatrix in the plasma, is developed. This question is not new and has been addressed by many authors. Force-free magnetic fields appear in many laboratory and astrophysical plasmas. These fields are defined by the solution of the problem ∇×B=ΛB with some field conditions B∂Ω on the boundary ∂Ω of the plasma region. In many physical situations, it has been noticed that Λ is not constant but may vary in the domain Ω giving rise to many different interesting physical situations. We set Λ=Λ(ψ) with ψ being the poloidal magnetic flux function. Then, an analytic method, based on a first-order expansion of ψ with respect to a small parameter α, is developed. The Grad–Shafranov equation for ψ is solved by expanding the solution in the eigenfunctions of the zero-order operator. An analytic expression for the solution is obtained deriving results on the transition through resonances, the amplification with respect to the gun inflow. Thus, the formation of Spheromaks or Protosphera structure of the plasma is determined in the case of nonconstant Λ.

Analysis of the isotropic and anisotropic Grad–Shafranov equation

Pressure anisotropy is a commonly observed phenomenon in tokamak plasmas, due to external heating methods such as neutral beam injection and ion-cyclotron resonance heating. Equilibrium models for tokamaks are constructed by solving the Grad–Shafranov equation; such models, however, do not account for pressure anisotropy since ideal magnetohydrodynamics assumes a scalar pressure. A modified Grad–Shafranov equation can be derived to include anisotropic pressure and toroidal flow by including drift-kinetic effects from the guiding-centre model of particle motion. In this work, we have studied the mathematical well-posedness of these two problems by showing the existence and uniqueness of solutions to the Grad–Shafranov equation both in the standard isotropic case and when including pressure anisotropy and toroidal flow. A new fixed-point approach is used to show the existence of solutions in the Sobolev space $H_0^1$ to the Grad–Shafranov equation, and sufficient criteria for their uniqueness are derived. The conditions required for the existence of solutions to the modified Grad–Shafranov equation are also constructed.

Diffusive time evolution of the Grad–Shafranov equation for a toroidal plasma

We describe the evolution of a plasma equilibrium having a toroidal topology in the presence of constant electric resistivity. After outlining the main analytical properties of the solution, we illustrate its physical implications by reproducing the essential features of a scenario for the upcoming Italian experiment Divertor Tokamak Test Facility, with a good degree of accuracy. Although we find the resistive diffusion time scale to be of the order of $10^4$ s, we observe a macroscopic change in the plasma volume on a time scale of $10^2$ s, comparable to the foreseen duration of the plasma discharge by design. In the final part of the work, we compare our self-consistent solution to the more common Solov'ev one, and to a family of nonlinear configurations.

Simple, general, realistic, robust, analytic tokamak equilibria. Part 1. Limiter and divertor tokamaks

Tokamak equilibria have been derived that are analytic solutions to the Grad–Shafranov equation. This paper, Part 1, describes a wide range of such equilibria including smooth limiter surfaces, double- and single-null divertor surfaces, arbitrary aspect ratio, elongation, triangularity and beta. Part 2 generalizes the analysis to include edge pedestals and toroidal flow.

On quasisymmetric plasma equilibria sustained by small force

We construct smooth, non-symmetric plasma equilibria which possess closed, nested flux surfaces and solve the magnetohydrostatic (steady three-dimensional incompressible Euler) equations with a small force. The solutions are also ‘nearly’ quasisymmetric. The primary idea is, given a desired quasisymmetry direction $\xi$ , to change the smooth structure on space so that the vector field $\xi$ is Killing for the new metric and construct $\xi$ –symmetric solutions of the magnetohydrostatic equations on that background by solving a generalized Grad–Shafranov equation. If $\xi$ is close to a symmetry of Euclidean space, then these are solutions on flat space up to a small forcing.

The impact of superconductivity and the Hall effect in models of magnetised neutron stars

Equilibrium configurations of the internal magnetic field of a pulsar play a key role in modelling astrophysical phenomena from glitches to gravitational wave emission. In this paper, we present a numerical scheme for solving the Grad–Shafranov equation and calculating equilibrium configurations of pulsars, accounting for superconductivity in the core of the neutron star, and for the Hall effect in the crust of the star. Our numerical code uses a finite difference method in which the source term appearing in the Grad–Shafranov equation, which is used to model the magnetic equilibrium is non-linear. We obtain solutions by linearising the source and applying an under-relaxation scheme at each step of computation to improve the solver’s convergence. We have developed our code in both C++ and Python, and our numerical algorithm can further be adapted to solve any non-linear PDEs appearing in other areas of computational astrophysics. We produce mixed toroidal–poloidal field configurations, and extend the portion of parameter space that can be investigated with respect to previous studies. We find that in even in the more extreme cases, the magnetic energy in the toroidal component does not exceed approximately 5% of the total. We also find that if the core of the star is superconducting, the toroidal component is entirely confined to the crust of the star, which has important implications for pulsar glitch models which rely on the presence of a strong toroidal field region in the core of the star, where superfluid vortices pin to superconducting fluxtubes.

Analytical studies of PROTO-SPHERA equilibria

Analytical solutions of the Grad–Shafranov equilibrium equation in simply connected plasma configurations, comprised of toroidal magnetic surfaces and open surfaces connected to electrodes, are reviewed and generalised. The Grad–Shafranov equation is linearised introducing assumptions on plasma current and pressure, which preserve regularity of solutions on the symmetry axis, as required for a simply connected geometry. Particular solutions are found by separation of variables both in cylindrical coordinates and in spherical ones. Equilibria that model local or global features of PROTO-SPHERA plasmas are constructed by combining a few particular solutions.