Relaxed Plasma State in Compact Tori

Abstract

Ever since J. B. Taylor (Phys Rev Lett 33:1139, 1974) published his theory with a model to derive the Taylor state \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{\nabla } \times \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{\text{B}} = {{\upmu}}\overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{\text{B}} \) and to explain the reversed field pinch and spheromak configuration, magnetic helicity has been believed to play an important role as a global invariant in the process of self-organization and relaxation for magnetized plasmas. In the present work, the lowest eigenvalues μmin, which associated with relaxed force-free compact toroidal plasma devices are calculated numerically. We prove the applicability of the collocation method through out a developed numerical programme to investigate such task. We calculate μmin for toroidal compact tori (CT) with arbitrary cross-sections and aspect ratios approaches to unity (\( {{\upalpha}} = {\text{R/a}} \to 1 \), R and a are the major and some sort of minor radius respectively). For a perfect conducting CT wall, we were able to obtain new results concerning satisfaction of the toroidal flux vanishing boundary condition \( \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{\text{B}} \cdot \overset{\lower0.5em\hbox{$\smash{\scriptscriptstyle\rightharpoonup}$}}{\text{n}} = 0 \). Besides, the magnetic field topology inside the CT are well represented. The plots showed a good fulfilment of the boundary condition along the whole boundaries of different cross sections. Dependence of μ on the aspect ratio is also obtained. Several runs of the programme for various wave numbers k showed that μa is very insensitive to the choice of k.