Relaxation models for single helical reversed field pinch plasmas

In this paper, a relaxation theory for plasmas where a single dominant mode is present [Bhattacharjee et al., Phys. Rev. Lett. 45, 347 (1980)], is revisited. The solutions of a related eigenvalue problem are numerically calculated and discussed. Although these solutions can reproduce well, the magnetic fields measured in experiments, there is no way within the theory to determine the dominant mode, whose pitch is a free parameter in the model. To find the preferred helical perturbation, a procedure is proposed that minimizes the "distance" of the relaxed state from a state which is constructed as a two region generalization of the Taylor's relaxation model [Taylor, Phys. Rev. Lett. 33, 1139 (1974); Rev. Mod. Phys. 58, 751 (1986)] and that allows current discontinuities. It is found that this comparison is able to predict the observed scaling with the aspect ratio and reversal parameter for the dominant mode in the Single Helical states. The aspect ratio scaling alone is discussed in a previous paper [Paccagnella, Nucl. Fusion 56, 046010 (2016)] in terms of the efficient response of a toroidal shell to specific modes (leaving a sign undetermined), showing that the ideal wall boundary condition, a key ingredient in relaxation theories, is particularly well matched for them. Therefore, the present paper altogether [Paccagnella, Nucl. Fusion 56, 046010 (2016)] can give a new and satisfactory explanation of some robust and reproducible experimental facts observed in the Single Helical Reversed Field Pinch plasmas and never explained before.