Equilibrium properties of flowing plasmas have attracted much attention in the fusion research area, especially in terms of H-mode and high β plasma confinement.
In the natural world, a high β plasma with high-speed flow is observed in Jupiter's magnetosphere [1]. Plasmas in Jupiter's magnetophere rigidly rotate with the planet's daily rotation , and suffer the centrifugal force. The mechanism of plasma confinement in Jupiter's magnetosphere is still not understood well. In this study, with the use of plasma relaxation theory, it is shown that rigid rotation and high β plasma confinement is compatible. The axisymmetry in magnetic field or velocity field makes canonical angular momenta of electron and ion fluid invariants, and this restriction condition enables relaxed states to confine high β plasmas by imposing the electromagnetic. The results of numerical analysis will be shown.
Since above equilibrium equation has only toroidal flow, it has a same structure as Grad-Shafranov (GS) equation for static equilibrium. While GS theory is clearly understood, including poloidal flow in ideal-MHD equilibrium makes the problem rather complicated. For compressible flow, the equilibrium equation transits between hyperbolicity and ellipticity according to flow speed and has a singularity where poloidal flow speed coincides with poloidal Alfvén speed [2]. For incompressible flow, equilibrium equation becomes always elliptic, however, still has a singularity. It is shown that this singularity corresponds to the Alfvén resonance intensively studied in linear theory. Moreover, it is shown that the singularity has an origin in alighnment of poloidal flow and poloidal magnetic field, i.e., degeneracy of characteristics. When the singularity occurs, the equilibrium equation has a singular solution. These are shown in 1-D analysis of the equilibrium equation.
Since a singular equilibrium solution does not exists in physical systems, it is expected that Hall effect as singular perturbation to ideal-MHD removes this singularity by changing characteristics. In fact, it is shown that the Hall-MHD equilibrium equation has no singularity, and has a regular solution with trans-Alfvénic flows.

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