Equilibrium analysis of four-field reduced model on single-helicity and incompressible MHD based on energy-Casimir method

A four-field reduced model on single-helicity and incompressible MHD in cylindrical geometry was derived, and the Hamiltonian structure of the model was clarified [1]. By setting the first variation of the energy-Casimir functional zero, four equations expressing equilibria are derived. It is found that these equations can be summarized in a single elliptic partial differential equation for a magnetic flux function, which is similar to the Grad-Shafranov equation. By solving the equilibrium equation under an appropriate boundary condition, helically-symmetric MHD equilibria can be obtained. Four arbitrary functions of the magnetic flux function exist in the Casimir invariant, and these appear in the equilibrium equation. The relations of the arbitrary functions to the well-known quantities such as the pressure and the safety factor are examined. For cylindrically symmetric equilibria without a plasma flow, only two arbitrary functions remain in the equilibrium equation, and their physical meanings can be understood in terms of the pressure and the safety factor.

M.F. was supported by JSPS KAKENHI Grand Number JP24K06993.

[1] M. Furukawa and M. Hirota, Physics of Plasmas 32, 012111 (2025).