Description
The Hamiltonian structure of the four-field reduced model of single-helicity and incompressible MHD in cylindrical geometry was found in [1]. An extremum of the summation of the Hamiltonian and the Casimir invariants, or the energy-Casimir functional, corresponds to an equilibrium [2]. The extremum is obtained by setting the first variation of the energy-Casimir functional zero, which yields four equations for this model. The four equations can be summarized in a single elliptic partial differential equation for a magnetic flux function. In principle, the equilibrium equation, together with appropriate boundary conditions, can be solved like the Grad-Shafranov equation. The solution can be a helically symmetric MHD equilibrium.
The Casimir invariant includes four arbitrary functions of the magnetic flux function. For cylindrically symmetric equilibria, which are trivial ones, we found that the arbitrary functions and its derivatives can diverge at rational surfaces where the single helicity dynamics resonates. This divergence separates the plasma region into multiple regions, where the helical components of the magnetic flux function must be connected properly across the boundaries of the regions. We have obtained helically perturbed equilibria by solving a linearized equilibrium equation with the connection conditions using the same arbitrary functions of the Casimir invariants for the cylindrically symmetric equilibrium. The numerical code is now extended to solve the equilibrium equation without linearization to obtain helical equilibria.
M.F. was supported by JSPS KAKENHI Grand Number JP24K06993.
[1] M. Furukawa and M. Hirota, Physics of Plasmas 32, 012111 (2025).
[2] V.I. Arnol’d, J. Appl. Math. Mech. 29, 1002–1008 (1965).