Gyrokinetic Magnetohydrodynamics and the Associated Equilibrium

A proposed scheme for the calculations of gyrokinetic MHD and its associated equilibrium is discussed related a recent paper on the subject [1]. The scheme is based on the time-dependent gyrokinetic vorticity equation and parallel Ohm's law, as well as the associated gyrokinetic Ampere's law. This set of equations, in terms of the electrostatic potential, $\phi$, and the vector potential, ${\bf A}$, supports both spatially varying perpendicular and parallel pressure gradients and their associated currents. The MHD equilibrium can be reached when $\phi \rightarrow 0$ and {\bf A} becomes constant in time, which, in turn, gives $\nabla \cdot ({\bf J}_\parallel + {\bf J}_\perp) = 0$ and the associated magnetic islands. Examples in simple cylindrical geometry will be given. [1] W. W. Lee,"Magnetohydrodynamics for collisionless plasmas from the gyrokinetic perspective," Phys. Plasmas {\bf 23}, 070705 (2016).