Revisiting linear gyrokinetics to recover ideal magnetohydrodynamics and missing finite Larmor radius effects

The linear gyrokinetics theory in the axisymmetric configuration is revisited. It is found that the conventional gyrokinetic theory needs to be repaired in order to recover the linear magnetohydrodynamics from the gyrokinetics and to obtain the finite Larmor radius effect on the magnetohydrodynamic modes in an ordering-consistent manner. Two key inclusions are: (1) the solution of the equilibrium gyrokinetic distribution function is carried out to a sufficiently high order; (2) the gyrophase-dependent part of the perturbed distribution function is kept. The new gyrokinetic theory developed in this paper can be used to extend directly the magnetohydrodynamic stability analysis to the gyrokinetic one without invoking the hybrid kinetic-fluid hypothesis.