Kinetic wave equations for drift and shear Alfvén waves in axisymmetric toroidal plasmas

The conductivity tensor operator obtained by integrating the linearized Vlasov equation along the unperturbed orbits in the drift approximation is specialized to frequencies well below the ion cyclotron frequency. The diamagnetic contribution and the contribution of the particle drift motion are included, and finite Larmor radius effects are retained to all orders. Under appropriate assumptions of broad validity we obtain a set of coupled integrodifferential wave equations for the amplitudes of the Fourier components of the fields in the poloidal direction. These equations describe drift and shear Alfvén waves in axisymmetric toroidal geometry, and are in a form well suited for efficient numerical discretization. We have verified that they reproduce correctly several known results based on the gyrokinetic theory under the appropriate conditions. The essential equivalence between the traditional Vlasov approach and the gyrokinetic formalism is thereby established. The range of validity of the approximations required to obtain this result is explicitly stated, and the physical origin of the contributions to the oscillating current and charge densities are clearly identified.