Influence of gyroradlus and dissipation on the Alfvén wave continuum

It is well known that in ideal magnetohydrodynamics there is a continuous spectrum of real frequencies associated with a singularity of the shear Alfvén waves on the surface k_|| v_A= ω. It is also known that the introduction of first-order gyroradius effects eliminates the continuum. In the present work the influence of the full gyroradius response and of dissipation on the continuum is examined. In the absence of dissipation it is first confirmed that if only first-order gyroradius effects are incorporated, the continuum disappears. However, when the full gyroradius response is included, this discrete spectrum vanishes, and a new continuum (associated with singularities at k_|| V_A = 0) appears. The introduction of collisional dissipation removes the original MHD continuum leaving discrete modes whose frequency tends to zero with the collision rate as v^1/3. Collisions also remove the new continuum of the full gyroradius model leaving discrete modes whose frequency tends to zero as (log v) ^-1. Collisionless Landau damping has a similar effect.