On the Alfvén resonance and its existence

In this Brief Communication, the continuous spectra of ideal magnetohydrodynamics are revisited in order to point out and discuss equilibrium configurations in which the Alfvén continuum is characterized by a nonlogarithmic spatial singularity in the velocity fluctuation of the plasma. This occurs in a compressible sheet pinch with a shearless equilibrium magnetic field directed parallel to a specified magnetic surface wave vector. It is shown that the component of the velocity vector parallel to the direction of inhomogeneity is bounded at the Alfvén resonance layers. The velocity vector possesses spatial singularities, but the singularities are restricted to its component perpendicular to both the direction of the spatial inhomogeneity and the equilibrium magnetic field. The singularities are found to be delta functions. Comments are also made on the Alfvén resonance in the incompressible plasma and its relationship to the continuous spectra of the compressible plasma.