Radial dependence of magnetohydrodynamic continuum modes for an incompressible plasma

The spatial structure of the magnetohydrodynamic shear Alfvén and cusp continuum modes in axisymmetric toroidal geometry is analyzed for an incompressible plasma. The normal component ξ_{ψ} of the plasma displacement is found to have an oscillatory type of singularity, ξ_{ψ} ∼ (ψ - ψ_0)^{δ} ∼ sin(|δ| ln|ψ-ψ_0| + {const}), or, for example, in the case of up/down symmetry, a purely logarithmic behavior. This result was also obtained in the more general case of a compressible plasma (Salat, A. and Tataronis, J. A. 1999 Phys. Plasmas 6, 3207). However, in that work the incompressible limit could not be taken because the continuum equations were written in terms of a variable that becomes ill-defined in the limit. The present ab initio derivation shows that, in general, the incompressible limit in the previous work, γ →∞, where γ is the ratio of the specific heats, does give the correct spatial dependence. Furthermore, we prove that it is possible to recover the oscillatory type of singularity for the cylindrical screw pinch from the logarithmic singularity of an up/down symmetric torus.