High-n ideal and resistive shear Alfvén waves in tokamaks

Ideal and resistive MHD equations for the shear Alfvén waves are studied in a low-β toroidal model by employing the high-n ballooning formalism. The ion sound effects are neglected. For an infinite shear slab, the ideal MHD model gives rise to acontinuous spectrum of real frequencies and discrete eigenmodes (Alfvén-Landau modes) with complex frequencies. With toroidal coupling effects due to nonuniform toroidal magnetic field, the continuum is broken up into small continuum bands and new discrete toroidal eigenmodes can exist inside the continuum gaps. Unstable ballooning eigenmodes are also introduced by the bad curvature when β > β_c. The resistivity (η) can be considered perturbatively for the ideal modes. In addition, four branches of resistive modes are induced by the resistivity: (1) resistive entropy modes which are stable with frequencies going to zero with resistivity as η ^{1 3}; (2) tearing modes which are stable (Δ′ < 0) with frequencies approaching zero as η ^{3 5}; (3) resistive periodic shear Alfvén waves which approach the finite frequency end points of the continuum bands as η ^{1 2}; and (4) resistive ballooning modes which are purely growing with growth rate proportional to η ^{1 3}β ^{2 3} as η → 0 and β → 0.