An analytic kinetic description of the toroidicity-induced Alfvén eigenmode (TAE) is presented. The theory includes electron parallel dynamics nonperturbatively, an effect that is found to strongly influence the character, and damping of the TAE−contrary to previous theoretical predictions. A parallel conductivity model that includes collisionless (Landau) damping on the passing electrons and collisional damping on both trapped and passing electrons is used. Together, these mechanisms damp the TAE more strongly than previously expected. This is because the TAE couples (or merges) with the kinetic Alfvén wave (KAW) within the gap region under conditions that depend on the gap size, the shear, the magnitude of the conductivity, and the mode numbers. The high damping could be relevant to recent experimental measurements of the TAE damping coefficient. In addition, the theory predicts a ‘‘kinetic’’ TAE, whose eigenfreqeuency lies just above the gap, whose existence depends on finite conductivity, and that is formed by the coupling of two KAW’s

The theory of toroidicity-induced AlfvCn eigenmodes (TAE) and kinetic TAE (KTAE) is generalized to arbitrary mode numbers for a large aspect ratio low-beta circular tokamak. The interaction between nearest neighbors is described by a three-term recursion relation that combines elements from an outer region, described by the ideal magnetohydrodynamic equations of a cylinder, and an inner region, which includes the toroidicity and the nonideal effects of finite ion Larmor radius, electron inertia, and collisions. By the use of quadratic forms, it is proven that the roots of the recursion relation are stable and it is shown how perturbation theory can be applied to include frequency shifts due to other kinetic effects, Analytic forms are derived which display the competition between the resistive and radiative damping, where the radiation is carried by kinetic AlfvCn waves. When the nonideal parameter is small, the KTAE modes appear in pairs. When this parameter is large, previously found scaling for the single gap case is reproduced analytically.where 4 (r;m,n)=[m/q(r)-n]2/ RE is the square of the parallel wave number, q(t) is the safety factor, R. the major radius, and u*(r) the Alfven speed. In the limit that the toroidicity r/R,+O, this resonance normally gives rise to continuum damping and AlfvCn wave heating.i3-l6 As is well known, the kinetic (nonideal) effects of electron inertia, finite ion Larmor radius rL , or collisions resolve the continuum resonance: the singularity is replaced by mode conversion to the kinetic Alfven wave ( KAW).'7*'8 As long as the KAW is not reflected back to the mode conversion layer, ideal MHD and kinetic theory predict the same damping rate. However, as has been emphasized in works with TAE3P5>7Jg-'1 and KTAE3*9-'1 modes, the continuum resonance is also resolved by finite toridicity, which creates gaps in the cylindrical continua. Consequently, there results a "competition" between the nonideal effects and toroidicity in the gap region. This competition is apparent in the form of the nonideal parameter v cc n2r#o/r) 3 [Eq. (58)] which appears in the inner layer equations developed in Sec. V (and also in Refs. 9-11). Essentially, the mode frequency no longer intersects the continuum as in the typical treatment of continuum damping. Rather, the nonideal effects are important because of the proximity of the mode frequency to the continuum of the ideal theory. The MHD continuum can only be reached by tunneling, for the TAE, because the mode frequency lies inside the gap, as discussed in Sets. V and VI. In addition, the toroidal interaction of KAW gives rise to a new type of standing wave, KTAE,%" which was treated in the resistive limit in Ref.3. These have mode frequencies just outside the gap of the ideal theory.Near the gap, r=rm,", where q(r) = (m-k 1/2)/n, neighboring poloidal harmonics m and m+ 1 will satisfy the resonance condition =4 (rm,,;w2=f$ (r,,,;m,~b~(r,,,> m+ l,nb~(r,,,) =f&,,.This degeneracy makes the poloidal coupling introduced by toroidal effects 3970