Two‐dimensional analytic theory for toroidal Alfvén eigenmodes

A two‐dimensional analysis of the toroidal Alfvén eigenmodes (TAE) is presented, based on an integrodifferential equation describing the shear Alfvén perturbation of a toroidal plasma equilibrium in terms of coupling among the toroidal Alfvén continua with the usual gap structure. Using a method similar to the Van Kampen–Case analysis for the Vlasov equation, exact analytic expressions are derived for the dispersion function and the two‐dimensional eigenmode structure. The dispersion function is expressed in terms of Cauchy‐type integrals, which explicitly expresses the global character of TAE modes and facilitates the calculation of their damping. The continuum‐damped TAE modes are shown to be, in general, not true eigenmodes of the toroidal plasma equilibrium, but rather resonances corresponding to zeros of the analytic continuation of the dispersion function onto unphysical sheets of its Riemann surface. Approximate but explicit expressions for the dispersion relation and the eigenfunction are also obtained in the limit of vanishing inverse aspect ratio.