Two-dimensional analytic theory for toroidal Alfvén eigenmodes

Abstract

A two-dimensional analysis of the toroidal AlfvCn eigenmodes (TAE) is presented, based on an integrodifferential equation describing the shear AlfvCn perturbation of a toroidal plasma equilibrium in terms of coupling among the toroidal AlfvCn 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.

Energy leakage as an attenuation mechanism for vertical kink oscillations in solar coronal wave guides

“…We note the following: (a) in Ye et al (1993) TAEs have a different boundary condition than the one we implemented; (b) Ruderman & Roberts (2002), Van Doorsselaere et al (2004a), Van Doorsselaere et al (2004b), and Andries et al (2005) basically deal with kink modes and their work can be related to what we observe. Our conclusion that energy leakage from the loop into the ambient medium is an efficient mechanism responsible for attenuation of vertical kink oscillations complements the results of these studies.…”

“…The literature on wave attenuation is vast (e.g., Ye et al 1993;Ofman & Aschwanden 2002;Ruderman & Roberts 2002;Van Doorsselaere et al 2004a;Van Doorsselaere et al 2004b;Andries et al 2005). The proposed models differ to various extents in the boundary conditions (bc) they use.…”

“…The proposed models differ to various extents in the boundary conditions (bc) they use. For instance, Ye et al (1993) applied bc that sets the radial electric field E r = 0 on the loop's surface. This is proper for the Toroidal Alfvén Eigenmode (TAE).…”

A computational approach to continuum damping of Alfvén waves in two and three-dimensional geometry

“…13 where a two-dimensional analysis of toroidal Alfv en eigenmodes (TAE) has been done using an integro-differential equation and a formal analogy to the Vlasov equation. In Ref.…”

Singular eigenfunctions for shearing fluids I

“…In other words, it is a kind of wave-mean flow resonance, a phenomenon which occurs in a variety of other fluid contexts. It is also closely related to waveparticle resonance in plasma physics (Van Kampen & Felderhof, 1967) and the Alfdn resonances of MHD (Ye et al 1993); as a mathematical problem it shares many common features with scattering and transport theory Feldman, 1966a and1966b;Case and Zweifel, 1967). We use these analogies to employ methods developed in plasma physics and elsewhere to formulate the singular stability analysis for Rayleigh's problem (sections 2 and 3).…”