Abstract
Tokamak flux surfaces are dominated by irrational surfaces, which brings the mathematical problem how to satisfy double periodicity in toroidal and poloidal angles for flute like perturbation expressed by the eikonal form. Ballooning transform is essential method to analyze 2D mode structure in toroidal geometry not only to ideal Ballooning mode, but also for high n toroidal drift waves and Alfven eigenmodes. Fundamentals of ballooning mode structure in toroidal plasma is discussed for the application to MHD instabilities and toroidal drift waves. After an introduction of ballooning transform from real geometry to covering space in section 6.1, the method to satisfy double periodicity flute like perturbation is discussed using the eikonal form in the flux coordinates using Poisson sum and its relation to translational symmetry in 6.2. The 2D hallooning transform and twisted radial Fourier transform are discussed in 6.3. Trapped and passing mode structures in 2D wave equation are discussed using the WKBJ formulation in 6.4. Poisson sum, Bloch theorem and WKBJ solutions are given as Columns.
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