THE THEORETICAL FOUNDATION OF 3D ALFVEN RESONANCES: NORMAL MODES

We consider the resonant coupling of fast and Alfven magnetohydrodynamic (MHD) waves in a 3D equilibrium. Numerical solutions to normal modes (proportional to exp(-iwt)) are presented, along with a theoretical framework to interpret them. The solutions we find are fundamentally different from those in 1D and 2D. In 3D there exists an infinite number of possible resonant solutions within a "Resonant Zone," and we show how boundary conditions and locally 2D regions can favor particular solutions. A unique feature of the resonance in 3D is switching between different permissible solutions when the boundary of the Resonant Zone is encountered. The theoretical foundation that we develop relies upon recognizing that, in 3D, the orientation of the resonant surface will not align in a simple fashion with an equilibrium coordinate. We present a method for generating the Alfven wave natural frequencies for an arbitrarily oriented Alfven wave, which requires a careful treatment of scale factors describing the background magnetic field geometry.