A Transmission Line Model for the Spherical Beltrami Problem
Abstract: We extend a previously introduced model for finding eigenvalues and eigenfunctions of PDEs with a certain natural symmetry set based on an analysis of an equivalent transmission line circuit. This was previously applied with success in the case of optical fibers ,  as well as in the case of a linear Schroedinger equation ,  and recently in the case of spherical symmetry (Ball Lightning) . We explore the interpretation of eigenvalues as resonances of the corresponding transmission line model. We use the generic Beltrami problem of non-constant eigen-vorticity in spherical coordinates as a test bed and we locate the bound states and the eigen-vorticity functions.
Summary (1 min read)
- Several types of solutions are reviewed in  and .
- Nowadays, the Beltrami equation serves as a model of a) “Force-Free Fields” for which the magnetic part of the Lorentz force disappears (J // B) in equilibrated plasma and b) for parallel E // B components when applied to the vector potential directly (in which case the authors get ArBAE ),(, tλω =−= ).
- In the next paragraph the authors turn to equation (1) and attempt to locate eigen-modes in terms of the orthogonal basis of vector spherical harmonics.
2. Spherical expansion of the Beltrami equation
- In what follows the authors restrict attention to purely radial functions for the eigen-vorticity λ(r).
- The authors note however that in the more general case they may allow for violations of condition (1b).
- In electromagnetic applications the authors may instead assume the presence of static charges satisfying the Lorenz gauge AA ∇=∇−=∂ − λλφ 1cct .
- From now on the authors will deal solely with the first condition (1a) for the rotation of A.
4. Trapped modes and bound states
- The form of the propagation constant leads to the possibility of “trapped” waves and their respective eigenmodes.
- This situation is reminiscent of the bound states of quantum mechanical systems with the function )(2 rλ playing a role analogous to that of the potential in the radial Schroedinger equation.
- Evanescent modes are then analogous to the scattered wave functions above a threshold.
- The authors will next evaluate one such example of an eigenvorticity function supporting a trapped mode.
- The authors showed that a general class of solutions of the spherical Beltrami problem can be found in terms of vector spherical harmonics of which the resulting system is equivalent with a generic lossless transmission line.
- When applied to a specific choice of eigen-vorticity function it allows finding the set of parameters necessary for each mode to exist.
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