Helicon waves in a flaring magnetic field

Helicon wave (whistler waves bounded transversely by a magnetic field surface) propagation is investigated for a cylindrically symmetric curved (flaring) boundary using a finite-element method where the propagation region is divided into a sequence of truncated cones. In each conical segment, a local spherical coordinate system is used with the origin at the apex of the cone. A vector wave equation for the fields is formulated for a cold plasma and reduced, in spherical coordinates, to a pair of coupled partial differential equations (PDEs) for two scalar functions. The PDEs are separable for the azimuthal eigenvalue m = 0. The θ-dependence is a Legendre function, with non-integer eigenvalues determined by the cone angle. The dependence on the radial coordinate, x, satisfies a fourth-order ordinary differential equation (ODE). A straightforward numerical integration of the equation from x1 to x2 fails for large values of x2 − x1, because of the existence of an exponentially growing solution to the equation. Consequently, a different approach is needed. Four independent solutions, valid for 0 6 x < ∞, in the form of power series in x (PS) are obtained which are each asymptotically proportional to e as x → ∞. For x large, four asymptotic expansions (AE) are obtained in the form of series in 1/x times e±ix or e±x . The problem then is to find four linear combinations of the four PSs, each of which approaches one of the AEs in the limit x → ∞ by first deriving a double-integral representation of each PS, valid for all x, and then using its x → ∞ limit to match each AE to a linear combination of the PSs. For special values of ` (` = 4N − 1 or 4N , N an integer) closed-form solutions (polynomials times exponentials) result that are exact representations of the solution for all x. Solutions are computed and illustrated for an outgoing wave using the closed-form solution for special values of `, and as a function of ` and x using the PS and AE, which have a large range of overlap. The propagating solutions of the ODE are also obtained using a WKB method and are used to calculate the propagation in a slowly diverging parabola of revolution.