Cold plasma waves expressed in terms of axisymmetric magnetic and current flux functions

Cold plasma waves with time dependence exp(-iωt) propagating in a background magnetic field are traditionally analyzed by assuming a plane-wave decomposition exp(ik·x) resulting in a dispersion relation prescribed by the determinant of a 3x3 matrix operating on the wave vector electric field E_w. However, in many situations the wave source is localized at a point in 3D space so plane waves are not the natural basis set. A spatially localized source can be considered to be axisymmetric since a non-axisymmetric source could be constructed as the sum of adjacent axisymmetric sources. Because magnetic fields have zero divergence, the general wave magnetic field can be expressed using two scalar functions; this is in contrast to the general wave electric field E_w where three scalar functions are required. The most basic axisymmetric wave magnetic field is B_w = ∇ψ × ∇φ + iχ∇φ where φ is the azimuthal angle in a cylindrical coordinate system {r,φ,z} having origin at the wave source and the z-axis is along the background magnetic field. Here lengths are normalized to c/ω and, except for factors of 2π, ψ(r,z) is the wave poloidal magnetic flux and χ(r,z) is the wave poloidal current (including displacement current). Cold plasma waves in a background magnetic field can then be expressed as a pair of coupled second-order partial differential equations in ψ(r,z) and χ(r,z) where these equations contain the parameters S, P, and D having the standard (Stix) definitions. Solutions to these equations will be discussed with emphasis on the ω >> ω_pi, ω_ci regime, i.e., whistler and above regime where ion motion can be ignored.