The Hare and the Tortoise Revisited from the Point of View of Whistler Wave Ducts

Instead of the standard cold plasma wave method where coupled equations are derived in terms of the wave electric field, a cylindrical full-wave method is used where the wave magnetic field is the fundamental parameter. The wave magnetic field is expressed using a formalism analogous to the magnetic formalism used in the Grad-Shafranov equation. Using the wave magnetic field instead of the wave electric field reduces the description to a pair of coupled differential equations involving the wave poloidal electric current (both displacement and particle) and the wave poloidal magnetic flux. Numerical and analytic solution of these coupled wave poloidal current and wave poloidal flux equations show that a duct forms at a cylindrical density depression. Detailed examination of the solution shows that in the density gradient at the duct edge, a radially outward propagating fast wave (the hare) mode converts into a radially inward propagating slow wave (the tortoise) while simultaneously a radially outward propagating slow wave (tortoise) mode converts into a radially inward propagating fast wave (hare). The result is that there are both fast and slow radial standing waves in the duct. Associated with these radial standing waves in the duct is a radially decaying, spatially oscillating evanescent standing wave external to the duct with radial wavelength intermediate to the wavelengths of the fast and slow waves in the duct. The consequence of standing waves existing at all radii is that there is no radial Poynting flux and so there is no radial power loss. This perfect radial confinement of wave energy in the duct explains why whistler waves make multiple bounces from one terrestrial hemisphere to the other with minimal attenuation.