An eikonal expansion of the Vlasov–Maxwell equations valid near cyclotron resonance

In the usual formulations of geometrical optics, the physics of the medium enters the equations through a conductivity tensor operator σ. An essential assumption in the subsequent expansion is that the magnitude of σH, the Hermitian part of σ, is much smaller than σA, the anti-Hermitian part. In a finite temperature plasma with ωpe∼‖Ωe‖, this condition is always violated sufficiently close to cyclotron resonance, even though in many cases the waves are weakly damped and k is slowly varying. Simultaneously expanding the Vlasov equation and Maxwell equations and taking explicit account of the relative magnitude of the electric field components in the ordering scheme yields a formalism in terms of real rays, real eikonal functions, and slowly varying amplitude that is valid at cyclotron resonance. It is assumed that ωpe∼‖Ωe‖∼ω are large, that ω≃‖Ωe‖, and that vek/ω is small. It is shown that when the waves are weakly damped at cyclotron resonance, the ray trajectories are, to leading order, exactly those of cold plasma theory. A Poynting theorem that can be put into the standard form is obtained by taking the expansion to first order. Care must be taken, however, to include finite temperature effects in calculating the zero-order electric field. The theory is also applied to ion cyclotron resonance.