Derivation of the mode conversion-tunneling equation from the Vlasov equation

Instead of using the uniform warm plasma dispersion relation with an inverse Fourier transform to form the mode conversion-tunneling equation, the equation is developed directly from the Vlasov equation and Maxwell’s equations. The plasma is assumed uniform parallel to B0. The expansion parameters are the Larmor orbit and the scale length, keeping terms to order ρ2L and L−1 = (1/ωc)(dωc/dx). For the special case with ω≃2ωc in a single species plasma, the asymptotic form of the mode conversion-tunneling equation is unchanged, but localized third and first derivative terms appear even as absorption vanishes.