Meta-equilibrum state of multi-species ambipolar diffusion and its relevance to Polar Summer Mesopheric Echoes

Normal mode analysis of Hill's three-species ambipolar diffusion equations [Hill, R.J. (1978), JGR-Space Phys. 83, 989] for an assumed spatially periodic inhomogeneity yields a fast and a slow time scale. For Fourier wavenumbers relevant to Bragg backscattering responsible for Polar Mesospheric Summer Echoes (PMSE) the fast time scale is of the order of milliseconds and the slow time scale is of the order of 10's of seconds or longer. When initial conditions are taken into account, the appropriate linear combination of fast and slow solutions manifests a relaxation on the fast time scale to a meta-equilibrium state which persists for times of the order of the slow time scale. Once this fast-slow interaction structure leading to the meta-equilibrium state is identified, the meta-equilibrium state can be determined using simple algebraic formulae rather than having to solve the time-dependent differential equations. The relaxed state can be given an intuitive interpretation as the resting point of a pseudo-particle falling into a valley along a track in a two-dimensional pseudo-space. The coordinates of the pseudo-particle are the perturbed densities in Hill's coupled diffusion equations. The location of the resting point at the bottom of the valley provides the magnitude of the electron perturbation at the Bragg wavelength, i.e., the quantity responsible for PMSE. Application of the algebraic solutions to the PMSE problem reveals the possibility of both overshoots and undershoots in the presence of electron heating by HF waves. Overshoots occur when the ratio of aerosol-bound electrons to gas-phase electrons is small while undershoots occur in the opposite limit.