## 1. Introduction

[2] Much of magnetospheric dynamics is driven by inputs, both steady and variable, from the solar wind, and hence understanding the propagation of changes, on all spatial and temporal scales, from the magnetopause or from the magnetotail inward through the magnetosphere and down to the ionosphere is of central importance. As a first step in describing how an imposed change at the magnetopause or in the magnetotail (e.g., a sudden enhancement of reconnection) produces its signal at the ionosphere (via Fourier decomposition of the input, propagation of each frequency component at its own velocity, and Fourier reconstitution of the output), *Song and Vasyliūnas* [2002] developed a simple model of propagating imposed perturbations along magnetic field lines from the outer magnetosphere to the ionosphere. Although their model includes all frequency/time regimes, from near-speed-of-light propagation well above the plasma frequency to MHD waves well below the ion gyrofrequency, it assumes a collisionless plasma and hence is applicable in the spatial domain only to the magnetosphere, excluding the ionosphere.

[3] In this paper, we extend the physical model developed by *Song and Vasyliūnas* [2002] and add the effects of collisions between plasma and neutral particles, using collision rates appropriate to the ionosphere. The model is thus applicable continuously over the entire spatial range, from the magnetosphere down to and including all the layers of the ionosphere. The basis of the model is a three-fluid description, applicable to the partially ionized medium of the ionosphere-thermosphere system and including the collisionless magnetosphere as a limiting case: electrons, ions, and neutral particles, with interspecies collisions among all three included. Previously, *Song et al.* [2005] derived the three-fluid Ohm's law to describe the steady-state structure of the ionosphere coupled to the magnetosphere and the thermosphere, and with it they derived, for given magnetospheric boundary conditions, the continuous change of the plasma bulk velocity with height, from the magnetosphere to the lower ionosphere, as collisions become more and more important. Here we extend our previous work from steady state to wave propagation.

[4] Our analysis brings together several aspects which previously have been discussed mostly in isolation. A multifluid structured ionosphere has been treated in many ionospheric models [e.g., *Akasofu and DeWitt*, 1965; *Boström*, 1974; *Roble et al.*, 1998; *Kelley*, 1989; *Richmond et al.*, 1992], including attempts at describing time-dependent (but not wave) behavior for the closed field line regions of region 2 currents [e.g., *Peymirat et al.*, 1998]. Waves as an essential element in magnetosphere-ionosphere coupling have been treated by *Lysak* [1990, 1999], *Strangeway* [2002], and others, although the effects of the neutral motion on the wave propagations have not been included. Effects of neutral motion, on the other hand, on the high-frequency wave propagations have long been known in radio science [e.g., *Sen and Wyller*, 1960; *Booker*, 1984]. To the conventional magnetospheric description of magnetosphere-ionosphere coupling we add neutral atmosphere effects and a structured as well as dynamically responding ionosphere, to the (by now) conventional three-fluid treatment of the ionosphere we add time-dependent global propagation effects from the magnetopause, and to radio science we extend the neutral effects on wave propagation to lower frequencies. Although individual equations from the set we analyze have appeared in previous works, systematic application of the entire set over a broad frequency range appears to be novel. In particular, we are aware of no previous derivation of the complete dispersion relation for parallel propagation near and below the ion gyrofrequency.

[5] This is a very complicated physical/mathematical system, a realistic treatment of which remains a formidable task for the future. In this paper we obtain solutions for a highly idealized situation. We restrict ourselves to waves at a given frequency, leaving for future work the analysis of frequency and time integration. In section 2, we begin with the three-fluid treatment, three-fluid generalized Ohm's law, plasma momentum equation, and neutral momentum equation. We then apply these equations to the magnetosphere-ionosphere system, for now under the simplifying approximation of local uniformity. In section 3, we derive the (complex) dispersion relation for incompressible parallel propagation. The detailed derivation of the dispersion relation is given in Appendix A. In section 4, we focus on the low-frequency range and on parameters at some characteristic altitudes within the ionosphere, where most of the neutral-atmosphere effects on magnetosphere-ionosphere coupling occur, and calculate wave propagation and attenuation properties as functions of parameters at each height. In section 5, we give a simple physical interpretation of some remarkable wave properties described by the mathematical dispersion relation, discuss the relative roles of resistivity and neutral drag, and examine the applicability of the assumed locally uniform-medium approximation.