Three-fluid Ohm's law

We develop a three-fluid theory, including electrons, ions, and neutral particles and their collisions, to describe the interaction between the magnetosphere, ionosphere, and thermosphere, We derive Ohm's law including the collisions between ions, electrons, and neutrals. In particular, our analysis includes the effects of electron-ion collisions. The electron and ion equations are solved in a completely coupled manner. The complete form of three-fluid Ohm's law in the plasma frame is obtained for a steady-state, uniform, three-fluid flow. This form of Ohm's law can be simplified to obtain the ideal MHD frozen-in condition when all collisions are neglected and the generalized Ohm's law when the collisions with neutrals are neglected. In our form the neutral velocity in the conventional collisional Ohm's law is replaced by the plasma velocity. More physical insights can be obtained from this form of Ohm's law in the context of magnetosphere-ionosphere coupling. This form describes the magnetosphere-ionosphere coupling without involvement of the neutral wind. The ionospheric velocity continuously deviates from the electric drift velocity from high altitudes to low altitudes. In addition, the plasma momentum equation for steady-state uniform three-fluid flow is derived. It describes ionosphere-thermosphere coupling. The Ohm's law in the neutral wind frame, or the conventional form of Ohm's law, can be derived by combining the Ohm's law in the plasma frame with the momentum equation. The conductivities become the same as the conventional ones under a few approximations. Combination of the two forms of Ohm's law provides the relationship between the electric field and the ionospheric velocity. When the neutral wind stays still? this relationship can be used to describe magnetosphere-ionosphere coupling.