Multifluid Equations for MHD

A set of magnetohydrodynamic equations is developed for multiple ion species in the limit of small Larmor radius. The derivation proceeds to explicitly calculate the Lorentz force resulting from the small ion drift velocities. The net effect is that in this limit all the ions move perpendicular to the magnetic field with the same $E\times B$ speed, but are free-streaming relative to one another in the parallel direction. The ions couple one to another in the parallel direction from changes in the magnetic field direction, leading to a collision-like term that, however, maintains a constant total kinetic energy. The equations governing parallel (centrifugal) acceleration, which may be an important process for ionospheric outflow, are then derived. The dispersion relation for a two-species, isotropic fluid with arbitrary mass and charge fractions, as well as electron pressure, is derived and the resulting wave modes are analyzed. A planar Alfvén mode separates from other, generally compressible, modes. It becomes unstable when the ram pressure of the streaming exceeds the firehose limit. The remaining modes satisfy a sixth order equation in the phase speed when counterstreaming and electron pressure are allowed. Without counterstreaming, three stable modes always exist, with two counter-propagating waves each, regardless of the presence of electron pressure. For the streaming case there are three modes, with two asymmetrically propagating waves each, whose behavior can be quite complicated, especially near the firehose instability limit. An example global magnetosphere simulation of a geomagnetic storm is presented using the derived multifluid formalism.