On the ideal two-fluid plasma equations and magnetohydrodynamics

The ideal two-fluid plasma equations, obtained from truncating moments of the Vlasov-Boltzmann equation, are increasingly used to describe an ion-electron plasma whose transport phenomena occur on a time scale slower, and length scale longer than those of particle collisions. A similar treatment under more stringent constraints gives the well-known single-fluid, ideal magnetohydrodynamic (MHD) equations for low-frequency macroscopic processes. Here we perform a sequence of formal asymptotic expansions for the dimensionless, ideal two-fluid plasma equations, with respect to limiting values of the speed-of-light c, the ion-to-electron mass ratio M and the plasma skin depth d_S. Three different closed MHD systems result including the well-known Hall-MHD and single-fluid MHD equations.  It is shown that  while the  zeroth-order description corresponding to the infinite c limit (with M, d_S fixed)  is strictly charge neutral, it  nonetheless uniquely determines  the perturbation charge non neutrality at first order.  Further, the additional M -> ∞ limit is found to be not required to obtain the single-fluid MHD equations, despite being essential for the Hall-MHD system.