On the magnetohydrodynamic limits of the ideal two-fluid plasma equations

The two-fluid plasma equations describing a magnetized plasma, originating 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 a length scale longer than those of particle collisions. A similar treatment under more stringent constraints gives the single-fluid magnetohydrodynamic (MHD) equations for low-frequency macroscopic processes. Since both stem from kinetic theory, the two-fluid plasma and MHD equations are necessarily related to each other. Such a connection is often established via ad hoc physical reasoning without a firm analytical foundation. Here, we perform a sequence of formal 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 dS. Several different closed systems of equations result, including separate systems for each limit applied in isolation and those resulting from limits applied in combination, which correspond to the well-known Hall-MHD and single-fluid ideal MHD equations. In particular, it is shown that while the zeroth-order description corresponding to the c→∞ limit, with M and dS fixed, is strictly charge neutral, it nonetheless uniquely determines the perturbation charge non-neutrality at the first order. Furthermore, 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. The hierarchy of systems presented demonstrates how plasmas can be appropriately modeled in situations where only one of the limits applies, which lie in the parameter space in between where the two-fluid plasma and Hall-MHD models are appropriate.