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

Abstract

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.

Impulse-driven Richtmyer-Meshkov instability in Hall-magnetohydrodynamics

“…As a result, one obtains the following nondimensional ideal Hall-MHD equations [19] that govern the evolution of the initial impulse for incompressible ions and compressible electrons,…”

“…The Larmor radius d L is the normal radius of the helix along which an ion moves about background magnetic field lines while β measures the ratio of kinetic to magnetic energy. Following Shen [19], it is useful to define one more related parameter, the normalized ion skin depth d S , given by…”

“…which is independent of the applied magnetic field strength. Moreover, the regular MHD equations are retrieved by taking the limit of d S → 0 of the Hall-MHD system [19]. Clearly for any fixed β, the MHD limit is equivalently achieved by taking d L → 0.…”

“…More recently, Bond et al [12] investigated computationally the RMI without initial magnetic field using the ideal two-fluid plasma equations. In order of decreasing complexity, Shen et al [19] showed that the ideal two-fluid plasma equations, the Hall-MHD and regular MHD models, are connected via a series of limiting processes with respect to the appropriately scaled parameters including the speed of light, the ion skin depth, and ion-to-electron mass ratio.…”

“…The present study concentrates on the Hall-MHD system, which is a reduced single-fluid model applicable in the regime where the speed of light is large compared to particle thermal speeds and the electron mass is negligible compared to the ion mass. Such system is categorized as a magnetohydrodynamic model for its property of charge neutrality due to effectively infinite speed of light [19]. Nevertheless, the Hall-MHD model captures some features of the more general two-fluid plasma equations over small length scales, such as the Larmor radius and skin depth for ions, that are missing in the ideal MHD equations.…”