A Six-moment Multi-fluid Plasma Model

We present a six-moment multi-fluid model, which solves the governing equations for both ions and electrons, with pressure anisotropy along and perpendicular to the magnetic field direction, as well as the complete set of Maxwell equations. This set of equations includes the Hall effect, different temperatures for different species and pressure anisotropy. It is more comprehensive than the five-moment equations with isotropic pressures and significantly less expensive than the ten-moment equations with a full pressure tensors. Similarly to the five- and ten-moment equations, the wave speeds are naturally limited by the speed of light, which eliminates the issue of unlimited whistler wave speeds present in Hall magnetohydrodynamics (MHD). It is also possible to simulate multiple negatively charged fluids, which cannot be done in MHD models. The six-moment model is a reasonable description of the plasma outside magnetic reconnection regions and therefore well-suited to be coupled with an embedded particle-in-cell model that covers the reconnection region. Our numerical implementation uses a point-implicit scheme for the stiff source terms, and we use a second-order accurate Rusanov-type scheme with carefully selected wave speeds. For the plasma variables and the magnetic field the maximum wave speed is based on the fast magnetosonic speed of MHD with anisotropic pressures that we derive. For the electric field related variables the speed of light is used. The divergence of the magnetic field and Gauss's law are controlled with a hyperbolic-parabolic scheme. We present a number of numerical tests to demonstrate that this numerical model is robust without being excessively diffusive.