Hybrid Modeling of Plasmas

Space plasmas are often modeled as a magnetohydrodynamic (MHD) fluid. However, many observed phenomena cannot be captured by fluid models, e.g., non-Maxwellian velocity distributions and finite gyro radius effects. Therefore kinetic models are used, where also the velocity space is resolved. This leads to a six-dimensional problem, making the computational demands of velocity space grids prohibitive. Particle in cell (PIC) methods discretize velocity space by representing the charge distribution as discrete particles, and the electromagnetic fields are stored on a spatial grid. For the study of global problems in space physics, such as the interaction of a planet with the solar wind, it is difficult to resolve the electron spatial and temporal scales. Often a hybrid model is then used, where ions are represented as particles, and electrons are modeled as a fluid. Then the ion motions govern the spatial and temporal scales of the model. Here we present the mathematical and numerical details of a general hybrid model for plasmas. All grid quantities are stored at cell centers on the grid. The most common discretization of the fields in PIC solvers is to have the electric and magnetic fields staggered, introduced by Yee [IEEE Transactions on Antennas and Propagation 14:302–307 1966]. This automatically ensures that ∇ ⋅ B = 0, down to round-off errors. Here we instead present a cell centered discretization of the magnetic field. That the standard cell centered second order stencil for ∇ × E in Faraday’s law will preserve ∇ ⋅ B = 0 was noted by Tóth [Journal of Computational Physics 161:605–652 2000]. The advantage of a cell centered discretization is ease of implementation, and the possibility to use available solvers that only handle cell centered variables. We also show that the proposed method has very good energy conservation for a simple test problem in one-, two-, and three dimensions, when compared to a commonly used algorithm.