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. This automatically ensures that div(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 rot(E) in Faraday's law will preserve div(B)=0 was noted by Toth. 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 three dimensions, when compared to a commonly used algorithm.
Deep Dive into Hybrid modeling of plasmas.
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. This automatically ensures that div(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 rot(E) in Faraday’s law will preserve div(B)=0 was noted by Toth. 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 three dimensions, when compared to a commonly used algorithm.
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 [17]. 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 [14]. The advantage of a cell centered discretization is ease of implementation, and the possibility to use available solvers that only provide for 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.
We have N I ions at positions r i (t) [m] with velocities v i (t) [m/s], mass m i [kg] and charge q i [C], i = 1, . . . , N I . By spatial averaging1 , we can define the charge density ρ I (r, t) [Cm -3 ] of the ions, their average velocity u I (r, t) [m/s], and the corresponding current density J I (r, t) = ρ I u I [Cm -2 s -1 ]. Electrons are modelled as a fluid with charge density ρ e (r, t), average velocity u e (r, t), and current density J e (r, t) = ρ e u e . The electron number density is n e = -ρ e /e, where e is the elementary charge. If we assume that the electrons are an ideal gas, then p e = n e kT e , so the pressure is directly related to temperature (k is Boltzmann’s constant).
The trajectories of the ions are computed from the Lorentz force,
where E = E(r, t) is the electric field, and B = B(r, t) is the magnetic field. 2
A brief overview of hybrid codes can be found in [16]. A more complete survey can be found in [10]. Most hybrid solvers for global simulations have the following assumptions in common.
Quasi-neutrality, ρ I + ρ e = 0, so that given the ion charge density, the electron charge density is specified by ρ e = -ρ I .
Ampere’s law whithout the transverse displacement current (also called the Darwin approximation, or the nonradiative limit) provides the total current, given B, by
where µ 0 = 4π • 10 -7 [Hm -1 ] is the magnetic constant (ǫ 0 µ 0 c 2 = 1), and from the total current we get the electron current, J e = J -J I , and thus the electron velocity, since the quasi-neutrality implies that u e = J e /ρ e = (J I -J)/ρ I .
- Massless electrons, m e = 0, lead to the electron momentum equation
where the force terms C can be due to collisions, such as electron-ion collisions, electron-neutral [13] collisions, or anomalous, i.e. representing electron-wave interactions [1]. In our numerical experiments we have assumed that C = ′. This provides an equation of state (Ohm’s law) for the electric field
with J from Ampere’s law. So the electric field is not an unknown. Whenever it is needed, it can be computed.
Faraday’s law is used to advance the magnetic field in time,
The electron pressure is isotropic (p e is a scalar, not a tensor).
For the electrons, the remaining degree of freedom is the pressure, p e . Note that p e only affects the ion motions through the electric field. The evolution of the magnetic field is not affected since we have ∇ × ∇p e = 0 in Faraday’s law. There are several ways to handle the electron pressure [15, p. 8790],
- Assume p e is constant, or zero [5].
Assume p e is adiabatic (small collision frequency). Then the electron pressure is related to the electron charge density by p e ∝ |ρ e | γ , where γ is the adiabatic index. Commonly used values are γ = 5/3 [1,8], and γ = 2 [12,2].
- Solve the massless fluid energy equation [11,8],
Here we assume that p e is adiabatic. Then the relative change in electron pressure is related to the relative change in electron density by
where the zero subscript denote reference values. From charge neutrality and p e = n e kT e we have that
a constant that is evaluated us
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