Model

We used the KEMPO1 [Omura and Matsumoto, 1993], which is a one-dimensional electromagnetic particle code. To make the essential physics clear, we further simplify the simulation model by eliminating electromagnetic part of the simulation code. Namely, we only solve Poisson's equation to obtain electrostatic fields the spatial profile of the charge density. We assume a one-dimensional space with the x-axis along which the static magnetic field is taken.

We assume three species of charged particles. Two electron beams and one ion beam. We assume these beams take densities n₁, n₂ and n_i, respectively. The first electron beam with n₁ take a drift velocity V_d/2, and the second electron beam with n₂ and the ion beam with n_i take a drift velocity -V_d/2. We take this frame of reference so that the maximum particle velocity in the simulation system becomes minimum, which gives a better numerical accuracy for a fixed time step \Delta t. It is noted that the frame of reference does not change the physics. We also assume these beams take shifted-Maxwellian velocity distributions with thermal velocities V_e for the electron beams and V_i for the ion beam, respectively.

The charge neutrality condition of the plasma gives, n₁ + n₂ = n_i . In the following simulation runs, we varies the density n₁ and n₂ keeping n_i constant. We define a density ratio R = n₁/n_i, and indicate it as a parameter of each simulation runs. Another parameter to be varied is the thermal velocity of the ion beam V_i, which control the Landau damping of the ion acoustic waves in the simulation system.

We list the fixed parameters for the simulation runs. Physical parameters are normalized to the initial electron thermal velocity and the electron plasma frequency.

Thermal velocity of electrons: V_e = 1.0
Plasma frequency of electrons: \Pi_e = 1.0
Mass ratio of electron and ion: m_e/m_i =0.01
Plasma frequency of ions: \Pi_i = 0.1
Drift velocity: V_d = 20.0
Time step: \Delta t = 0.05
Grid spacing: \Delta x = 1.0
Number of grid points: N_x = 1024
Number of superparticles for each electron beam: 16,384
Number of superparticles for an ion beam: 131,072
Number of time steps: N_t = 8192

The simulation system has a spatial scale of 1024 \lambda_D, where \lambda_D is the initial Debye length. We assume the periodic bounary condition for the simulation system. Since we assume a relatively warm ion beam, the thermal kinetic energy of the ion beam is the major energy in the system. To suppress the thermal fluctuations of the ion beam, we have to assign a much larger number of superparticles in the system. Otherwise, the thermal fluctuation due to a small number of superparticles may distroy the ESW. We varied the density ratio R from 0.1 to 1.0. When R=1.0, the model is reduced electron-ion stream model where Buneman instability is expected. For the thermal velocity of ions, we assume V_i = 0.1 and 2.0.