The Particle-in-Cell (PIC) method (Birdsall and Langdon. Plasma Physics via Computer Simulation. McGraw-Hill, 1985) is based on the kinetic plasma model described by the Vlasov equation. This method is a particle simulation which means that the Vlasov equation is not solved directly. Rather, the model sets out to compute the motion of individual particles under the effect of the self-consistent electric and (in the electromagnetic case) magnetic field and any externally applied fields.
In the current code we wanted to focus on electrostatic phenomena on the ion plasma period time scale. As a consequence, the electron density was assumed to be Boltzmann distributed, $n_e = n_0{\exp}(e{\phi}/{\kappa}T_e)$, maintaining full nonlinearity in the exponential term when solving for the electric potential. Furthermore, any magnetic field generated by the plasma itself was assumed to be negligible.
The code describes the plasma in two dimensions in space, but retaining all three velocity components. This made it possible to study the effects of any externally applied magnetic field.
Ion phase-space vortices are formed by the nonlinear saturation of the ion-ion two stream instability, excited by injecting an ion beam at the boundary of a stationary background plasma. Our main interest has so far been to study the formation and evolution of these structures in two dimensions for cases of both finite and infinite beam width. It has been shown that an externally applied magnetic field parallel to the beam can increase the lifetime of these structures considerably. For high injected beam velocities in magnetized plasma we observe the excitation of electrostatic ion-cyclotron instabilities.
We believe the present results can be relevant for interpreting data from the ionospheres and magnetospheres of magnetized planets, such as the Earth. Observations of electrostatic solitary structures in the Earth's magnetotail has been reported from the GEOTAIL satellite (Krasovsky, Matsumoto and Omura, J. Geophys. Res.,1997; Omura, Kojima, Miki, Mukai, Matsumoto and Anderson. J. Geophys. Res.,1999).
The code described above is currently being expanded to three spatial dimensions. Among the many interesting phenomena we then hope to be able to study, is the kinetic drift wave instability.