Charge separation and velocity shear at a plasma edge in the finite gyro-radius guiding center approximation

A numerical code has been developed to study the nonlinear evolution of the Kelvin-Helmholtz instability, and the existence and evolution of a charge separation at a plasma edge. The finite gyro-radius guiding center approximation, which also includes the polarization drift, is used to describe the ions. A kinetic equation is used for the electrons. The code applies a method of fractional steps which has been previously applied with success to the Eulerian Vlasov codes. In the physical model we are studying, the finite Larmor radius correction allows for a charge separation to exist, and the polarization drift, which has different sign for ions and electrons, has a tendency to accentuate a charge separation in a time varying electric field. We present results for the case where the plasma layer is in two dimensions, and the direction of magnetic field is varied very close to the normal to the plane of the plasma (θ close to 90°). In this case the evolution of the system shows a behavior in accordance with some basic physics associated with the set of equations describing the behavior of a guiding center plasma in a strong magnetic field, namely the energy condensing in the lowest k modes (inverse cascades), while the system is evolving from an initially unstable flow with shear, through a stage showing complex structures with vortices, to a final more stable shear dominated flow. That the charge separation and the flow rearrange themselves via inverse cascades to adjust to a preferred state, as determined by boundary conditions and by the conservation properties associated with the model equations is one of the main points of the present results. The sheared flow we are studying is generated at a plasma edge, and the formation and existence of a charge separation and an electric field at the plasma edge is studied self-consistently with the velocity shear. Getting the value of θ away from 90°, the instability decreases. At θ = 88°, the nonlinear evolution of the Kelvin-Helmholtz instability shows a spectrum which is turbulent and is dominated by higher harmonics, saturates at lower level, and has little effect on the electrons and ions density profiles. At θ = 85°, the system is essentially stable for the set of parameters we are studying. The transition of the spectrum of the nonlinear solution from a turbulent spectrum to a spectrum dominated by the fundamental mode is studied.