A numerical simulation is performed to study the effect of a viscous dissipation term on the generation of shear flow at a plasma edge in the guiding center approximation. The guiding center model includes the effects of finite Larmor radius corrections and polarization drift. The numerical code applies the method of fractional steps to the fluid guiding center equations. We attempt to discriminate between the smoothing of the microstructure by a small viscous dissipation term to control numerical instabilities, and the modification of the macroscopic physical results introduced by this small viscous dissipation term. The finite Larmor radius effect allows for a charge separation to exist, which can be further accentuated by the polarization drift. A difference in the viscous term between electrons and ions can add to the charge separation effect at the plasma edge, which can modify the physical results. The numerical calculation is effected using a slab model, periodic in one direction and finite in the other direction, with an inhomogeneous density of guiding centers to simulate a plasma edge. The evolution of the system shows the potential evolving to a shape characterized by the longest wavelength associated with the transverse dimension of the system, an evolution characteristic of an inverse cascade. We present an analysis of the effect of different values of the viscous term on the time evolution of this guiding center system, and on the formation and existence of a charge separation and an electric field at the edge of a plasma and the associated shear in the E × B flow.

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 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.

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For the first time, beatwave simulations relevant to the UCLA experiment ͑University of California at Los Angeles͒ ͓see Clayton et al., Phys. Rev. Lett. 70, 37 ͑1993͒, and also Phys. Plasmas 1, 1753 ͑1994͔͒ have been made with a relativistic Eulerian Hilbert-Vlasov code for a realistically high ratio of driver frequency to plasma wave frequency ͑ 0 / p Ϸ30͒. Some of the more striking features that have emerged from the Hilbert-Vlasov simulations are discussed in this paper, with particular emphasis on particle dynamics in phase space with beam injection, and action transfer results obtained from the derivation of the integrated Manley-Rowe relations derived for a finite causal system.